Practicality of Quantum Information Processing by Hoi-Kwan Lau A

Practicality of Quantum Information Processing

Hoi-Kwan Lau

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto

Abstract

Practicality of Quantum Information Processing

Hoi-Kwan Lau Doctor of Philosophy Graduate Department of Physics University of Toronto 2014

Quantum Information Processing (QIP) is expected to bring revolutionary enhancement to various technological areas. However, today’s QIP applications are far from being practical. The problem involves both hardware issues, i.e., quantum devices are imperfect, and software issues, i.e., the functionality of some QIP applications is not fully understood.

Aiming to improve the practicality of QIP, in my PhD research I have studied various topics in quantum cryptography and ion trap quantum computation. In quantum cryp-

tography, I ﬁrst studied the security of position-based quantum cryptography (PBQC). I discovered a wrong assumption in the previous literature that the cheaters are not allowed to share entangled resources. I proposed entanglement attacks that could cheat all known PBQC protocols.

I also studied the practicality of continuous-variable (CV) quantum secret sharing (QSS). While the security of CV QSS was considered by the literature only in the limit of infinite squeezing, I found that finitely squeezed CV resources could also provide finite secret sharing rate. Our work relaxes the stringent resources requirement of implementing

QSS.

In ion trap quantum computation, I studied the phase error of quantum information induced by dc Stark effect during ion transportation. I found an optimized ion trajectory for which the phase error is the minimum. I also defined a threshold speed, above which ion transportation would induce significant error.

ii In addition, I proposed a new application for ion trap systems as universal bosonic simulators (UBS). I introduced two architectures, and discussed their respective strength and weakness. I illustrated the implementations of bosonic state initialization, transformation, and measurement by applying radiation ﬁelds or by varying the trap potential.

When comparing with conducting optical experiments, the ion trap UBS is advantageous in higher state initialization eﬃciency and higher measurement accuracy. Finally, I proposed a new method to re-cool ion qubits during quantum computation. The idea is to transfer the motional excitation of a qubit to another ion that is prepared in the motional ground state. I showed that my method could be ten times faster than current laser cooling techniques, and thus could improve the speed of ion trap quantum computation.

iii Acknowledgements

First of all, may I express my sincere gratitude to my supervisors: Prof. Daniel James and Prof. Hoi-Kwong Lo. I really appreciate that they agreed to co-supervise my PhD thesis; this was an adventurous decision as they work in different directions. I am very fortunate to have had numerous inspiring discussions with my supervisors, through which I have learnt a lot in both physics knowledge and the ways to tackle problems. I am obliged for their patience and tolerance, particularly they relentlessly gave me friendly advices even when I held a strong opposing opinion. Besides, I would like to thank for their effort and consideration in helping me in job-hunting and in training me to be a better physicist and person. In summary, my PhD period would not be so fruitful and enjoyable without the kindness of my supervisors. I would like to thank my supervisory committee members, Joseph Thywissen and John Sipe, and my examiners, Jonathan Dowling and John Wei, for their valuable comments and beneficial advices. I also want to thank my colleagues, including Eric Chitambar, Serge Fehr, John Gaebler, Barry Sanders, Marcos Villagra, Christian Weedbrook, and Andrew White, for their illuminating discussions and useful helps. In addition, I want to thank Mr. David Tang and Ms. Kristy Kwok for their hospitality during my stay in Toronto. It is my honour to acknowledge the support from the Kwok Sau Po Scholarship, the E. F. Burton Fellowship, the Lachlan Gilchrist Fellowship Fund, and the Queen Elizabeth II Graduate Scholarship in Science and Technology. Finally, I would like to thank my beloved wife, Dan Sun, for agreeing to accompanying me in the rest of my life.

iv Contents

Glossaries and Acronyms xi

1 Introduction 1

2 Position-Based Quantum Cryptography 6 2.1 Introduction ...... 6 2.2 Classical Position-Based Cryptrography ...... 8 2.2.1 Individually Cheating ...... 10 2.2.2 Collaborative Cheating ...... 10 2.3 PBQCProtocols ...... 10 2.3.1 ProtocolA...... 11 2.3.2 ProtocolB...... 12 2.4 Cheating in N =2Case ...... 12 2.4.1 CheatingagainstProtocolA ...... 12 2.4.2 CheatingagainstProtocolB ...... 15 2.5 Cheating in N > 2Case ...... 17 2.5.1 CheatingagainstProtocolA...... 17 2.5.2 CheatingagainstProtocolB ...... 20 2.6 Principle of the cheating schemes ...... 22 2.6.1 ProtocolA...... 22 2.6.2 ProtocolB...... 23 2.7 ModiﬁedPBQCProtocol ...... 24 2.8 Summary ...... 26

3 Quantum Secret Sharing with Continuous-Variable Cluster States 28 3.1 Introduction...... 28 3.2 Background ...... 30 3.2.1 QuantumSecretSharing ...... 30

v 3.2.2 Continuous-Variable Cluster States ...... 32 3.2.2.1 Nullifier representation ...... 32 3.2.2.2 Wigner function representation ...... 33 3.2.2.3 Correlations of measurement ...... 34 3.2.2.4 Cluster-class state ...... 35 3.3 CCQuantumsecretsharing ...... 35 3.4 CQQuantumsecretsharing ...... 38 3.4.1 Equivalence of CQ Quantum Secret Sharing and QKD ...... 40 3.4.2 SecretSharingRate ...... 41 3.4.3 Simplification ...... 45 3.4.3.1 MixedStateApproach ...... 45 3.4.3.2 Classical Memory ...... 46 3.4.3.3 LocalMeasurement...... 48 3.4.3.4 Simplified CQ Protocol ...... 50 3.5 QQQuantumsecretsharing ...... 51 3.6 Conclusion ...... 53

4 Motional States of Trapped Ions 56 4.1 TrapandIonMotion ...... 57 4.2 InternalStructureandLaserOperation ...... 59 4.2.1 Transition...... 59 4.2.2 Measurement ...... 61 4.3 IonMotioninTrapPotential ...... 61 4.3.1 Generalized Harmonic Oscillator ...... 63 4.3.2 InteractionPicture ...... 67

5 Decoherence Induced by dc Electric ﬁeld During Ion Transport 69 5.1 Speedofiontrapquantumcomputer ...... 69 5.2 Motionofion ...... 71 5.3 PhaseshiftduetodcStarkeﬀect ...... 72 5.4 Minimum possible phase shift ...... 73 5.5 Threshold speed of transporting ion qubits ...... 76 5.6 Non-encodingStateExcitation ...... 77 5.7 SummaryandDiscussion...... 80

6 Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap 82 6.1 Introduction ...... 82

vi 6.2 Layoutofthesystem ...... 84 6.3 Universal Bosonic Simulation ...... 85 6.4 Laser Implementation of Basic Operations ...... 87 6.4.1 DisplacementOperator...... 89 6.4.2 Phase-ShiftOperator ...... 90 6.4.3 SqueezingOperator...... 90 6.4.4 NonlinearOperator...... 90 6.4.5 BeamSplitter...... 91 6.5 Readout ...... 95 6.5.1 Adiabaticpassage...... 95 6.5.2 ResonantPulses...... 99 6.5.2.1 Post-selection Method ...... 100 6.5.2.2 Multiple Electronic State Method ...... 103 6.6 Initialization ...... 104

7 Ion Trap Bosonic Simulator 2: Ions in Separate Trap 107 7.1 Introduction...... 107 7.2 Model ...... 108 7.3 SingleModeOperations ...... 109 7.3.1 DisplacementOperator ...... 110 7.3.2 SqueezingOperator ...... 111 7.3.3 Phase-ShiftOperator ...... 112 7.3.4 NonlinearOperator ...... 113 7.4 Two-modeOperation ...... 114 7.4.1 IonTransportandPick-up ...... 119 7.4.2 Accuracyofbeamsplitter ...... 120 7.5 Initialization and readout ...... 122 7.6 Conclusion ...... 123

8 Rapid ion re-cooling by swapping beam splitter 126 8.1 Introduction ...... 126 8.2 Model ...... 127 8.3 Cooling ...... 130 8.4 SwappingBeamSplitter ...... 131 8.4.1 Ansatz...... 133 8.5 Groundstatequbitpair ...... 133 8.6 Transportbetweentraps ...... 136

vii 8.7 Implementationofpotential ...... 137 8.8 Anharmonicity ...... 138 8.9 Fluctuation ...... 139 8.10Conclusion...... 140

9 Summary 142 9.1 Prospects ...... 145

A Appendix 148 A.1 SecurityofModiﬁedProtocol ...... 148 A.1.1 Security Against Attacks with One Entangled Qubit ...... 148 A.1.2 Security Against Attacks with One Entangled Qutrit ...... 151 A.2 ExampleofCCQSS ...... 154 A.2.1 Example1:(2,3)-CCprotocol ...... 154 A.2.1.1 Parties 1,2 collaboration...... 155 { } A.2.1.2 Parties 2,3 collaboration...... 156 { } A.2.2 Example2:(3,5)-CCprotocol ...... 157 A.2.2.1 Parties 1,2,3 collaboration...... 158 { } A.2.2.2 Parties 1,3,4 collaboration...... 159 { } A.3 ExamplesofCQQSS ...... 161 A.3.1 Example1:(2,3)-CQprotocol ...... 161 A.3.1.1 Parties 1,2 collaboration ...... 162 { } A.3.1.2 Parties 2,3 collaboration ...... 164 { } A.3.2 Example2:(3,5)-CQprotocol ...... 164 A.3.2.1 Parties 1,2,3 collaboration ...... 165 { } A.3.2.2 Parties 1,3,4 collaboration ...... 167 { } A.4 ExampleofQQQSS ...... 169 A.4.1 Example1:(2,3)-QQprotocol ...... 169 A.4.2 Example2:(3,5)-QQprotocol ...... 169 A.5 Example of Application: Demonstration of Hong-Ou-Mandel Eﬀect ... 171 A.6 Moving multiple ions by harmonic trap ...... 174 A.6.1 MotionalExcitation ...... 175 A.6.2 Example:Twoionsinatrap...... 178

Bibliography 181

viii List of Figures

2.1 Space-time diagram of the one dimensional position-based cryptography scenario...... 9 2.2 Illustration of the prover’s location ...... 10 2.3 Circuit for teleportation...... 14 2.4 Scenarioofthree-veriﬁerPBQCprotocol ...... 17

3.1 Strategy for computing the secret sharing rate using CV QKD techniques. 41

4.1 LayoutofPaultrap...... 58 4.2 Energy levels of Raman transition ...... 60 4.3 Energy levels under ﬂuorescence measurement ...... 61 4.4 Layout of scalable ion trap quantum information processor ...... 62

5.1 Time variation of optimal trajectories ...... 75

6.1 Layout of bosonic simulator with multiple ions in a single trap ...... 84 6.2 Schematic description of Raman transition...... 88 6.3 Minimum frequency difference between Raman field and unwanted mode transition...... 92 6.4 Probability fluctuation of phonon state under beam splitter operation with differentLamb-Dickeparameter...... 93 6.5 Energy levels of ion during measurement process...... 96 6.6 List of elements in circuit diagram of boson simulation...... 101 6.7 Circuit diagram of phonon non-resolving measurement using the post- selectionmethod ...... 102 6.8 Probability of the first 30 Fock states in the post-selected branch. . . . . 103

6.9 Circuit diagram of phonon number resolving measurement for nmax =2 . 105

7.1 Conﬁguration of ion trap UBS architecture involving ions in separated traps.109 7.2 Trap potential change for single mode operations ...... 110

ix 7.3 Procedureofaphononbeamsplitter ...... 115 7.4 Fidelity of phonon state after 50:50 beam splitter operation ...... 121

8.1 Outline of the cooling process ...... 128 8.2 Time variation of mean phonon number, ion separation, and local trap parametersduringSBS...... 134 8.3 Procedureofforminggroundstatequbitpair...... 135 8.4 Variations of potentials during diabatic ion separation ...... 136 8.5 Motional excitation of qubit induced by Coulomb anharmonicity . . . . . 139 8.6 Motional excitation caused by random ﬂuctuation of potential ...... 141

A.1 Schematic representation of the cluster states for the CC protocol. . . . . 155 A.2 Secret sharing rate of (2,3)- and (3,5)-CC protocols ...... 160 A.3 Schematic representation of the cluster state for (2,3)- and the (3,5)-CQ protocols...... 161 A.4 Secret sharing rate of (2,3)- and (3,5)-CQ protocols ...... 168 A.5 Entanglement extracted from CV cluster states in QQ QSS ...... 170 A.6 Circuit diagram of the Hong-Ou-Mandel eﬀect demonstration ...... 172

x Glossaries and Acronyms

access structure The set of authorised subsets of parties in a secret sharing protocol, see Sec. 3.1 (p. 28). adversary structure The set of unauthorised subsets of parties in a secret sharing protocol, see Sec. 3.1 (p. 28).

CC Quantum secret sharing scheme that shares classical secret through secure quantum channels, see Sec. 3.2.1 (p. 30). cheater (PBQC) An unauthorised party who attempts to mimic the correct response of the prover in PBQC, see Sec. 2.1 (p. 7). CM Centre-of-mass phonon mode, see Sec. A.5 (p. 171). CPHASE Controlled-phase gate, see Sec. 3.2.2 (p. 29). CQ Quantum secret sharing scheme that shares classical secret through insecure quantum channels, see Sec. 3.2.1 (p. 30). CV Continuous-variable (p. 29). dc Direct current (p. 69). dealer The party who encodes a secret into some information carrier according to a secret sharing protocol, see Sec. 3.1 (p. 28).

EPR Einstein-Podolsky-Rosen state (p. 38), see, e.g., Ref. [34].

GHZ Greenberger-Horne-Zeilinger state, see Eq. (2.2) in p. 12 and Ref. [15].

HOM Hong-Ou-Mandel eﬀect (p. 104), see Ref. [86].

KMW Kielpinski-Monroe-Wineland architecture of ion trap quantum computer, see Sec. 4.3 (p. 61).

LDA Lamb-Dicke approximation, see Sec. 6.4 (p. 89).

MBQC Measurement-Based Quantum Computation (p. 17), see Refs. [147, 148].

xi nulliﬁer See Sec. 3.2.2.1 (p. 32).

PBC Position-Based Cryptography, see Sec. 2.1 (p. 6). PBQC Position-Based Quantum Cryptography, see Sec. 2.1 (p. 7). prover The person whose position is to be veriﬁed in PBQC, see Sec. 2.1 (p. 6). PVM Projection-valued measurement (p. 97).

QC Quantum computer (p. 56). QIP Quantum Information Processing (p. 1). QKD Quantum Key Distribution (p. 2). QQ Quantum secret sharing scheme that shares quantum secret through insecure quantum channels, see Sec. 3.2.1 (p. 30). QSS Quantum Secret Sharing, see Sec. 3.1 (p. 28). rf Radio frequency (p. 58). RWA Rotating wave approximation, see Sec. 6.4 (p. 89).

SBS Swapping Beam Splitter, see Sec. 8.1 (p. 127). state-averaging See p. 43.

UBS Universal Bosonic Simulator, see Sec. 6.1 (p. 83). UQC Universal quantum computer (p. 82).

veriﬁer Trusted reference station in PBQC, see Sec. 2.1 (p. 6).

xii Chapter 1

Introduction

At the beginning of the 20th century, physicists formulated the theory of quantum mechanics. This new theory successfully explains numerous phenomena related to the low energy “quantum” particles, such as atoms, electrons, and photons, whose properties cannot be fully understood by using the classical theory of physics. However, quantum mechanics also predicts that quantum particles would exhibit counter-intuitive behaviours. For example, unlike any everyday object whose physical attribute is definite, a quantum particle may be present in a superposition state that simultaneously possesses multiple different attributes. For instance, an electron in a superposition state can be located inside two different quantum wells at the same time. Besides, unlike a classical object that can be repeatedly measured without being disturbed, merely one shot of measurement would erase the original state of a quantum particle. Furthermore, spatially separated quantum particles can exhibit a correlation called entanglement that violates some classical concepts of causality [57]. Such correlation cannot be explained by any classical local hidden-variable theory [24, 25], thus entanglement is generally believed to be a genuine physical effect. The extraordinariness of quantum mechanics imposes difficulties for philosophers to interpret the foundation of our world; on the other hand, scientists see the opportunities and ask: Can we use these extraordinary quantum systems to build devices, which can outperform current devices whose functionality is based on classical physics? The study of Quantum Information Processing (QIP) aims to answer this question. The principle of QIP is to use the states of quantum particles to encode information; then the logical processing of information is conducted by applying physical operations to transform the quantum states. In the past two decades, numerous QIP applications have been discovered that are superior over their classical counterparts [143]. One QIP application is to improve the security of communication. Because quantum information

1 Chapter 1. Introduction

must be disturbed by a measurement, any eavesdropping activities would leave a trace that could be detected by the authorised parties. Using this property, a quantum key distribution (QKD) protocol was proposed [26] and was proved to be secure against any kind of eavesdropping [126, 120, 162], while such an unconditional security is impossible for any classical communication scheme. Another application of QIP is to improve the efficiency of computation. Since the size of the Hilbert space of a quantum system scales exponentially with the number of particles involved, a quantum computer can process an exponential amount of data in each operation. By using such capability, quantum computational algorithms were invented and shown to be advantageous over any known classical algorithm in numerous tasks, such as factoring larger numbers [161], searching for a particular data from a library [73], and simulating complicated physical systems [61, 116, 41]. Recently, quantum communication has been realised between stations that are over 100 km apart [171], and quantum computation involving 100 operations has been demonstrated [104]. Nevertheless, current QIP applications are still far from being practically useful. The problem involves both “hardware” and “software” issues. Firstly, it is difficult to build a QIP device that functions as desired. Due to respective physical properties, each quantum system suffers from limitations on the operations that can be efficiently implemented. For example, due to the weak interaction between photons, entanglement operations in optical quantum computers have to be implemented by non-deterministic methods whose success rate is low. Besides, realistic quantum devices are imperfect and are exposed to environmental influence; even a tiny flaw of control or a little background noise would significantly affect the volatile quantum states, and thus contaminate the encoded information. Furthermore, in spite of significant breakthroughs in recent years, the speed of state-of-the-art quantum operations are generally much slower than that of classical devices, whose technologies are already mature. All of these hardware issues would hinder a realistic quantum device from attaining the theoretical efficiency of quantum computational algorithms, and the ideal security of quantum communication protocols. Apart from the hardware issues, the practicality of some QIP applications is still controversial. For example, it is not uncommon that seemingly promising QIP proposals were later found to be not functioning as predicted. The problems usually originate from making false assumptions due to classical perceptions. For instance, after long debate quantum bit commitment protocols were proved to be insecure against simple tricks of entanglement attacks [119, 125]. Furthermore, some theoretical proposals of QIP require stringent resources that could not be achieved by technologies in foreseeable future. All of

2 Chapter 1. Introduction

these software issues hinder QIP applications from being realised and practically applied. To improve the practicality of a QIP application, both the hardware and software issues have to be settled. This can be done by exploring possible implementation flaws and invalid theoretical assumptions that would affect the functionality of the application, modifying the device architecture and the protocol procedure in order to improve the application’s efficiency and feasibility, and proposing new QIP applications that are practically useful and could be implemented with near future technology. In my PhD research, I particularly study two of the most mature QIP applications: quantum cryptography, which has already been commercialised and elementary real-life applications have been demonstrated [1]; and the ion trap quantum computer, which is widely recognised as a promising implementation [182] that holds several records of quantum computation (see Ch. 4 for more details). Most of the results of my work is included in this thesis. The abstract of each chapter is presented as follows: In Chapter 2, I present my work with Hoi-Kwong Lo about Position-Based Quan- tum Cryptography (PBQC). PBQC aims to send a secret message to a person that the security of the message is guaranteed by the person’s geographical location. Since the concept of PBQC was developed as early as in 2002 [96], it was generally believed to be unconditionally secure. However, we discovered a loophole that allows the dishonest parties to obtain the exact secret message if they possess entangled quantum resources. We found that all PBQC protocols, at the time our work was conducted, were insecure against our attack. We also calculated the minimum amount of entanglement required for a hacking. Based on the idea of our work, PBQC was later shown to be generally insecure [43]. The result of this work was presented by Hoi-Kwong Lo in the CIFAR QIP meeting 2010 1, and was published as Ref. [108]. In Chapter 3, I present my work with Christian Weedbrook about quantum secret sharing (QSS). We studied the performance of a QSS scheme when a continuous-variable cluster state is employed as the resources. In the literature, the security of continuous variable QSS is guaranteed only in the scenario that the quantum state involves infinite energy, which is unrealistic. In our work, we quantified the amount of leaked secret information when finite energy states are employed. We also developed strategies to protect the security of QSS against such information leakage. Our results relax the stringent requirement of resources that can be used for secure QSS. Our work was presented in the APS DAMOP meeting 2013 2, and was published as Ref. [109].

1‘Position-based quantum cryptography: Eﬃciency of cheating strategies’, Canadian Institute for Advanced Research (CIFAR) Quantum Information Processing meeting, Toronto, November 2010 2‘Quantum secret sharing with continuous variable cluster states’, American Physical Society (APS) Division of Atomic, Molecular, and Optical Physics (DAMOP) Meeting 2013, Quebec City, Canada,

3 Chapter 1. Introduction

In Chapter 4, I introduce the background of the ion trap system that I studied, and the mathematical techniques that I employed to describe the dynamics of trapped ions. In Chapter 5, I present my work with Daniel James in examining the speed limit of a scalable ion trap quantum computer. The quantum information stored in ions would be influenced by direct current (dc) Stark effect if the ion transportation is too fast. We formulated the relation between the transportation speed and the magnitude of the Stark effect. Then we suggested an optimised transportation trajectory that minimises the influence. From the results, we defined a threshold speed of ion transportation, above which dc Stark effect would significantly alter the encoded quantum information. Our work was presented as a poster in the SQuInT meeting 2011 3, and was published as Ref. [106]. In Chapters 6 and 7, I present my work with Daniel James in proposing a new QIP application for the ion trap system: a universal bosonic simulator (UBS). The idea is that the motional state of trapped ions exhibits bosonic behaviours, so the system can be used to study the physics of other bosonic systems, such as optical systems. When comparing to conducting experiments directly on optical systems, our UBS has the advantages that state preparation can be more flexible, measurements can be more accurate, and nonlinear bosonic interaction is tuneable and can be arbitrarily strong. We have proposed two trapped ion UBS architectures. The architecture in Chapter 6 involves multiple ions trapped in a single harmonic potential. The initialisation and transformation of bosonic states can be conducted by applying radiation fields with precisely tuned frequencies. Although the quality of operation is substantially reduced when more than 4 bosonic modes are simulated, this architecture can be realised with present technology, and thus be useful for demonstrating simple but important bosonic phenomena. This work was presented as a poster in the ICAP meeting 2012 4. In Chapter 7, I introduce another UBS architecture that involves separately trapped ions. The initialisation and transformation of bosonic states can be achieved by varying the trap potential. This architecture is more scalable because the accuracy of mode operations will not reduce as the scale of simulation increases. The result of this chapter was also presented as a poster in the ICAP meeting 2012, and was published as Ref. [107]. In Chapter 8, I present my own work about a new method to re-cool ions during quantum computation. Currently, ion re-cooling process is considered as the speed bot-

June 2013 3‘Dephasing of trapped-ion qubit due to Stark shift during shuttling’, 13th annual Southwest Quan- tum Information and Technology (SQuInT) meeting, Boulder, U.S.A., February 2011 4‘Proposal for ion trap bosonic simulator’, International Conference on Atomic Physics (ICAP) 2012, Palaiseau, France, July 2012

4 Chapter 1. Introduction tleneck of an ion trap quantum computer. Re-cooling ions by state-of-the-art laser cooling takes an order of magnitude longer time than other quantum logical operations [78]. My method could resolve this speed bottleneck because laser cooling is not involved during the quantum computation. The principle of my method is to ﬁrst prepare some coolant ions in the ground state. When a qubit ion has to be re-cooled, it collides with a coolant ion. I show that if the collision is well-controlled, the motional excitation of the qubit ion will be completely transferred to the coolant ion. By using this method, a qubit can be re-cooled ten times faster than by laser cooling, thus the clock rate of an ion trap quantum computer can be improved. The result of this work was presented in the APS March meeting 2013 5, and was submitted to the Physical Review A [105]. In Chapter 9, I summarise my thesis and discuss several possible research directions in the future.

5‘Rapid laser-free ion cooling by controlled collision’, American Physical Society March Meeting 2013, Baltimore, U.S.A., March 2013

5 Chapter 2

Position-Based Quantum Cryptography

2.1 Introduction

In everyday life, we constantly place trust on spatial locations. For instance, when we deposit money in a bank, we seldom request the teller to prove his/her identity as a bank employee. This is because we believe the area behind the bank counter is a secure region where an imposter is rather difficult to get into, and more importantly, we have verified by our eyes that the “teller” is indeed inside this secure region. However, there are more cases that we cannot verify a person’s spatial location directly by our eyes. For example, if a postman is delivering a registered parcel, the recipient may be located in a distant building of where our vision is blocked by concrete walls. At this time, we may want a method to verify if the postman is delivering the parcel to the location we want. Location verification is believed to be achievable by a class of informatics method referred as position-based cryptography (PBC) [42]. The principle of PBC is to set up several trusted reference stations, called verifiers, to send out messages to the person, called prover, who is supposed to be in a designated position. In an ideal PBC protocol, the messages contain a challenge that the correct response can be made by only the prover at (or in the small neighbourhood of) the designated location. Thus the prover’s credential of spatial location can be verified by making the correct response. PBC was first studied in classical settings, of which the challenges are prepared as classical messages. Unfortunately, unconditionally secure 1 classical PBC has been proven

1Here I deﬁne the unconditional security as: no matter what kind of resources and cheating strategies are employed by the dishonest parties, they cannot obtain any information about the secret message

6 Chapter 2. Position-Based Quantum Cryptography

to be impossible [42, 43]. More explicitly, for any classical challenges sent by the veriﬁers, a group of cheaters, none of whom is inside the secure location, can reproduce the exact response as the prover. The main problem of the classical PBC was once (in 2010) believed to be that classical challenges can be cloned and re-sent by the cheaters.

If the problem is the duplicability of classical messages, a natural improvement of PBC is to employ quantum messages as the challenges. It is a well-known property that an arbitrary quantum state cannot be deterministically cloned with unity ﬁdelity [185, 52]. Based on this property, the quantum extension of some cryptographic tasks can be unconditionally secure, such as quantum key distribution [26, 60, 126, 120, 162] and quantum secret sharing [83, 51]. The possibility of Position-Based Quantum Cryp- tography (PBQC) was ﬁrst studied by Kent under the name of ‘quantum tagging’ as early as in 2002 [96], and the idea was later revisited independently by Chandran et al. [43] and Malaney [123, 122] at around 2010.

In the contrary to the authors’ claims of unconditional security, both the PBQC protocols suggested by Chandran et al. and Malaney were in fact insecure. As discussed by Kent, Munro, and Spiller [97], and in our work [108] (our publication that contains the material of the current chapter), cheaters can use entanglement resources and non-local quantum operations to produce the same response as if the prover in the secure region. By generalising the cheating strategy in [97, 108], every PBQC protocol was later shown to be insecure [39] and can be cheated eﬃciently [23].

In this chapter, I summarise my work in 2010 about the security of PBQC, which the material was published as Ref. [108]. In Sec. 2.2, I introduce the idea of position-based cryptography and discuss the insecurity of the classical protocols. In Sec. 2.3, I review the PBQC protocols in Refs. [39, 123], which are all known protocols when my work was conducted. In Sec. 2.4, I present the cheating strategy for the case that the verifiers and the prover are collinear located. In Section 2.5, I present the cheating strategy for a more general case that the verifiers are distributed in three-dimensional space. Discussion about the insecurity of the PBQC protocols and the loopholes of the claimed security proof are discussed in Sec. 2.6. In Sec. 2.7, I outline a modified protocol, of which the security analysis is given in Sec. A.1. I conclude this chapter in Section 2.8 with a remark about the prospect of PBQC.

without being detected by the honest parties.

7 Chapter 2. Position-Based Quantum Cryptography

2.2 Classical Position-Based Cryptrography

To appreciate the motivation of PBQC, it would be better to first understand the idea of classical PBC. For simplicity, I assume that all parties have synchronized clocks and work with a flat Minkowski space-time. A prover is supposed to be located at the position P surrounded by a finite secure region, to where no cheaters can access. To conduct a PBC protocol, N verifiers are established, at locations V1,...,VN , around P . For simplicity, I hereafter assume P is equidistant to all verifiers, but PBC can be modified to incorporate non-equidistant verifiers by changing the sending time of the messages. The simplest implementation of a PBC protocol involves two verifiers that are collinear with the secure region, and the prover is supposed to be in the middle between the verifiers. The layout is shown in Fig. 2.1 when t = 0. Before the PBC starts, the verifiers have to decide a challenge for the prover, and the information of the challenge is divided and distributed to each verifier. In the classical case, the challenge can be a secret encoded by classical secret sharing [31, 160]. As an example, the challenge is that the prover has to report a number, which is the sum of the numbers possessed by each verifier. When PBC starts, the divided information is sent simultaneously from the verifiers. The functionality of PBC is based on the fact that information cannot travel faster than the speed of light. As shown in the space-time diagram in Fig. 2.1, the signals are gathered at the soonest when the prover is located at the mid-point between the verifiers (point P in Fig. 2.1). Then the prover immediately computes the answer of the challenge, and reports the answer 2. In the original PBC [42], the prover’s location can be verified by the time that the response reaches the verifiers. In the honest case, i.e., the prover is located at the mid-point, response is made at t = d/c and will reach the verifiers at t =2d/c. I note that although the verifiers are collinear for N = 2, or coplanar for N = 3, all three coordinates of the prover can be verified. This is because the prover can take the shortest time to make the response, only if he is located at the mid-point of the straight line (for N = 2) or the centre of the triangle (for N = 3). I also note that P should be surrounded by the verifiers, even if each verifier can choose a different time of sending message. Otherwise, there must be places surrounded by V1,...,VN such that shorter or equal time is needed to receive all information. The idea is illustrated on Fig. 2.2.

2Unless speciﬁed, I assume hereafter that the operation time of the detectors, the computational devices, and the emitters are negligible when comparing with the travel time of the challenge.

8 Chapter 2. Position-Based Quantum Cryptography

t 2l t 2d/c

(d+l)/c

(d-l)/c

0 V BP B V

Figure 2.1: Space-time diagram of the one dimensional position-based cryptography scenario. The verifiers are separated by a distance 2d, and the secure region (shaded area) is in the middle with length 2l. Solid lines denote space-time trajectory of the information, which can be quantum or classical, while double lines denote that of classical information only. The protocol starts at t = 0, when message of the challenge is sent by the verifiers. At t = (d l)/c (squares), the messages reach the boundary of the secure region, the closest to the− mid-point the cheaters can locate. At t = (d l)/c (circles), the message − or response exits the secure region. In the honest case, i.e., the prover is located at the mid-point, response is made at t = d/c and reach the verifiers at t = 2d/c. If there is only one cheater at B1, the correct response is made at t = (d + l)/c and reach V2 at time later than 2d/c (trajectory is shown as dot-dashed line).

9 Chapter 2. Position-Based Quantum Cryptography

V V P

Figure 2.2: At a particular time t1, the front of signals sent from V1,V2,V3 are represented by solid, dashed, and short-dashed lines respectively. While signals reach P at t = t1, another position P2 inside the triangle of three veriﬁers (framed by dotted lines) can obtain all information before t1.

2.2.1 Individually Cheating

The classical PBC is secure if there is only one cheater. In the case shown in Fig. 2.1, let

us assume the cheater is located at B1. The cheater can intercept the message from V1 at t =(d l)/c, but the message from V reaches at t =(d + l)/c at the earliest. Therefore, − 2 the response can be made only at t = (d + l)/c, and the response will reach V2 at time 2l/c later than in the honest case. Thus the veriﬁers can detect the presence of cheater.

2.2.2 Collaborative Cheating

However, the classical PBC is insecure if more than one collaborating cheaters are involved. In the case shown in Fig. 2.1, one cheater is located at B and B . At t =(d l)/c, 1 2 − the cheater at B1 (B2) intercepts the message from V1 (V2). A copy of the message is saved before it is forwarded to another cheater. At t = (d + l)/c, both cheaters gather both piece of message. Therefore the correct response can be made, and the response will reach the veriﬁers at the same time as if produced by the prover inside the secure region.

2.3 PBQC Protocols

The crucial step in the collaborative cheating of classical PBC is to faithfully clone the message from the verifier. The aim of PBQC is to solve this problem by using the quantum property of no-cloning. While the layout and procedure of PBQC is the same as classical PBC, the only difference is that the verifiers encodes the challenge information in a (or part of an entangled) quantum state, and the challenge for the prover is a

10 Chapter 2. Position-Based Quantum Cryptography

transformation or measurement of the state. The information to be reported can be the transformed quantum state or the measurement result. In the following, I describe the protocols suggested by Chandran et al. [43] (hereafter referred as Protocol A) and Malaney [123, 122] (hereafter referred as Protocol B).

2.3.1 Protocol A

The idea of Protocol A is to send the basis of measurement and the encoded qubit separately from different verifiers. Security of this protocol relies on the idea that a quantum state can be measured perfectly (obtain the information encoded in a quantum state deterministically with 100% accuracy) only if the correct measurement basis is known. The procedures of Protocol A is shown as follows. Step 1. The verifier at V encodes a classical message u 0, 1 as a qubit u , where 1 ∈{ } | i 0 and 1 are respectively the +1 and 1 eigenstate of the Pauli Z operator 3. Inspired | i | i − by the BB84 QKD protocol [26], the message is encrypted by applying the transformation Hq on the qubit, where H is the Hadamard gate 4, and q is a random bit valued 0 or 1.

Step 2. The V1 veriﬁer generates N 2 random bits q2, q3,...,qN 1, and decide a bit − − qN by the relation q = q + q + + q mod 2 . (2.1) 2 3 ··· N

The bits q2, q3,...,qN are respectively distributed to the verifiers at V2,...,VN . The classical message u is also sent to other verifiers. The communication between verifiers is assumed to be secure, for example QKD system is employed.

Step 3. Verifiers compromise to send the messages to prover at the same time t = t0. The V verifier sends the encoded qubit Hq u , while other verifiers send the classical bits 1 | i qi. Step 4. Upon receiving all information, the prover at P adds up all classical bits to obtain q. The qubit can then be decrypted by applying Hq. The decrypted state is measured in Z basis to obtain the encoded message. The prover immediately reports the results to all verifiers.

Step 5. If qi’s are random, missing any one classical bit would cause half chance of wrong measurement basis. veriﬁers validate the identity of the prover by checking if the reported message matches the encoded message. By checking the arrival time of the response, the location of the prover is also veriﬁed.

3Throughout this chapter, the eigenvalues and measurement outcomes of a Pauli operator are +1 or -1. 4i.e., H 0 = + and H2 = 1, see Ref. [143] for more information | i | i

11 Chapter 2. Position-Based Quantum Cryptography

2.3.2 Protocol B

The idea of Protocol B [123] is to encode information into a maximally entangled state. The state is then distributed to verifiers, and encrypted by each verifier with a random local transformation. The classical information about the transformation will be sent from the verifiers. Security of this protocol relies on the intuition that the state cannot be decrypted and then perfectly measured before all local transformation information is gathered. The procedure of Protocol B is shown as follows. Step 1. N bits of message is encoded as a N qubit Greenberger-Horne-Zeilinger (GHZ) state 1 GHZ = ( a1 a2 ... aN 1 a1 1 a2 ... 1 aN ) , (2.2) | i √2 | i| i | i±| ⊕ i| ⊕ i | ⊕ i where a ,...,a 1, 0 ; denotes addition with modulus 2. Each qubit of the state 1 N ∈ { } ⊕ is distributed to a separate verifier.

Step 2. After encrypted by a local transformation Ui, the veriﬁers send their qubit to the prover. The prover will store the entangled state in a perfect quantum memory.

Step 3. At an agreed time t = t0, each veriﬁer sends the classical information about

the transformation Ui to the prover. Step 4. After the classical information is received, the prover decrypts the state back to a N-qubit GHZ state. The encoded message is then obtained by quantum measurement. The result is immediately announced. Step 5. The measurement result is probably wrong if the state is measured before decryption. Hence the identity of the prover can be authenticated from the announced result, and the prover’s location can be veriﬁed by the time of response.

2.4 Cheating in N =2 Case

Contrary to the claim(s) of unconditional security, both Protocols A and B are, in fact, insecure. I first demonstrate the cheating strategy for the case with two verifiers. I will consider the scenario shown in Fig. 2.1. The general strategy for the case with more verifiers will be discussed in the next section.

2.4.1 Cheating against Protocol A

Protocol A was once believed to be unconditional secure. In fact, a detailed claim of security based on complementary information tradeoﬀ was given in Ref. [43]. The intuition behind the claimed security proof is that, if the correct basis is not known, any

12 Chapter 2. Position-Based Quantum Cryptography

measurement on the encrypted qubit would inevitably disturb the state. Therefore the measurement outcome would be wrong with non-zero probability. The problem of this claim is the implicit assumption that no prior entanglement is shared by the cheaters. In fact, quantum teleportation can be conducted by appropriately measuring the qubit and the shared entangled resources [27]. The teleportation measurement would not extract any information about the quantum state. Thus the security claim by Chandran et al. is wrong in this case, i.e., measurement can cause no distortion on the qubit, while the no-cloning theorem is still obeyed. The main idea of

my cheating scheme in the N = 2 case is to teleport the encrypted qubit from B1 to B2 for measurement in the correct basis. Detailed procedure of the cheating strategy is as follows. Step 1. Before the cheaters move to the destination, they come together and each pick a qubit from a Bell state

1 1 1 Φ00 ( 00 + 11 )= ( ++ + )= ( +i i + i +i ) . (2.3) | i ≡ √2 | i | i √2 | i |−−i √2 | − i | − i

5 Their quantum memory is assumed to be perfect, so that the state remains coherent until measurement. Step 2. At t = 0, V veriﬁer sends out the qubit Hq u , and V veriﬁer sends out the 1 | i 2 classical bit q = q that contains the basis information. At t = (d l)/c, B cheater 2 − 1 captures the qubit and B2 cheater obtains the classical bit. To avoid suspicion of the prover, I assume the cheaters have the power to send dummy qubit and basis information to the prover, and block the prover’s subsequent response. I hereafter neglect the role of the prover in the cheating procedure.

Step 3. B1 cheater immediately teleports the captured qubit to B2 cheater by performing a Bell measurement on the captured and the Bell state qubit. The circuit of the Bell measurement is given in Fig. 2.3. The measurement outcomes of the encrypted qubit, s1, and Bell state qubit, s2, are then sent to B2 cheater. Step 4.At the same instance t = (d l)/c, the teleported qubit being held by B − 2 cheater becomes

(1 s2)/2 (1 s1)/2 q X − Z − H u . (2.4) | i Let us consider if q = 0, the state becomes

(1 s2)/2 (1 s1)/2 u(1 s1)/2 X − Z − u =( 1) − u (1 s )/2 . (2.5) | i − | ⊕ − 2 i

5 i = ( 0 i 1 )/√2 is the 1 eigenstate of the Y operator. |± i | i± | i ±

13 Chapter 2. Position-Based Quantum Cryptography

X Z

Figure 2.3: Circuit for teleporting an unknown qubit ψ [143, 27]. Measurement is | i denoted as squares, the measurement basis is represented by the character inside the squares. Because no-signalling theorem has to be obeyed, the teleported state is trans- (1 s2)/2 (1 s1)/2 formed by a byproduct, UΣ = X − Z − , which depends on random measurement outcome s1 and s2.

Since B2 cheater knows the basis is Z, and the state in Eq. (2.5) is an eigenstate of the Pauli Z operator, a perfect measurement can be conducted with the outcome ( 1)us . − 2 On the other hand, if q = 1, the state becomes

(1 s2)/2 (1 s1)/2 [u (1 s1)/2](1 s2)/2 HZ − X − u =( 1) ⊕ − − H u (1 s )/2 . (2.6) | i − | ⊕ − 1 i

Since B2 cheater knows the basis is X, and the state in Eq. (2.6) is an eigenstate of the Pauli X operator, a perfect measurement can be conducted with the outcome ( 1)us . − 1 B2 cheater immediately sends the measurement outcome to B1 cheater. I note that al-

though the measurement outcome of B2 cheater contains information about the outcome

of B1 cheater, no superluminal communication can be implemented by using quantum

teleportation. This is because B1 cheater cannot choose the measurement result deterministically.

Step 5. At t =(d + l)/c, both B1 and B2 cheaters know all the measurement results, i.e., s ,s , ( 1)us or ( 1)us , as well as the correct measurement basis, q. The secret 1 2 − 1 − 2 u can then be inverted by both cheaters. B1 (B2) cheater reports the result to V1 (V2) veriﬁer. As shown in Fig. 2.1, both veriﬁers will receive the correct response at t =2d/c. The cheaters use the same amount of time to produce the same correct response as the prover, Protocol A is therefore insecure. The crucial process in the cheating strategy is the teleportation in Step 3. The

teleported state received by B2 cheater will be acted upon by one of the four teleportation byproduct, I,X,Z and XZ. Since the encrypted state is an eigenstate of either X and Z, the teleported state would either be orthogonal to or the same as (up to an irrelevant overall phase) the encrypted state. In other words, the teleportation does not change the basis of the state; this is the reason that the cheating strategy works. Therefore,

14 Chapter 2. Position-Based Quantum Cryptography

B2 cheater, who has the basis information, can simply measure the qubit in that basis without disturbing the state. Subsequently, after hearing the actual Bell measurement outcomes from B1 cheater, B2 cheater will be able to tell what the encrypted state is. For this reason, cheating can be successful with certainty. The cheating strategy can also be understood in another viewpoint. Since the measurements by B1 and B2 cheaters commute, it is also legitimate to consider that B2 cheater performs the measurement before B1 cheater does. In this case, B2 cheater measures the Bell state qubit in the correct basis. By the Einstein-Podolsky-Rosen eﬀect, the Bell state qubit being held by B1 cheater will be projected to either the same state

or the opposite state to the encrypted qubit. Then the duty of B1 cheater is to perform a parity check on the two qubits. While a parity check is impossible for arbitrary basis, it is possible for the basis only considered by Protocol A: Pauli X and Z basis. In this case, it happens that the operator

XX commutes with ZZ, so B1 cheater can perform a parity check by simply doing a Bell measurement. More explicitly, the Bell measurement will project the qubits to one of the four states in Eq. (2.3) and

1 1 1 Φ01 ( 00 11 )= ( + + + )= ( +i +i + i i ) , (2.7) | i ≡ √2 | i−| i √2 | −i | − i √2 | i | − − i 1 1 i Φ10 ( 01 + 10 )= ( ++ )= − ( +i +i + i i ) , (2.8) | i ≡ √2 | i | i √2 | i−|−−i √2 | i | − − i 1 1 i Φ11 ( 01 10 )= − ( + + )= ( +i i i +i ) . (2.9) | i ≡ √2 | i−| i √2 | −i − | − i √2 | − i−|− i

If the qubits are in the Z basis and same (diﬀerent) in parity, the measurement results can be only Φ and Φ ( Φ and Φ ). Similarly, if the qubits are in the X basis | 00i | 01i | 10i | 11i and same (diﬀerent) in parity, the measurement results can be only Φ and Φ ( Φ | 00i | 10i | 01i and Φ ) 6. For these reasons, Protocol A can be cheated successfully with certainty. | 11i

2.4.2 Cheating against Protocol B

In the two-veriﬁer case, 2 bits of information, ab = 00, 01, 10, 11 , are encoded into one { } of the four Bell states Φ [123], i.e., Eqs. (2.3) and (2.7)-(2.9). After encrypted by | abi random local transformation U1 and U2, the state is sent to P . At t = 0, V1 (V2) veriﬁer

broadcasts U1 (U2). If the prover is at P , correct response will reach veriﬁers at t =2d/c. To cheat this protocol, each cheater captures and decrypts a qubit when the decryption information is received. The decrypted state is then teleported to one cheater for

6I note that Bell measurement can also be checked the parity for two qubits in Y basis.

15 Chapter 2. Position-Based Quantum Cryptography

Bell state measurement. The step by step procedure is shown as follows.

Step 1. Before the PBQC process, the cheaters share a Bell state as in Eq. (2.3), and store it in good quantum memory.

Step 2. The cheaters capture the state sent by the veriﬁers. The state is then stored in another quantum memory.

Step 3. At t =(d l)/c, B (B ) cheater receive the classical information of U (U ) − 1 2 1 2 from the V1 (V2) veriﬁer. The stored state can then be decrypted by the inverse of the local transformation, i.e., U † and U †. The encoded state Φ is hence recovered. 1 2 | abi

Step 4. Immediately after the decryption, B2 cheater teleports the encoding qubit to

B1 cheater by using the pre-shared Bell state. To explicitly describe the change of the state during the teleportation, I here employ the stabilizer formalism as introduced in Ref. [2]. Before the teleportation, the stabilisers of the states are

K =( 1)aZ Z ; K =( 1)bX X ; K = Z Z ; K = X X , (2.10) 1 − 1 2 2 − 1 2 3 3 4 4 3 4

where qubits 1 and 2 are respectively the encoded qubit captured by B1 and B2 cheater;

qubits 3 and 4 are respectively the pre-shared Bell state qubit of B1 and B2 cheater. B2 cheater ﬁrst applys a CNOT gate on his qubits, the stabilisers then become

K =( 1)aZ Z ; K =( 1)bX X X ; K = Z Z Z ; K = X X . (2.11) 1 − 1 2 2 − 1 2 4 3 2 3 4 4 3 4

Qubit 2 is then measured in the X basis and the result is s2; qubit 4 is measured

in the Z basis and the result is s4. These outcomes are then sent to B1 cheater. The stabilizers after the measurement is

a b K′ =( 1) s Z Z ; K′ =( 1) s X X ; K′ = s Z ; K′ = s X . (2.12) 1 − 4 1 3 2 − 2 1 3 3 2 2 4 4 4

Qubit 2 and 4 are obviously no longer entangled as they are measured. K1′ and K2′

show that qubits 1 and 3 become a Bell state Φ ′ ′ , where a′ = a + (1 s )/2 and | a b i − 3 b′ = b +(1 s )/2. So B cheater can measure the state perfectly by Bell measurement, − 2 1 the outcomes a′ and b′ are then sent to B2.

Step 5. At t =(d + l)/c, both cheaters obtain information from each others. The en-

coded message a and b can be calculated from a′, b′, s2 and s3. The cheaters immediately report the result, which will reach the veriﬁers at t =2d/c. Since the cheaters can make the correct response within the same time as the prover, Protocol B is thus insecure.

16 Chapter 2. Position-Based Quantum Cryptography

V V

B B P l B d

Figure 2.4: Locations of the three verifiers, three cheaters, and the prover are shown as black dots. The shaded region represents the restricted area surrounding P . Without cheating, information flows along solid lines; if cheating presents, information flows along solid lines outside the restricted area and follows dotted lines in the restricted area. As the path of V2 P V1 is longer than V2 B2 B1 V1, the process of cheating costs shorter time→ than→ the honest case. → → →

2.5 Cheating in N > 2 Case

The above cheating strategy can be generalised to the situation with more than two verifiers. I first consider the case with three verifiers; the case with more verifiers will be discussed later. For simplicity, I assume the location of verifiers, V1,V2,V3, are the vertices of an equilateral triangle. The location of the prover, P , is the centre of the triangle, with a distance d from each verifier. P is surrounded by a restricted area with radius l. In this case, three cheaters are sufficient to cheat both Protocol A and B. I assume the cheaters are located at B1, B2, B3, which are just outside the restricted area and along the straight lines linking P with V1,V2,V3. The layout of the scenario is shown in Fig. 2.4.

2.5.1 Cheating against Protocol A

In this protocol, V verifier prepares the encoded state u and encrypts it as Hq u . The 1 | i | i encrypting information q2 and q3, where q = q2 + q3 is distributed to the verifiers at V2 and V3 respectively. At t = 0, the V1 verifier sends the encrypted state, and V2 and V3 cheaters send q2 and q3, to the prover. They expect the result of u will return at t =2d/c. In this case, the cheaters are not going to teleport the qubit as there is only one qubit but two separate pieces of encrypting information. Instead they need a method to share the encrypting information. I find this task can be accomplished by techniques in measurement-based quantum computation (MBQC) [147, 148]. The steps of the cheating

17 Chapter 2. Position-Based Quantum Cryptography

Table 2.1: Tables of stabilizers in diﬀerent cases of qi’s. K1 is the stabilizer of GHZ state compatible with the measurement basis. K1′′, K2′′, and K3′′ are stabilizers after the measurement according to the cheating scheme.

q2 q3 K1 K2′′ K3′′ K1′′ 0 0 X1X2X3 s2X2 s3X3 s2s3X1 0 1 Y X Y s X s Y s s Y − 1 2 3 2 2 3 3 − 2 3 1 1 0 Y Y X s Y s X s s Y − 1 2 3 2 2 3 3 − 2 3 1 1 1 X Y Y s Y s Y s s X − 1 2 3 2 2 3 3 − 2 3 1

scheme are shown as follows. Step 1. Before the PBQC process, each cheater share a qubit from a 3-particle GHZ state, i.e, 1 Φ000 = ( 000 + 111 ) . (2.13) | i √2 | i | i Step 2. The veriﬁers send out their information at t =0. At t =(d l)/c, B cheater − 1 captures the encrypted state, and B2 and B3 cheaters get q2 and q3.

If qi = 0, Bi cheater measures the GHZ qubit in X basis; otherwise he measures in Y basis. As a property of the GHZ state, if both B2 and B3 cheater measure in the same basis, the GHZ qubit holding by B1 cheater becomes an eigenstate of Pauli X operator, otherwise it is an eigenstate of Y operator. To explain this property more explicitly, let us consider the stabilisers of the GHZ state in Eq. (2.13) are

K1 = X1X2X3 ; K2 = Z1Z2 ; K3 = Z1Z3 . (2.14)

When qubit 2 (3) is measured in Y , the stabiliser K1 has to be modiﬁed by multiplying with K2 (K3). The resultant stabilisers in each case is shown in Table 2.1. After measurement, the compatible operator in a stabiliser is replaced by a number representing the 7 measurement result . Upon all the combination of q2 and q3, qubit 1 will be stabilised by X (Y ) if qubit 2 and 3 are measured in the same (diﬀerent) basis.

Step 3. Immediately after the measurement, B1 cheater applies a Hadamard transformation H followed by a π/4-gate, S 8, on the GHZ state qubit. This operation is to transform the eigenstates of X to that of Z, and the eigenstates of Y to that of X, with the same eigenvalues. After this operation, the GHZ qubit of B1 cheater will become a

Z eigenstate if q2 + q3 is even, otherwise the qubit is a X eigenstate. At the the same

7More information about the stabiliser formalism is referred to Refs. [143] and [2]. 8S + = +i ; S2 = Z. | i | i

18 Chapter 2. Position-Based Quantum Cryptography

time t =(d l)/c, B cheater also receives the encrypted qubit. It is easy to see that the − 1 two qubits of B1 cheater are in the same basis, i.e. both qubits are eigenstates of either

X or Z. B1 cheater then performs a parity check by conducting a Bell measurement. Step 4. The cheaters share their measurement outcomes and basis information. Since the mutual distance between B1, B2 and B3 is √3l, the cheaters can obtain all the information at t = (d +(√3 1)l)/c. From the information of B and B cheaters, − 2 3 the actual state of the GHZ qubit of B1 cheater is known from Table 2.1. With the information of parity from the Bell measurement outcome, the state of the encoded qubit is obtained.

Step 5. If the correct result is immediately reported by the cheaters at t =(d+(√3 − 1)l)/c, the response will reach the veriﬁers at t = (2d+(√3 2)l)/c, which is earlier than − the expected response from the prover at t = 2d/c. The cheaters can simply delay the report for an appropriate time, in order to match the time span in honest case. Hence the protocol is cheated.

I note that the time is shortened because information takes 2l/c time to travel from

B1 to B2 in the honest case, while only √3l/c is needed if there are cheaters. In general if the cheater’s locations do not form an equilateral triangle, the cheating scheme may still process faster than the honest case, provided that P is not on the same straight line as any two veriﬁers. This is because in the honest case, the information has to be sent from a vertex to the centre of triangle, and then resent to another vertex, while information from the cheaters can be sent through a shortcut, as shown in Fig. 2.4.

The above cheating scheme can be generalized to cases with N > 3 veriﬁers; at most N cheaters are needed in each case. Before the PBQC process, the cheaters create a N particle GHZ state, of which the stabilisers are

K1 = X1X2 ...XN , Ki = Zi 1Zi , (2.15) −

for i =2, 3,...,N. The ith cheater picks the ith qubit from the GHZ state, and travels

to a position Bi between P and Vi.

When each cheater at B2,...,BN receives the basis information, the GHZ qubit is

measured in X basis if qi = 0, or Y basis if qi = 1. The result is sent to other cheaters. If

even number of q’s are equal to 1, the qubit of B1 is will be in the X basis, otherwise it becomes an eigenstate of the Y basis. Such phenomenon can be understood as the following. In the even number case, the Y measuring qubits can be grouped in pairs. Let us consider, for example, the ﬁrst pair is the qubits m and n. In order to construct a stabiliser that

19 Chapter 2. Position-Based Quantum Cryptography

is compatible to the measurement basis 9, the ﬁrst stabiliser is modiﬁed by multiplying

those in Eq. (2.15) as K1′ = K1Km+1Km+2 ...Kn = X1 ...YmXm+1 ...Xn 1YnXn+1 ... . − − This new stabiliser K1′ is compatible to the measurement basis of the qubits m and n. The stabiliser can be similarly modiﬁed to make compatible to other pairs of Y mea-

suring qubits. The final stabiliser will consist of a X1 operator, Y operator for those Y measuring qubits, and X operator for the X measuring qubits. When all qubits except qubit 1 are measured, qubit 1 will then become an eigenstate of X, with an eigenvalue depending on measurement outcomes of other qubits. On the other hand in the odd number case, the Y measuring qubits can also be grouped in pair with one singled out. The stabiliser can be modified as the even number case for the pairs. For the singled out qubit, say qubit r, the stabiliser has to be modified

as K′ = K K ...K = Y ...Y ... . Then the ﬁnal stabiliser will consist of a Y 1 1 2 r − 1 r 1 operator, Y operator for those Y measuring qubits, and X operator for the X measuring qubits. When all qubits except qubit 1 are measured, qubit 1 will then become an eigenstate of Y , with an eigenvalue depending on measurement outcomes of other qubits.

Identical to the N = 3 case, B1 cheater applies a SH gate onto his GHZ state qubit, he then obtains an eigenstate of X operator if q is odd, or an eigenstate of Z operator if q is even. He then check the parity of the GHZ state qubit and the encrypted qubit by measuring in the Bell basis; the measurement outcome is then shared with other cheaters. In the present case of N > 3, cheaters do not receive all information at the same time, but it is easy to check that even the slowest piece of information should arrive at the

latest as in the honest case. The information provided by B2,...,BN cheaters determines

the actual state of the GHZ state qubit of B1 cheater, and the parity check measurement

of B1 cheater reveals the identity of the encrypted qubit. Hence the value of encoded message is obtained and sent to the veriﬁers by the cheaters. The whole process takes time fewer than or the same as in the honest case.

2.5.2 Cheating against Protocol B

In Protocol B with three veriﬁers, three bits of information is encoded as one of the eight

tripartite GHZ states [123] characterized by parameters b1, b2, b3:

1 b1 Φb b b = ( 0 b2 b3 +( 1) 1 1 b2 1 b3 ) , (2.16) | 1 2 3 i √2 | i| i| i − | i| ⊕ i| ⊕ i

9If a stabiliser is compatible to a measurement basis, it is a tensor product of the measurement operator.

20 Chapter 2. Position-Based Quantum Cryptography

where b , b , b 0, 1 . This state is stabilised by 1 2 3 ∈{ }

K =( 1)b1 X X X ; K =( 1)b2 Z Z ; K =( 1)b3 Z Z . (2.17) 1 − 1 2 3 2 − 1 2 3 − 1 3

The ith GHZ qubit is distributed to the Vi veriﬁer. The qubit is then encrypted by an

arbitrary local transformation Ui, and all encrypted qubits are subsequently sent to P .

At t = 0, veriﬁers send classical information of Ui to P . In the honest case, the prover

decrypts and measures the state to get b1, b2, b3. Correct results should return to the veriﬁers at t =2d/c. In this case, three cheaters are suﬃcient to cheat. The idea is the same as in the N = 2 case. The cheaters decrypt and teleport the captured qubits to one cheater. Then the encoded information can be obtained by conducting the GHZ state measurement. The cheating strategy is as follows.

Step 1. Before the PBQC process, a Bell state in Eq. (2.3) is shared between B1 and

B2 cheaters, as well as between B1 and B3 cheaters. I denote the Bell state qubits of B1

and B2 (B1 and B3) cheaters as qubit 4 and 5 (6 and 7). Step 2. The cheaters capture the encrypted qubits. At t =(d l)/c, cheaters receive − the classical information of Ui. The qubits are then decrypted and the encoding state is obtained.

Step 3. B2 and B3 cheaters teleport the encoded qubits by using the teleportation

protocol described in Fig. 2.3. Afterwards, the qubits of B1 cheater is stabilized by

b1 b2 b3 K′ =( 1) s s X X X ; K′ =( 1) s Z Z ; K′ =( 1) s Z Z . (2.18) 1 − 2 3 1 4 6 2 − 5 1 4 3 − 7 1 6

These stabilisers indicate that qubits 1, 4 and 6 become a GHZ state Φb′ b′ b′ , where | 1 2 3 i b′ = b +(1 s s )/2, b′ = b +(1 s )/2, b′ = b +(1 s )/2. B cheater then conducts 1 1 − 2 3 2 2 − 5 3 3 − 7 1 a GHZ state measurement to reveal b1′ , b2′ , and b3′ . The result is sent to other cheaters. Step 4. Information exchange among cheaters is ﬁnished at t = (d +(√3 1)l)/c. − The encoded message b1, b2, b3 can easily be inferred from the measuring outcomes. Just as the cheating of Protocol A, the cheaters delay the report for an appropriate time, so the results can reach the veriﬁers at t = 2d/c. Since the response is the same as if the prover is at P , Protocol B is hence cheated. For N > 3, the procedure of cheating is more or less the same. Each cheater shares

a Bell state with B1 cheater. After the encoded qubit is captured and decrypted, it is

teleported to B1 cheater. A N-qubit GHZ state measurement is applied on the teleported state. The measurement results are shared among cheaters, and the encoded message can be obtained. It is easy to see that the cheaters can produce the same correct results

21 Chapter 2. Position-Based Quantum Cryptography

at the same time as the prover at P .

2.6 Principle of the cheating schemes

2.6.1 Protocol A

Before discussing the reason that cheating is possible, I note that the strategy discussed in Sec. 2.4 and 2.5 does not cheat only Protocol A that the encrypted states are BB84 states (eigenstates of X and Z operators), but it can also cheat a more general protocol that the encrypted states are the eigenstates of any Pauli (X, Y , or Z) operators 10. In the two-verifier case, this new protocol can be cheated by the same procedure as described in Sec. 2.4. There are two main reason for this: First, the teleportation byproducts, I,X,Z, and XZ, does not change the basis of the eigenstates of Pauli operators. Second, Bell measurement can be acted as a parity check for two states that are the eigenstates of the same Pauli operator. In the case of more than two verifiers, the strategy can be modified to cheat a more general protocol that the encoded qubit is encrypted as ... u , where is a Clifford CN C2| i Ci operator that transforms an eigenstate of Pauli operator to that of (possibly another) Pauli operator. The classical information of each will be sent to the prover from the Ci Vi verifier. Instead of using a N-particle GHZ state, the cheaters share a 4N 3 particles chain − 11 cluster state for conducting MBQC. B1 cheater picks the end qubit of the chain, while other cheaters pick four consecutive qubits from the chain. As stated in Ref. [148], arbitrary rotation of the state can be conducted by measuring three consecutive cluster state qubits in appropriate directions, while the state can be teleported to the next qubit in the chain by measuring the fourth qubit in the X basis. To cheat the current protocol, each cheater conducts MBQC on the four cluster state qubits to implement on the Ci cluster state qubit of B1 cheater. Therefore after all measurements, the qubit will be in

the same basis as the incoming qubit. Then B1 cheater measures the encoded qubit and cluster state qubit in the Bell basis for a parity check. I note that all cluster state qubits can be measured at the same time, because the measurements are local operations and obviously independent to each other. In summary, this protocol aims to encrypt the quantum message by using Cliﬀord operators, but it can be cheated by using MBQC. The principle for the success of the

10This more general protocol includes Protocol A because Hadamard operator is a Cliﬀord operator. 11A chain cluster state can be constructed by preparing a chain of + qubits, and then applying Controlled-phase gate on two consecutive qubits. | i

22 Chapter 2. Position-Based Quantum Cryptography

cheating strategy is related to several properties of Cliﬀord and Pauli operators. Firstly, the byproducts of MBQC are Pauli operators [148]. More explicitly, the cluster state qubit of B cheater, ψ , after MBQC is given by [148] 1 | outi

N ψ = (U ) 0 , (2.19) | outi Σi Ci | i i=2 Y

where UΣi is the Pauli byproduct depending on the measurement outcomes of Bi cheater. The next useful property is that Cliﬀord gates are, by deﬁnition, maps a Pauli operator to another Pauli operator, i.e.,

= ′ , (2.20) CP P C where and ′ are some Pauli operators. By using this properties, ψ can be ex- P P | outi pressed as ψ = ... U 0 , (2.21) | outi CN C2 Σ| i

where UΣ is a Pauli operator. Finally, I observe that any Pauli operator does not change the basis on 0 states, but it only ﬂips or add an unimportant global phase on the qubit, | i i.e., ψ = eiφ ... 0 or 1 . (2.22) | outi CN C2| i where eiφ = 1 or i. As a result, ψ will be in the same basis as the encrypted state ± ± | outi ... u . Because the two qubits are eigenstates of a Pauli operator, their parity CN C2| i can be checked by a Bell measurement. After sharing the measurement information, all cheaters will know the encoded message u and will report to veriﬁers. The whole process takes the same or shorter time than the honest case, thus the protocol is insecure.

2.6.2 Protocol B

The security argument of Protocol B is based on the quantum no-cloning theorem [123]. However, it is not necessary to clone the state in order to remotely conduct a perfect measurement. The problem of Protocol B is that each codeword of a GHZ state is related to other codewords by bit-ﬂips and phase-shifts. Since the random byproducts of teleportation are Pauli operators, the teleported state is only bit-ﬂipped or phase-shifted but remaining in the codespace. As a result, a standard GHZ state measurement can perfectly measure the teleported state.

23 Chapter 2. Position-Based Quantum Cryptography

2.7 Modiﬁed PBQC Protocol

In the previous section, I have discussed that the generalised Protocol A is insecure because the encoded state is encrypted by Clifford operators. To resist the cheating strategy discussed in Sec. 2.4 and 2.5, a natural modification to Protocol A is to encrypt the encoded state by non-Clifford operation. One example is to transform the encoded state as,

θ θ 0 ψ = cos 0 + sin eiφ 1 , (2.23) | i → | 0i 2| i 2 | i θ θ 1 ψ = sin 0 cos eiφ 1 , (2.24) | i → | 1i 2| i − 2 | i where the polar angles are randomly picked from 0 θ π and 0 φ 2π. The states ≤ ≤ ≤ ≤ ψ and ψ are the 1 eigenstates ofn ˆ(θ,φ) ~σ, where ~σ = Xxˆ + Y yˆ + Zzˆ is the Pauli | 0i | 1i ± · vector, andn ˆ(θ,φ) is a unit vector pointing along the polar angles θ and φ. I refer this modiﬁed protocol as Protocol A′ hereafter. A similar protocol was also proposed in Ref. [97].

While the configuration is the same as Protocol A, the only difference in Protocol A′ is that the encoded state prepared by V cheater is U ...U u , where each U is a single 1 N 2| i i qubit rotation. The classical information of Ui is delivered to Vi verifier, and will be sent to the prover. It is not difficult to check that the cheating strategy demonstrated in

Section 2.4 and 2.5 cannot cheat Protocol A′ perfectly. In the two-veriﬁer case, suppose B cheater captures the state ψ at t =(d l)/c, and teleports to B cheater immediately. 1 | ii − 2 Although B2 cheater knows the basis from V2 veriﬁer, the teleported state is in general neither parallel nor anti-parallel with ψ , i.e. for a non-trivial teleportation byproduct, | ii the matrix element

(1 s2)/2 (1 s1)/2 ψ X − Z − ψ =0or1 , (2.25) h | | i 6 where s1 and s2 are the measurement outcomes of the teleportation measurement. There- fore B2 cheater cannot perfectly measure the qubit before hearing the measurement outcomes from B1 cheater, but the information of B1 cheater arrives B2 at the earliest t =(d + l)/c. Even if B2 cheater measures the qubit immediately at t =(d + l)/c, correct feedback will reach V1 no earlier than t = 2(d + l)/c, which costs more time than the honest case. The security of Protocol A′ is hence enforced.

Similarly in the case of more veriﬁers, the cluster state qubit of B1 cheater will be transformed by MBQC as N ψ = (U U ) 0 . (2.26) | outi Σi i | i i=2 Y

24 Chapter 2. Position-Based Quantum Cryptography

Since U ’s do not belong to the Cliﬀord group, ψ is generally not in the same basis as i | outi ψ . Furthermore, even if the states are in the same basis, parity check by Bell measure- | 0i ment is not possible as the states are not eigenstates of Pauli operators. Therefore, B1 cheater cannot make a perfect measurement before receiving the information from other cheaters. Thus the total time to make the correct response is slower than the honest case. In practice, neither the quantum operations, quantum channel, nor measurements are noiseless, incorrect response can be given even in the honest case. The total error rate of a practical PBQC system ought to be lower than the rate of fail cheating, i.e. probability that the cheaters make incorrect response. Otherwise failure of cheating may be regarded as error caused by noise, and the PBQC protocol becomes insecure.

I now discuss the successful cheating rate of Protocol A′ under various simple cheating schemes in the two-veriﬁer case. First of all, I consider B1 cheater measures the encrypted qubit in a random basis once received. Since θ and φ are random, I can assume every measurement is made in the Z basis without loss of generality. Let us assume the measurement outcome is 0 . The probability that the encrypted state is ψ ( ψ ) is equal | i | 0i | 1i to 0 ψ 2 = cos2(θ/2) ( 0 ψ 2 = sin2(θ/2)). After receiving the basis information |h | 0i| |h | 1i| from B cheater, if θ <π/2, ψ is more likely to be the encrypted state; otherwise if 2 | 0i θ >π/2, ψ is more likely. So the cheaters report ψ when θ <π/2, and ψ when | 1i | 0i | 1i θ > π/2. After simple calculations, the total probability for the cheaters to make the correct response is 75%.

Next, I consider cheaters conduct the teleportation cheating scheme in Sec. 2.4. B2 cheater measures the teleported state in the basis ψ , ψ . Consider if the result is {| 0i | 1i} ψ . After knowing the teleportation measurement outcomes s and s , the cheaters | 0i 1 2 announce the more probably correct result, i.e.

(1 s2)/2 (1 s1)/2 2 v = ψ , ψ max( ψ X − Z − v ) . (2.27) | i | 0i | 1i |h 1| | i| By averaging over θ and φ, the rate of obtaining a correct guess can be found as

1 1 + max( ψ X ψ 2, ψ X ψ 2) + max( ψ Z ψ 2, ψ Z ψ 2) 4π |h 0| | 0i| |h 0| | 1i| |h 0| | 0i| |h 0| | 1i| Z h + max( ψ XZ ψ 2, ψ XZ ψ 2) dΩ . (2.28) |h 0| | 0i| |h 0| | 1i| i The numerical result of the above integral is about 85%. In general, B2 cheater can measure in basis other than ψ , ψ . I numerically ﬁnd that 85% is the highest {| 0i | 1i} probability that the cheaters can get. In the case of more veriﬁers, MBQC requires

25 Chapter 2. Position-Based Quantum Cryptography

more measurements and thus produces more measurement byproducts, so the successful cheating rate is anticipated to be lower than 85%.

The above analysis shows that Protocol A′ is secure against the entanglement attack strategy proposed in this chapter. I note that the protocol is in fact secure against arbitrary attack if the cheaters share only a pair of entangled qubit or qutrit. Detailed security proof is given in Appendix A.1. Historically, our results gave the ﬁrst lower bound of entanglement resources that is required to successfully cheat a PBQC protocol: at least two Bell pairs.

2.8 Summary

In this chapter, I have presented my work in position-based quantum cryptography, which has been published as Ref. [108]. I have reviewed all known PBQC protocols in 2010, and introduced the respective cheating strategies, by using which cheaters can always produce the same response as the honest prover. The main reason for the successful cheating is that the cheaters can share entangled resources, of which this possibility is not considered in the proposals of PBQC protocols [43, 122]. I have further discussed the principle and limitations of my cheating strategies. I have also proposed a modified protocol that is secure if the cheaters share only a pair of maximally entangled qubit or qutrit. Nevertheless, the modified protocol was soon shown to be insecure. In fact, any PBQC protocol can be cheated with arbitrarily high successful probability by using nonlocal quantum measurement [172, 43, 188] or port teleportation [89, 88, 23]. In other words, PBQC is not unconditionally secure, even if all the apparatus of the verifiers and the honest prover is perfect. In practice, the apparatus must be imperfect that leaves extra room for cheating. For example, practical detectors are not 100% efficient and the quantum channel is lossy, so in some rounds of challenge the honest prover may not receive any signal. The cheater can make use of this property to improve their successful cheating rate. More explicitly, the cheaters may report only in the rounds that the encrypted qubit is perfectly measured, while they claim a loss in the rounds of measurement in a wrong basis. The verifiers would have a high probability to receive correct response, and falsely regard the presence of a honest prover. In addition, the imperfection of apparatus also limits the case that PBQC is practically useful. For example, the quantum and classical operation time is finite instead of infinitesimally short, and the quantum channel may not be vacuum that quantum

26 Chapter 2. Position-Based Quantum Cryptography

signals are not transmitted at the speed of light. These extra time consumptions have to be smaller than the response time difference that verifies the prover’s location. In other words, PBQC can verify the location of the prover within a sufficiently large area, through which the light travelling time, l/c, is longer than the time consumption due to the apparatus imperfection. Let us consider the refractive index of a standard optical

fiber is c′ c/1.5, so the extra time consumption is δt d/3c. Therefore the preci- ≈ ≈ sion range of the verifiable position is at least one third of the verifier-prover separation. When applied to verify the location of human-size object, this limitation requires a high density of verifier station, say one per 10 m2, which is not very practical. Because the scheme is not unconditionally secure, and it suffers from various technical difficulties in implementation, I conclude that PBQC is not a practical quantum information application. Nevertheless, there are still academically interesting questions that remain open. For example, all cheating strategy may fall in a communication complexity that cheating consumes much more resources than the honest execution of the protocol [40]. If this claim is valid, PBQC can be conditionally secure, i.e., when the cheaters have only bounded amount of entanglement. Besides, in the most efficient cheating strategy known today, the amount of required entanglement scales polynomially as the degree of freedom of the encrypted message. On the other hand, the provable minimum amount of entanglement that is necessary for cheating scales linearly as the degree of freedom of the encrypted message. The optimal amount of entanglement for cheating remains unknown. Answering this question would deepen our understanding about the capability of entanglement in communication tasks. In addition, according to the analysis of all known cheating strategies, infinite entanglement is required if the message is encrypted in a state with infinite degree of freedom. In practice, such a state is not difficult to realise; the quantum states used in continuous- variable communication schemes [34], such as coherent states and squeezed states, have infinite degree of freedom. It is interesting to study if the known cheating strategies can be extended to cheat continuous-variable PBQC scheme. These studies will be related to nonlocal measurement and port teleportation of continuous-variable quantum states, of which the techniques is not know at this moment.

27 Chapter 3

Quantum Secret Sharing with Continuous-Variable Cluster States

3.1 Introduction

Secret sharing is a cryptographic task aiming to distribute a secret amongst a group of parties. It starts from a party, known as the dealer, who encodes the secret into some information carrier according to a secret sharing protocol. A good protocol should allow each set of parties in the access structure, which is the set of all authorised subsets of parties, to faithfully reconstruct the secret; while other parties in the adversary structure, which is the set of all unauthorized parties, are denied any information about the secret. Classical secret sharing protocols have been proposed [31, 160] where classical information is encoded by mathematical transformations. The protocols can be proven to be information-theoretically secure, i.e. if the communication channels between the dealer and the parties are secure, no information about the secret can be obtained by the adversary structure even when they have unlimited computational power. Following the rapid development of quantum information, the extension of secret sharing to the quantum regime has received much theoretical attention [83, 51, 71, 170]. The objective of quantum secret sharing (QSS) is to use the quantum correlations in well-constructed entangled states to securely transmit a set of classical or quantum information to only the access structures. As the involved parties are supposed to be spatially well separated, an optical system is the most suitable implementation for QSS due to its excellent mobility. Several proof-of-principle experiments have already been demonstrated [167, 47, 102]. However, constructing a large-scale optical QSS state is technically challenging, because the nonlinear interaction between photons is weak, and

28 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States some QSS protocols require more than two quantum levels [51, 71], where the commonly employed polarisation encoding is not applicable. Recently, Markham and Sanders [124] proposed a unified QSS approach based on qubit cluster states [147, 148], which could be constructed efficiently using only linear optics and post-selection [142, 37, 56]. Cluster states have another advantage that an N-mode cluster is well characterised by N stabilisers or an N-vertex connected graph, in contrast to a general quantum state that have to be expressed in an exponential number of superpositions. Therefore, the theoretical construction and the security analysis of a cluster state QSS scheme could be simplified. The idea of cluster state QSS has also been extended to odd-dimensional states (qudits) in Ref. [95]. In this chapter, I present the work of Christian Weedbrook and myself in extending cluster state QSS into the continuous-variable (CV) regime. While many quantum information protocols can be optically implemented by using discrete- or continuous-variable formalism, CV systems have the advantages that multi-partite entangled states can be produced deterministically, and the measurement is high in fidelity using present technology. In particular, CV cluster states are proposed to be useful resources to conduct measurement-based universal quantum computation [128, 176]. A CV cluster state can be efficiently implemented in an optical system by various approaches including the conventional method of controlled-phase (CPHASE) operation [128, 190], linear optics with offline squeezing [173], optical parametric oscillator [130, 129, 64], and quantum nondemolition gate [131]. Recently, CV cluster states involving as much as 10000 optical modes have been demonstrated experimentally [165, 189, 187]. For simplicity, I consider that the CV cluster states are prepared by the conventional method of CPHASE operation, though the states can be equivalently prepared by other approaches, and my result is independent of the method of state preparation. The main objective of this work is to investigate how CV cluster states can be used to securely share quantum and classical secrets. Instead of directly extending the qudit approach to the d limit, a CV cluster state is critically different from its discrete- →∞ variable counterparts in the sense that constructing a perfect (infinitely squeezed) CV cluster state is practically impossible. I find that when realistic finitely squeezed cluster states are instead utilised, QSS is still possible but the security is inevitably reduced, i.e., the secret is not precisely recovered by the access structure while partial information is leaked to unauthorised parties. I suggest benchmarks to evaluate the performance of each of the QSS tasks. For the sharing of classical information, I calculate the amount of secure key, which is used to encode the secret, that can be distilled from each cluster state. A procedure is provided to transform the distilled state to the standard form

29 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

that can be analysed by the techniques in CV quantum key distribution (QKD). For the sharing of quantum information, I estimate the number of cluster states required to establish a high fidelity teleportation channel to transmit the secret state. The amount of entanglement is quantified by the logarithmic negativity. In both tasks, I give two examples to demonstrate the decoding and security analysis procedures. As I want to focus the discussion on the application of the quantum correlations of CV cluster states, the states received by the parties are assumed to be the same as when prepared by the dealer, i.e., all quantum channels are ideal (noiseless and lossless). Detections are also assumed to be perfect in fidelity. This chapter is outlined as follows. In Sec. 3.2, I introduce QSS and classify it into three tasks. The physical and mathematical background of CV cluster states is also reviewed. In Sec. 3.3 and 3.4, I respectively analyse the security of classical information sharing when the cluster state is delivered through secure and insecure channels. In Sec. 3.5, I discuss the performance of quantum state sharing. I conclude in Sec 3.6 with a short discussion. I note that most the material in this chapter has been published in Ref. [109]. I denote the quantities of the access structure by the subscript A, that of the adversary structure or other unauthorised parties by E, and that of the dealer by D. I pick ~ =1 in the following calculations, and all logarithms are to base 2.

3.2 Background

3.2.1 Quantum Secret Sharing

In the literature, the idea of QSS has been developed to serve one of the following three tasks [124]: CC: Classical information is shared among parties by distributing QSS states through private (secure) channels, which are invulnerable to eavesdropping. The role of quantum resources is to substitute the mathematical correlations in classical secret sharing protocols by the quantum correlations in a QSS state. CQ: Classical information is shared among parties by distributing QSS states through public (insecure) channels, which are open for eavesdropping. The quantum correlation in the QSS states is used both to detect the disturbance of eavesdropping and to share the secret. When compared with the hybrid approach that incorporates both classical secret sharing and QKD, the CQ scheme can reduce the cost of communication [83]. QQ: Also known as ‘quantum state sharing’, a secret quantum state is shared among

30 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

parties by distributing QSS states through public channels. The QQ scheme can be implemented by either encoding the quantum secret into a QSS state, or by using a QSS state to distribute entanglement between the dealer and the access structure for teleporting the secret state [48]. Because the former approach would allow the eavesdropper to access the secret by capturing all the QSS states, I consider in my work the later approach that the secret can be sent after the security of the teleportation channel is verified. The three tasks form a hierarchy of the required resources: a QQ state can perform all three tasks, and a CQ state can be used for CC, while the reverse is not always true. In principle, constructing a QQ state is versatile, but the amount of required resources and infrastructure can be optimised according to the properties of the shared information and the channels. For CC and CQ, I consider that both the dealer and the access structure measure the cluster states after state distribution. Because of the entanglement, random but strongly correlated measurement outcomes will be obtained, from which the dealer and the access structure can distill secure keys. The keys can then be used to encrypt the classical secret, which will be shared through public classical channels. Therefore the secret sharing rate, i.e., the amount of classical information securely shared in each round of QSS, is determined by the net amount of secure key distilled from each cluster state. For QQ, I consider that the dealer and the access structure extract entangled states from the cluster states. After accumulating enough extracted states, entanglement distillation can be conducted to distill a more entangled state, through which the secret state can be teleported from the dealer to the access structure with higher fidelity. I note that in all the QSS tasks, the objective of the dealer is to securely transmit the secret to the access structure, although the identities of the access structure are not revealed until all QSS states have been received. Because, in a secure protocol, the mutual information between the dealer and the access structure is larger than the information obtained by the adversary structure, the access structure’s identities can be authenticated using parts of the shared information. The dealer should then trust the access structure and co-operate in subsequent post-processing of the shared QSS states. I also note that in the limit of infinite squeezing, our cluster state scheme is not as general as the QQ scheme proposed in Ref. [170]. However our scheme is interesting because all three kinds of QSS are considered in a unified approach, and the resource state is a cluster state that can be efficiently constructed and easily analysed.

31 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

3.2.2 Continuous-Variable Cluster States

As an analog to the discrete-variable cluster state, which is formed by preparing all qudits in an eigenstate of the generalised Pauli X operator and then applying the CPHASE gate, a CV cluster state is formed by first preparing all quantum modes as squeezed vacuum states and applying CV CPHASE gates, i.e., Cˆ = exp i qˆ qˆ . A n-mode CV cluster { Aij i j} state can be characterised by a n-vertices graph, where each quantum mode acts as a vertex v , where = v , and a CPHASE operation is applied across each edge i ∈ V V { i} e , where = e = v , v , with weight [95]. The CV cluster state Ψ is ij ∈ E E { ij { i j}} Aij | i defined as n Ψ := exp i qˆ qˆ ψ ⊗ . (3.1) | i { Aij i j}| 0i e Yij ∈E In the infinitely squeezed case, ψ is given by | 0i

ψ = 0 , wherep ˆ 0 =0 , (3.2) | 0iinﬁnite | ip | ip

while in the ﬁnitely squeezed case,

√σ σ2q2/2 ψ = e− q dq , (3.3) | 0iﬁnite π1/4 | iq Z where q is the eigenstate ofq ˆ with eigenvalue q; σ is a parameter characterising the | iq degree of squeezing.

3.2.2.1 Nulliﬁer representation

Apart from the ket vector representation, an infinitely squeezed cluster state can be characterised by its stabilisers [2, 176]. A stabiliser Sˆ of a state ψ is defined as the | i operator of which ψ is an eigenstate with +1 eigenvalue, i.e., Sˆ ψ = ψ . Analogous to | i | i | i the discrete-variable cluster state, a n-mode infinitely squeezed CV cluster state has at most n independent stabilisers, although any sum and product of the stabilisers is a new stabiliser. The whole set of independent stabilisers uniquely specifies the cluster state [74]. In a CV system, it is sometimes more convenient to study with the nullifiers than the stabilisers. A nullifier Nˆ is defined as an operator of which ψ is an eigenstate with | i eigenvalue 0, i.e., Nˆ ψ = 0. There are infinitely many choice of nullifiers because any | i sum and product of nullifiers is another nullifier. For an infinitely squeezed CV cluster

32 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

state, I choose a set of nulliﬁers, which is hereafter referred as the standard set, as [74]

Nˆ =p ˆ qˆ , (3.4) i i − Aij j j X∈N where the position operators are summed over the neighbours of the vertex i in the graph, i.e., j (i, j) . The standard nullifiers can be constructed by the following ∈E procedure. Before the CPHASE operations, the squeezed vacuum modes are nullified byp î’s. The CPHASE operation between the mode i and j transforms the nullifiers as i qˆ qˆ i qˆ qˆ pˆ e Aij i j pˆ e− Aij i j =p ˆ qˆ . From this procedure, it can be easily seen that all i → i i −Aij j standard nullifiers commute and are linearly independent.

3.2.2.2 Wigner function representation

As an extension to the nullifier representation, the Wigner function is a good representation of the quantum correlation of finitely squeezed CV cluster states. The Wigner function of a single mode CV stateρ ˆ is defined as [74]

1 ∞ x x W (q,p) := exp(ipx) q ρˆ q + dx , (3.5) 2π − 2 q 2 q Z−∞ D E

where the deﬁnition can be trivially generalised to multi-mode states. The Wigner function of n ﬁnitely squeezed vacuum states is given by

1 n p2 W (q, p)= exp( σ2q2) exp i , (3.6) 0 πn − i i −σ2 i i Y and that of a ﬁnitely squeezed CV cluster state is

1 n N 2 W (q, p) W (q, N)= exp( σ2q2) exp i , (3.7) c ≡ 0 πn − i i − σ2 i i Y T T T where q = (q1,...qn) , p = (p1,...pn) , and N = (N1,...Nn) ; Ni is the standard nulliﬁer in Eq. (3.4) with the operators replaced by the respective scalar variables; the initial degree of squeezing of each mode i is σ . In the inﬁnite squeezing limit, i.e., σ i i → 0 i, the exp( σ2q2) terms would converge to some constants, while the exp( N 2/σ2) ∀ − i i − i i terms would become Dirac delta functions, i.e.,

n W (q, p) δ(N ) . (3.8) inﬁnite ∝ i i Y

33 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

3.2.2.3 Correlations of measurement

Let us consider an infinitely squeezed CV cluster state that is locally measured by the operators Mˆ , where Mˆ is a linear combination ofq ˆ andp ˆ , i.e., homodyne detection { i} i i i in a rotated basis. I define the measurements are compatible to the nullifiers, if there

exists a linear combination of Mˆ i’s that equals to a linear combination of the standard ˆ ˆ nullifiers, i.e., i=1,n kiMi = i=1,n liNi for some real ki’s and real li’s. Measurement compatible to nullifiers is important in my studies, because k Mˆ is a nullifier, P P i=1,n i i and so the measurement outcomes, Mi, would be correlated asP i=1,n kiMi = 0. Such correlation is originated from the entanglement in the cluster stateP.

Similar quantum correlations of measurements prevail in finitely squeezed CV cluster states, but the accuracy depends on the degree of squeezing. Let us consider a finitely squeezed CV cluster state that is measured by the same set of measurement operators Mˆ , the expectation value of the measurement outcomes are statistically correlated as { i} in the infinitely squeezed case, i.e.,

kiMˆ i = liNˆi =0 . (3.9) i=1,n i=1,n D X E D X E However, the variance is ﬁnite, i.e.,

2 2 ∆ kiMˆi = ∆ liNˆi i=1,n i=1,n D X E D X E 2 l2σ2 = l N W (q, p)dnqdnp = i i , (3.10) i i c 2 i=1,n i=1,n Z X X but scales as σ2 that is small. The correlation (and variance) comes from the exp( N 2/σ2) i − i i terms in Eq. (3.7), which are narrow-width Gaussian functions.

In subsequent discussions, I regard a quantum correlation is “strong” if the collective variance of the local measurement outcomes is small; otherwise the correlation regarded as “weak”. The modes are regarded strongly correlated if their local operators produce strong correlations. My QSS scheme is secure if the access structure is stronger correlated to the secret than the unauthorised parties. According to Eq. (3.10), the local measurement operators exhibit strong correlations if they linearly combine as a nulliﬁer

and σi’s are small. There the secret is usually encoded in nulliﬁers in my scheme.

34 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

3.2.2.4 Cluster-class state

A class of states that shares similar properties as the CV cluster state can be constructed by applying local Gaussian operators onto Ψ . These operators linearly transform the | i quadrature operators in nullifiers, as well as the quadrature parameters in the Wigner function, asq ˆ a qˆ+ b pˆ+ c andp ˆ a qˆ+ b pˆ+ c for some real constants a, b, c that → q q q → p p p obey the uncertainty principle. General linear transformations can be implemented by only three kinds of basic operators [117]: displacement, squeezing, and Fourier (phase- shift) operator. A displacement operator Dˆ(α) shifts a nullifier by a constant factor, i.e., the components in nullifiers are transformed asq ˆ qˆ + √2Re(α) andp ˆ pˆ + √2Im(α). All → → the displacements do not affect the measurement basis nor the variance of the quantum correlations; only the expectation values of the measurement results are changed. A squeezing operator Sˆ(γ) = exp( ir(ˆqpˆ +p ˆqˆ)/2) scales the quadrature operators as − r Sˆ†qˆSˆ γqând Sˆ†pˆSˆ p/γˆ , where γ = e . Linear coefficients ofx ˆ andp ˆ in the nullifiers → → i i will be altered that may change measurement basis in general. A Fourier operator 2 2 Fˆ(θ) = exp( iθ(ˆq +ˆp )/2) transforms the quadrature operators as Fˆ†qˆFˆ = cos θqˆ+sin θpˆ − and Fˆ†pˆFˆ = sin θqˆ+cos θpˆ. The Fourier operator changes the local measurement bases − that exhibit the quantum correlation. I note that all CV cluster-class states are Gaussian states, which are states that the Wigner function is a, possibly multi-variable, Gaussian function.

3.3 CC Quantum secret sharing

In the CC setting of QSS, the dealer is connected to the n parties through secure quantum channels. A classical secret value s is encoded by displacing certain modes i of the cluster state by some function f (s). The value of f (s), the strength of the CPHASE , and i i Aij the neighbours of the cluster are designed for speciﬁc access and adversary structures. N A CV cluster state can be used for CC QSS if for each access structure, there is a nulliﬁer containing both s and the local quadrature operators of only that access structure, i.e., there exists real numbers li such that

liNˆi = kjMˆj + g(s) , (3.11) i j A X X∈ where kj are real numbers; Mˆ j are linear combinations of the local quadrature operators of access structure parties; g(s) is a nontrivial function of s. On the other hand, every

35 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

adversary structure cannot construct a nulliﬁer that contains both s and only their local operators. In the case of inﬁnite squeezing, the access structure can obtain g(s), and thus s, by locally measuring their modes according to Mˆ j. The scheme is secure if the reduced Wigner function of the adversary structure is independent of s.

In the case of finite squeezing, the access structure also measures according to Mˆ j. Their results are strongly correlated to s, but some information about the secret is leaked to the adversary structure due to weak correlations. The security of the QSS scheme can be analysed by comparing the amount of information obtained by the access structure and the adversary structure. The information obtained by the access structure is quantified by the mutual information, I(D : A), between the dealer and the access structure [143]. Let the dealer chooses a secret value s according to a probability distribution (s). The access structure would PD not obtain exactly the same value due to the finite squeezing. The conditional probability of obtaining a result s′ follows A D(s,s′). The total probability of the access structure’s P | result is then given by

A(s′)= D(s) A D(s,s′)ds . (3.12) P P P | Z The mutual information I(D : A) is deﬁned as [143]

I(D : A)= H(A) H(A D) , (3.13) − | where H(A) is the entropy of the access structure’s result, which is deﬁned as

H(A)= (s′) log (s′)ds′ ; (3.14) − PA PA Z H(A D) is the conditional entropy of the access structure when s is known, which is | deﬁned as [143]

H(A D)= D(s) A D(s,s′) log A D(s,s′)dsds′ . (3.15) | − P P | P | Z On the other hand, the adversary structure can unite their modes through ideal quantum channels, and can conduct any operation allowed by physics. The amount of information leaked to the adversary structure, I(D : E), is capped by the Holevo bound χ [143, 84], i.e.,

I(D : E) χ = S(ˆρE) D(s)S(ˆρE D(s))ds , (3.16) ≤ − P | Z

36 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

where S(ˆρ) is the von Neumann entropy;ρ ˆE D(s) is the state obtained by the adversary | structure if s is prepared by the dealer;ρ ˆE is the average state obtained by the adversary structure, viz.

ρˆE = D(s)ˆρE D(s)ds . (3.17) P | Z As CV cluster states are Gaussian, the reduced state of the adversary structure is also Gaussian. The von Neumann entropy of Gaussian states can be calculated by using their covariance matrix V , which is deﬁned as V := ∆x , ∆x /2 [176]. If the Wigner ij h{ i j}i function of an r-mode Gaussian state is known, V can be obtained through the relation [176] T 1 exp( 1/2(x x¯) V − (x x¯)) W (x)= − − − , (3.18) (2π)r√det V

T 2r where x = (q1,p1,...,qr,pr) ; x¯ = xW (x)d x. Covariance matrices can be characterised by their symplectic spectrum ν , which is equal to the eigenspectrum of the R { k} matrix iΩV [176], where | | 1 if i =2k 1, j =2k , − Ωi,j = 1 if i =2k, j =2k 1 , (3.19) − − n 0 else,

for k =1,...,r. The von Neumann entropy is calculated by

S(ˆρ) = g(νk) , (3.20) i X 1 1 1 1 where g(ν) := ν + log ν + ν log ν . (3.21) 2 2 − − 2 − 2

In the case that the covariance matrices ofρ Ê andρ Ê D are independent of s, their | respective von Neumann entropies are also so. Then the Holevo bound can be simplified as,

I(D : E) S(ˆρE) S(ˆρE D) . (3.22) ≤ − | The minimum secret sharing rate in each round of the protocol is thus

K = I(D : A) I(D : E) . (3.23) cc −

After m rounds of state distribution, mKcc secret keys can be distilled from the strongly correlated random numbers s and s′ for sharing the classical secret [20].

As examples, I demonstrate in Appendix A.2 the procedure for calculating Kcc of

37 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

the (2,3)- and the (3,5)-protocol. I note here a few interesting findings regarding the examples. Firstly, the secret sharing rate is positive when the squeezing parameter σ is smaller than some threshold. The threshold for both protocols we considered are around σ 1. The results show that a secure CC QSS can be implemented with only weakly ≈ squeezed resources, hence the requirement of infinitely squeezed states can be relaxed. Besides, on the contrary to common beliefs that a CV state with σ = 1 cannot transmit secure information, the CC secret sharing rate is non-zero in some scenarios even when σ 1. The result is not surprising in cluster state QSS, because implementing ≥ a CPHASE requires the initial modes to be squeezed [173]. In fact, a two-mode cluster state can be easily shown to be local-unitarily equivalent to a finitely-squeezed Einstein- Podolsky-Rosen (EPR) state unless σ [34]. →∞ In addition, although any access structure collaboration can obtain s in the infinite squeezing case, surprisingly different collaborations obtain different secret sharing rate in the finite squeezing case. This heterogeneity is related to the entanglement structure of the cluster state. In practice, the dealer has to consider the disadvantage of certain collaborations when applying CC QSS.

3.4 CQ Quantum secret sharing

In the CQ setting of QSS, the dealer is connected to the parties through insecure quantum channels, so that unauthorised parities, which include adversary structure and eavesdropper who does not involve in the protocol, can manipulate all the modes sent from the dealer. The CC protocol mentioned in Sec. 3.3 is insecure in this setting, because the unauthorised parties can capture and measure the modes to obtain s. This eavesdropping can be intractable if the adversary structure resends to the access structure an infinitely squeezed state with the same s encoded, so that the access structure have the same measurement results as the adversary structure. Here I modify the CC protocol for the CQ setting. I first present an entanglement- based protocol, and discuss how it can be reduced to a mixed-state protocol that reduces the resource requirement. Instead of constructing a n-mode cluster state and encoding a classical secret s into the state, the dealer prepares an (n+1)-mode standard cluster state (cluster states with standard nullifiers in the infinite squeezing case), where n of the modes are delivered to the parties while the dealer keeps the remaining one, denoted as mode D. A good CQ protocol should produce a much stronger quantum correlation between the dealer and the access structure than that between the dealer and the unauthorised parties.

38 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

Here I make two assumptions to simplify the the security analysis, but these assumptions will be relaxed at the end of this section without compromising the secret sharing rate. Firstly, I assume the access structure parties are connected by secure and ideal quantum channels, so the modes can be sent to one party, say party h, with perfect fidelity. I also assume both the dealer and the access structure have quantum memories, so the cluster states delivered in each round are stored with perfect fidelity for subsequent quantum operations and measurements. After each round of state distribution, a strongly correlated entangled state is shared between the dealer and the access structure. Let us consider the strong correlation is represented by the two nullifiers,p ˆ Qˆ andq ˆ Pˆ , which are linear combinations of D − A D − A the standard nullifiers. QÂ and PÂ are linear combinations of only the access structure parties’ localq ˆ andp ˆ operators. By applying a global operation, UÂ, on all the modes at party h, QÂ and PÂ are transformed to single mode operatorsq ˆh andp ˆh. As a result, the strong correlations with mode D are transferred to mode h. After all the rounds of cluster state distribution, the stage of parameter-estimation ensues. The dealer or party h randomly selects half of the shared modes for measurement, and the selection is announced. Both the dealer and party h measures the selected modes in either thex ˆ andp ˆ basis. The measurement outcomes are announced for characterising the unmeasured states. In the infinitely squeezed case, the estimated parameters should indicate that the state between mode D and mode h is maximally entangled 1. The dealer and party h measure each residual modes randomly in either theq ˆ orp ˆ basis, the basis is then announced. Each measurement outcome is a random number on the real axis, and the outcomes are the same if the measurement bases are matching, i.e., one party measures inq ˆ while the other measures inp ˆ. The common random numbers can be used as secure keys to encode the secrets. Because the state was maximally entangled, no information is leaked to unauthorised parties. In the finitely squeezed case, although mode D and mode h are strongly correlated, the state of adversary structure is still weakly correlated with mode D. As to be discussed, local quantum operation is applied on each residual mode to rectify the covariance of the state according to the estimated parameters. The dealer and party h then measure each mode in either thex ˆ orp ˆ basis, and announce the basis. Unlike the infinitely squeezed case, post-processing is required to extract secure keys from the correlated measurement outcomes due to two reasons. First, even if the measurement basis is matching, the mea-

1Here I refer to ‘maximally entangled’ as an entangled state originating from two inﬁnitely squeezed modes.

39 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

surement outcomes of the dealer and the access structure are merely strongly correlated but not exactly equal. Besides, partial information about the outcomes is leaked to the unauthorised parties due to the non-uniform distribution of the measurement outcomes, and the weak entanglement between the unauthorised parties’ and the dealer’s modes. In the following, I employ the security analysis techniques from CV QKD [3] to estimate the minimal amount of secure key that can be distilled from each cluster state, which is equivalent to the secret sharing rate in each round of CQ QSS.

3.4.1 Equivalence of CQ Quantum Secret Sharing and QKD

I now show why CQ QSS and CV QKD can be analysed by using the same techniques.

Let us consider that before the CPHASE operation, mode D is squeezed with σD while all other modes are squeezed with σ. Suppose mode D is connected to N neighbours after the cluster state formation, the reduced Wigner function of mode D is

2 2 σDqD 2 σσDe− pD WD(qD,pD)= exp . (3.24) 2 2 2 2 π N + σ σ − σD + N/σ D p W is the same as the reduced Wigner function of a two-mode cluster state , where D |CN i mode D is connected to a mode u that is squeezed with σ/√N. Because both the CQ cluster state and are pure, the amount of entanglement between mode D and the |CN i cluster state modes to be delivered is the same as the amount between the two modes in . |CN i As in common security analysis of QKD, I grant the unauthorised parties the full power to manipulate the modes sent from the dealer. Then there will be no diﬀerence for the dealer to prepare the CQ cluster state or , because the unauthorised parties |CN i can transform the delivered cluster state modes to mode u or vice versa. Then the CQ state delivery is equivalent to the following scenario: The dealer ﬁrst prepares and |CN i delivers mode u through an insecure quantum channel. The unauthorised parties capture mode u, entangle it with ancillae, and forward some modes to the access structure.

The access structure’s modes are then gathered at party h. After the operation UÂ, modes other than mode h are still weakly correlated with mode D. For simplicity, these weak correlations are neglected in our analysis, i.e., all modes except h are traced out. This action only reduces the quantum correlation between the dealer and the access structure, thus the security is not unphysically improved. Now the CQ protocol is effectively reduced to a CV QKD protocol: The dealer first prepares a two-mode Gaussian state, , and delivers one mode. The access structure finally gets a mode h that |CN i

40 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

QKD QSS

D D u Local unitary equivalent

Delivered mode Modes delivery captured by adversary

D u

Decoding

Deliver strong correlation to mode h h D

Dealer Tracing out modes besides h Unauthorized parties

Authorized parties D h

Figure 3.1: Strategy for computing the secret sharing rate using CV QKD techniques. Strongly (weakly) correlated modes are linked by solid (dotted) lines. The procedure of CQ QSS is shown on the right while that of QKD is shown on the left. The key idea is that both QKD and QSS have the same initial (pure entangled state with parts delivered) and ﬁnal resources (two strongly correlated modes between the dealer and the authorised parties.) remains strongly correlated with mode D, but the quantum correlation is reduced due to the entanglement with the environment controlled by the unauthorised parties. The degradation of quantum correlations in the encoding and the decoding processes in CQ QSS can be analogous to the loss and noise when transmitting an EPR state through an imperfect channel in QKD. The whole idea is summarised schematically in Fig. 3.1.

3.4.2 Secret Sharing Rate

In the uniﬁed picture of CV QKD, a ﬁnitely squeezed two-mode squeezed state is prepared by the dealer and delivered to the authorised party through an imperfect channel [68, 176].

41 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

Both parties measure some of the delivered states to estimate the covariance matrix, V , of the unmeasured states. By using the fact that Gaussian states minimise the distillable secure key for every state having the same V [184, 68], the assumption that the unmeasured states are Gaussian upper-bounds the information leakage to unauthorised parties. Because a Gaussian state is completely characterised by its covariance matrix, the secure key rate can be deduced from only V . For realistic channels that are usually symmetrical for quadraturesx ˆ andp ˆ, V can be expressed in a standardised form as

V I cZ V = , (3.25) cZ V ′I !

where I and Z are the 2 2 identity and Pauli Z matrices respectively; V is the variance × of the undelivered mode of the dealer; V ′ is the variance of the mode received by the authorised party; c accounts for the correlation between the two modes. V can be characterised by only V and two channel parameters: the transmittance, τ, and the noise, χ, which are deﬁned by the relations

2 1 c = τ(V ); V ′ = τ(V + χ) . (3.26) r − 4

To calculate the minimum secret sharing rate of the CQ QSS, in the parameter- estimation stage the dealer and the access structure construct the covariance matrix by measuring some of their modes. In order to employ the analysis of CV QKD, subsequent local operations are applied on the unmeasured modes to transform their covariance matrix to the standard form. This can be done in three steps: Firstly, any covariance matrix can be transformed by local unitaries as [54] 2

VD 0 0 c1

 0 VD c2 0  . (3.27)  0 c2 VA 0     c1 0 0 VA      Next, the access structure squeezes the modes to balance the oﬀ diagonal terms, i.e., the

2 This can be done by applying FˆA(π/2) onto the standard form I in Ref. [54]. I consider both c1 and c2 are positive, which is the case that strong correlation is retained in the cluster state.

42 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

covariance matrix becomes

VD 0 0 √c1c2 0 V √c c 0  D 1 2  . (3.28)  0 √c1c2 Vq 0     √c1c2 0 0 Vp      3 Finally, the variances ofq ˆA andp ˆA can be balanced by a ‘state-averaging’ process . Let us consider that the dealer randomly divides the unmeasured states into two sets, and the choice of division is announced. In one set, the dealer applies a Fourier operator, Fˆ ( π/2), on each mode that transforms the quadrature operators as qˆ pˆ and D − D → D pˆ qˆ . In the other set, the access structure applies Fˆ ( π/2) on each mode that D → − D A − generates the transformationq ˆ pˆ andp ˆ qˆ . Subsequently, the choice of A → A A → − A division is discarded. The state will be transformed as

1 1 ρˆ Fˆ ( π/2)ˆρ Fˆ† ( π/2) + Fˆ ( π/2)ˆρ Fˆ† ( π/2) , (3.29) DA → 2 D − DA D − 2 A − DA A −

and the covariance matrix becomes

VD 0 √c1c2 0

 0 VD 0 √c1c2  − , (3.30)  √c1c2 0 (Vq + Vp)/2 0     0 √c1c2 0 (Vp + Vq)/2   −    which is in the standard form. The analogous τ and χ can then be obtained according to Eq. (3.25). For pedagogic purposes, I demonstrate in Sec. A.3 the procedure of getting the standardised covariance matrix for diﬀerent collaborations in the (2,3)- and (3,5)-CQ protocols. I assume the QSS protocol is direct reconciliation, i.e., the measurement result of the dealer is the secret value that has to be estimated by the access structure, but the secret sharing rate of a reverse reconciliation protocol can be easily calculated by similar procedure [3]. The secret sharing rate, KCQ, is given by the secure key rate in the analogous CV QKD protocol as [3]

K = I(D : A) I(D : E) , (3.31) CQ − 3In spite of the simplicity, this action would sacriﬁce some quantum correlations, and hence reduce the secret sharing rate. As my aim is to demonstrate the possibility, but not the optimality, of performing CQ QSS with CV cluster states, the employment of state-averagings is appropriate within the scope of the current work.

43 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

where I(D : A) is the mutual information between the dealer and access structure; the information obtained by the unauthorised parties is given by I(D : E), which is capped by the Holevo bound. The mutual information I(D : A) can be calculated by comparing the variance of mode h with and without knowing the measurement results of mode D. In terms of the analogous channel parameters, the mutual information is given by [3]

1 V + χ I(D : A)= log . (3.32) 2 χ + 1 4V In direct reconciliation protocols, the Holevo bound of the unauthorised parities’ information is deﬁned as I(D : E)= S(E) S(E D) , (3.33) − | where S(E) is the von Neumann entropy of the unauthorised parties’ state; S(E D) is the | conditional von Neumann entropy if the measurement result of the dealer is known. As the unauthorised parties can control the environment that puriﬁes the whole system, the entropy of the unauthorised parties is the same as that of the system DA, i.e., S(E) =

S(DA). The entropy can be calculated by using Eq. (3.20), i.e., S(DA)= g(ν+)+ g(ν ) − [176], where the symplectic spectrum of V , ν+, ν , is given by { −}

1 2 2 ν = (V + V ′) 4c (V V ′) . (3.34) ± 2 − ± − p Similarly, because the state of system AE is pure after system D is measured, the conditional entropy S(E D) is the same as S(A D). The covariance matrix of system A | | after the measurement of the dealer is given by [59, 62]

V c2/V 0 VA D = − , (3.35) | 0 V ′ !

where the symplectic eigenvalue is

c2 1 ν = V V = τ (V + χ) + χ . (3.36) c ′ ′ − V V s s Hence we get S(E D)= g(ν ). | c After m rounds of state distribution, the dealer and the access structure can distill

mKCQ secret keys from the strongly correlated measurement outcomes for sharing the classical secret [20].

44 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

In the (2,3)- and (3,5)-protocols I have analysed, secure keys can be distilled if the squeezing parameter is smaller than a threshold limit 4. The threshold values are about σ 1 in the examples I considered. Just as in the CC case, our results show that ≈ nonzero secret sharing rate can be obtained with ﬁnitely squeezed resources. Disparity of the secure key rate in diﬀerent collaborations is also observed in CQ protocols, which is also due to the structure of the entangled state. Besides, the secret sharing rate is nonzero in some cases even when σ 1. As I have discussed in the CC case, a two-mode ≥ cluster state is still entangled even if the initial state is not squeezed. The entanglement between the dealer and party h imposes strong quantum correlations from which secure keys can be extracted.

3.4.3 Simpliﬁcation

The above security analysis of CQ protocols is studied under three assumptions: (i) the resource is a (n + 1)-mode cluster state; (ii) quantum memory is available to store the delivered modes until all states are received; (iii) ideal quantum channels are available between the access structure parties. Here I show that these assumptions can be relaxed without compromising the security.

3.4.3.1 Mixed State Approach

Because the distributed modes are unaﬀected by any local operation of the dealer, the state obtained by the parties is independent of whether mode D has been measured. Therefore, instead of preparing a (n + 1)-mode cluster state Ψ and measuring mode D | i afterwards, the dealer can simulate the consequence of the measurement by distributing a n-mode state that is the same as the measured Ψ . More explicitly, let us consider | i if the dealer intends to measure in theq ˆ basis, then he can instead prepare the pure state ( s ) Ψ , where s is aq ˆ eigenstate with the eigenvalue s. Similarly in thep ˆ h |qD | i | iqD D measurement rounds, the dealer can prepare ( s ) Ψ , where s is thep ˆ eigenstate h |pD | i | ipD D with the eigenvalue s. Other parties cannot distinguish the mixed state from Ψ if s is | i picked according to the probability distributions

σD σ2 s2 (s)= W (s,p )dp = e− D ; (3.37) PqD D D D √π Z s2 σ2 +N/σ2 e− D p (s)= WD(qD,s)dqD = . (3.38) P D √π σ2 + N/σ2 Z D 4Except in the 2, 3 collaboration in (2,3)-protocol that securep keys can be distilled for any squeezing parameter, because{ the} entanglement of the adversary structure is completely removed.

45 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

These are the probability distribution of dealer’s measurement outcomes on Ψ , which | i are obtained by integrating the Wigner function in Eq. (3.24). In the simulatedq ˆ measurement rounds, the infinitely squeezed ( s ) Ψ is charac- h |qD | i terised by the nullifiers Nˆ q =p ˆ A s qˆ , (3.39) i i − iD − Aij j j X∈N where i = 1,...,n; AiD = 0 if the mode i is not a neighbour of mode D. The nullifiers ˆ q ˆ Ni is the same as Ni except the operatorq ˆD is replaced by the simulated measurement outcome s. In the finitely squeezed case, the state can be characterised by the Wigner function WqD (q1,p1,...,qn,pn), which is obtained by tracing out pD and replacing all qD by s in the Wigner function of Ψ . Because W is the same as W in Eq. (3.7) with | i qD c the nullifiers Nˆ q, ( s ) Ψ can be constructed by displacing a finitely squeezed cluster i h |qD | i state. In the simulatedp ˆ measurement rounds, the infinitely squeezed ( s ) Ψ is charac- h |pD | i terised by the nullifiers

ˆ p ˆ p ˆ ˆ N1 = AjDqˆj s ; Ni = Ni Ni 1 , (3.40) − − − j X∈N where j = 2,...,n. In the ﬁnitely squeezed case, the state can be characterised by the

Wigner function WpD (q1,p1,...,qn,pn), which is obtained by tracing out qD and replacing all p by s in the Wigner function of Ψ . However, W cannot be represented by W D | i pD c with the nulliﬁers Nˆ p, therefore ( s ) Ψ is generally not a cluster state. Nevertheless, 1 h |pD | i ( s ) Ψ is a Gaussian state that can be eﬃciently prepared by squeezed vacuum states, h |pD | i displacement operators, and linear optical elements [173].

3.4.3.2 Classical Memory

When estimating the secret sharing rate, I require the delivered modes to be rectiﬁed so that the covariance matrix is in the standard form. The modes are stored in quantum memories until the covariance matrix is constructed from parameter-estimation. Here I

show that the measurement probability distribution of the transformed stateρ ˆ′′ can be obtained by: ﬁrst measuring the original stateρ ˆ, and then subjecting the measurement results to classical manipulations. Therefore the delivered modes can be measured before the parameter-estimation stage, quantum memory is thus not necessary. The rectifying process involves two stages: local-squeezing and state-averaging. After

state-averagings,ρ ˆ′′ becomes a mixture of Fˆ ( π/2)ˆρ′Fˆ† ( π/2) and Fˆ ( π/2)ˆρ′Fˆ†( π/2). D − D − h − h − By deﬁnition, the Wigner function ofρ ˆ′′, W ′′, can be written as the sum of the Wigner

46 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

function ofρ ˆ′, W ′, as

1 1 W ′′(q ,p , q ,p )= W ′(p , q , q ,p )+ W ′(q ,p ,p , q ) . (3.41) D D h h 2 D − D h h 2 D D h − h

Let us consider that the dealer measuresρ ˆ′′ inq ˆD and party h measures inq ˆh, the

probability of obtaining measurement outcomes y1 and y2, ′′ (y1,y2), is given by PqD,qh

′′ (y1,y2) = W ′′(y1,pD,y2,ph)dpDdph PqD,qh Z 1 1 = W ′(p , y ,y ,p )dp dp + W ′(y ,p ,p , y )dp dp 2 D − 1 2 h D h 2 1 D h − 2 D h 1 Z 1 Z = ′ ( y1,y2)+ ′ (y1, y2) , (3.42) 2PpD,qh − 2PqD,ph −

where the last equality involves renaming of variables; ′ is the joint xˆD, xˆh mea- PxD,xh { } surement probability ofρ ˆ′. Similarly, the probability of another strongly correlated measurement, pˆ , pˆ , can be expressed as { D h} 1 1 ′′ (y1,y2)= ′ (y1,y2)+ ′ (y1,y2) . (3.43) PpD,ph 2PqD,ph 2PpD,qh

These two relations indicate that the measurement probability distributions after state-averagings are not diﬀerent from mixing some measurement probability distributions before state-averagings. Let us consider that the dealer and party h randomly measureρ ˆ′ inx ˆ andp ˆ basis. Half of the qˆ , pˆ outcomes and half of the pˆ , qˆ { D h} { D h} outcomes are picked to mimic the qˆ , qˆ measurement ofρ ˆ′′. For the qˆ , pˆ half, all { D h} { D h} thep ˆ outcomes are multiplied by 1 and then regarded asq ˆ outcomes; for the pˆ , qˆ h − h { D h} half, all thep ˆ outcomes are multiplied by 1 and then regarded asq ˆ outcomes. After D − D combining these two sets of data, the probability m of getting y ,y is given by P 1 2

m 1 1 (y1,y2)= ′ ( y1,y2)+ ′ (y1, y2) , (3.44) P 2PpD,qh − 2PqD,ph − which is the same as Eq. (3.42). The pˆ , pˆ measurement probability ofρ ˆ′′ can be { D h} mimicked by similar procedures. In the local-squeezing stage, the dealer and party h apply local squeezing operations,

SˆD(γD) and Sˆh(γh) , to balance the variance of mode D and the coherent terms. The Wigner function of the stateρ ˆ is transformed as

qD qh W (q ,p , q ,p ) W ′(q ,p , q ,p )= W ,γ p , ,γ p . (3.45) D D h h → D D h h γ D D γ h h D h

47 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

Let us consider thatρ ˆ′ is measured inq ˆD andp ˆh basis, the probability of obtaining

the outcomes y1,y2 is changed to

y1 qh q ,p (y1,y2) ′ (y1,y2) = W ,γDpD, ,γhy2 dpDdqh P D h → PqD,ph γ γ Z D h γ y = h 1 ,γ y , (3.46) γ PqD,ph γ h 2 D D where is the probability distribution when measuringρ ˆ. Similarly, the probability P distribution ofp ˆD andq ˆh measurement is changed to

γD y2 pD,qh (y1,y2) pD,qh (γDy1, ) . (3.47) P → γh P γh

Measurement results of qˆ , qˆ and pˆ , pˆ are sifted as they are merely weakly corre- { D h} { D h} lated. In fact, physically squeezing the state is not necessary because the transformations in Eqs. (3.46) and (3.47) can be conducted by classically scaling the measurement outcomes.

Consider that everyq î measurement outcome is scaled by 1/γi, and everyp î measurement outcome is scaled by γ . The old probability, ,ofaq ˆ measurement outcome lying in the i P range [y,y + dy], is equal to the new probability, s, of a scaled outcome in the range of P [γy,γy + γdy]. Thus we have the relation (y)dy = ′(γy)γdy forq ˆ measurement, and P P similarly (y)dy = ′(y/γ)dy/γ forp ˆ measurement. By eliminating the common factors P P and redefining variables, we get

s γh y1 s γD y2 qD,ph (y1,y2)= qD,ph ,γhy2 and pD,qh (y1,y2)= pD,qh γDy1, . P γD P γD P γh P γh (3.48) The above probability distributions are the same as Eq. (3.46) and (3.47).

3.4.3.3 Local Measurement

I have assumed the access structure parties have forwarded their modes to a single party for global operations. Here I show that the measurement results of the dealer and the access structure remains strongly correlated even if the access structure conducts local measurements only. Let us recall that the strong correlation is represented by the nulliﬁersp ˆ Qˆ and D − A qˆ Pˆ . Because they are linear combinations of standard nulliﬁers, both the operators D − A

48 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

QˆA and PˆA are sums of local operators, i.e.,

n n ˆ q ˆ q ˆ p ˆ p QA = kj Mj ; PA = kj Mj , (3.49) j j X X q p ˆ q ˆ p where kj and kj are real coeﬃcients; Mj and Mj are rotated quadrature operators

of mode j. After agreeing the measurement basis to be QˆA or PˆA, the access structure ˆ q ˆ p parties homodyne detect their modes according to the basis Mj or Mj . The measurement results are then shared among access structure through secure classical channels. Without loss of generality, I consider that the access structure has chosen to mea-

sure QˆA. The resultant outcome QA is a linear combination of the local measurement q n q q outcomes Mj , i.e., QA = j kj Mj . The strong correlation is observed from the joint probability distribution of PpD and QA, which will be shown the same as the joint probability distribution of pD and qh. Let us consider that the Wigner function of the state of the dealer and the access structure, WDA(qD,pD, qA, pA) where qA and pA are the quadrature variables of the access structure, is obtained by tracing out the unauthorised parties’ contributions in Wc in Eq. (3.7). When rewritten in terms of the new variables M q = M q and ∗M q = M q , the Wigner function becomes { j } {∗ j }

q q W (q ,p , q , p ) W ′ (q ,p , M , ∗M ) , (3.50) DA D D A A ≡ DA D D where M q is the complementary variable of M q, i.e., the corresponding operators satisfy ∗ j j [Mˆ q, Mˆ q]= i. The choice of Mˆ q is not unique, but we can pick the set that Pˆ can j ∗ j {∗ j } A be written as a linear combination of Mˆ q. ∗ j I construct another set of variables Q = Q , Q ,...,Q and P = P ,P ,...,P , { A 2 m} { A 2 m} where Q (P ) involves linear combinations of M q ( M q ) only; and the corresponding { j } {∗ j } operators obey the commutation relations: [Qˆj, Pˆl] = iδjl,[Qˆj, Qˆl] = 0, and [Pˆj, Pˆl] = 0. Such a construction of variables is possible as there exists unitary operators that q transform Mˆ q to Qˆ and ∗Mˆ to Pˆ while preserving the commutation relations. In terms of Q and P , the Wigner function can be rewritten again as

q q W ′ (q ,p , M , ∗M ) W ′′ (q ,p , Q, P ) . (3.51) DA D D ≡ DA D D

The local measurement outcomes follow a classical probability distribution ′′ , PDA which is obtained by tracing out the complementary components, i.e.,

q m q m ′′ (p , M )= W ′ dq d (∗M )= W ′′ dq d P , (3.52) PDA D DA D 1 DA D Z Z

49 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

q where the last equality is imposed because P is a linear combination of ∗M1 only . The probability distribution of QA is obtained by tracing out the other independent variables in ′′ , i.e., PDA (p , Q )= ′′ dQ dQ .... (3.53) PDA D A PDA 2 3 Z On the other hand, let us consider that the access structure parties’ modes are transferred to party h. The strong quantum correlation is transferred to mode h by applying the decoding sequence, i.e., a global operation Uˆ , which transforms Qˆ Uˆ † Qˆ Uˆ =q ˆ A A → A A A h and Pˆ Uˆ † Pˆ Uˆ =p ˆ . Other operators are transformed as Qˆ yˆ and Pˆ zˆ . A → A A A h j → j j → j Using the deﬁnition in Eqs. (3.50) and (3.51), the Wigner function becomes

W (q ,p , q, p) W ′′ (q ,p , q ,p , y, z) , (3.54) DA D D → DA D D h h

T T where y =(y2,...,ym) and z =(z2,...,zm) . Because Qˆj and Pˆj commute with both

QˆA and PˆA, the transformed operatorsy ˆj andz ˆj do not contain any attributes of mode

h. The joint probability distribution ofp ˆD andq ˆh is obtained from the Wigner function after tracing out the modes other than mode D and h, as well as the complementary variables qD and ph, i.e.,

m 1 m 1 (p , q )= W ′′ dq dp d − yd − z . (3.55) PDA D h DA D h Z The probability distribution in Eqs. (3.53) and (3.55) are deduced by different procedures. The former one is deduced by first obtaining the classical probability distribution of all local measurements, and then extracting the probability distribution of the classical variable QA; the later one is deduced by first achieving the Wigner function of the transformed quantum state, and then obtaining the measurement probability of the operator qˆh. However, Eqs. (3.53) and (3.55) are mathematically equivalent because their overall derivations are the same: tracing out all quadrature variables in the Wigner function except those specifying the strong correlation. Similar analysis can be applied to the correlation between qD and PA. As a result, the access structure can obtain the same covariance matrix as I have previously discussed. Hence the secret sharing rate remains unchanged even if the access structure’s modes are measured locally.

3.4.3.4 Simpliﬁed CQ Protocol

By incorporating the above ideas, the CQ protocol can be simpliﬁed as follow: a (n +1) cluster state or a n mode mixed state is prepared by the dealer and delivered. Parties

50 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

in the access structure have agreed on the measurement basis in each round, local measurements are conducted on each received modes. The classical measurement results are shared among the access structure through secure classical channels. Both the dealer and the access structure announce half of the results to estimate the covariance matrix; while the other half is scaled and mixed so the covariance matrix is in the standard form. The variance and the analogous channel parameters are then recognised for calculating the secret sharing rate. Finally secure keys are distilled from the strongly correlated measurement outcomes, the key are then used for sharing classical secrets. I end this section with two comments. Firstly, although the quantum channels for delivering cluster states are assumed to be ideal, I believe a modiﬁed version of my protocol would allow secure CQ QSS with realistic (lossy and noisy) channels. The covariance matrix of the delivered modes can still be obtained by parameter-estimation, and subsequent classical manipulations can always scale the measurement results to obey the standard covariance matrix. Secondly, in all the examples I have studied, the entanglement with the unauthorised parties’ modes only add noise to the access structure parties’ modes, while the analogous transmittance remain 1. This result is surprising in the scenario of CV QKD, because imperfection is always simulated by adding noise into a beam splitter which reduces transmittance. I believe this phenomenon originates from the distinctive entanglement structure in the resource states: a cluster state is employed in CQ QSS, while the information carrier in CV QKD is an EPR state.

3.5 QQ Quantum secret sharing

In the QQ setting of QSS, the dealer shares a secret quantum state among parties by delivering a multipartite entangled state. The channels connecting the dealer and the parties can be insecure, so the unauthorised parties can manipulate all the delivered states. In an ideal QQ protocol, the access structure can recover the secret state with perfect ﬁdelity, while the unauthorised parties cannot get any information about the state due to the quantum no-cloning theorem [158]. My QQ QSS scheme is a generalisation of the CQ protocol. The dealer prepares a (n + 1) mode cluster state, of which n modes are distributed to the parties while one is kept by the dealer. After forming the collaborations, the access structure parties forward their modes to party h. I assume the parties are connected by secure quantum channels, so the access structure can combine their modes without being eavesdropped. A global operation is then applied to extract a strongly entangled state between mode D and a

51 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

single mode h.

For an infinitely squeezed QQ cluster state, the strong correlation between the dealer and the access structure is represented by the nullifiersp ˆ Qˆ andq ˆ Pˆ . After D − A D − A all modes are gathered in party h, a decoding operation UÂ is applied to transfer the quantum correlation to mode h, i.e., Qˆ Uˆ † Qˆ Uˆ =q ˆ and Pˆ Uˆ † Pˆ Uˆ =p ˆ . I A → A A A h A → A A A h note that both the nullifiers, QÂ and PÂ, and the operation UÂ are the same as that in the corresponding CQ protocol in Sec. 3.4.

The transformed nullifiers,p ˆ qˆ andq ˆ pˆ , indicate that the dealer and party h are D − h D − h sharing an infinitely squeezed two-mode cluster state, which is a CV maximally entangled state. By jointly measuring the secret state and the two-mode cluster, the dealer can teleport the secret state to mode h. After appropriate error correction according to the dealer’s measurement results, party h can revert the secret state with perfect fidelity.

In the finite squeezing case, party h conducts the same UÂ to transform the strong correlation to mode h. However, mode D and mode h are not maximally entangled, because their states are finitely squeezed and are weakly entangled to other modes. Con- ducting teleportation using the non-maximal entanglement will reduce the fidelity of the teleported state. The inaccurately shared secret state may indicate a reduction of security of the QQ QSS, because some information about the secret state would be leaked through the measurement results announced by the dealer, and through the states held by the adversary structure that are weakly entangled with the teleported state.

Instead of conducting teleportation after each round of QQ QSS, I consider the extracted state is stored in quantum memories. After several rounds of the QQ protocol, a more entangled state can be distilled from the stored extracted states through CV entanglement distillation [63, 38, 58, 132, 4]. Although distilling a maximally entangled CV state is impossible due to the inﬁnite required energy, the enrichment of entanglement can enhance the ﬁdelity of the teleportation.

The amount of entanglement of the distilled state is determined by that of each extracted state, as well as the number of extracted states accumulated in the quantum memory. We quantify the amount of entanglement by the logarithmic negativity E [174], which is the upper bound of the distillable entanglement. The logarithmic negativity of a stateρ ˆ is deﬁned as E(ˆρ) = log ρˆTA , (3.56) || ||1 where the superscript T denotes a partial transpose of the density matrix; is A || · ||1 the trace norm. Ifρ ˆ is a two-mode Gaussian state with a covariance matrix V , the

52 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States

logarithmic negativity can be calculated as

E(ˆρ)= F (˜νk) , (3.57) Xk where F (x) = log(2x) if x < 1/2, and F (x) = 0 if x 1/2; ν˜ is the symplectic − ≥ { k} spectrum of V˜ , which is deﬁned as [34]

I 0 I 0 V˜ = V . (3.58) 0 Z ! · · 0 Z !

The covariance matrix V can be obtained by randomly measuring some of the stored states in eitherq ˆ orp ˆ.

Because logarithmic negativity is additive [174], at least E0/E extracted states with logarithmic negativity E is required to distill a two-mode squeezed vacuum state with

logarithmic negativity E0. As examples, I demonstrate in Appendix A.4 the procedure of extraction, and the calculation of extracted entanglement in each round of the (2,3)- and the (3,5)-protocols. Just as in the CC and CQ cases, finite entanglement can be extracted when the squeezing operator is smaller than a threshold, and the amount of extracted entanglement is different for different collaborations. I note that logarithmic negativity is additive but not strongly superadditive [184], so the amount of entanglement may be overestimated if the access structure’s modes in different rounds are entangled [146], i.e., when the unauthorised parties conduct coherent attacks on the delivered modes. In that case, the amount of entanglement should be characterised by other strongly superadditive entanglement measures, such as distillable entanglement and squashed entanglement [184]. However, logarithmic negativity is applicable in the current case because the quantum channels are assumed to be ideal, i.e., the access structure is expected to get the same states as prepared by the dealer, which are individually prepared in each round. The adversary structure parties get only the information about the shared secret through their modes obtained in each round, which is effectively a collective attack.

3.6 Conclusion

In this work, I extended the uniﬁed cluster state quantum secret sharing framework proposed in [124, 95] into the continuous-variable regime. I proposed that all three tasks of quantum secret sharing can be implemented by CV cluster states. Although a QQ

53 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States protocol can be used to conduct CC and CQ, simplifications in the later two scenarios can reduce the requirement of resources. For a CC protocol involving n parties, only n-mode cluster states are needed, and the states can be measured once received. A CQ protocol requires either a mixture of two n-mode Gaussian states or a (n + 1)-mode cluster state. The states can be locally measured once it is received. A QQ protocol requires (n + 1)-mode cluster states. The states have to be transferred to one party and accumulated in quantum memories for entanglement distillation. On the contrary to discrete-variable systems, where no known physical principle hin- ders the creation of a maximally entangled state, the creation of a maximally entangled CV state requires infinite energy, and is thus not practical. Finitely squeezed states are realistic substitutes for the maximally entangled resources, but the non-maximal entanglement would leak information about the shared secret to the unauthorised parties. I proposed computable measures to account for the security of each of the three tasks of quantum secret sharing. The secret sharing rate of a CC protocol is the difference between the mutual information between the dealer and the access structure, and the adversary structure’s information that is capped by the Holevo bound. The secret sharing rate of a CQ protocol can be computed by calculating the secure key rate of the analogous QKD protocol. The performance of a QQ protocol can be determined by the amount of extracted entanglement between the dealer and the access structure. Although I have demonstrated the analysis of only the (2,3)- and the (3,5)- protocols that are both threshold protocols [51], the technique is applicable to non-threshold protocols because the security analysis involves only the variance of measurement results of the dealer and the access structure. In fact, the security of more general continuous- variable CQ and QQ protocols can be analysed by using my techniques, i.e., transferring the strong correlation to mode h and then compute the covariance matrix between mode D and mode h, even if the resource state is not a continuous-variable cluster state. To the best of my knowledge, the current work is the first one showing that quantum secret sharing is feasible with finitely squeezed CV resources. A finitely squeezed cluster state can be deterministically constructed by using only squeezed vacuum states and linear optics, which are practically available resources in nowadays laboratory; our work significantly lowers the required technological level for implementing quantum secret sharing. However, there are more theoretical investigations have to be done before our scheme is practically useful. An important remaining question is to determine if the performance of quantum secret sharing is seriously worsened under the presence of environmental noise and apparatus imperfections. Besides, the above calculations of secret sharing rate are calculated at the asymptotic limit, which infinite rounds of state distri-

54 Chapter 3. Quantum Secret Sharing with Continuous-Variable Cluster States bution are assumed to have been conducted. In practice, the access structure and the dealer only share finite number of cluster states, which the secret sharing rate would be affected. Nevertheless, as I borrow the security analysis techniques from CV QKD, which works well in noisy and finite-key circumstances, it is likely that the realistic performance of quantum secret sharing can be analysed by using a similar formalism as presented in this work.

55 Chapter 4

Motional States of Trapped Ions

Since it was realized that quantum algorithms can speed up complicated computational tasks, such as factorizing a large integer, which cannot be efficiently performed by known classical algorithms, building a quantum computer (QC) has become one of the ambitious goals in modern physics [143, 5]. Among current proposals for physical implementations of a QC, in many ways the ion trap proposal of Cirac and Zoller [50] seems the most auspicious at the moment. By exploiting well-developed techniques from quantum optics and atomic physics, entanglement of up to 14 ions [77, 136], high fidelity gates and readout [139, 111], and long coherence time quantum memory for more than 10 s [103] have been recently demonstrated. Simple quantum algorithms including Deutsch-Jozsa algorithm [75] and quantum teleportation [151, 21], as well as the verification of the Bell inequality [156], have been successfully realized in ion-trap QC. In the conventional approach of ion trap QIP, two metastable electronic states are employed as a two-level system (qubit) to encode quantum information, while the motional degree of freedom is acted only as a bus for transmitting quantum information. Recently, directly implementing QIP on the motional states has received much attention. There are two reasons for this trend. Firstly, the advanced trapping technology has reduced the heating rate on an trapped ion to lower than 0.1 quanta per millesec- ond, which implies the motional coherent time is longer than 1 millisecond [36]. When comparing with the characteristic time scale of ion motion, which is at the microsecond range in state-of-the-art MHz trap, thousands of coherent operation can be conducted to implement meaningful quantum information process. Secondly, each motional degree of freedom of an ion is quantised as a harmonic oscillator that exhibits bosonic behaviour. In contrast to the electronic states that behave like Fermions, the motional states could function differently in quantum information processing and thus increase the capability of ion trap systems. Additionally, more quantum information can be encoded in a harmonic

56 Chapter 4. Motional States of Trapped Ions

oscillator than in a qubit. In my work, I have studied several aspects related to the quantised ion motion in quantum information processing. In Ch. 5, I study the relation between ion transportation speed and dc Stark effect. From the relation, I deduce the threshold speed above which quantum information will be affected by dc Stark effect during ion transport. In Ch. 6 and Ch. 7, I introduce two architectures of universal bosonic simulator, which can simulate the evolution of any optical system by manipulating the quantised ion motion. In Ch. 8, I invent a new method to remove the unwanted thermal motion of a qubit by collision. Such a method is one order of magnitude faster than the conventional laser cooling methods. Before discussing my works in detail, in this chapter I describe the basics of the ion trap system that will be considered. In Sec. 4.1, I review the configuration of ion traps and the mathematical tools for describing the quantised motion of trapped ions. In Sec. 4.2, I discuss the internal states and their interaction with laser fields. In Sec. 4.3, I introduce a theory that describes the exact evolution of trapped ion motional states under a general harmonic potential. This theory is the key component in my works, because it allows us to develop processes to manipulate ion motion much faster than conventional approaches.

4.1 Trap and Ion Motion

In spite of having similar internal structure to atoms, ions are more controllable due to their net electric charge. Positively charged ions are usually employed in QIP due to its stability against lost of charge. As the ion’s wave function is much shorter than the typical length of trapping apparatus, an ion is usually treated as a point charge. The necessary condition to stably trap a point charge is to construct an electric potential V (~r), so that at the vicinity of the ion, the second derivative of the potential is positive for all the three dimension, i.e.,

∂2 ∂2 ∂2 V > 0 ; V > 0 ; V > 0 . (4.1) ∂x2 ∂y2 ∂z2

However, the potential created by any electrostatic distribution cannot simultaneously satisfy all of the three conditions. This is because 2V = 0 due to the Maxwell’s equation ∇ in vacuum. The key to bypass this limitation is to use an oscillating, instead of a static, electric ﬁeld. Fig. 4.1 shows the design of the Paul trap, which its inventor Wolfgang Paul

57 Chapter 4. Motional States of Trapped Ions

+ + +

Figure 4.1: Layout of Paul trap. Radio frequency potential is applied on the dark electrodes, in order to trap the ions along the x axis. Positive electrostatic (dc) potential is applied on the grey electrode segments, in order to trap the ions (little circles with positive sign) in the the middle of the trap. More details can be referred to Fig. 3 in Ref. [110].

was awarded the Nobel Prize due to this work [145]. The trap consists of four parallel segmented electrodes. In the direction parallel to the electrodes (x direction in Fig. 4.1, which is usually referred as “axial”), electrostatic potential is applied to stably conﬁne the charges. While in the directions perpendicular to the electrodes (y and z directions in Fig. 4.1, which are usually referred as “radial”), a sinusoidal potential oscillating at radio frequency (rf), ωrf, is applied to the electrodes of one of the diagonals (dark electrodes in Fig. 4.1). The overall potential experienced by a point charge can be approximated as

V (~r)= U x2 + U cos(ω t)(y2 z2) . (4.2) dc ac rf −

For an ion with mass m experiencing such potential, the y-direction motion follows the equation of motion: my¨(t)= U cos(ω t)y(t) . (4.3) − ac rf The motion in the z-direction follows the same equation except the a π/2 phase shift in the radio frequency potential. After scaling, this equation can be transformed to the

Mathieu equation [110]. For some range of Uac and ωrf, the solution of Mathieu equation is bounded. Thus ions are stably trapped in the radial as well as the axial direction. Before quantum information processing, ions are cooled to near the ground state by Doppler and resolved sideband cooling [181]. For a singly trapped ion, around its

58 Chapter 4. Motional States of Trapped Ions

classical position the static axial trap potential and the eﬀective radial trap potential can be approximated by harmonic wells. The Hamiltonian of the ion motion is thus

pˆ2 pˆ2 pˆ2 1 1 1 Hˆ = x + y + z + mν2xˆ2 + mν2yˆ2 + mν2zˆ2 , (4.4) m 2m 2m 2m 2 x 2 y 2 z where νx depends on the axial static potential; νy and νz depend on Uac and ωrf [110]. By this Hamiltonian and canonical quantisation, the ion motion is quantised as three independent harmonic oscillators. Each quanta of motional excitation in one direction is regarded as a phonon. Because the phonons share the same form of Hamiltonian as photons, which is also a harmonic oscillator, phonons are expected to exhibit bosonic behaviours, and each motional degree of freedom is analogously referred as a mode. Here I have introduced the quantisation of single ion motion. In future chapters, I will discuss that if a trap contains more than one ions and they are close enough that Coulomb interaction become significant, then the phonon mode has to be redefined as the collective motion of multiple ions. In experiments, the effective radial trapping frequency is about ν 2π 10 MHz, y,z ≈ × which is roughly one order of magnitude higher than the axial trapping frequency ν x ≈ 2π 1 MHz. In my work, I focus on the motional state on the axial direction, while × the radial motion is assumed to be tightly confined and would not be excited during operation on the axial modes.

4.2 Internal Structure and Laser Operation

The type of ions involved in quantum information processing should have simple atomic structure in order to be manipulable by laser operation. The singly charged alkaline earth ions are particularly suitable because they have only one outer most electron, so their electronic structure resembles that of a hydrogen atom. In order to provide a long storage time of quantum information, encoding states should be forbidden from dipole transition. Examples of suitable electronic states include the 4S and 3D states in 40Ca+ ion, which the decay time is about 1 second [90], and the hyperﬁne states of 9Be+ ion, which the decay time is longer than centuries [66].

4.2.1 Transition

The transition between the encoding states requires non-dipole transition, such as quadrupole transition or Raman transition [110]. In experiments, Raman transition is often employed

59 Chapter 4. Motional States of Trapped Ions

Figure 4.2: Energy levels of a Λ-type system (left side). The quantum information encoding states g and e are dipole transition forbidden. The auxiliary state d is dipole transition| allowedi | betweeni both g and e , with transition frequency ω and| iω | i | i 1 2 respectively. Under Raman transition caused by two detuned laser ﬁeld, the electron transits between g and e as in a two level system (right side), while the auxiliary state | i | i is barely populated.

due to the strong coupling strength and the high stability of lasers. The implementation of a Raman transition requires an auxiliary state that is dipole-transition allowed with both encoding states; the energy level of an example of such system is shown in Fig. 4.2. Two laser fields are applied that are both ∆ detuned from each of the dipole transition frequency. If ∆ is much larger than the Rabi frequency of each laser field, the auxiliary state is barely populated, while the electron transits between the encoding states just as in a two-level system. Unless specified, throughout this thesis I will consider the internal states of a trapped ion as a two-level system, and the transition between the encoding states is effectively driven by a single light field. For a field with frequency ω, wave vector ~k, and phase φ with respect to some phase reference, the Hamiltonian is given by

Hˆ = ~Ω e g exp(i~k.rˆ ωt + φ) + h.c. , (4.5) | ih | −

where Ω is the Rabi frequency 1. The spatial extent of the light field will couple with the ion motional states via the term ~k.rˆ, wherer ˆ is the position vector operator of the ion position. The coupling strength of a mode depends on (i) the intensity of light field, which determines the Rabi frequency; (ii) the direction of the light field, contributes to the intersection angle in ~k.rˆ; and (iii) the frequency of the light field, ω, which would induce strong coupling when it is in resonant with the transition and the mode frequency. In future discussions, I will

1 For a Raman transition driven by two field both with Rabi frequency Ω0, the effective Rabi frequency 2 ~ ~ is Ω = Ω0/∆. The effective transition frequency, wave vector, and phase are ω1 ω2, k1 k2, and φ2 φ1 respectively [110]. − − −

60 Chapter 4. Motional States of Trapped Ions

click No click |d |d ed ed

|e |e |g |g Figure 4.3: Left: If the electron is in e , photons will be scattered. Right: If the electron is in g , no photon is scattered. | i | i

consider the light field is tuned to be off-resonant from the radial mode, so these modes will not be affected by the drive and could be neglected. In other words, I consider only the ion’s axial motion, while the radial modes will remain in the ground state.

4.2.2 Measurement

Apart from transition, the electronic states can also be measured by applying a laser field. The frequency of the field is tuned to resonant to the transition between e and an | i excited state d , i.e., ω . If the state is in e , the laser field will interact with the dipole | i ed | i moment between e and d , so some photon of the field is scattered. On the other hand, | i | i if the state is in g , the laser field would not interact with the electronic state. The | i idea is shown schematically in Fig. 4.3. Therefore the electronic state can be projectively measured by detecting the scattered photons (fluorescence) by nearby photon-detectors.

4.3 Ion Motion in Trap Potential

As previously described, each ion can be treated as a two-level system 2 with three harmonic oscillators attached. For a large scale quantum information process, multiple trapped ions are required to operate collaboratively. This is usually achieved by conﬁning a chain of ions along the axis of a single linear trap [50]. However, this method cannot be scaled to involve much more than ten ions, otherwise the unwanted interaction may become serious [181]. A scalable architecture ion trap quantum information processor is proposed in Refs. [181, 99] (This architecture is referred as the Kielpinski-Monroe-Wineland (KMW) architecture throughout the thesis). The layout of this architecture is shown in Fig. 4.4. This

2More internal levels could be involved to form a qudit system, but each additional internal level requires an extra laser of diﬀerent frequency, which is resources consuming in practice.

61 Chapter 4. Motional States of Trapped Ions

+ + + Electrodes

+ +

Interaction region Memory region

Figure 4.4: Layout of scalable ion trap quantum information processor. Ions are initially stored in storage traps (memory region). To conduct quantum logic operations, ions from diﬀerent traps are transported to and combined into a single trap (interaction region). Both ion transportation and combination are implemented by varying the electrostatic potential of the segmented electrodes. More details can be referred to Fig. 1 in Ref. [99].

architecture consists of interconnected traps, each of which is assigned for a specific function. In each trap serving as a memory, a chain of a small number of ions, say less than ten, is stored. During quantum information processing, ions are transported to different traps through linear traps and junctions. The ion transportation has been demonstrated experimentally [155, 33, 175, 81, 30, 159]. When two chains of ions have to interact, they are combined into a single interaction trap. After transportation or combination, the ions can be cooled by sympathetic cooling to the ground motional state, so subsequent quantum logic operations can be conducted precisely [22, 115]. The KMW architecture is regarded as a promising approach due to various advantages. For example, each trap contains only a small number of ions, therefore the quality of quantum logic operation is independent of the number of ions in the processor. Be- sides, the operation in one trap is barely affecting the ions in other traps because they are spatially well separated. Therefore logic operations can be conducted in parallel, hence the number of operational cycles of a computational task can be reduced. The key feature that allows these advantages is ion transportation. It can be implemented by tuning the trapping potential through varying the voltage of the segmented electrodes around the ions. The utility of a KMW quantum information processor could be facilitated by rapid and precise manipulation of trapping potential. Such a degree of control has been realized in recent experiments [33, 175]. Although the trapping potential can be very complicated, it can be approximated by harmonic wells around the vicinity

62 Chapter 4. Motional States of Trapped Ions

of each ion. The quantised ion motion can then be well described by the evolution of harmonic oscillators. In the following, I present the exact solution of a wave function under a general harmonic potential. As we will see in future chapters, the exact solution allows us to design quantum logic and operational processes that are fast and do not induce additional heating.

4.3.1 Generalized Harmonic Oscillator

The most general harmonic oscillator consists of a harmonic well that both the potential strength and the position of trap centre are time dependent. The wave function ψ(t) | i obeys the Schr¨odinger equation

pˆ2 1 i~∂ ψ(t) = Hˆ ψ(t) + mν2(t)(ˆx s(t))2 ψ(t) . (4.6) t| i | i ≡ 2m 2 − | i This generalized harmonic oscillator has been investigated in the context of driven Fock states [118] and the variation of quadrature operators in a driven oscillator [100]. Here I formulate it in the context of a trapped ion to give a clear physical understanding about the quantum and classical evolution of the system. Firstly, I decouple the classical motion and quantum fluctuation from the total motional state. Let us define the state of the quantum fluctuation, χ(t) , as | i

χ(t) ˆ†(x ,p ) ψ(t) , (4.7) | i ≡ D c c | i

where xc(t) and pc(t) are real functions of time; the displacement operator is deﬁned as

ˆ(x ,p ) exp (i(x pˆ p xˆ)/~) . (4.8) D c c ≡ c − c

The quantum ﬂuctuation obeys the equation

i~∂ χ(t) = Dˆ †Hˆ Dˆ iDˆ †(∂ Dˆ) χ(t) (Hˆ + Hˆ ) χ(t) . (4.9) t| i − t | i ≡ 1 2 | i

Hˆ1 is the collection of all terms containing the ﬁrst order of position and momentum operators, i.e., Hˆ = pˆ + qˆ, where 1 V F p = c x˙ ; =p ˙ + mν2(t) x s(t) . (4.10) V m − c F c c − If xc and pc are chosen respectively as the ion’s classical position and momentum, i.e.,

63 Chapter 4. Motional States of Trapped Ions

they obey the classical equation of motion

p x˙ = c ;p ˙ (t)= mν2(t)(x s(t)) , (4.11) c m c − c −

then and , and thus Hˆ vanish. The dynamics of the quantum ﬂuctuation is solely V F 1 determined by Hˆ2 that involves only the second order terms of position and momentum operators, viz.

pˆ2 1 i∂ χ(t) = Hˆ χ(t) + mν2(t)ˆx2 χ(t) . (4.12) t| i 2| i ≡ 2m 2 | i

Hˆ2 is the general Hamiltonian of a harmonic oscillator with time dependent well strength, ν2(t), but a ﬁxed well centre at x = 0. Now the action of the above operations becomes clear: separating the classical and quantum attributes of motion. The classical motion is completely described by the classical equation of motion, and it is controllable by adjusting the well centre s(t). On the other hand, the quantum ﬂuctuation of motion evolves as a time dependent quantum harmonic oscillator that is independent of s(t). I note that the above procedure of classical-quantum motion separation is also applicable for other (non-harmonic) potentials.

The exact solution of a general time dependent harmonic oscillator was proposed by Lewis and Riesenfeld. They deduce the solution by considering the dynamic invariant operator [113, 45] :

˙ 2 (bpˆ mbxˆ) 1 2 2 1 ˆ(t)= − + mν xˆ = ~ν Aˆ †(t)Aˆ(t)+ . (4.13) I 2m 2 0 0 2 where b(t) is a dimensionless real auxiliary function that satisﬁes the equation

ν2 ¨b + ν2(t)b 0 =0 . (4.14) − b3

ν0 is a characteristic frequency of the problem that could be taken as the static harmonic frequency before or after the potential variation. The operators Aˆ(t) and Aˆ †(t) are the raising and lowering operators of the eigenstates, λ , t , of ˆ(t), i.e. | n i I

ˆ(t) λ , t = λ λ , t , (4.15) I | n i n| n i Aˆ(t) λn, t = √n λn 1, t , (4.16) | i | − i Aˆ †(t) λn 1, t = √n λn, t , (4.17) | − i | i

64 Chapter 4. Motional States of Trapped Ions

with λn being the corresponding eigenvalues. The dynamic invariant is deﬁned in such a way that its total time derivative vanishes, i.e., its Heisenberg equation of motion becomes

d ˆ(t) i~∂ ˆ(t) + [ˆ(t), Hˆ (t)]=0 . (4.18) dtI ≡ tI I

As the system evolves, the values of λ remain unchanged, and the eigenstates λ , t are n | n i always orthogonal during the evolution, i.e.,

λ , t i~∂ Hˆ (t) λ , t =0 , if n = m . (4.19) h m | t − | n i 6 Then the evolution operator of quantum ﬂuctuation from time t to t′ can be deduced as 3 ∞ i(n+ 1 )(Θ(t) Θ(t′)) Uˆ (t, t′)= e− 2 − λ , t λ , t′ . (4.20) χ | n ih n | n=0 X The phase Θ is chosen as t ν0 Θ(t)= dt′′ , (4.21) b2(t ) Z0 ′′ i(n+ 1 )Θ(t) such that the states e 2 λ , t are solutions of Eq. (4.12). | n i When the harmonic well is static, i.e. ν is a constant, the general real solution of Eq. (4.14) is [113]

ν b (t)= 0 cosh δ + sinh δ sin(2νt + ϕ) , (4.22) static ν r p where δ and ϕ are constant parameters. In my work, I am mainly interested in the operations that the trapping potential are steady at the beginning and the end, i.e.,

ν(t < ti)= ν0 ; ν(t > tf )= νf (4.23)

where ti and tf are the starting and ending time of the operation. In general, the values of δ and ϕ have to be determined by integrating Eq. (4.14), but for simplicity I can set

both δ = 0 and ϕ = 0 at the beginning, so that b(t < ti) = 1. I pick the initial situation as a reference. The annihilation operator of the initial oscillator is defined as mν i aˆ = 0 xˆ + pˆ . (4.24) 2~ √2m~ν r 0 3Throughout this thesis, an evolution operator is considered in the Schrödinger picture, while the “evolution operator” in the interaction picture is referred as the S-matrix that will be defined later.

65 Chapter 4. Motional States of Trapped Ions

Since the invariant operator ˆ(t) is identical to Hˆ (t) at t = t , we have I i

Aˆ(ti) =a ˆ ; (4.25)

and thusa ˆ λn, ti = √n λn 1, ti . (4.26) | i | − i

After the operations, the lowering operator becomes [113]

Aˆ(tf )= η(tf )â + ζ(tf )â† , (4.27) where 1 1 b˙ 1 1 b˙ η(t)= + b i , ζ(t)= b i . (4.28) 2 b − ν0 ! 2 b − − ν0 ! The absolute magnitudes of η(t) and ζ(t) satisfy the normalisation condition η 2 ζ 2 = | | −| | 1. The action of the harmonic potential variation can be represented by the evolution ˆ ˆ of the annihilation operator, i.e., Uχ†aÛχ, which can be obtained from the relationship between the raising operator and the annihilation operator. According to Eqs. (4.16) and (4.26), we have

Aˆ i(Θ(tf ) Θ(ti)) ˆ ˆ (tf )= e− − Uχ(tf , ti)ˆaUχ†(tf , ti) . (4.29)

By linearly combining the above equation and its complex conjugate, the evolution of the annihilation operator in the Heisenberg picture is given by

i(Θ(t ) Θ(t )) i(Θ(t ) Θ(t )) Uˆ †(t , t )ˆaUˆ (t , t )= η∗(t )e− f − i aˆ ζ(t )e f − i aˆ† . (4.30) χ f i χ f i f − f

Now I can combine the evolution of both the classical motion and quantum ﬂuctuation into a complete evolution of the wave function. Let us assume that both the harmonic

well and the ion are initially resting at the origin, i.e., s(ti) = 0 , xc(ti) = 0,and pc(ti) = 0.

By substituting the deﬁnition of Uˆχ into Eq. (4.7), the evolution operator of the total wave function, i.e., Uˆ(t, t ) ψ(t ) = ψ(t) , is given by i | i i | i

Uˆ(t, t )= ˆ(x (t),p (t))Uˆ (t, t ) . (4.31) i D c c χ i

Let us deﬁne a complex displacement β as

mν 1 β(t)= 0 x (t)+ i p (t). (4.32) 2~ c 2~mν c r r 0

66 Chapter 4. Motional States of Trapped Ions

Then the displacement operator in Eq. (4.8) can be rewritten as

Dˆ(x ,p ) Dˆ(β) = exp(βaˆ† β∗aˆ†) . (4.33) c c ≡ −

I note that now the annihilation operator corresponds to the total wave function instead of the quantum ﬂuctuation. The transformation of the annihilation operator is then given by

i(Θ(t ) Θ(t )) i(Θ(t ) Θ(t )) Uˆ †(t , t )ˆaUˆ(t , t )= η∗(t )e− f − i aˆ ζ(t )e f − i aˆ† + β(t ) . (4.34) f i f i f − f f

If b(t) is known, a closed form of the classical displacement can be obtained as [100]

tf m i(Θ(t ) Θ(t )) i(Θ(t ) Θ(t )) 2 iΘ(t) β(t )= i η∗(T )e− f − i + ζ(T )e f − i b(t)ν (t)s(t)e dt . f 2~ν r 0 Zti (4.35) Eq. (4.34) shows that the most general harmonic potential causes only two eﬀects on the state: a squeezing operation, as indicated by the ﬁrst two terms in the right hand side; and a displacement operation, as represented by the last term in the right hand side.

4.3.2 Interaction Picture

I have discussed the evolution of a general harmonic oscillator in the Heisenberg picture, it may be useful to consider the evolution also in the interaction picture, which is commonly employed in quantum optics. The interaction picture wave function is deﬁned as

exp iHˆ (t t )/~ ψ (t) = ψ(t) , (4.36) − 0 − i | I i | i

where Hˆ0 is an arbitrarily chosen reference harmonic oscillator

pˆ2 1 Hˆ + mν2xˆ2 . (4.37) 0 ≡ 2m 2 0

The “evolution operator” of an interaction picture wave function is the S-matrix, i.e.,

ψ (t ) = ˆ(t , t ) ψ (t ) , (4.38) | I f i S f i | I i i

where the S-matrix is deﬁned as ˆ(t , t ) exp(iHˆ (t t )/~)Uˆ(t , t ). Therefore the S f i ≡ 0 f − i f i

67 Chapter 4. Motional States of Trapped Ions

annihilation operator is transformed in the interaction picture as

iν0(t t ) aˆ ˆ†(t , t )ˆa ˆ(t , t )= Uˆ †(t , t )ˆaUˆ(t , t )e f − i . (4.39) → S f i S f i f i f i

From now on I denote the notation “ ” as the above transformation. → By combining Eqs. (4.39) and (4.34), the interaction picture evolution of the annihilation operator is given by

i(Θ(t ) Θ(t ) ν0(t t )) i(Θ(t ) Θ(t )+ν0(t t )) iν0(t t ) aˆ η∗(t )e− f − i − f − i aˆ ζ(t )e f − i f − i aˆ† + βe f − i . (4.40) → f − f

68 Chapter 5

Decoherence Induced by dc Electric ﬁeld During Ion Transport

5.1 Speed of ion trap quantum computer

In spite of the rapid advancement of the ion trap system, we are still far from having a quantum computer (QC) with the computing power higher than (or even comparable to) its classical counterparts. Apart from the problem that only a small number of entangled qubits have been realised, the speed of quantum operations is another issue that limits the clock rate of an ion trap QC. For example, let us consider a QC is built to run the Shor algorithm. It would be of great practical interest only if it could break a RSA classical cryptography code in a short time, say a few hours. Then each quantum logic gate has to be performed at the time scale of µs 1 [65]. When employing fault- tolerant techniques, each quantum logic gate consists of several concatenated rounds of physical quantum operations on ion qubits, including transportation, cooling, and laser interaction [181, 99]. Assessing the true time speciﬁcations for a QC in a complicated problem depends on paradigm choices, such as circuit-based versus measurement-based QC, the error correction code used, and the implementation of the algorithm to be performed. But we can assert quantum computation is useful and promising if the speed of each physical quantum operation is of nanosecond scale [164]. In this chapter, I present my work with Daniel James in investigating the eﬀect of direct current (dc) Stark shifts during ion trap QC operation according to the KMW architecture [181, 99] discussed in Sec. 4.3 . During quantum computing, ions are moved from the storage region to the interaction region, and transported back to storage region

1As shown on Table 1 of [65], the most eﬃcient quantum algorithm for factorizing a 640 bit number requires 3000 (640)2 =1.2 109 quantum operations. × ×

69 Chapter 5. Decoherence Induced by dc Electric ﬁeld During Ion Transport

after operations. The ions can be transported by changing the electric potential of each trap that induces an eﬀective non-equilibrium electric ﬁeld.

Fast transportation of an ion requires a large electric field, which will result in a va- riety of potentially detrimental effects. One problem is that a large electric field is less stable; the field fluctuations will heat up the ion to motional excited states [155]. Cycles of sympathetic cooling are required to bring the ion back to its motional ground state for precise logic operations, but the operation is time-consuming and thus not preferable in high-speed quantum computation. The motional heating effect is anticipated to be reduced by improving experimental techniques, such as using surface traps and coating the electrodes of the trap [155], or transporting the ions under trajectories with minimal vibrational quanta excitation [168]. Besides, some proposals for entanglement gates remain effective even though the ion has a small motional excitation [163, 91].

More seriously, a large electric field will induce a dc Stark effect onto the internal electronic states of ions. Due to the detuning of energy levels and mixing of eigenstates, information-encoded quantum state will be altered by being (i) phase-shifted, and (ii) excited out of the computational basis. Although this issue was first discussed more than 10 years ago by Wineland et al. (see Ref. [181] p.310), its significance was neglected at that time due to the low operational speed they envisioned. This effect will become important as the trapped ion shuttling speed is becoming faster; ignoring this effect may cause decoherence on the quantum information. The aim of my work is to find out the relation between the total effects by dc Stark shift, the size of trap, trajectory and total time of the flight of ion qubits. From the relation, I define the ‘threshold speed’ of ion transportation, above which the influence of dc Stark effect becomes significant.

This chapter is organized as follows. In Sec. 5.2, I use the tools presented in Sec. 4.3 to calculate the classical trajectory of an ion being transported by a displacing harmonic well. Because the ion’s classical motion is determined only by the dc electric field, I could obtain from the acceleration the net electric field experienced by the ion. In Sec. 5.3, the total phase-shift induced by dc Stark effect is calculated. In Sec. 5.4, I minimize the phase-shift with respect to the ion trajectory. A threshold speed is deduced from the relation between the minimum phase shift and the optimized time of flight. In Sec. 5.5, I calculate the threshold speed for various choices of ion qubits. In Sec. 5.6, I discuss the significance of state excitation caused by dc Stark effect. I summarize the results in Sec. 5.7 with some discussion. I note that most of the material in this chapter has been published as Ref [106]. For simplicity, I do not consider the junctions where ions are transferred between different traps; the ion is assumed to move along a linear trap only.

70 Chapter 5. Decoherence Induced by dc Electric ﬁeld During Ion Transport

5.2 Motion of ion

In the axial direction, the ion is weakly trapped by an electrostatic ﬁeld. Close to a minimum of the axial electrostatic potential, the Hamiltonian of the system can be approximated by a harmonic oscillator 2. During shuttling, the strength of the electrostatic ﬁeld is changed so that the potential well is displaced. In order to avoid parametric excitation that complicates the problem, let us suppose the the strength of harmonic well

is tuned to be constant throughout the process, i.e., νx(t) = ν0. The time-dependent Hamiltonian is given by

pˆ2 1 Hˆ (t) + mν2 [ˆx s(t)]2 . (5.1) M ≡ 2m 2 0 −

Let the transportation lasts from t =0 to t = T . The general evolution operator of the Hamiltonian in Eq. (5.1) is given by Eq. (4.31). The constant trap frequency gives a constant auxiliary function, i.e., b(t) = 1, and as a result Uˆχ is reduced to a phase-shift operator, i.e., ˆ ˆ iν0t Uχ†(t)ˆaUχ†(t)=ˆae− . (5.2)

If the harmonic potential is centered initially at x(0) = 0 and the ion is prepared in the ground motional state, i.e. Ψ(0) = 0 , Eqs. (4.31) and (5.2) implies that the ion | i | i remains in a coherent state throughout the whole process, i.e. Ψ(t) = α(t) . The | i | i displacement α(t) can be obtained from Eq.(4.35) with b(t) = 1 and ν(t)= ν0. After an integration-by-parts, the displacement can be found as

t mν0 iν0t iν0t1 α(t) s(t) e− s˙(t )e dt . (5.3) ≡ 2~ − 1 1 r Z0 The ﬁrst term on the right hand side represents the position of the potential well centre, while the second term, which represents the displacement of the ion with respect to the well, contributes to the motional heating when ignored. I note that a s(t) can always be found to produce any ﬁnal displacement (classical position and momentum) [168, 46]. Time variation of the expectation value of the ion position (classical trajectory of the ion), q(t), can be calculated as

t q(t)= s(t) s˙(t ) cos [ν (t t )] dt . (5.4) − 1 0 − 1 1 Z0

2Here I only consider harmonic potentials, more general trapping potential and detailed dynamics of trapped ion are discussed in [168].

71 Chapter 5. Decoherence Induced by dc Electric ﬁeld During Ion Transport

~ The net dc electric ﬁeld experienced by the ion, ξ = ξ~ex where ~ex is the unit vector along x direction, is directly proportional to the ion’s classical acceleration (second derivative of the position expectation value) by the Newton’s third law, viz.,

m mν2 t ξ(t)= q¨(t)= 0 s˙(t ) cos[ν (t t)]dt . (5.5) e e 1 0 1 − 1 Z0 5.3 Phase shift due to dc Stark eﬀect

In this section, I study the total phase shift during the transportation process. The effect of the applied electric field on the ion’s internal structure is described by an additional Hamiltonian Hˆ = d~ ξ~, where d~ is the dipole operator. Since the electric field varies Stark − · much slower than the electronic states evolution, which the time scale is characterised

by 1/ωnm, the dc Stark shift energy at time t can be obtained by the time independent perturbation theory [6], viz.,

2 (0) (2) (2) eξ(t) m xˆ n En(t) En + En (t); En (t)= | h | | i| (5.6) ≈ ~ωnm m=n X6 where electric field points to x direction only; ~ω ~ω ~ω is the energy difference nm ≡ n − m between internal states n and m . | i | i Suppose f and i are the computational basis states 3. The ion is initially encoded | i | i with some quantum information as the state α i + β f . Because the dc Stark energy | i | i of each internal state is different, a relative phase between the computational states will be induced after transportation, i.e.,

α i + β f α i + βeiφ f , | i | i→ | i | i up to an unimportant global phase. According to Eqs. (5.5) and (5.6), the extra phase factor φ is given by

(2) (2) T E (t) E (t) m2 m xˆ i 2 m xˆ f 2 φ i − f dt = |h | | i| |h | | i| ζ[q(t)] , (5.7) ≡ ~ ~ ~ωim − ~ωfm 0 m=i m=f ! Z X6 X6 where T ζ[q(t)] = q¨(t)2dt . (5.8) Z0 All terms except ζ depend on only the atomic structure of the ion, so a good choice of ion 3The relative phase due to the energy diﬀerence of the states is eliminated in the interaction picture.

72 Chapter 5. Decoherence Induced by dc Electric ﬁeld During Ion Transport

qubit would give a small magnitude of these terms. ζ is a functional of the ion trajectory, q(t), which is independent of the choice of ion species and the computational states.

5.4 Minimum possible phase shift

As shown in Eqs. (5.7) and (5.8), the phase shift is linearly proportional to the trajectory

functional ζ. Among all the possible trajectories, there is an optimal one, q0, which

produces the minimum ζ, and hence the minimum phase shift, φmin. I note that ζ is independent of the ion species, so its minimum value, ζ[q0], is only a function of the transportation length, L, and the time of the ﬂight, T .

The minimum ζ[q0] can by found by using calculus of variation (see, e.g. [7]). I ﬁrst set the speed of the ion, γ(t), as an independent parameter function. Its relationship with the ion position, i.e., γ(t) =q ˙(t), (5.9) will be treated as a constraint of the minimisation. Then ζ can be rewritten as

T ζ˜[q, q,˙ γ, γ,˙ t]= γ˙ 2(t)+ µ(t)[q ˙(t) γ(t)] dt , (5.10) − Z0 where µ(t) is the Lagrange multiplier for the constraint in Eq. (5.9). Although the minimisation of ζ and ζ˜ are equivalent, it is beneﬁcial to conduct calculus of variation on ζ˜ because it is a functional of q, γ, and their ﬁrst order time derivative only (i.e. no high order derivatives). The Euler equations with respect to q and γ are given by

µ(t) 2¨γ(t) = 0 (5.11) − µ˙ (t) = 0 . (5.12)

Incorporating these equations and the constraint Eq. (5.9), we ﬁnd q0 obeys

.... q0 (t)=0 . (5.13)

I assume the ion is displaced by distance L after the process, the initial and ﬁnal position can be set as q(0) = 0 and q(T )= L. Additionally, I require the ion remains in the motional ground state before and after the shuttling, so the initial and ﬁnal velocity both vanish, i.e.q ˙(0) =q ˙(T ) = 0. Putting these conditions into Eq. (5.13), the optimal

73 Chapter 5. Decoherence Induced by dc Electric ﬁeld During Ion Transport

trajectory can be found as

t3 t2 q (t)= L 2 +3 , (5.14) 0 − T 3 T 2 2 3 which gives ζ(q0)=12L /T . Hence the minimum phase shift is

12m2L2 m xˆ i 2 m xˆ f 2 φmin = 3 |h | | i| |h | | i| . (5.15) ~T ~ωim − ~ωfm m=i m=f ! X6 X6

Contrary to the adiabatic transportation [156], I note that the ion does not stay the equilibrium position of the potential well s(t) throughout the transportation. As expected by the classical intuition, the ion ‘sloshes’ when the well is displaced. Transporting the ion in the desired trajectory has to be achieved by tuning the trapping electric ﬁeld carefully to balance this sloshing. The position of the well can be obtained by combining

Eq. (5.4) with its second derivative. The displacement of the well s0(t) which produces the optimal trajectory of ion is given by:

q¨ (t) t3 t2 t 6 s (t)= q (t)+ o = L 2 +3 12 + . (5.16) 0 0 ν2 − T 3 T 2 − ν2T 3 (ν T )2 0 0 0

Both q0(t) and s0(t) are shown schematically in Fig. 5.1. The position of the potential well jumps sharply at t = 0 and t = T . This means that the trapping electric fields are sharply changed at the beginning and at the end. However, the sudden variation of electric field does not cause a significant dc Stark effect. This is because the energy perturbation, within the dipole approximation, only depends on the strength of electric field but not its time derivative.

Although the phase shift is minimum when the ion travels in the optimal trajectory,

but trajectories other than q0 may be employed due to experimental convenience. Here I investigate the robustness of the optimality, by studying if the value of ζ dramatically increases when the ion takes another trajectory. Firstly, I consider a more experimental realizable trajectory that the electric ﬁeld is tuned gradually, i.e. no sharp jumps of electric potential. I construct a trajectory that is a ﬁfth order of time, so that there are six parameters to incorporate the constraints of the continuity of ion motion, i.e., q(0) = 0, q(T )= L,q ˙(0) =q ˙(T ) = 0, as well as the continuity of harmonic well position, i.e., s(0) = 0, s(T )= L. Such an ion trajectory is given by

q(t)= L(6t5/T 5 15t4T 4 + 10t3/T 3) . (5.17) −

74 Chapter 5. Decoherence Induced by dc Electric ﬁeld During Ion Transport

Figure 5.1: Time variation of the optimal trajectory of ion, q0(t) (solid line), and the optimal trajectory of potential well, s0(t), for trapping frequencies ν0 = 3/T (dashed), ν0 =5/T (dotted), and ν0 = 10/T (dot-dashed).

2 3 This trajectory gives ζ = 17.1L /T , which is larger than ζ[q0] but remains in the same order of magnitude. Secondly, I consider the trajectory of the experimental setting of Rowe et al. [156], where the location of the trapping potential sR varies as

πt s (t)= L sin2 , (5.18) R 2T and the frequency of the potential well is ν = 2π 2.9 MHz. By obtaining the ion R × 2 3 trajectory qR from Eq. (5.4), ζ is found to be ζ(qR) = 24.3L /T , which is only about a double of ζ[q0]. The ζ in these two examples are at the same order as ζ[q0], therefore

φmin is a useful reference for the phase shift induced during ion transportation. I note that if the anharmonicity of the potential well is ignored, following the above trajectories an ion can be transported in an arbitrary short time. The transportation would induce minimal ﬁnal motional excitation even though the speed is much higher than the adiabatic limit. 4

4To compare the adiabatic and non-adiabatic transportation speed, let us consider the adiabatic trajectory s(t)= tL/T . Adiabaticity requires the motional excitation about the trap centre to be small at any time during the transportation, i.e., α(t) s(t) 2 1. According to Eq. (5.3), the adiabaticity 2 2m L | − | ≪ 9 2 requirement can be written as ~ν0 T 1. As an example, for a trap with length L 100µm, a Be+ ion in a MHz trap has to be transported≪ at a time T 0.66 ms. In a recent experiment,≈ a trajectory ≫

75 Chapter 5. Decoherence Induced by dc Electric ﬁeld During Ion Transport

5.5 Threshold speed of transporting ion qubits

In this section, I calculate the threshold speed of ion transportation, above which φmin becomes signiﬁcant, i.e., the magnitude is too large to be corrected by quantum error correction. Since the threshold error rate of quantum error correction codes are about

1% (see e.g. [149]), I set the upper limit of φmin as π/100. I assume the length of the trap being travelled is L = 100µm, which matches setting of current experimental setup [156]. Two popular types of ion qubits are investigated. The first one is 40Ca+ ion [151], where the computational basis states are electronic states S , M = 1/2 = i and | 1/2 J − i | i D M = 1/2 = f . The second type is 9Be+ ion [103, 21, 156], where the compu- | 5/2 J − i | i tational states are hyperfine states F =2, m =0 = i and F =1, M =1 = f . | i | i | F i | i Now I calculate the term inside the bracket in Eq. (5.15). Values of the matrix elements can be calculated from the tabulated parameters in Ref. [8]. I demonstrate here the calculation of the terms belonging to i while the procedure for f is similar. | i | i Let the state i belongs to an energy level k. It can be expressed as a superposition of | i states with definite magnetic quantum number r , i.e., | i

i = A r . (5.19) | i ir| i r k X∈ I pick the intermediate states m as the energy eigenstates with a deﬁnite magnetic | i quantum number, so the summation of m involves summing all the states in a particular | i energy level l, and then summing over all energy levels. The transition energy ~ωim is the energy diﬀerence between levels k and l, which the value can be found in Ref. [8]. When calculating the matrix elements of the dipole operator, I extract the reduced matrix elements as

m xˆ i 2 = k Qˆ1 l 2 A 2 C 2 , (5.20) |h | | i| |h || || i| | ir| | mr| m l r k m l X∈ X∈ X∈ where Qˆ1 is the rank 1 irreducible tensor operator, k Qˆ1 l 2 is the reduced matrix |h || || i| 5 element between energy levels k and l, Cmr is the Clebsch-Gordan coefficients between m and r 6. The reduced matrix elements can be calculated from the line strength S | i | i lk derived from Eq. (5.3) has been employed in transporting a 9Be+ ion across a 370 µm trap [33]. While the final excitation is insignificant (about 0.1 quanta), the transportation is conducted in 8 µs, which is two orders of magnitude faster than the adiabatic limit. 5Not to confuse with the reduced density matrix, which is the quantum state of a subsystem after tracing out some degree of freedom of a quantum system. 6Expressions of the Clebsch-Gordan coefficients are given in, e.g. Appendix 4, p.1000 in Ref. [9].

76 Chapter 5. Decoherence Induced by dc Electric ﬁeld During Ion Transport

provided in Ref. [8]. By deﬁnition,

S = g p d~ m m d~ p = e2g k Qˆ1 l 2 , (5.21) lk k h | | i·h | | i k|h || || i| m l X∈ where gk is the degeneracy of the level k. The last relation is derived from the orthonor- mality condition of Clebsch-Gordan coefficients. For a 40Ca+ qubit, only the states m = P , M = 1/2 and P , M = 1/2 | i | 3/2 J − i | 1/2 J − i can yield non-zero dipole matrix elements. By using the formalism mentioned above, I find the minimum phase is 2 Ca 18 [L ] φ =9.86 10− , (5.22) min × [T 3] where the square bracket denotes the value of quantities in S.I. unit. Both computational states of a 9Be+ qubit are hyperfine states of the ground electronic state, so m are states | i on P1/2 and P3/2 levels only. I find the minimum phase is

2 Be 25 [L ] φ =2.6 10− . (5.23) min × [T 3]

For L = 100 µm and φmin . π/100, the threshold time of ﬂight for ion qubits are

Ca Tmin,100µm & 14.6 ns (5.24) Be Tmin,100µm & 0.044 ns . (5.25)

Be Ca Tmin,100µm is 3 orders of magnitude smaller than Tmin,100µm because the energy between two qubit states, EZ , is Zeeman energy that is small when compared with the energy difference between atomic energy levels, EA. If no magnetic field is applied and the tiny hyperfine splitting is neglected, the value of the bracket in Eq. (5.15) vanishes since both qubit states are atomic ground states. The first non-vanishing term would be suppressed 6 9 + by a factor of E /E 10− . Thus the phase shift of a Be qubit is much smaller than Z A ≈ that of a 40Ca+ qubit.

5.6 Non-encoding State Excitation

Apart from shifting the phase, the dc electric ﬁeld also excites the electron into states outside the computational basis. I start the analysis by writing the dc Stark eﬀect

77 Chapter 5. Decoherence Induced by dc Electric ﬁeld During Ion Transport

Hamiltonian in the interaction picture as

m Vˆ (t)= V q¨(t)eiωrnt r n , (5.26) I e rn | ih |

where m m r Hˆ n V q¨(t)= r d~ xˆ n q¨(t) . (5.27) h | Stark| i ≡ e rn − e h | · | i By applying time dependent perturbation theory, the ﬁrst three terms of the Dyson series

of the propagator UˆI (t) are given by

t ′ i m iωrnt Uˆ (t) Iˆ V q¨(t′)e dt′ r n I ≈ − ~ e rn | ih | rn Z0 2 X t t′ ′′ ′ 1 m i(ω ′ t +ω ′ t ) V ′ V ′ q¨(t′′)¨q(t′)e r n rr dt′′dt′ r n .(5.28) −~2 e2 rr r n | ih | rn ′ 0 0 X Xr Z Z The two most significant terms are: the first order term with ω = ω 7; and the second r 6 n 8 order term with ωr = ωn . Although the time integrals cannot be solved without the exact form of q(t), the significance of each term can be compared by the estimated magnitude. After simple integration by parts, the integral in the first order term in Eq. (5.28) will be

T i i T ... q¨(t)eiωrntdt = − q¨(T )eiωrnT q¨(0) + q (t)eiωrntdt . (5.29) 0 ωrn − ωrn 0 Z Z 15 1 Since the typical order of atomic transition frequency ωrn is 10 s− and the timescale 9 of ion transport is about 10− s, the last term on the right hand side can be neglected as it is much smaller than the term in the left hand side 9. As I am interested only in the order of magnitude, it is appropriate to consider the bracket in the right hand side has the same magnitude as L/T 2. The magnitude of the second order term in Eq. (5.28) is estimated by a trick. It can be recognised that the term for r = n is responsible for the phase-shift that is studied | i | i in Sec. 5.5. Therefore we know the time integral can be approximated by iζ[q]/ωr′n, which the value is at the order of 10L2/T 3. Here I consider the fact that the dominating

Vr′n are those of the low lying energy states, and the matrix elements of the states in the same energy levels are in the same order of magnitude. The ratio of the ﬁrst and second

7The states in the same energy level have the same parity and so Vnn =0 8Because ωrr′ + ωr′n = 0 is required, otherwise there will be a rapidly oscillating term in the integral and the contribution... of the term is reduced by an order of perturbation. 9assuming q q/T¨ , the right hand side term is 1/ωrnT smaller, see Ref. [6] ≈

78 Chapter 5. Decoherence Induced by dc Electric ﬁeld During Ion Transport

order term can be estimated as

first order ~eT . (5.30) second order ≈ 10m V L | rn| 4 For a qubit moving in a 100 µm trap within 10 ns, the above ratio is about 10− . Therefore the second order term in Eq. (5.28) dominates the state excitation. Let us consider the dc Stark effect has excited the electron to a non-computational state b , i.e., | ii Uˆ (t) Ψ = 1 ǫ 2 Ψ + ǫ b . (5.31) I | i − | i| | i i| ii s i i X X The probability of exciting a non-computational state is ǫ 2. For a 40Ca+ qubit, the i | i| amplitude of the second order term in Eq. (5.28) is the sameP order of magnitude as the phase shift φ, so the probability of non-computational state excitation is about φ2, which is much smaller than the phase error φ. Therefore the threshold speed of a calcium ion QC is determined by the phase-shift effect, i.e. Eq. (5.22). For a 9Be+ qubit, the second order term in Eq. (5.28) vanishes if hyperfine splitting is neglected. This is because both computational states consist of only one orbital state, l =0, m =0 , i.e. | l i 1 3 1 1 1 1 3 1 1 1 2, 0 F = , − I , S + , I , − S l =0, ml =0 , (5.32) | i √2|2 2 i |2 2i √2|2 2i |2 2 i ⊗| i √3 3 3 1 1 1 3 1 1 1 1, 1 F = , I , − S , I , S l =0, ml =0 , (5.33) | i 2 |2 2i |2 2 i − 2|2 2i |2 2i ! ⊗| i

where the first and second number of a state denotes respectively the angular momentum and magnetic quantum number; the state with the subscript I (S) corresponds to the nuclear (electronic) spin. Since the orbital state has no degeneracy, non-computational states are off-resonant and barely excited by dc Stark effect. If hyperfine splitting is included, the second order term in Eq. (5.28) is finite but its 6 magnitude is suppressed by a factor of E /E 10− . According to previous analysis, Z A ≈ the first order term may then become dominant. However, this term could be suppressed by tuning the trajectory of the ion qubit. Let us recall that I have approximated the bracket in Eq. (5.29) by the value L/T 2. However, if a trajectory is chosen so that the initial and final acceleration of the ion qubit are both zero, then the bracket vanishes and 1 6 the first order will be reduced by a factor of (ωrnT )− , which is about 10− for T = 1 ns.

Although the optimal trajectory q0(t) does not satisfy this criteria, a modified trajectory can be constructed without significantly sacrificing the optimality. One possible modi-

79 Chapter 5. Decoherence Induced by dc Electric ﬁeld During Ion Transport

fication is to include two buffer periods, each lasts for time T ′, at the beginning and at the end of the transportation. During the beginning buffer period, the ion acceleration is increased from zero toq ¨0(0); similarly in the ending buffer period, the acceleration is decreased fromq ¨0(T ) to 0 at the end of the flight, while the remainder of the trajectory still follows q0(t). I find that if T ′ is much larger than 1/ωrn but much smaller than T , then the first order term in Eq. (5.28) is greatly suppressed, while the phase shift is scaled by only a factor of order unity. Therefore the second order term in Eq. (5.28) of a 9Be+ qubit remains dominant. Just as the 40Ca+ qubit, the amplitude of non-computational states is at the same order as the phase shift of the computational states, so the probability of excitation is much smaller than the phase-shift error. Therefore, the threshold speed of 9Be+ qubit is still given by Eq. (5.25).

5.7 Summary and Discussion

In this chapter, I have studied the influence of dc Stark effects on the quantum information encoded in an ion qubit. I find that phase-shift is the most significant error, while non- computational state excitation is a less important effect. The magnitude of the phase-shift scales as quadratically as size of traps and inversely cubic as the shuttling time. For an ion qubit being transported across a 100 µm trap, the threshold time of flight of a 40Ca+

ion, Tmin,100µm, is about 10 ns unless the induced phase is larger than 1%. On the other hand, a 9Be+ qubit is more resistant to the dc Stark effect. I find that the induced phase is 6 order smaller than that of the 40Ca+ ion, so the threshold transportation time of a 9Be+ qubit is as short as 50 ps. In principle, the average Stark shift might be assessed by Ramsey interferometry, and then corrected by unitary transformations. In practice, however, the ion is undergoing a complex trajectory involving acceleration and deceleration, moving in straight lines, turning around bends and through junctions, disengaging the individual ion from the storage register and the logic trap; all of these effects will be too complicated to track and calculate the Stark shift accurately. The uncertainties will turn the phase-shift and excitation into errors. Quantum error correction will perforce be needed, and the requirements of fault-tolerant quantum computing (in particular, ensuring the error be below some threshold) will place a speed limit on the operation of the QC. Magnitudes of the errors depend on the setup of the ion trap system. In any case, our result is a useful reference to the speed limit. As an illustration, suppose the overall Stark shifts can be evaluated, with accurately tracking the trajectory, well-controlled electric field, and

80 Chapter 5. Decoherence Induced by dc Electric field During Ion Transport other very precise experimental techniques, up to 90% accuracy for a particular ion trap QC. According to Eq. (5.22) and (5.23), this 10% of apparatus uncertainty will impose a transportation time limit that is about half of the threshold time I have calculated 10. A possible way to lower this threshold is to reduce the size of the trap. But this 2/3 method is inefficient because Tmin only scales at L , and the reduction of trap size may cause more serious heating on the qubits. Another method is to encode quantum information into the decoherence free subspace of two ions [114, 99, 153], i.e. 0 i f , | i→| 1 2i 1 f i . If the ions are transported in the same trajectory, the phase-shift on each | i→| 1 2i ion would be decoupled from the quantum information encoded in the logical qubit. In addition, there are alternative scalable ion trap QC architectures that require much fewer two qubit operations, hence fewer ion transportation (the measurement-based ion trap QC, c.f. [164, 147]), or even no transportation of ions (the ion-photon network model, c.f. [32, 133, 55]). The dc Stark effect is thus insignificant in these architectures.

10Due to the T 3 dependence of the induced phase, √3 0.1 46%. ≈

81 Chapter 6

Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

Apart from the dramatical speed-up of quantum algorithms running on a universal quantum computer (UQC), we can also take advantage of the capability of quantum mechanics by constructing a less versatile,i.e., without implementing the full set of universal logical gates with accuracy above the fault tolerant threshold, quantum device to perform spe- ciﬁc tasks. One such proposal is to use well-controlled quantum systems to simulate other physical systems that are too complicated, e.g. due to the exponentially large Hilbert space required, to be simulated on classical computers [61]. Because the precision requirement of a quantum simulator is less stringent than a UQC [41], quantum simulation is conceived to be realized by technology close to the state-of-the-art. Due to its technological maturity, the trapped ion system has been proposed and employed as the platform for physical simulations of, for example, relativistic quantum eﬀects [70], quantum phase transitions [72], and the evolution of open systems [169] 1. In the following two chapters, I will propose two ion trap architectures that simulate bosonic quantum systems.

6.1 Introduction

Coherently manipulated photons have been proposed to be a good candidate for test- ing the foundation of quantum mechanics [178], performing quantum computations [101, 144], conducting high precision measurements [140], and many other applications. How- ever, because of poor sources, detection ineﬃciencies, and weak photon-photon interac- tions, implementing these proposals for large-scale devices is very diﬃcult. It would be

1A good review of trapped-ion quantum simulator experiments is Ref. [92].

82 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

fundamentally and practically interesting if we could build a quantum simulator of photons by using another well-controlled system with the same (bosonic) behaviour. More specifically, a universal bosonic simulator (UBS) should be able to reproduce the evolution of a bosonic system under the most general form of Hamiltonian. This requirement is not too stringent as the evolution can be approximated to arbitrary accuracy by a sequence of basic operators that belong to a universal set [116]. Lloyd and Braunstein [117] suggested that the simplest universal set of basic operators comprised: all single mode linear operators; at least one multi-mode operator; and at least one nonlinear element. Efficiently performing only these basic operations is necessary and sufficient for implementing a UBS.

Ion traps are a suitable candidate for implementing a UBS, in which a high degree of controllability has been demonstrated [78]. The motion of laser-cooled ions is quantum in nature, and the excitations of the motional states, i.e. phonons, exhibit bosonic behaviour. The collective displacement and momentum of the ions are analogous to the quadratures of light fields. Any arbitrary motional state can be created by combining techniques such as sideband transition [181], parametric amplification [80], and adiabatic passage [49]; in particular, the creation of Gaussian states [127] and non-classical states [127, 134] from the ground state have been experimentally demonstrated. When applied to non-ground states, some of these techniques can achieve single phonon linear or nonlinear operations. Interaction between phonon modes at the few-quanta level has also been observed. For example, nonlinear beam splitting on a single ion has been performed by applying a Raman field [112]; coupling two phonon modes have also been demonstrated through the Coulomb interaction between two separately trapped ions [36, 79], or two ions in the same trap [154, 141].

In this chapter, I revisit the idea of trapped-ion bosonic simulation that was first proposed by Wineland et. al. [180, 112]. Their proposal consists of only one trapped ion, which could provide at most three motional modes for simulating three bosonic modes. Aiming for a larger scale of simulation, I extend their idea by considering a trap with multiple ions. The number of bosonic modes available in the simulator is proportional to the number of ions. I will show that the basic operators of Hamiltonian can be realised by applying finely-tuned laser field to couple the motional and the internal states of ions. As an example, I outline the procedure for implementing Hong-Ou-Mandel effect on phonons.

83 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

xi xi+1 ν0

Figure 6.1: Layout of the bosonic simulator consisting of multiple ions in a single harmonic well, with trap frequency ν0. The ions are aligned along the axial direction (x in the figure), while radial motion is strongly confined (not shown in the figure). Before the simulation, the ions are cooled to the ground motional state, and the classical position of each ion is the equilibrium point between the harmonic trapping force coming from the electrodes and the mutual Coulomb force between ions.

6.2 Layout of the system

I consider the UBS to be composed of N ions trapped in a linear trap. All the ions are prepared in the ground electronic state. The ions are weakly trapped along the axial direction, while the effective radial potential is strong enough that the ions are aligned linearly but not in zigzag configuration. The layout of our system is shown schematically in Fig. 6.1. The Hamiltonian that governs the ion motional states can be approximated as [90] N pˆ2 N pˆ2 1 N e2 1 Hˆ = i + Vˆ i + mν2xˆ2 + , (6.1) 2m ≡ 2m 2 0 i 4πǫ xˆ xˆ i=1 i=1 i>j 0 i j X X X − wherep î andx î are the momentum and position operator of the ith ion; ν0 is the harmonic frequency of the trap; Vˆ is the total potential that includes the harmonic trap potential and the Coulomb potential between ions.

The position operatorx î can be expressed as the sum of the classical equilibrium (0) 2 position, xi , and the position operator of the quantum fluctuation,q î . The spread of quantum fluctuation is much smaller than the classical separation between ions at 2 (0) (0) low temperature, i.e. qî / xi xj=i 1. Let us Taylor-expand the Hamiltonian in h i | − 6 | ≪ Eq. (6.1) and apply the quadratic approximation, i.e., collecting terms up to the second order of q’s, the Hamiltonian will become a coupled quantum harmonic oscillator, viz.

N 2 N 2 N 2 ˆ pî ˆ pî 1 ∂ V H + V2 + qîqˆj (0) (0) , (6.2) ≈ 2m ≡ 2m 2 ∂x ∂x qi=qj =0 i=1 i=1 i,j i j X X X

3 where Vˆ2 is the quadratic approximated Hamiltonian . After diagonalizing Vˆ2, the Hamil-

2For simplicity, I have neglected the identity operator associated to the classical position. Formal treatment of the separation of classical and quantum contribution can be given in Ref. [90] or Sec. 4.3 3the partial derivative of V is constructed by treating Vˆ andq ˆ in Eq. (6.2) as scalar function and

84 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

tonian in Eq. (6.2) becomes a sum of independent harmonic oscillators that represent the collective motion of the ions, i.e., phonon modes, viz.

N ˆ2 N N Pk 1 2 2 1 H = + mν Qˆ = ~ν (ˆa† aˆ + ) . (6.3) 2m 2 k k k k k 2 Xk=1 Xk=1 Xk=1 The position operator of the kth phonon mode is given by

Qˆk = αkiqˆi , (6.4) i=1 X

T ∂2V ˆ where (αk1,αk2,...,αkn) is the kth eigenvector of the matrix (0) (0) ; Pk is the ∂xi ∂xj qi=qj =0 ˆ 2 ∂2V conjugate momentum of Qk; mνk are the eigenvalues of (0) (0) , where νk is the ∂xi ∂xj qi=qj =0 frequency of the kth mode. If there are N phonon modes in this system, N bosonic modes can be simulated. The annihilation and creation operators of the kth phonon mode are deﬁned as

1 mνk ˆ 1 ˆ 1 mνk ˆ 1 ˆ aˆk = Qk i Pk ;a ˆk† = Qk + i Pk . (6.5) √2 ~ − mνk~ √2 ~ mνk~ r r r r I refer interested reader to Ref. [90] for further details of the derivations.

6.3 Universal Bosonic Simulation

The most general bosonic behaviour can be simulated if and only if the simulator can be engineered to evolve under the most general Hamiltonian, which can be expanded as a series of products of annihilation and creation operators as:

ˆ (1) (2,0) (1,1) H(âi, aî†, t)= Ai (t)âi + Aij (t)âiaˆj + Aij (t)âiaˆj† + ... + h.c. . (6.6) X This Hamiltonian is time dependent, consists of superpositions of non-commuting terms, and involves high order products of operators. Reproducing such a Hamiltonian requires complicated engineering of the simulator, which is difficult in practice. Fortunately, Lloyd and Braunstein [117] suggested three tricks to simplify the implementation of the general evolution. The first trick is to divide the evolution into short intervals, so the evolution operator

parameters, and treating the classical and quantum displacement, x(0)’s and q’s, as independent variables.

85 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

can be approximated as

Uˆ(t) exp( iHˆ (t )δt) exp( iHˆ (t )δt) .... (6.7) ≈ − 1 − 2

At each short time interval δt, the evolution is determined by a time independent Hamil- tonian at that time instance. Instead of engineering the time dependent Hamiltonian, the evolution can be simulated by applying a sequence of time independent Hamiltonian. The evolution simulated by this method is more accurate as δt shrinks. The aim of this trick is to get rid of the time dependence of the Hamiltonian. The second trick is to get rid of the superposition of non-commuting Hamiltonian. This can be done by applying the Suzuki-Trotter expansion: for any Hamiltonian that can be written as Hˆ = Aˆ+Bˆ, the evolution can be approximately constructed by applying the components Aˆ and Bˆ in sequence, i.e.,

exp( i(Aˆ + Bˆ)δt) exp( iAδtˆ ) exp( iBδtˆ )+ O(δt2) . (6.8) − ≈ − −

This approximation is more accurate as δt shrinks. The third trick is to implement highly nonlinear operation, i.e., evolution under the Hamiltonian with higher than third orders of annihilation and creation operators, by Hamiltonian with linear and less nonlinear operations. The key idea is that if two Hamiltonians, Aˆ and Bˆ, are applied in appropriate sequence, the ﬁnal evolution will be determined by the commutator of Aˆ and Bˆ, i.e.,

exp(iAδtˆ ) exp(iBδtˆ ) exp( iAδtˆ ) exp( iBδtˆ ) = exp([A,ˆ Bˆ]δt2)+ O(δt3) , (6.9) − − where the higher order contribution, O(δt3), can be neglected when δt is small. Lloyd and Braunstein [117] showed that if both Aˆ and Bˆ are higher than the third order, then a Hamiltonian with arbitrary high order can be constructed by this method. By using these three tricks, Lloyd and Braunstein showed that the evolution of arbitrary multi-mode Hamiltonian can be efficiently simulated by applying in sequence only a basic set of operations. These operations include (i) displacement operator, which can be implemented by applying a Hamiltonian with the first order ofa ˆ anda ˆ†; (ii) squeezing operator; (iii) phase-shift operator, which can be implemented by a Hamiltonian with second order ofa ˆ anda ˆ†; (iv) beam splitter, which can be implemented by a Hamiltonian aˆ1† aˆ2+ h.c.; and (v) a nonlinear operator that is generated by higher than second order terms ofa ˆ anda ˆ†. The first four operators are regarded as the basic set of Gaussian operations, which can transform a Gaussian Wigner function to any Gaussian Wigner

86 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

function (c.f. Sec. 3.2.2). They are also referred as the basic set of active linear elements in quantum optics, which transforms an annihilation operator to any superposition of annihilation and creation operators. The addition of the nonlinear operator, which is also a non-Gaussian operator, is to break the Gaussianity of the operation 4.

6.4 Laser Implementation of Basic Operations

In this section, I present the strategies required to implement each of the five basic operators on the phonon modes of the ion trap bosonic simulator. The principle is to apply laser field that couples the internal states and the motional states of the ion. Different motional operations can be implemented by changing the frequency and magnitude of the laser field. Because the transition between internal state is undesirable in this situation, two laser fields, where the frequency difference is much smaller than the internal state transition frequency, is applied to induce Raman transition (see Fig. 6.2). Let us consider the ~ ~ frequency, wavevector, and phase of the first (second) laser field are ω1 (ω2), k1 (k2),

and φ1 (φ2) respectively. The effective frequency, effective wavevector, and phase of the resultant Raman field are then ω = ω ω , ~k = ~k ~k , and φ = φ φ . As our 1 − 2 1 − 2 1 − 2 consideration is focused on the axial motion, the radial components of the Raman field are neglected. This is possible because the radial direction is tightly trapped so that the effective oscillation is off-resonant with the Raman field, and the angle between the laser

ﬁelds can be tuned so that ~k = kx~x is along the x axis. The Hamiltonian generated by the Raman ﬁeld is given by

iφ iωt VˆR = ~Ωe exp (ikxqˆ) e− + h.c. , (6.10)

where Ω is the effective Rabi frequency. If the Rabi frequency of each laser field is Ω0, 2 5 then Ω = Ω0/∆, where ∆ is the detune of ω1 from ωgd . Let us assume the Raman field is applied on the ith of the N ion in the chain, while other ions are unaffected 6. The position operator in Eq. (6.10) is then belonging to the

ith ion, i.e.,q î. As seen in Eq. (6.4), the motion of ith ion involves differently in the collective motion of each mode, because of the inhomogeneity of eigenvector coefficients.

4This is analogous to the implementation of a discrete-variable UQC, which requires all the Clifford gate (analogous to Gaussian operation) and one non-Clifford gate (analogous to non-Gaussian operation) 5I have defined the energy difference between internal state m and n to be ~ωmn. 6Such a single ion addressing can be realised by using composite| i pulses| i [78].

87 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

(a) (b)

Figure 6.2: (a) Two laser fields with different frequency and wavevector are applied on an ion while other ions are assumed to be unaffected. Single ion addressing is possible by using screening or composite pulses. (b) Internal level diagram of the ion. The laser fields interact with the ion through the dipole transition between the internal states g and d , but the laser frequencies are far-detuned from the transition frequency so that| i | i no internal state is excited.

By inverting Eq. (6.4), the position operators can be expressed as

qî = βikQˆk , (6.11) Xk=1 where the matrix β is the inverse of the matrix α . The coefficients of the matrices { ik} { ki} can be found in Ref. [90]. By substituting Eq. (6.5) into Eq. (6.10), the Raman field Hamiltonian can be written as

N ˆ ~ iφ iωt VR = Ωe exp i ηik(ˆak +ˆak† ) e− + h.c. , (6.12) ! Xk=1

where the Lamb-Dicke parameters, ηik, are deﬁned as

~ η = k β . (6.13) ik x ik 2mν r k

Typical ion trap experiments operate at the Lamb-Dicke regime, where η 1 7. The ≪ Hamiltonian in Eq. (6.12) can be Taylor-expand in terms of η’s. In the interaction picture with respect to the steady state Hamiltonian in Eq.(6.3), the Raman ﬁeld Hamiltonian

7Motional and internal states can still be manipulated by laser ﬁeld beyond the Lamb-Dicke regime if the state-dependency of interaction coeﬃcients is considered [177].

88 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

can be expressed as a series,

N (I) iφ i(ν +ω)t i(ν ω)t Vˆ ~Ωe I + i η (â e− k +â† e k− ) R ≈ ik k k Xk=1 (i)2 N i(νk+νk′ +ω)t i(νk νk′ ω)t + ηikηik′ (âkaˆk′ e− +âk† aˆk′ e − − 2 ′ k,kX=1 i(νk νk′ +ω)t i(νk+νk′ ω)t +âkaˆk† ′ e− − +âk† aˆk† ′ e − )+ ... + h.c. . (6.14) As we can see from this expression, the contribution of each term is determined by an oscillating parameter, which the frequency depends on the Raman field frequency and the mode frequencies. Each of the term can be made dominant by tuning the Raman field frequency to be resonant, i.e., the oscillating term becomes a constant. The contribution of other terms in the same order can be neglected according to rotating wave approximation (RWA), while higher order terms are neglected because they scale as higher power of small Lamb-Dicke parameters, i.e., Lamb-Dicke approximation (LDA). In the following subsections, I will specifically discuss the implementation of each of the basic operation. The validity of the approximations will be discussed later.

6.4.1 Displacement Operator

A displacement operator, ˆ(α) = exp(αaˆ† α∗aˆ), transforms the annihilation operator D − as

aˆ ˆ†(α)â ˆ(α)=â + α , (6.15) → D D 2~ √ ~ where mν Re(α) is the displacement of the collective position of the mode; 2 mνIm(α) is the displacementq of the collective momentum of the mode. A displacement operator of the kth mode can be implemented by applying a Hamil- tonian with the first order ofa ˆk anda ˆk† [127, 134]. By applying a Raman field with the frequency νk to the ith ion (assume the Lamb-Dicke parameter is nonzero for the mode), the dominating terms in Eq. (6.14) will be

(I) iφ iφ Vˆ i~η Ω(e aˆ† e− aˆ ) . (6.16) R ≈ ik k − k

After time t, the kth mode will be transformed as Eq. (6.15), where the displacement is iφ given by α = ηikΩe t. The value of the displacement is tuneable by changing the phase and duration of the Raman ﬁeld, as well as the eﬀective Rabi frequency.

89 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

6.4.2 Phase-Shift Operator

A phase-shift operator ˆ(θ) = exp( iθaˆ†aˆ) transforms the annihilation operators as P −

iθ aˆ ˆ†(θ)ˆa ˆ(θ)=ˆae− , (6.17) → P P

where θ is the phase shift. A phase-shift operator of the kth mode can be implemented by applying a Hamiltonian with the dominant terma ˆk† aˆk. This Hamiltonian can be realised by applying a Raman field with ω = 0, i.e., both laser fields have the same frequency, to the ith ion. Then Eq. (6.14) will become (I) 2 Vˆ ~η Ω(â† aˆ +â aˆ† ) . (6.18) R ≈ − ik k k k k After time t, the kth mode will be transformed as Eq. (6.17), where the phase shift is given by θ = 2η2 Ωt. The value of the displacement is tuneable by changing the − ik duration of the Raman field and the effective Rabi frequency.

6.4.3 Squeezing Operator

2 2 A squeezing operator Sˆ(g) = exp (g∗aˆ gaˆ† )/2 transforms the annihilation operator − as g aˆ Sˆ†(g)ˆaSˆ(g) = cosh g aˆ sinh g aˆ† , (6.19) → | | − g | | | | where g is the complex squeezing parameter. A squeezing operator can be implemented by applying a Hamiltonian that involves 2 2 second order terms of the annihilation and the creation operators, i.e.a ˆ anda ˆ† . A squeezing operator on the kth mode can be realised by applying a Raman ﬁeld with frequency 2νk to the ith ion. Then Eq. (6.14) will become

2 (I) ηikΩ iφ 2 iφ 2 Vˆ ~ (e aˆ† + e− aˆ ) . (6.20) R ≈ − 2 k k

After time t, the kth mode will be transformed as Eq. (6.19), where the squeezing parameter is g = iη2 eiφΩt, which is tuneable by changing the duration, the phase, and − ik the eﬀective Rabi frequency of the Raman ﬁeld.

6.4.4 Nonlinear Operator

Nonlinear operators transform an annihilation operator to an operator involving quadratic and higher order terms ofa ˆ anda ˆ†. It can be achieved by applying a Hamiltonian that is

90 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

at least the third order ofa ˆ anda ˆ† [117]. For example, when a Raman ﬁeld with ω =3νk is applied, the Hamiltonian in Eq. (6.14) will be dominated by

3 (I) ηikΩ iφ 3 iφ 3 Vˆ i~ (e aˆ† e− aˆ ) . (6.21) R ≈ − 6 k − k

6.4.5 Beam Splitter

A beam splitter transforms the annihilation operators of two modes to be a superposition of each other, i.e.,

iφ′ iφ′ aˆ cos θaˆ + sin θe aˆ ;a ˆ sin θe− aˆ + cos θaˆ , (6.22) 1 → 1 2 2 → − 1 2

where θ is some real number. Such an operation can be implemented by an operator iφ′ iφ′ = exp(θ(e aˆ† aˆ e− aˆ† aˆ )) . B 1 2 − 2 1 The realisation of a beam splitter in trapped ion systems is first proposed in Ref. [180]. The proposed setup contains a single ion being trapped in an inhomogeneous three dimensional harmonic trap, so that the phonon mode frequencies are different in x, y, and z direction. A beam splitter between two of the modes can be realised by applying Raman radiation field with the frequency equal to the difference of the mode frequencies. Here I extend this idea to a linear ion chain. By applying a Raman field to the ith ion with frequency ω = ν ν , the effective Hamiltonian in Eq. (6.14) will be dominated k − l by ~ (I) ηikηilΩ iφ iφ V (e aˆ† aˆ + e− aˆ†aˆ ) . (6.23) R ≈ − 2 k l l k After applying the field for time t, the beam splitter operation will be implemented with iφ′ iφ θ = ηikηilΩt/2 and e = ie . The quality of this operation is determined by the validity of LDA and RWA, in other words how dominating the terms are in Eq. (6.23) when comparing to other terms in Eq. (6.14). More explicitly, the zeroth order term in Eq. (6.14) does not alter the performance of beam splitter because it only contributes to a global phase. The first order terms will exert a displacement operation on each of the mode, but the coefficient is multiplied by an off-resonant oscillation term. The variance of such erroneous displacement scale as η Ω/δ 2, where the detuning δ = ω ν for the mth mode. For a perfect beam | im | | − m| splitter operation that the Raman pulse is exactly resonant with the frequency difference of the kth and lth mode, the most seriously affected mode will be the one, say mth mode, that ν is the closest to ν ν . I have plotted the minimum value of (ν ν ) ν m k − l | k − l − m| among all k,l,m in Fig. 6.3(a). If the variance of the erroneous displacement has to be

91 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

0.3 | 0.08 k ν ) − u ))| 0.06 l ν

0.2 ν − − k v ν ν 0.04 )−( v ν −

0.1 Minimum u

ν 0.02 |( Minimum |( 0.0 0.00 2 4 6 8 10 2 4 6 8 10 Number of ions Number of ions (a) (b)

Figure 6.3: (a) The minimum possible value of (a) (νk νl) νm , (b) (νk νl) (νu νv) , for diﬀerent number of ions in the linear trap. All| the− above− values| of| frequencies− − − are in| the unit of ν0.

less than 1%, the following criterion is implied:

η Ω . 0.1 min (ν ν ) ν . (6.24) m × | k − l − m|

To implement a beam splitter, the Hamiltonian in Eq. (6.23) is required to switch on for about t 1/η2Ω, where the η ’s are assumed to have the same order of magnitude. ≈ ik In practice, t is desired to be shorter, preferably at the range of 100 µs 8. With the criterion (6.24), the value of η can be estimated as

η & 105/min (ν ν ) ν . (6.25) | u − v − k|

If the number of modes to be simulates is less than 7, Fig. 6.3(a) shows that the minimum value of (ν ν ) ν is one order smaller than the trap frequency ν . For state-of-the-art | u− v − k| 0 value of ν =2π 106 MHz, η should be about 0.1. 0 × Fig. 6.4 shows the numerical simulation of the evolution of state 1, 0 and 0, 1 , | i | i where the first (second) index denotes the Fock state of centre-of-mass (stretching) mode 9 of a two ion system. I vary the parameter η, where η = √2η1 = √2η2, and select the appropriate Ω to retain the same evolution time. As the above estimation suggests, a smaller η causes more serious mode population fluctuation due to the first order term in Eq. (6.14). Therefore an appropriately large η should be chosen to implement a high

8Heating rate of trapped ion is as low as 70 quanta per second [36]. To suppress the error to less than 1%, i.e. heating rate is less than 0.01 quanta, the experiment has to be ﬁnished in about 100 µs. 9 In centre-of mass mode, the ions always move in the same direction, i.e., in Eq. (6.4), α11 = α12 = 1/√2; in stretching mode, the ions always move in opposite direction, i.e., α = α =1/√2. 21 − 22

92 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

0.8

0.6

Probability 0.4

0.2

0 0 20 40 60 80 100 Time (μ s)

Figure 6.4: Probability of 1, 0 (solid lines) and 0, 1 (dashed lines) after the beam | i | i splitter Hamiltonian (Eq. (6.14) with ω = ν2 ν1) is applied on the initial state 1, 0 . The curves correspond to η =0.2 (red), η =0.−1 (blue), η =0.05 (black). | i quality beam splitter operation.

If ηkΩ satisﬁes Eq. (6.24), increasing ηk can also speed up the beam splitter operation.

However, a large ηk also enhances the magnitude of the higher order terms in Eq. (6.14), which would bring nonlinearity to the beam splitter. Particularly, the third order terms is suppressed because it is scaled by the small Lamb-Dicke parameter, and it has to be off-resonant. As discussed previously, the off-resonant effect can reduce a first order term to be less significant than the second order resonant terms in Eq. (6.23), so the third order term is expected to be less significant than the forth order term. For the forth order terms, while most of them are also off-resonant, there are some, like ak† akak† al, could be in resonant with the terms in Eq. (6.23). These terms would produce a nonlinear phase shift, which the magnitude scales as η 2 and the phonon number of the involved mode, | | i.e., a† a . In other words, the nonlinear error is more serious if more phonon is involved h u ui in the quantum simulation. For a simulation with a high phonon number, the η’s should be reduced. Therefore the restriction in Eq. (6.25) has to be removed. Let us recall that Eq. (6.25) is imposed because I require the first order term to be suppressed by the RWA. In fact, this restriction can be relaxed if the first order terms can be suppressed by other methods. One method is to add another Raman field which has the same phase and frequency but opposite wave vector. The physical idea is to set up a standing wave and the ion locates at a node of the radiation. Then the ions should not experience a state-dependent force. More

93 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

explicitly, the total Hamiltonian of the two Raman ﬁelds becomes

(I) ~ iφ i(kx ωt) i( kx ωt) VR2 = Ωe e − + e − − + h.c. N (i)2 ~ iφ I i(νk+νk′ +ω)t i(νk νk′ ω)t Ωe + ηikηik′ (âkaˆk′ e− +âk† aˆk′ e − − ≈ 2 ′ k,kX=1 i(νk νk′ +ω)t i(νk+νk′ ω)t +âkaˆk† ′ e− − +âk† aˆk† ′ e − )+ ... + h.c. . (6.26) Because the first order terms cancel exactly, the condition in Eq. (6.24) is no longer required to eliminate the first order term by off-resonance. Therefore, the value of ηΩ can be increased for faster operation while η remains small for effective LDA.

Without the first order terms, the next major error source is the off-resonant second order terms. These terms will induce beam splitting or two-mode squeezing operation on other phonon modes. Suppose the uth and vth modes are affected, the variance of the beam-splitting angle or the squeezing parameter oscillating scales as χ = η η Ω/δ 2, | u v | where δ =(ν ν ) (ν ν ) is the detuning; the sign + ( ) corresponds to the case of k − l − u ± v − erroneous squeezing (beam splitter). Therefore the beam splitter would negligibly affect other modes if χ 1 for any combination of k,l,u,v. If the smallest χ required is about ≪ 1%, the following criterion is imposed

0.1 min (ν ν ) (ν ν )) & η2Ω , (6.27) | k − l − u − v |

where all η’s are again assumed to be the same order of magnitude; and I have checked that the contribution from the erroneous two-mode squeezing is less signiﬁcant than the erroneous beam splitter. Since the operation duration of a beam splitter is about 1/η2Ω, Eq. (6.27) lower bounds the operation time. The minimum value of (ν ν ) (ν ν )) | k − l − u − v | is plotted in Fig. 6.3(b) for diﬀerent number of ions. For example when N = 10, the time required for a beam splitter on any two modes is at least 150 µs.

As a summary, a beam splitter can be implemented by applying a Raman field with frequency tuned to the mode frequency difference of two modes. The operation time of such approach is limited by Eq. (6.25) that guarantees the validity of RWA. Faster operation can be implemented if an additional Raman field is applied, then the accuracy criterion is improved to Fig. 6.3(b). I note that further speed up of beam splitter operation may be possible if multiple pulses is applied, where erroneous Hamiltonian could be eliminated by carefully choosing the pulse sequence [78].

94 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

6.5 Readout

For most of the investigation about bosonic systems, readout of boson number is particularly important. In the current proposal of ion trap bosonic simulator, the phonon states are measured by first coupling with electronic states and then conducting measurement on the electronic states. Here I propose two ways to couple the phonon and electronic states: by adiabatic passage and resonant flipping. I find that the adiabatic passage approach can mimic the photon number non-resolving detectors in optical experiments, but the operation has to be sufficiently slow to preserve the adiabaticity. On the other hand the resonant flipping method can be much faster, but it can be applied with a priori information about the phonon number.

6.5.1 Adiabatic passage

The idea of using adiabatic passage to create non-classical motional states of trapped ions was first proposed by Cirac, Blatt and Zoller [49]. Inverting their protocol can map the motional state to the electronic state of an ion. Let us consider a stimulated Raman field with the effective frequency ω is interacting with the encoding states, g and e , | i | i of the ion, the interaction potential is given by

n (I) ~ iωt iω0t iω0t ~ iωt iνkt iνkt iω0t iω0t VT = Ωe− (σ+e +σ e− )+i Ωe− ηk(ak† e +ake− )(σ+e +σ e− )+h.c. , − − k=1 X (6.28) where ω0 is the energy diﬀerence between the encoding states. If the kth mode is to be measured in the Fock state basis, the ω is tuned to ω = ω ν +∆(t). ∆(t) is time varying 0− k but the amplitude is always small when comparing to νk. The coupling of electronic states to other phonon modes, as well as the carrier transition between the electronic states, are oﬀ-resonant and so their contribution is suppressed. The Hamiltonian can be approximated as involving only the dominating term, i.e.,

(I) ~ i∆(t)t i∆(t)t VT = i Ωηk(σ+ake− σ ak† e ) . (6.29) − −

The energy levels and the Raman ﬁeld frequencies in an adiabatic transfer are shown in Fig. 6.5(a). If ∆(t) is slowly varying, the system will remain in the same level throughout the whole process. The energy states change adibatically as follows,

g 0 g 0 ; g n e n 1 ; e n g n +1 , (6.30) | i→| i | i→| − i | i→| i

95 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

(a) (b)

Figure 6.5: Energy levels of an ion during measurement processes. The dashed line is a virtual energy level that is detuned from an electronic state. The dotted lines are sideband energy levels. (a) Energy levels during adiabatic transfer. The Raman ﬁeld frequency is tuned from slightly smaller than the red sideband (light grey) to slightly higher than the red sideband (dark grey). (b) Raman ﬁeld frequencies of the red sideband transition (red arrows) and blue sideband (blue arrows). That of the carrier transition is shown in Fig. 4.2.

where the second index in the bra denotes the phonon number state.

A phonon number non-resolving detector can be implemented by separating the 0 | i from other phonon number state. I assume all ions are initially in g . Let us consider a | i general motional state of the kth mode is ψ = ∞ α m . When adiabatic passage | i m=0 m| i is applied, the total state transforms as P

g (α 0 + α 1 + α 2 ...) α g 0 + e (α 0 + α 1 + ...) . (6.31) | i 0| i 1| i 2| i → 0| i | i 1| i 2| i

The electronic state is then measured by standard ﬂuorescence measurement techniques discussed in Sec. 4.2.2. From Eq. (6.31), the measurement of e is equivalent to the | i measurement of nonzero phonon state, which is an phonon analogous of a photon number non-resolving detector.

Incorporating with carrier pulses (The pulse that transits electronic states g and | i e while leaving motional states unchanged. More will be discussed in Sec. 6.5.2.) and | i an additional meta-stable state r , the method of adiabatic passage can be extended to | i projectively measure any Fock state. More explicitly, after the operation in Eq. (6.31), carrier transition and subsequently another round of adiabatic passage are applied. The

96 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

state will be transformed as

∞ ∞ α g 0 + e α n α e 0 + g α n 0| i | i n+1| i → 0| i | i n+1| i n=0 ! n=0 ! X X ∞ g (α 1 + α 0 )+ e α n . (6.32) → | i 0| i 1| i | i n+2| i n=0 ! X Repeating this process for k times, all the components with phonon number n 6 k will be coupled to g , while those with n>k will be coupled to e . Finally, adiabatic passage | i | i is conducted to push all states with n

g k g 0 ; g n r k n (for nk) . (6.33) | i→| i | i→| − i | >ki→| − i

Measuring the electronic state g , e , r corresponds to the measurement with the | i | i | i projection-valued measurement (PVM)

k k , I , I , (6.34) {| ih | k}

where Ik). To the best of our knowledge, such a measurement scheme does not exist in quantum optics experiments. The scheme would be a useful tool in analysing the boson number population of a general motional state, and also in quantum information tasks that involve post-selection of phonon number states.

The successful rate of adiabatic state transfer is higher for a larger Landau-Zener parameter, i.e. Γ 1 (see e.g. [183] for reference), where ≫ (η Ω)2 Γ= k 1 . (6.35) ∆(˙ t) ≫

Suppose the detuning varies linearly as time from ∆ to ∆ , then ∆(˙ t)=2∆ /T for a − 0 0 0 total transfer time T . A large Γ can be achieved by setting a long T or a small ∆0.

The magnitude of ∆0 has to be bounded to avoid mixing of energy eigenstates. Let us ﬁrst consider the lower bound. The Hamiltonian in Eq. (6.29) at t = 0 is equivalent to the interaction picture of the following Hamiltonian

~ ~ H = ∆0 e e + i Ωηk(σ+ak σ ak† ) , (6.36) | ih | − −

of which the dressed energy eigenstates, expressed in terms of g n and e n , are given | i | i

97 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

√n +1ηk~Ω En En = g n +1 + i ± e n , (6.37) ± 2 2 2 2 | i (En ) +(n + 1)(ηkΩ) | i (En ) +(n + 1)(ηkΩ) | i ± ± p p 1 2 2 where En = (∆0 ∆0 + 4(ηkΩ) ) are the eigenenergies. Since adiabatic passage ± 2 ± transfer states from En to En+ , it can be used to transfer g n + 1 to e n if | p−i | i | i | i 2 2 En g n +1 and En+ e n are close to 1. This implies the condition |h −| i| |h | i|

∆ √n η Ω , (6.38) 0 ≫ max k where nmax is the maximum phonon number involved in the simulation. Combining with the criterion of adibaticity, Eq. (6.35), the total transfer time should obey T ≫ 1/∆0.

On the other hand, ∆0 has to be much smaller than νk, in order to prevent the mixing of g n and e n due to the carrier transition terms, i.e., ~Ω(σ + σ+), that | i | i − are of the zeroth order of η (to be discussed in Sec. 6.5.2). A prudential choice is to set ∆ η ν . The duration of an adiabatic state transfer can then be estimated by ≈ k k using typical parameters in state-of-the-art ion-trap experiments, i.e., ν 106 MHz and ≈ η 0.1. If each “ ” relationship is assumed to denote a diﬀerence of one order of ≈ ≫ 3 magnitude, then T 10− s. ≈ Changing the time dependence of ∆(t) causes little eﬀect on the required time T . Alternatively, T can be suppressed by an order of magnitude if the Rabi frequency is also time dependent. One of the scheme is proposed by Allen and Eberly in Ref. [10, 44], which the Rabi frequency and detuning vary as

T 1 T 1 Ω(t) = Ω sech (t ) ; ∆(t) = ∆ tanh (t ) , (6.39) 0 − 2 t 0 − 2 t 0 0

where t0, Ω0 and ∆0 are respectively the characteristics time, Rabi frequency, and detuning of the process. The successful transfer rate of using this scheme is higher than the stardard linear-varying detuning, because the initial state mixing is small due to the small initial Rabi frequency. Besides, ∆ varies slowly at the vicinity of ω ω ν so ≈ 0 − k the Landau-Zener parameter is large that exhibits high adiabaticity. If Eq. (6.29) is the exact expression for the Hamiltonian, the probability of the state transfer from g n to | i e n 1 will be given by [44] | − i π πt P =1 sech2 ∆ t cos2 0 nη2Ω2 ∆2 . (6.40) − 2 0 0 2 0 − 0 q 98 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

2 2 2 The cos term is bounded by 1 when nη Ω0 > ∆0, therefore a high transfer rate is determined by a small sech term, i.e. a large ∆0t0. On the other hand, the cos term 2 2 2 will become cosh, which is a diverging function, when nη Ω0 < ∆0. Therefore the Rabi frequency has to be reasonably large, i.e., Ω0 & ∆0/ηnmax. However a too large Ω0 will

break the RWA of the zeroth order term in Eq. (6.28). Therefore the parameters Ω0, ∆0,

and t0 have to be optimised for a fast and accurate adiabatic transfer. For a bosonic

simulator with two ions in a trap with ν =2π MHz, an adiabatically transfer for nmax =3 with 0.99 ﬁdelity takes only 80 µs 10.

6.5.2 Resonant Pulses

When the Raman ﬁeld frequency is tuned exactly to the electronic state transition frequency, Eq. (6.28) will be dominated by the zeroth order term, viz.

(I) ~ VT Ω(σ+ + σ ) . (6.41) ≈ −

which will induce Rabi transition between the electronic states, by the motional state is not aﬀected if the RWA is eﬀective. This operation is referred as the carrier transition. The motional and electronic states of an ion are coupled by tuning the laser frequency to be resonant to motional sideband frequencies. When the frequency is tuned to the red sideband, i.e., ω = ω ν , under RWA the Hamiltonian in Eq. (6.28) is dominated by 0 − k (I) ~ VT i Ωηk(σ+ak σ ak† ) . (6.42) ≈ − −

The evolution operator of this Hamiltonian can be reduced to a tensor product of transformations within the subspaces g n +1 , e n , viz. {| i | i}

g 0 g 0 (6.43a) | i→| i g n +1 cos Ωη √n +1t g n +1 + sin Ωη √n +1t e n (6.43b) | i→ k | i k | i e n sin Ωη √n +1t g n +1 + cos Ωη √n +1t e n . (6.43c) | i → − k | i k | i On the other hand, the blue side-band transition is activated when ω = ω0 + νk. Under RWA, the Hamiltonian in Eq. (6.28) is dominated by

~ V = i Ωηk(σ ak σ+ak† ) . (6.44) − −

10 The parameters for this transfer are η =0.2, Ω0 =2.25MHz, and ∆0 =0.1ν

99 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

Under this Hamiltonian, the dressed states transform as

e 0 e 0 (6.45a) | i→| i g n cos Ωη √n +1t g n + sin Ωη √n +1t e n +1 (6.45b) | i→ k | i k | i e n +1 sin Ωη √n +1t g n + cos Ωη √n +1t e n +1 . (6.45c) | i → − k | i k | i The Raman ﬁeld frequencies of the resonant pulses are shown in Fig. 6.5(b)

The major diﬀerence between the resonant-pulse method and adiabatic passage is that the transition amplitude depends on the phonon number. As an example, let us consider the state that is a superposition of 0 , 1 , 2 . When g 1 is transferred to e 0 | i | i | i | i | i by a red sideband pulse with t = π/(2Ωηk), other photon states are transited as

√2π √2π g 0 g 0 ; g 1 e 0 ; g 2 cos g 2 + sin e 1 . (6.46) | i→| i | i→| i | i→ 2 ! | i 2 ! | i

Since 2 will be coupled to both g and e after the pulse, afterwards measurement | i | i | i on the electronic state cannot provide precise information about the phonon number of the initial state. If the maximum phonon number in the state is known, in this case n = 2, the ground phonon state could still be singled out to g by applying a sequence max | i of sideband diﬀerent phases [78]. In general if no a priori information about the phonon state is known, singling out a Fock state is non-trivial. In the following, I will describe two tricks to apply the resonant pulse to conduct projective measurement on particular a phonon state k . | i For demonstrating the ideas in future discussions, I introduce the circuit diagram in Fig. 6.6.

6.5.2.1 Post-selection Method

The ﬁrst trick is to use a sequence of measurement to remove unwanted superpositions, and then post-select the desired outcomes. Let us consider if 0 is to be post-selected | i from a state ψ = α 0 + α 1 + α 2 . After applying a red sideband pulse with | i 0| i 1| i 2| i t = π/(2Ωη ), a ﬂuorescence measurement on e state is conducted. Depending on the k | i

100 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

B R g,e g,e g,e (a) Blue sideband transition cou- (b) Red sideband transition cou- (c) Carrier transition with pling with g n e n + 1 pling with g n e n 1 angle (θ, φ) between g , e . transition.| Thei operation ↔ | followsi transition.| Thei operation ↔ | follows− i The operation| followsi | i Eq. (6.45) with parameter θ =Ωηkt Eq. (6.43) with parameter θ =Ωηkt Eq. (6.41) with parameter and φ depends on phase of the Ra- and φ depends on phase of the Ra- θ = Ωt and φ depends on man field. man field. phase of the Raman field.

BS BS g g (d) Evolution of the elec- (e) Beam splitter generated by Raman ﬁeld cou- (f) Fluorescence tronic state (solid line) and pling to g . The operation follows Eq. (6.22) measurement on the | i the phonon state (dashed with angle θ = ηuηvΩt and φ depends on phase electronic state g . line). of the Raman ﬁeld. | i

Figure 6.6: List of elements in circuit diagram of boson simulation. measurement outcome, the state becomes

√2π α0 0 + cos α2 2 (electronic state is not e ) ; (6.47) | i 2 ! | i | i √2π α1 1 + sin α2 2 (electronic state is e ) , (6.48) | i 2 ! | i | i up to some normalization constant. If the electronic state is e , the ion will recoil when | i scattering the measurement pulse; the gain in momentum will add significant noise to the motional state. In other words, the phonon information is lost if positive outcome is obtained and the simulation has to be terminated. On the other hand, the motional state is insignificantly affected if the electronic state is not in e , the simulation can then | i be further proceeded. The aim of this step is to remove the amplitude of 1 from the | i superposition. The next step is then to remove 2 by a transition pulse with duration | i π/(2√2Ωη ) that flips g 2 to e 1 . A negative result of e measurement is then a k | i | i | i projective measurement on 0 . The procedure is outlined as circuit diagram in Fig. 6.7. | i The above scheme can be extended to measuring 0 from a wider range of phonon | i state. The most direct method is to remove the amplitude of n in an ascending order | i of n, but the number of measurement is equal to nmax. Alternatively, let us consider that each round of resonant transition and measurement post-selection does not only remove

101 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

R g,e

click e

R g,e

click e

Figure 6.7: Circuit diagram of phonon non-distinguishing measurement using the post- selection method. If all electronic state measurements are negative, the initial motional state is projected onto the ground state.

the superposition of a particular m , but also weakens the amplitude of other states | i by a factor of cos(√mπ/2√n). The cumulative effect of the suppression factors are not evenly distributed; some Fock states remain prominent after a few rounds of operation. A shortcut of measurement sequence is to remove those prominent states instead of following the ascending order of phonon number. Let us consider if the PVM g 0 g 0 | ih | is to be imposed on a state with nmax = 30. Let the state after post-selection to be ψ Oˆ ψ . For a high fidelity measurement, the post-selection process would map any | i→ | i state to be close to g 0 . The fidelity of this measurement method can be defined as | i

= g 0 Oˆ( g I)( g I)Oˆ† g 0 , (6.49) F h | | i ⊗ h | ⊗ | i where the averaged input state is given by I, which is the identity operator of the subspace with n 30. If six rounds of measuring sequence is to remove the state 1 to 6 , then the ≤ | i | i measurement ﬁdelity is =0.62. On the other hand, if the six rounds of measurement is F to remove 1 , 2 , 3 , 4 , 7 , 13 , then the measurement ﬁdelity is improved to =0.99. | i | i | i | i | i | i F In Fig. 6.8, I show the probability of remaining in the state after a measurement sequence, if the initial state is a Fock state. Lastly, the above method can be further generalised to measure any Fock state m . | i Any Fock state can be transferred to the ground state by applying alternative red side-

102 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

Figure 6.8: Probability of the first 30 Fock states in the post-selected branch after re- moving the 1 (hollow square), 1 and 2 (hollow circle), 1 , 2 , 3 (filled triangle), 1 , 2 , 3 , |4i, 5 , 6 (filled diamond),| i |1 i, 2 , 3 , 4 , 7 , |13i (filled| i | i square, very close |toi zero).| i | i | i | i | i | i | i | i | i | i | i band and carrier pulses, i.e.

g m red e m 1 carrier g m 1 ... g 0 . (6.50) | i −−−→| − i −−−−−→| − i | i

Other Fock state components will be transformed to some superpositions of g n +1 | i and e n , except g 0 . e is then measured and the state is retained if negative result is | i | i | i obtained, in order to remove the phonon states associated with e . Conducting the above | i post-selection measurement scheme is then equivalent to measuring the PVM m m on | ih | the original state. Following the same principle, a PVM of any motional state, ψ ψ , | ih | can be measured, because an arbitrary motional state can be mapped to the ground state by a sequence of resonant pulses [69]. I note that this post-selection method preserves the total probability of measuring any state m , in other words the accumulative probability of measuring m from ψ = | i | i | i α n remains α 2. Furthermore, if this phonon mode is entangled with other n=0 n| i | m| modes as Ψ = α n φ , the measurement sequence preserves the associated P | i n=0 n| i| ni component in other degree of freedom, i.e., φ . The above measurement scheme is P | mi useful to investigate quantum information protocols that involves entanglement and post- selection, such as the linear optics quantum logic gate [101].

6.5.2.2 Multiple Electronic State Method

A more powerful phonon number-resolving measurement can be achieved by using multiple electronic states. Although the number of phonon state can be measured is restricted

103 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

by the number of meta-stable state available in an ion, this method is robust and achievable by current technology, Besides, it is particularly useful for small scale bosonic simulation, such as the demonstration of the Hong-Ou-Mandel (HOM) eﬀect that will be discussed in the next section. As an illustration, let us consider three electronic states g , e , and r are available | i | i | i for the laser manipulation. If a motional state with nmax = 2 is to be measured, a red sideband pulse of the g e transition is ﬁrst applied with the duration t = | i↔| i π/(2√2Ωηk). The Fock state components will transform as

π π g 0 g 0 ; g 1 cos g 1 + sin e 0 ; g 2 e 1 . (6.51) | i→| i | i→ 2√2 | i 2√2 | i | i→| i Then a red sideband pulse of the e r transition with duration t = π/(2Ωη ) is | i↔| i k applied and transforms the states as

g 0 g 0 ; | i → | i π π π π cos g 1 + sin e 0 cos g 1 + sin e 0 ; 2√2 | i 2√2 | i → 2√2 | i 2√2 | i e 1 r 0 . (6.52) | i → | i

Finally, a red sideband pulse of the g r transition with duration t = (√2 | i↔| i − 1)π/(2√2Ωηk) is applied, which transforms

π π g 0 g 0 ; cos g 1 + sin e 0 e 0 ; r 0 r 0 . (6.53) | i→| i 2√2 | i 2√2 | i→| i | i→| i After the above operations, each phonon state, 0 , 1 , 2 , is then associated to an | i | i | i electronic state, g , e , r respectively. Therefore the measurement of electronic states | i | i | i is equivalent to a phonon number resolving detection. The procedure is outlined as a circuit diagram in Fig. 6.9. I note that the above sequence is applicable only if nmax = 2.

For a system with a larger nmax, components of higher number Fock states will distribute on the three electronic states, so a dedicated pulse sequence has to be designed for each nmax.

6.6 Initialization

Before the simulation, all the ions are prepared in the ground electronic state, g , by | i optical pumping, and the collective modes are cooled to the ground motional state, 0 , | i

104 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

R g,e

R e,r

R g,e

g,e,r

Figure 6.9: Circuit diagram of phonon number resolving measurement for nmax = 2 with three electronic states involved.

by laser cooling [181, 78]. As I have shown that the universal set of bosonic operators can be implemented in the UBS, arbitrary phonon state ψ can be initialised by applying | i a unitary transformation that maps Uˆ 0 = ψ . In Bosonic simulation, there are input | i | i states that are of particular interest. These include Gaussian states, such as coherent states, single mode squeezed states, and multimode squeezed states, and non-Gaussian states, such as Fock states and Schr¨odinger Cat state. For Gaussian states, coherent states can be initialised by applying the displacement operator in Sec. 6.4.1 to the ground motional state. Similarly, squeezed vacuum state can be initialised by applying the squeezing operator in Sec. 6.4.3 to the ground motional state. A multimode squeezed state can be realised by applying in beam splitter Sec. 6.4.5 to single mode squeezed states [176]. Fock states can be initialised by applying alternative sideband transitions. More explicitly, a blue sideband pulse with duration t = π/2ηΩ transforms g 0 to e 1 . Then | i | i a red sideband transition with duration t = π/2√2ηΩ transforms g 1 to e 2 . This | i | i cycle of sideband pulses retain the electronic state as g while a Fock state with n = 2 | i is created. Higher number Fock states can be created by continuing this sequence of blue and red sideband transition (with appropriate pulse duration, which depends on the initial phonon number state in that round). I note that if an odd number phonon state is to be created, one of the sideband pulses can be replaced by a carrier pulse. A Fock state with n = 2 has been realised in ion trap experiments [127]. Schr¨odinger Cat states can be created by applying displacement operators and carrier

105 Chapter 6. Ion Trap Bosonic Simulator 1: Multiple Ions in Single Trap

pulses [134]. Firstly, a carrier pulse is applied to create a superposition of electronic state, i.e., ( g + e )/√2. Then a displacement operator in effective with g is applied to | i | i | i transform the total state as ( g α + e 0 )/√2, where α is a coherent state of motion. | i| i | i| i | i Similarly, a displacement operator in effective with e is applied to transform the total | i state as ( g α + e α )/√2. Finally, a carrier pulse is applied to create superpositions | i| i | i|− i of electronic states, the total state is then given by g ( α + α )/2+ e ( α α )/2. | i | i |− i | i | i−|− i Obviously, the motional states associated to each of the electronic state are Schrödinger Cat states with different phases. Subsequent simulation operations can be applied in effective with g . At the end of the simulation, the electronic state of the ion is measured. | i By post-selecting the case that g is measured, the result of the simulation on α + α | i | i |− i can be obtained. I note that although both displacement operators are implemented by two laser fields with the frequency difference ν, the operation is electronic state dependent. Let us consider the operator in effective with e . The laser fields are roughly ∆ detuned from | i the transition between e and d , which is the intermediate state that facilitates Raman | i | i transition (c.f. Fig. 6.2(b)). The field frequencies are detuned from the transition between g and d by ∆+ ω ω , which is much larger than ∆. Therefore the effective coupling | i | i e − g to g is small, and hence the motional state associated to g is barely affected. The | i | i situation is similar for the displacement operator in effective with g . | i In the context of Gaussian state initialisation, the ion trap UBS described in this chapter offers no advantage over optical experiments, because the creation of Gaussian photon states has been realised with high fidelity. On the other hand, due to the weak nonlinear interaction of photons, non-Gaussian optical states have to be created by inefficient techniques, such as post-selection and using nonlinear optical elements. How- ever, Fock states and Schrödinger Cat states have been efficiently created in ion trap experiments [127, 134]. Therefore, realising an ion trap UBS allows us to study how non- Gaussian states evolve under general interaction, which is difficult to observe in optical experiments. In Appendix A.5, I outline the procedure of demonstrating the HOM effect.

106 Chapter 7

Ion Trap Bosonic Simulator 2: Ions in Separate Trap

7.1 Introduction

Although the number of ions in a trap is not fundamentally limited, the bosonic simulator architecture presented in Sec. 6 is not very scalable. This is because the running time of a bosonic simulation increases as more ions and modes involved, so the maximum scale of a bosonic simulation is limited by the heating and the decoherence rate of the motional states. The reason is that the addition of ions will narrow the frequency gap between phonon modes, so sideband transitions have to be conducted slowly to avoid significant errors. In addition, a measurement on one phonon mode via resonance fluorescence may cause significant heating of the ion chain, which distorts the states of other phonon modes. These shortcomings limit the number of modes and the population of phonons that can be simulated accurately. The problem of an excess of ions in a single trap also appears in ion trap quantum computing [78]. As discussed in Sec. 4.3, these problems can be solved by adopting the KMW architecture: chains of small number of ion qubits are stored in separate locations of an array of traps, so that the manipulation on one qubit negligibly affects the others, and the quality of individual logic operation is independent of the scale of quantum computation. Considerable advances in experimental realization of these ideas have been made in the past few years [150, 30, 33, 175]; in particular, entanglement gates have been performed on ions which were initially far-separated, and ions have been moved between traps. In this chapter, I propose to use the KMW idea to implement a UBS on trapped ions

107 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

system. I consider each ion to be stored in a separate harmonic trap, in which only one bosonic mode is present. All single mode operations can be conducted by either changing the storage trap potential or by laser manipulations. The linear beam splitter is based on Coulomb interaction, which is the same principle as the kinetic energy exchange in Refs. [36, 79]; the diﬀerence here is that the distance between ions is variable in order to speed up the process. Ions can be transported in speciﬁc trajectories that do not cause motional excitations [168, 46]. The advantages of the current scheme are that the quality of each operation is independent of the number of modes involved in the simulation, and the initialization and readout of any one mode will not distort the others. This architecture of bosonic simulator is thus more scalable. I begin by presenting the setup and the physical model of the proposal in Sec. 7.2. The implementation of single mode operations is introduced in Sec. 7.3. In Sec. 7.4, I show that a linear beam splitter can be implemented by precisely combining and splitting two traps through changing the quadratic and quartic potentials. The initialization and readout of phonon states are presented in Sec. 7.5. This chapter is summarized in Sec. 7.6 with some discussions. Most of the material of this chapter is included in a paper published by Daniel James and myself [107].

7.2 Model

I again assume ions are tightly trapped in the y and the z directions by a strong rf field while a weaker dc potential is applied along the x direction. I assume this configuration would effectively restrict the ion to move along only the x direction because the excitations in other directions are negligible. The configuration of the system is schematically shown in Fig. 7.1. Ions are trapped in an array of harmonic storage traps, and only one ion is present in each trap. The distance between the equilibrium position of two neighbouring traps is L, which is sufficiently large that Coulomb coupling between the ions can be neglected. Hence the total Hamiltonian of the system is given by pˆ2 1 Hˆ = n + mν2xˆ2 , (7.1) 0 2m 2 0 n n X where the subscript n denotes the quantities belonging to the ion in the nth trap;x ˆn is the position operator of the nth ion with respect to the equilibrium position of the nth

trap, i.e., position operator of the quantum ﬂuctuation;p ˆn is the momentum operator of

108 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

Figure 7.1: The conﬁguration of the current ion trap UBS is an array of storage traps. Only one ion is trapped in the axial harmonic potential in each trap. The traps are separated by a distance L, which is large enough to prevent disruption from the others. The position displacement of the ith ion, xi, is accounted with respect to the centre of the ith trap.

the nth ion. The annihilation operator of the phonon mode of the nth ion is given by

mν 1 aˆ = 0 xˆ + i pˆ . (7.2) n 2~ n 2~mν n r r 0 The ions are cooled to both the motional and electronic ground state before the simulation. The trap potentials will be varied, but the potentials should return to the original form in Eq. (7.1) after each operation. Each operation is characterized by the transformation of the motional state in the interaction picture. By generalising the discussion in Sec. 4.3.2, a multiple ion state is transformed in the interaction picture as

ψ (T ) = ˆ ψ (0) , (7.3) | I i S| I i where ˆ exp(iHˆ t/~)Uˆ is the S-matrix; Uˆ is the evolution operator in the Schr¨odinger S ≡ 0 picture; Hˆ0 is the multiple ion storage Hamiltonian in Eq. (7.1). The annihilation operator of each mode in the interaction picture is then transformed as

iω0T aˆ ˆ†aˆ ˆ = Uˆ † aˆ Uˆ e . (7.4) n → S nS S n S

I will omit the n in future discussions of single mode operation.

7.3 Single Mode Operations

As discussed in Sec. 6.3, any single mode operator can be achieved by alternatively applying the displacement operators, phase-shift operators, squeezing operators, and a

109 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

(a) (b) (c)

Figure 7.2: (a) A displacement operator is implemented by changing the trap centre of the harmonic well. (b) A phase-shift operator or a squeezing operator is implemented by varying the harmonic potential strength. (c) An extra quartic potential is applied to implement the nonlinear phase gate.

nonlinear operator [117, 176]. In Sec. 6.4, I have discussed that each of these operators could be implemented by applying laser ﬁelds with diﬀerent frequencies. However, the accuracy and speed of these laser-mediated operations are limited by the validity of the LDA and RWA [180, 112]. In this section, I consider an alternative approach that the operators are implemented by varying the harmonic trap potential or by perturbatively applying a quartic potential. Both the harmonic and the quartic potential can be realised in experiments [85]. In future discussions, I assume the operations are operating from t =0 to t = T . A summary of the operations is shown in Fig. 7.2.

7.3.1 Displacement Operator

Apart from the laser-mediated method presented in Sec. 6.4.1, another way to perform the displacement operator is to move the harmonic trap, i.e. replacing the static storage trap by a displaced harmonic well with constant trap strength. The Hamiltonian of such potential is the same as Eq. (5.1), i.e.,

pˆ2 1 Hˆ = + mν2(ˆx s(t))2 ; (7.5) D 2m 2 0 −

where s(t) speciﬁes the path of the trap centre. According to the discussion in Sec. 4.3.1 and Sec. 5.2, this potential implements a displacement operator on the motional states. I require the trap centre locates at the origin before and after the operation, i.e. s(0) = s(T ) = 0. Then every coherent state, χ , will be transformed as | i T mν0 iν0T χ χ ~ s˙(t) exp(iν0t)dt e− (7.6) | i→ − 2 0 r Z E up to a global phase that will not aﬀect the simulation result [106]. In the interaction picture, the annihilation operator transforms according to Eq. (7.4) is

mν T aˆ aˆ 0 s˙(t) exp(iν t)dt aˆ + α[s(t)] , (7.7) → − 2~ 0 ≡ r Z0

110 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

where the displacement α is a functional of s(t).

The s(t) that produces a speciﬁc displacement α0 is not unique. The appropriate s(t) can be obtained systematically by the inverse engineering method described in Ref. [168], or by a simpler method that bases on the linearity of the displacements and paths. The

later method is introduced as follows. First of all, we guess two arbitrary paths, s1(t)

and s2(t), that satisfy the boundary conditions s(0) = s(T ) = 0. According to Eq. (7.7), the paths will produce two displacements, α α[s ] and α α[s ]. α and α should 1 ≡ 1 2 ≡ 2 1 2 not be scaled by a real number, otherwise another path s3(t) has to be guessed. If the

requirement is satisﬁed, there must exist two real parameters, γ1 and γ2, such that

α0 = γ1α1 + γ2α2 . (7.8)

Due to the linearity of the functional α, the path s(t) = γ1s1(t)+ γ2s2(t) will give the

desired displacement α0.

7.3.2 Squeezing Operator

As discussed in Sec. 6.4.3, a squeezing operator can be realised by applying a Raman

ﬁeld with ω = 2ν0 to the ion. However, the operation time of this method should be

much longer than 1/ν0 for RWA to be valid. I here describe an alternative approach to realise a squeezing operator by varying the trap potential. Let us consider the storage potential is replaced by a harmonic well with varying trap strength. The Hamiltonian of such potential is given by pˆ2 1 Hˆ = + mν2(t)ˆx2 , (7.9) S 2m 2 The trap frequency is required to return to that of the storage trap after the operation,

i.e. ν(0) = ν(T )= ν0. According to Eq. (4.40), the operation transforms the annihilation operator as

i(Θ(T ) ν0T ) i(Θ(T )+ν0T ) aˆ η∗(T )e− − aˆ ζ(T )e aˆ† , (7.10) → − where there is no displacement because the harmonic well centre is ﬁxed. The above operation coincides with a squeezing operation if ζ(T ) = 0. | | 6 By comparing Eqs. (7.10) and (6.19), the magnitude of the squeezing parameter is related to the auxiliary function as η = cosh g , and the phase of the squeezing operator | | | | g η(T ) ζ(T ) 2i(Θ(T ) ν0T ) is given by g = ζ(T ) η∗(T ) e− − . I note that the evolution of the phonon mode | | | | is exactly described by the analytical solution of the time dependent harmonic oscillator, so the duration of the above squeezing operation is not limited by the validity condition of the RWA.

111 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

Here I describe a systematic process to deduce a ν(t) for implementing any squeezing parameter, g. By putting the general solution of b(t) in Eq. (4.22) into Eq. (4.28), it can be checked that η(t) = cosh(δ/2), and hence δ = 2 g . Therefore, obtaining a desired | | | | magnitude of the squeezing parameter is equivalent to obtaining a ν(t) that b(t) acquires a desired δ after the operation. Such a condition can be satisﬁed by a wide range of ν(t); a particular ν(t) can be obtained inversely from an ansatz of b(t). An example is

b (t)= cosh δ + sinh δ sin(2ν t)h(t)+(1 h(t)) , (7.11) S 0 − p where h(0) = 0 and h(T ) = 1. ν(t) should be continuous before and after the operation, so bS(t), b˙S(t), and ¨bS (t) have to be continuous at t = 0 and t = T . For instance, h(t) = 10(t/T )3 15(t/T )4 +6(t/T )6 meets the requirement. The time variation of ν(t) − can then be obtained by inputting bS (t) into Eq. (4.14). I note that the above method may not construct the correct phase of the desired squeezing operator, but the phase can be rectiﬁed by applying phase-shift operator after the squeezing operation.

7.3.3 Phase-Shift Operator

As discussed in Sec. 6.4.2, a phase-shift operator can be realised by applying a Raman

ﬁeld to the ion with ω = 0. The drawback of this method is that in order for thea ˆ†aˆ

term to be dominant, the operation time has to be much longer than 1/ν0 for an eﬀective RWA. I here describe an alternative implementation of a phase-shift operator that the operation time can be reduced to the same order as 1/ν0. The method is similar to the squeezing operation described in Sec. 7.3.2, i.e., by varying the harmonic potential of the trap. The only diﬀerence here is the boundary conditions are set as η(t T ) = 1 and ζ(t T ) = 0, in order to avoid parametric ≥ ≥ excitation. This criterion is equivalent to require the auxiliary function, bφ(t), to satisfy the boundary conditions

b (t 0) = 1 ; b (t T )=1 . (7.12) φ ≤ φ ≥

According to Eq. (4.40), such a harmonic potential variation will transform the annihilation operator as

i(Θ(T ) ν0T ) aˆ aeˆ − − , (7.13) → where there is no displacement because the harmonic trap centre is ﬁxed; Θ(T ) is de-

112 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

pending on the auxiliary function via Eq. (4.21). Obviously, the operation realises a phase-shift operator.

The above conditions do not impose a unique form of ν(t) and bφ(t), therefore we have the freedom to choose ν(t) in a manner that is convenient in practice. Alternatively, we can guess an appropriate b(t) and obtain the corresponding ν(t). A possible choice is b (t)=1 k exp( (t T/2)2/σ2) , where 1/σ T is the characteristics time scale of φ − − − ≪ the operation; k is chosen to produce the desired phase shift. There is no fundamental limitation on the magnitude of σ, so the phase-shift operator can be implemented

indeﬁnitely fast, even faster than 1/ν0 if the apparatus permits.

7.3.4 Nonlinear Operator

Nonlinear operators transform an annihilation operator to an operator involving quadratic

and higher order terms ina ˆ anda ˆ†. As discussed in Sec. 6.4.4, a nonlinear operator can

be achieved by applying a Hamiltonian that is at least third order ofa ˆ anda ˆ†, such as

a Raman field with ω = 3ν0. The major issue of this approach is that the validity of both the LDA and the RWA have to be satisfied, so the undesired component in the Hamiltonian are suppressed. The consequence is that the power of the radiation field is constrained, which limits the speed of the operation. Nevertheless, the current separate trap UBS architecture facilitates this laser-mediated nonlinear operator, because the mode spectrum is simplified as only one phonon mode is exhibited in each trap. Here I describe an alternative approach to realise a nonlinear operator by switching on an additional quartic potential, i.e.

Hˆ (t)= Hˆ + Vˆ (t) Hˆ + (t)ˆx4 . (7.14) 4 0 4 ≡ 0 F

In the interaction picture with respect to Hˆ0, the quartic potential becomes

(t)~2 ˆ I 2 2 2 i2ν0t 4 i4ν0t V4 = F 2 (6â† aˆ + 12â†aˆ +3)+ a† (4â†aˆ + 6)e +â† e + h.c. . (7.15) (2mν0) h i If the variation of (t) is sufficiently slow, the off-resonant terms can be eliminated by F the RWA; the only effective terms are

~2 (t) 2 2 ˆ † † HN F 2 (6ˆa aˆ + 12ˆa aˆ + 3) . (7.16) ≡ (2mν0)

Applying the quartic potential from t = 0 to T , the S-matrix of the operation in the

113 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

Schr¨odinger picture is given by

iµ(T )(6ˆa†2aˆ2+12ˆa†aˆ) ˆ = e− , (7.17) S4

where T (t′)~ µ(t)= F dt′ . (7.18) (2mν )2 Z0 0

I have neglected the unimportant global phase, and have employed the fact that [Hˆ0, HˆN ]= 0. Obviously, the addition of quartic potential realises a Kerr-like nonlinear phase shift. The speed of this operation is mainly determined by the validity of the RWA. Accord- ing to Ref. [67], applying the RWA is to collect the leading order terms in a series expansion of time-averaged Hamiltonians. The suﬃcient condition for a valid series expansion is that the largest eigenvalue of HˆN /~ should be much smaller than the oﬀ-resonant

frequencies, which are multiples of ν0 in our case. Although HˆN /~ has unbounded eigenvalues, the series expansion is still valid if the maximum phonon number in each mode, n , is small. To estimate the RWA validity condition, I approximate (t)~2/(2mω )2 max F 0 by ~µ(T )/T because Vˆ4 is slowly varying. The maximum eigenvalue of HˆN in the simu- 2 lation is hence nmaxµ(T )/T , which gives a valid RWA when

n2 µ(T ) max 1 . (7.19) ν0T ≪

7.4 Two-mode Operation

Because the ions are separately trapped in this UBS architecture, their modes are diﬃcult to be interact through laser operation. In this section, I describe a controlled collision process that can implement a phonon beam splitter. The beam splitter would transform the phonon modes of two separate ions according to Eq. (6.22). As discussed in Sec. 6.3, universal bosonic simulation can be conducted by using the beam splitter and the single mode operations discussed in Sec. 7.3. The whole beam splitter process is summarized schematically in Fig. 7.3. Here I consider the duration of the process is T , and it starts from t = T/2. Harmonic wells − with moving trap centres are applied from t = T/2 to transport the ions from the − storage traps to the pick-up distance and then switched oﬀ at t = T ′/2, a double well − potential is immediately switched on to relay the transportation. The separation of the double well shrinks and then expands. This action aims to bring the two ions to proximity, so the two phonon modes can interact through the Coulomb interaction between the ions.

114 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

III

Figure 7.3: Displacement of ions and variations of potentials during a phonon beam splitter. The origin is deﬁned as the mid-point between two storage traps. Step I, at T/2 t T ′/2, ions are transported by harmonic well from the storage traps to the − ≤ ≤ − pick-up positions. Step II, at t = T ′/2, a double well is switched on to pick-up the ions. − Step III, at T ′/2 t T/2, the separation of double well shrinks and then expands. − ≤ ≤ Step IV, at t = T ′/2 , the ions are brought back to the pick-up positions. Step V, at T ′/2 t T/2, the double well is switched oﬀ, and the harmonic wells pick-up the ions ≤ ≤ and bring them to the storage traps.

The double well potential ﬁnally separates the ions to the pick-up distance. It is then

switched oﬀ at t = T ′/2, while two moving harmonic wells are switched on immediately to transport the ions back to the storage traps at t = T/2. In this section I set the origin of position, X = 0, to be the mid-point between the two storage traps. I assume the system is both spatially and dynamically symmetric about the origin. To simplify the discussion, the classical motion and the quantum ﬂuctuations are separated as 1 Xˆ X¯ +ˆq ; Pˆ P¯ +ˆπ (7.20) i ≡ i i i ≡ i i where the subscripts i = 1, 2 denote the ions involved in the beam splitter operation;

Xˆ and Pˆ are the position and momentum operator of the total motional state; X¯i and P¯ mX¯˙ are the classical position and momentum of the ith ion;q ˆ andπ ˆ are the i ≡ i i i position and momentum operator of the quantum ﬂuctuation. The aim of the beam splitter is to transform the quantum ﬂuctuations of the two ions according to Eq. (6.22), whereas the ions will be classically stationary at the storage

1Formal procedure of separating the classical and quantum motion is referred to Sec. 4.3.1.

115 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

traps before and after the operation, i.e.

T T L T T L X¯ ( )= X¯ ( )= ; X¯ ( )= X¯ ( )= ; (7.21) 1 − 2 1 2 − 2 2 − 2 2 2 2 T T T T P¯ ( )= P¯ ( )= P¯ ( )= P¯ ( )=0 . (7.22) 1 − 2 1 2 2 − 2 2 2

At the storage traps, the quadrature operators (position and momentum operators) of the quantum ﬂuctuation are the same as that of the phonon modes deﬁned in Eq. (7.1), i.e.

qî storage =x î ;π î storage =p î . (7.23)

The core component of the beam splitter operation is a double well potential with varying well separation. It can be constructed by a quartic potential, A(t)X4, and a harmonic potential, B(t)X2, which can be realised in experiments [85]. The evolution of the total motional state, ψ , follows the Schr¨odinger equation i~∂ ψ = Hˆ (t) ψ , | i t| i BS | i where

ˆ2 ˆ2 2 P1 P2 2 2 4 4 e HˆBS(t)= + + B(t)(Xˆ + Xˆ )+ A(t)(Xˆ + Xˆ )+ . (7.24) 2m 2m 1 2 1 2 4πǫ (Xˆ Xˆ ) 0 2 − 1 I note that the ions cannot tunnel pass each other due to the strong Coulomb repulsion,

so X¯2 > X¯1.

In terms of the variables in Eq. (7.20), the Schr¨odinger equation becomes

i~∂ ψ˜ =( + ˆ + ˆ ) ψ˜ , (7.25) t| i H0 H1 HB | i where and ˆ collect the terms with the zero and the first order terms of the quadra- H0 H1 ture operators, and H˜ contains the rest; ψ˜ is the state of the quantum fluctuations. B | i The first term = P¯2/2m+P¯2/2m+A(t)(X¯ 4 +X¯ 4)+B(t)(X¯ 2 +X¯ 2)+e2/4πǫ (X¯ X¯ ) H0 1 2 1 1 1 1 0 2 − 1 is the total mechanical energy of the system; it contributes only an unimportant global phase. The second term ˆ vanishes if the classical equations of motion are satisfied, i.e. H1 e2X¯ X¯˙ = P¯ /m ; P¯˙ = 4A(t)X¯ 3 2B(t)X¯ + i . i i i − i − i 4πǫ X¯ (X¯ X¯ )2 0| i| 2 − 1 Because of the symmetry, we have X¯ (t) = X¯ (t) and P¯ (t) = P¯ (t). The classical 1 − 2 1 − 2 separation between the ions is defined as r X¯ X¯ , then the equation of motion is ≡ 2 − 1

116 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

reduced to the following:

2 A(t) 3 2B(t) e r¨ = r r + 2 . (7.26) − m − m 2πǫ0mr

If the quantum ﬂuctuation of position is much smaller than the separation of ions, i.e. δqˆ2 /r 1, then ˆ can be approximated by a quadratic Hamiltonian, viz h i ≪ HB p πˆ2 πˆ2 3 e2(ˆq qˆ )2 ˆ ˆ = 1 + 2 + A(t)r2 + B(t) (ˆq2 +ˆq2)+ 2 − 1 . (7.27) HB ≈ H2 2m 2m 2 1 2 4πǫ r3 0 The validity of this quadratic approximation will be discussed later.

Instead of analysing the motion of individual ions, it is simpler to study the collective modes of motion, i.e., the centre-of-mass mode (+ mode) and the stretching mode (- mode). The quadrature operators of the collective modes are deﬁned as

qˆ qˆ πˆ πˆ qˆ = 2 ± 1 ;π ˆ = 2 ± 1 . (7.28) ± √2 ± √2

In this new basis, Eq. (7.27) can be written as the Hamiltonian of two decoupled harmonic oscillators, i.e., πˆ2 πˆ2 ˆ + 1 2 2 1 2 2 2 = + mν+(t)ˆq+ + − + mν (t)ˆq , (7.29) H 2m 2 2m 2 − − where the mode frequencies are

3A(t) 2 2B(t) ν+(t)= r + ; (7.30) r m m 2 2 e ν (t)= ν+(t)+ 3 . (7.31) − s πǫ0mr

The annihilation operators of the modes are deﬁned as

mω 1 ˆ = ± qˆ + i πˆ . (7.32) A± 2~ ± 2~mω ± r s ±

Because the collective modes are always decoupled, the variation of the double well potential will induce only single mode squeezing or phase-shift on the collective modes. Because we expect no parametric excitation after a beam splitter, the double well operation should give only a phase shift. In general, satisfying such condition would require us to simultaneously solve the evolution of both modes. For simplicity, I assume the quartic

117 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

and the harmonic potentials are adjusted to produce a constant ν+, i.e.,

3A(t) 2B(t) r2(t)+ = ν2 (t)= ν2 . (7.33) m m + 0

According to Eq. (4.40), the + mode then remains unchanged after the operation, i.e. ˆ ˆ . A+ → A+ The remaining problem is to construct a double well variation that only phase-shifts i2θ the stretching mode, i.e. the annihilation operator transforms as ˆ e− ˆ , where A− → A− 2θ = Θ(T ) ν T according to Eq. (4.40). The phonon modes of individual ions would − 0 then transform as

1 i2θ ˆ1 ( ˆ+ e− ˆ ) = cos θ ˆ1 + i sin θ ˆ2 ; (7.34) A → √2 A − A− A A 1 i2θ ˆ2 ( ˆ+ + e− ˆ )= i sin θ ˆ1 + cos θ ˆ2 , (7.35) A → √2 A A− A A

iθ where an unimportant global phase e− has been neglected. This transformation can be rectiﬁed to the form in Eq. (6.22) by applying local phase-shift operators: before the transformation, mode 2 is shifted with a phase ieiφ′ ; and after the transformation, mode − iφ′ 2 is shifted with a phase ie− .

The pick-up distance should be sufficiently large, at where ν and ν+ are roughly the − same according to Eq. (7.31), i.e. ν (T ′/2) = ν (T ′/2) = ν0. To avoid - mode from − − parametric excitation, the ν (t) should produce an auxiliary function b(t) that satisfies − b( T ′/2) = b(T ′/2) = 1. Such a ν (t) can be produced by controlling the quartic and − − the harmonic potential strength, A(t) and B(t), while keeping ν+(t) as constant. The appropriate time variations of A(t) and B(t) exist and are not unique; they can be chosen in a manner that is convenient in practice. Here I outline a systematic procedure to inversely engineer the desired A(t) and B(t) from an intellectually guessed b(t). Let us consider b(t T ′/2) and b(t T ′/2) are the necessary conditions for the operation ≤ − ≥ to be a phase-shift operator, but not a squeezing operator, on - mode. An ansatz that these conditions are satisfied by construction is

t2/σ2 b (t)=1 ke− ; (7.36) B − where σ T ′ determines the time scale of the operation; the value k is chosen to generate ≪ the desired phase shift. With an ansatz of the auxiliary function, the corresponding ν (t) can be deduced − inversely from Eq. (4.14). The time variation of the ion separation, r(t), is then obtained

118 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap from ν2 (t) by Eq. (7.31). A constraint on A(t) and B(t) is obtained by using the classical − equation of motion Eq. (7.26) and the time variation of r(t). Together with Eq. (7.33), the unique form of A(t) and B(t) can then be deduced.

7.4.1 Ion Transport and Pick-up

Before and after the double well operation (step II-IV), the ions are transported back and forth between the storage traps and the pick-up distance (step I and V). If both the transporting harmonic potentials and the double well potential can be switched on and oﬀ quickly, the pick-up process can be conducted smoothly that the phonon states will not be disturbed. The pick-up distance is arbitrary; it can be chosen in a manner that is convenient in experiments. The ions’ classical velocity in the double well operation is determined by the choice of ˙ ˙ A(t) and B(t). The velocity at the pick-up distance, X¯ ( T ′/2) and X¯ (T ′/2), is obtained i − i by integrating the equation of motion Eq. (7.26) with the initial condition that velocity vanishes at the turning point, i.e.r ˙(0) = 0. In step I, the transporting harmonic wells ˙ should increase the classical velocity of the ions from 0 at the storage trap to X¯ ( T ′/2) i − at the pick-up distance. Similarly in step V, the transporting harmonic wells should ˙ reduce the classical velocity from X¯i(T ′/2) at the pick-up distance to 0 at the storage traps. During Step I and V, each ion is transported by a harmonic well with moving centre, i.e., Eq. (4.6). In this case, the trap strength should be ﬁxed, i.e., ν(t) = ν0, in order to avoid parametric excitation. As discussed in Sec. 4.3.1 and 5.2, a moving harmonic well will induce a classical displacement (position and momentum) on the ion. The displacement depends on the path of the trap centre, si(t). By putting the variables in Eq. (7.20) into Eq. (4.11), the classical equation of motion is

X¯¨ (t)= ν2 X¯ (t) s (t) , (7.37) i − 0 i − i where the exact total classical displacement is given in Eq. (5.3). Appropriate si(t), which produces the X¯ and P¯ that match the boundary conditions at t = T/2, T ′/2, T ′/2,T/2, i i − − can be obtained by the inverse-engineering or bang-bang method [168], and further optimised for extra constraints [106, 46]. Such heatingless rapid ion transportation has recently been realised by using segmented traps [175, 33]. During the transportation, the evolution of the quantum ﬂuctuation is determined by the Hamiltonian πˆ2 1 ˆ = i + mν2qˆ2 . (7.38) HT 2m 2 0 i

119 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

Obviously, the motional states will not be disturbed by ˆ and hence the transportation. HT All in all, the operation from step I to step V realizes a phonon beam splitter, i.e. Eq. (6.22), on the phonon modes in neighbouring storage traps.

7.4.2 Accuracy of beam splitter

Now let us consider the errors in the beam splitter operation. The transportation in step I and V would cause small error if the harmonic well is sufficiently precise. The pick-up process is also assumed to be fast enough that does not cause significant error. Such a high precision and rapid switching of trap potential has recently been realised in ion transportation experiments [175, 33]. Most of the error is expected to come from the double well process. One of the problem comes from the anharmonic terms in the Hamiltonian of quantum fluctuations. Let us recall that the Hamiltonian ˆ in Eq. (7.27) is constructed under quadratic approxi- H2 mation. In fact, the full Hamiltonian is given by

2 3 √2e qˆ 2 3 A(t) 4 2 2 4 ˆB = ˆ2 − + √2A(t)r(3ˆq qˆ +ˆq )+ (ˆq +6ˆq qˆ +ˆq ) , (7.39) 3 + + + H H − 2πǫ0r (r + √2ˆq ) − − 2 − − − which includes higher than second order (anharmonic) terms of quadrature operators. When comparing with ˆ , the magnitude of these anharmonic terms is suppressed by H2 the ratio between the spread of quantum fluctuation and the ion separation. This ratio can be characterised by the anharmonic factor: qˆ2 /r. According to the cases I have h i studied, if a µs range beam splitter is conductedp in a trap with ν0 in the MHz range, the shortest ion separation is about l 3 2e2/4πǫ mν2 that is in the µm range 2. On the 0 ≡ 0 0 ~ other hand, the spread of quantum fluctuationp is roughly, √n¯ /2mν0, which is about tens of nm if the average number of phonon in a mode,n ¯, is smallerp than or at the order of 10. The influence of the anharmonic terms is numerically assessed by simulating a 50:50

beam splitter with the bB(t) in Eq. (7.36). The evolution of the motional states is tracked by integrating the Schr¨odinger equation with the Hamiltonian in Eq. (7.39). For numerical eﬃciency, the interaction terms between the + and - modes are replaced by the expectation value, e.g.q ˆ2 qˆ2 . This is a good approximation in our case because + →h +i the back reaction scales as higher orders of the small anharmonic factor. 40 + I consider the ions are Ca and the trap frequency of the storage traps is ν0 = 2π 3 MHz. The pick-up position is set as 50 l0/√2 from the mid-point of the ions. The

2 I note that when two ions are placed in a single harmonic well, the ion separation is l0.

120 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

Figure 7.4: Fidelity of the phonon state n+ n after a 50:50 beam splitter operation 2 2 | i| −i with ν0 σ =2, 3, 4, 5, 7, 9. Total time for the double well to bring ions from and back to the pick-up position is 11, 13, 15, 17, 20, 22 1/ν respectively. The minimum separation × 0 between ions is about 1 l0 in all the six runs. The dotted line shows a benchmark of 0.99 ﬁdelity. operation speed is adjusted by tuning σ, and the value of k is chosen to generate the desired phase shift. I set the input states of both the CM mode and the stretching mode to be pure

Fock states, i.e. ψ( T/2) = n+ n , such that the final states should be the same as | − i | i| −i the input states up to a phase. Only the runs with n+ = n are shown in Fig. 7.4 for − comparison, because the fidelity of the states with inhomogeneous phonon number, i.e., n+ = n , is generally higher for the same maximum phonon number, i.e., max n+, n . 6 − { −} The double well process generates less than 1% error when ν2σ2 & 5 for n 8; the 0 max ≤ total process time is only about 17/ν 2.7 µs. In addition to the fidelity, I have also 0 ≈ examined the accuracy of the final state. In all cases simulated, the phase errors are less than 1%. These simulated results suggest that in a bosonic simulation with single digit number of phonons in each mode, a quality phonon beam splitter can be implemented within a few µs. The accuracy of the beam splitter is worsened, as expected, when more phonons are involved, because the anharmonic factor increases. Besides, a higher operation speed also exacerbates the error due to two reasons. Firstly, a faster beam splitter would require a stronger Coulomb force to interact the phonon states, so the ions have to be brought closer and hence the anharmonic factor would be larger. Secondly, when considering

121 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

Eq. (7.39) in the interaction picture with respect to ˆ , the terms that are third order H2 to qˆ2 /r are off-resonant. The contribution of these terms is suppressed by the RWA h i if theyp are slowly varying, then the effective Hamiltonian would be reduced to the fourth order of the anharmonic factor. On the other hand, the off-resonant terms are significant for a high speed operation because the RWA is less effective. I note that the actual anharmonic effect crucially depends on the time variation, as well as the physical implementation of the trap potential, e.g., the configuration of the electrodes. In practice, the anharmonicity can be suppressed by optimising the potential with a better design of trap and of the ansatz of b(t). In any case, the above numerical analysis provides evidence that the beam splitter operation is accurate even the potential is realistically anharmonic.

7.5 Initialization and readout

The single 3 mode state initialisation techniques presented in Sec. 6.6 is still applicable in the current separated trap architecture. In addition, Gaussian states can also be efficiently created by varying the trap potential, i.e., by implementing Gaussian operations on the motional ground state. However, the creation of non-Gaussian states by using the the nonlinear phase gate in Sec. 7.3.4 may require a complicated sequence of operations. Some non-Gaussian states, such as Fock states and Schrödinger’s cat states, can be more efficiently created by laser interaction. When comparing with the single trap architecture in Ch. 6 , the current separated trap architecture simplifies the laser-mediated state initialization. Since each trap contains only one mode, other modes would not be accidentally excited, so some conditions of RWA can be relaxed. Besides, it is no longer necessary to implement techniques, such as composite pulses and shielding, to prevent the laser operations from influencing other ions. Information about the phonon states can be extracted by the three measurement schemes suggested in Sec. 6.5: adiabatic transfer, post-selection techniques, and using multiple electronic states. In all of these schemes, the current separated trap architecture is more favourable than the architecture in Ch. 6. Because the ions are individually trapped, the recoil of an ion after fluorescence measurement does not distort other phonon modes. The spectral distribution of resonance is also simplified because each trap contains only one mode; the speed of sideband transition is hence increased as a stronger pulse

3in Sec. 6.6, the creation of multiple mode states involves laser-mediated beam splitter operation, which is not applicable in the current architecture.

122 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

can be applied without accidentally mixing other modes.

7.6 Conclusion

In this chapter, I have described a possible architecture to implement the universal bosonic simulator by using separately trapped ions. The excitation of an ion’s quantized motion can simulate a bosonic mode. Linear single mode operations can be realized by changing the strength and the centre of the trap harmonic potential. Nonlinear phase- shift operator can be implemented by exerting a perturbative quartic potential. Linear beam splitter is implemented by controlling the ion separation through varying a double well potential; the interaction between phonons ensues from the Coulomb interaction between the ions. By alternatively applying these operators, arbitrary bosonic evolution can be efficiently simulated [116, 117]. Provided that the harmonic potential implemented in experiment is sufficiently accurate and controllable, there is no fundamental limit on the speed of the single mode linear operators, and a linear beam splitter can be implemented within tens of 1/ν0 if the phonon number in each interacting mode is about or less than 10. I end this chapter with some discussions on the capabilities of the ion trap UBS. When comparing with conducting linear optical experiments, the ion trap UBS has both strength and weakness. Firstly, passive linear elements, which include beam splitters and phase-shift operator (wave plate), as well as displacement operator, have been accurately implemented in optical system. The phonon counterparts that require laser operations or manipulations of trap potential may include more error than in the optical system. On the other hand, applying a squeezing operation on an arbitrary optical state is difficult to implement, but squeezing an ion motional state can be achieved with the same level of accuracy as other linear elements. Furthermore, optical nonlinear effect is small and highly dependent on the material of the nonlinear optical element, while the nonlinear potential applied on phonon is tuneable and there is no fundamental limit of its strength. Therefore, in terms of the operation, ion trap UBS is not overwhelming advantageous over optical systems. The advantage of an ion trap UBS can be seen in the state initialisation and readout. Most of the quantum states prepared in optical experiments are Gaussian states, while the preparation of non-Gaussian state is inefficient, due to the large amount of post- selection required or the general weakness of optical nonlinearity. On the other hand, non-Gaussian states, such as Fock states and Schrödinger’s Cat state, have been realised in ion trap experiments with high fidelity [127, 134]. In general, arbitrary ion motional

123 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap

states can be deterministically created by applying a sequence of resonant pulses [69]. Such techniques would be difficult to realise in optical systems, even in foreseeable future. In most optical experiments, quantum states are measured by photon non-resolving detectors, most of which the efficiency is about 10 90% [76]. On the other hand, − both the electronic-motional state transition and electronic state measurement have been implemented with over 90% accuracy in ion trap experiments [139] 4. Furthermore, the readout scheme described in Sec. 6.5 could construct measurement schemes, such as quantum non-demolition measurement and arbitrary binary PVM, that are difficult to deterministically implement in optical systems. With the possibility of creating non- Gaussian states and new measurement schemes, ion trap UBS provides a new testbed for studying quantum optical problems and quantum information protocols. The two architectures introduced in Ch. 6 and the current chapter have their own strength and disadvantages; their practicality depends on the simulation to be conducted. All the necessary techniques for implementing the “multiple ions in single trap” UBS have been demonstrated in experiments. Besides, the accuracy of each operation is high because the intensity and frequency of laser field can be precisely manipulated, while the trap potential is kept constant throughout the simulation. However, the speed of each operation is limited by the validity of LDA and RWA, in order to engineer the desired form of interaction and to avoid unwanted influence on other modes. The condition of RWA is more stringent when more ions and modes are involved in the simulation. On the other hand, the “single ion in individual trap” approach is more scalable, because the quality of each operation is independent of the number of ions in the UBS. More explicitly, if a single mode bosonic operation is applied by exerting focused laser field, there is low probability that other modes are influence because the ions are separately trapped. Apart from the laser-mediated operation, each bosonic operation can also be realised by rapidly tuning the trap potential, which the speed is not limited by the validity of LDA and RWA. In addition, the measurement of each mode would not affect the state of other modes. This property facilitates the simulations that involves post-selection or monitoring during the process. The essential techniques for implementing this architecture is the high precision and speed manipulation of the trap potential, which has been realised in recent experiments. The quality of the bosonic operations implemented by these techniques has to be further investigated. In both UBS architectures, nonlinear operations can be simulated efficiently but rela-

4Because the motional state cannot be directly measured, the accuracy of sideband transition can only be observed indirectly. For a blue sideband transition, the accuracy can be deduced from the g population after the pulse. As shown in Ref. [93], the experimental result is less than 10% deviated from| i ideal.

124 Chapter 7. Ion Trap Bosonic Simulator 2: Ions in Separate Trap tively slow. A simulation can be much faster if it involves only linear operations. Various interesting bosonic phenomena can be investigated by using only linear operations, in particular the boson-sampling idea proposed by Aaronson and Arkhipov [17]. A boson- sampler is an array of linear bosonic elements with Fock state inputs and boson number non-resolving detection at the output. The authors claim that if there exists a classical algorithm that eﬃciently samples the probability distribution of the output detection, then the polynomial hierarchy would collapse to the third level, which is generally believed to be impossible in computer science [11]. In other words, if a boson-sampler were realised, it would be a machine that exhibits post-classical computing power. Aaronson and Arkhipov suggest that a meaningful demonstration of boson-sampler would require n = 10 to 50 bosons and about n2 to n5 log n modes. Due to the high scalability, fast and high quality operation, ability of deterministic Fock state preparation, and the ﬂex- ible and accurate measurement, ion trap bosonic simulator is a promising platform to implement boson-sampling, and thus to demonstrate the post-classical computing power.

125 Chapter 8

Rapid ion re-cooling by swapping beam splitter

8.1 Introduction

The ion trap system is regarded as the most advanced implementation of quantum computers (QC) [50, 32], where various building blocks [53, 143] , such as fast and precise quantum gates [35, 111], long time storage of quantum information [103], and high fidelity readout [139], have been demonstrated experimentally. In general, achieving high fidelity logical operations requires minimal motional excitation of the ions. However during computation, the ions are unavoidably heated up by, for example, fluctuations of the trap potential, imprecise ion transportation, and momentum gain from recoil during fluorescence readout. In practice, the ions have to be frequently re-cooled by sympathetic cooling [98], which takes about hundreds of microseconds (µs) [93]. The cooling time has recently been improved to tens of µs by using electromagnetically-induced-transparency techniques [138, 137, 115]. However, the cooling process remains a speed bottleneck of an ion trap QC because its duration is an order of magnitude longer than other operations [78]. This speed bottleneck should be resolved for a faster ion trap quantum computer for better preservation of quantum coherence and higher computational power. A recent proposal suggests that µs range cooling can be achieved by using sequences of strong laser pulses [121]. However this method is subjected to the limitation from the laser’s power. In this chapter, I describe a scheme that can rapidly re-cool a pair of ion qubit without applying laser cooling during the computation. The scheme is divided into three processes as shown in Fig. 8.1. Firstly, coolant ions, i.e. ions that will not be involved

126 Chapter 8. Rapid ion re-cooling by swapping beam splitter

in quantum logic gates and can be different in species from the qubits, are prepared in the motional ground state before quantum computation. Each coolant is stored in an individual harmonic well inside a linear trap. The second process, which is the core of the cooling scheme, is a swapping beam splitter (SBS). As an extension of the phonon beam splitter described in Sec. 7.4, here the SBS can swap the motional states of two ions even they have different masses. When a qubit has to be re-cooled, it is brought to the linear trap of a coolant. A SBS is then applied to transfer the motional ground state from the coolant to the qubit, thus the qubit is effectively cooled. If excessive coolants are prepared, a new coolant can be employed in each round of cooling, thus laser cooling is not required during the computation. The last process is to combine the individually cooled qubits to a ground state qubit pair, which can be done by a diabatic ion combination process. Both the SBS and the ion combination can be implemented by a controlled collision of the ions in a precisely manipulated double well trapping potential. I will show that these processes can take less than ten trap oscillation periods, so the total process durations are at the µs range for state-of-the-art MHz traps. The necessary rapid and precise control of a double well potential has been demonstrated by using micro-fabricated surface traps [33].

8.2 Model

As discussed in Sec. 4.3.1, if an ion is transported through a linear trap by a moving harmonic well with fixed strength, the ion’s quantum fluctuation is unchanged while only a classical displacement is induced. In other words, linear ion transportation can be diabatic, i.e., arbitrarily fast without causing any motional excitation, if the harmonic well centre is controlled so that the final displacement vanishes [168, 106, 46, 175, 33]. So let us consider the ground state coolants are restricted to move only in their respective linear traps. On the other hand, the heated qubit has to be transported, possibly through linear traps and junctions, to the coolant’s trap for cooling. The transportation would further heat up the qubits, but they will eventually be cooled. A SBS is implemented by a controlled collision of the two ions (see Fig. 8.1). The ions are radially tightly confined but weakly trapped in the axial (x) direction. The total axial motional state of the ions Ψ is governed by the equation i~∂ Ψ = ˆ Ψ , where | i t| i H| i

2 2 2 Pˆ 1 2 e ˆ = i + ξ2(t) Xˆ R (t) + . H 2m 2 i i − i ˆ ˆ i=1 i 4πǫ0(X1 X2) X − 127 Chapter 8. Rapid ion re-cooling by swapping beam splitter

(I) Heated ion Cooled ion Temperature in transient

(II)

(III)

Figure 8.1: Outline of the cooling process. Step I: Each heated (red) qubit (large ball) is transported to one of the coolants’ traps to collide with a ground state (blue) coolant (small ball). In a linear trap, the ions are transported by moving harmonic well until picking up by the double well potential. Step II: The double well potential varies the ion separation to implement the SBS. The qubit’s motional excitation is transferred to the coolant. Step III: Moving harmonic wells pick-up the ions from the double well. The heated coolant is to be discarded or re-cooled. Two individually cooled qubits can be combined into a single harmonic well for entanglement operation. The combination process can be fast and can cause negligible excitation.

128 Chapter 8. Rapid ion re-cooling by swapping beam splitter

Here Xˆ and Pˆ denotes the position and momentum operator of the total motional state. The qubit (ion 1) and the coolant (ion 2) can be different in mass, i.e., m = m . 1 6 2 Without making any assumption on the trap implementation, I characterize the local trap potential by a displaced harmonic well on each ion. The four local trap parame- 2 2 ters, ξ1 (t), ξ2 (t), R1(t), and R2(t), can be determined when expanding the global trap potential around the classical position of each ion. The realistic implementation of such potential will be discussed in the Sec. 8.7. Time variation of these parameters, which can be independently controlled by tuning the global potential, will be specified by four constraints that leads to a SBS operation. The motional excitation is characterised by the quantum fluctuation around the classical displacement of the ions. By definition, a cooling is complete only if the quantum fluctuation is brought to the ground state while the ion has no classical displacement.

Let us define the state of the quantum fluctuation as ψ Dˆ †(x ,p )Dˆ †(x ,p ) Ψ , | i ≡ 1 1 1 2 2 2 | i where Dˆ (x ,p ) = exp i(x Pˆ p Xˆ )/~ is the same displacement operator defined in i i i i i − i i Eq. (4.8); xi and pi are classical parameters that could be chosen as the classical position and momentum of ion i. After neglecting a constant term that contributes only a global phase, the state ψ obeys the equation | i

i~∂ ψ =(Hˆ + Hˆ ) ψ . (8.1) t| i 1 2 | i

Hˆ involves only the ﬁrst order position and momentum operators, i.e., Hˆ = pˆ + 1 1 V1 1 pˆ + qˆ + qˆ , where V2 2 F1 1 F2 2 i 2 pi 2 ( 1) e i = x˙ i ; i =p ˙i + ξi (t)(xi Ri(t)) + − 2 . (8.2) V mi − F − 4πǫ0r r x x > 0 is the ion separation. The above equations become the classical equation ≡ 1 − 2 of motion when = 0 and = 0, thus Hˆ = 0 for my choice of x and p . Vi Fi 1 i i The dynamics of the quantum ﬂuctuation is governed only by Hˆ2 that involves second and higher order of operators:

2 pˆ2 1 e2 Hˆ = i + ξ2(t)ˆq2 + (ˆq qˆ )2 + O(ˆq3) . (8.3) 2 2m 2 i i 4πǫ r3 1 − 2 i=1 i 0 X For clarity, I have recast the position and momentum operators of the quantum ﬂuc- tuation asq ˆ andp ˆ respectively, and the “motional state” is only referred to that of quantum ﬂuctuation hereafter. The Coulomb potential is Taylor-expanded with respect to (ˆq qˆ )/r. For the moment, quadratic approximation is applied on Hˆ , i.e., O(ˆq3) 1 − 2 2

129 Chapter 8. Rapid ion re-cooling by swapping beam splitter

is neglected. This approximation is applicable in our case because the spread of ions’ wavefunction is much shorter than the ion separation, i.e., qˆ2 r. The validity of h i i ≪ this approximation will be further examined in Sec. 8.8. p

8.3 Cooling

It is well known that energy can be transferred between harmonic oscillators in the presence of weak coupling [12]. Such effect is recently demonstrated with two separately trapped ions, of which the motional state is swapped by the Coulomb interaction [36, 79]. The cooling scheme discussed in this chapter employs a similar effect to transfer the motional excitation from the qubit to the coolant. The major difference here is that both the interaction and local trap strength are tuneable, and the evolution of the harmonic oscillators is solved exactly instead of perturbatively. As we will see, these tricks allow the current proposal to have the following improvement: (i) energy can be completely transfer between two oscillators even they have different masses; (ii) energy transfer can be diabatic by using strong coupling.

Within the quadratic approximation, the evolution under Hˆ2 is a two-mode squeezing operation on the ions’ motional state [117, 176]. The squeezing parameters depend on 2 the tuneable local trap strength, ξi (t), and on the Coulomb coupling that is determined

by r, which is controllable by adjusting Ri(t). I will show a systematic way to obtain the trap parameters that the two-mode squeezing becomes a SBS, i.e., after the process at 0

ˆ ˆ iθ ˆ ˆ iθ UT† aˆ1UT =ˆa2e ; UT† aˆ2UT =ˆa1e , (8.4)

where √miξi(0) 1 aî ~ qî + i ~ pî ; (8.5) ≡ r 2 s2 √miξi(0) !

Uˆt is the evolution operator at time t. The phase factor θ does not involve in the cooling 2 2 process. The local trap strength before and after the SBS are the same, i.e., ξi (0) = ξi (T ),

so the initial and ﬁnal states can be characterised by the samea ˆi. To see how a SBS cools the qubit, let us consider the initial motional state of the coolant is the ground state while that of the qubit is an arbitrary pure state, i.e., ψ(0) = | i f(α) α dα 0 , for some complex function f. This state lies in the eigensubspace | i1 ⊗| i2 of the coolant’s phonon number operator:a ˆ†aˆ ψ(0) = 0. According to Eq. (8.4), the R 2 2| i SBS transforms the eigenvalue equation asa ˆ†aˆ ψ(T ) = Uˆ Uˆ † aˆ†Uˆ Uˆ † aˆ Uˆ ψ(0) = 1 1| i T T 1 T T 1 T | i

130 Chapter 8. Rapid ion re-cooling by swapping beam splitter

Uˆ aˆ† aˆ ψ(0) = 0. This derivation implies that the qubit will result in the ground T 2 2| i motional state. Since the eigenvalue equation is valid for any complex function f, the SBS can cool a qubit with any initial motional state.

8.4 Swapping Beam Splitter

The construction of a SBS is clearer when considering the collective modes of the quantum ﬂuctuation. Let us deﬁne the quadrature operators of the centre-of-mass (+) mode and the stretching (-) mode as

1 m2 1 m1 qˆ (ˆq1 qˆ2);p ˆ = (ˆp1 pˆ2) . (8.6) ± ≡ √2 ± m1 ± √2 ± m2 r r

Then in the quadratic approximation as Hˆ2 can be re-written as

pˆ2 pˆ2 ˆ + 1 2 2 1 2 2 H2 + − + m1ν+(t)ˆq+ + m1ν (t)ˆq + qˆ+qˆ , (8.7) ≈ 2m1 2m1 2 2 − − E −

where the coupling strength between the modes is

2 2 2 ξ1 (t) m1 ξ2 (t) m1 e = + 1 3 , (8.8) E 2 − m2 2 − m2 4πǫ0r and the mode frequencies are

2 2 2 2 2 ξ1 (t) ξ2 (t) 1 1 2e ν (t)= + + 3 . (8.9) ± 2m1 2m2 m2 ∓ m1 4πǫ0r r r Since the Hamiltonian in Eq. (8.7) involves only the quadratic terms of the quadrature operators, the evolution is generally a two-mode squeezing operation. Here I discuss the necessary conditions that correspond to a SBS. Let us define the annihilation operators of the modes as m ν 1 aˆ 1 0 qˆ + i pˆ , (8.10) ± ≡ 2~ ± 2~m ν ± r r 1 0 where ν0 ξ1(0)/√m1 is the qubit’s initial trap frequency. The annihilation operators ≡ of the collective modes and the ions’ motional state are related asa ˆ = (â1 aˆ2)/√2. ± ± Since a beam splitter should not induce parametric excitation, the modes should only ˆ ˆ iθ±(T ) be phase-shifted after the process, i.e., UT† aˆ UT =a ˆ e− . Additionally, the beam ± ± splitter is a SBS, i.e. Eq. (8.4) is satisfied, if the modes acquire a π phase difference after

the operation, i.e., θ (T ) θ+(T )= π. − −

131 Chapter 8. Rapid ion re-cooling by swapping beam splitter

The system involves two coupled time dependent harmonic oscillators, of which the analytical general solution is diﬃcult to obtain. Here I provide a procedure to systematically deduce the time variation of the four local trap parameters that would yield a SBS. Firstly, I require the + and - modes are decoupled, i.e. = 0. According to Eq. (8.8), E this can be achieved by imposing Constraint 1 as:

2 2 m2 2 m2 e ξ2 (t)= ξ1 (t)+ 1 3 . (8.11) m1 m1 − 2πǫ0r The system is reduced to two decoupled harmonic oscillators, where the mode fre- 2 quencies can be individually controlled by tuning ξ1 (t) as well as Ri(t) that control r. While appropriate mode frequencies can be individually obtained by using the inverse- engineering method (to be discussed), one particular solution is to set one of the mode,

e.g. + mode, to be time independent, i.e., ν+(t)= ν0. According to Eqs. (8.9) and (8.11), this imposes the Constraint 2 as:

2 2 2 m1 e ξ1 (t)= m1ν0 1 3 . (8.12) − − m2 2πǫ0r r

Then + mode is not parametric excited and the phase is θ+(t)= ν0t. With Constraint 1 and Constraint 2 satisﬁed, the ion separation is then uniquely determined by ν2 (t) via the Constraint 3: −

2 2 2 e ν (t)= ν0 + 3 . (8.13) − √m1m2πǫ0r

The remaining problem is to find an appropriate ν2 (t) so that - mode is not parametric − excited and acquires the desired phase. Here I introduce an inverse engineering method to find such an ν2 (t). As a time dependent harmonic oscillators, the evolution of - − mode can be exactly solved by using the dynamic invariant formalism as discussed in Sec. 4.3.1. At any time t, the annihilation operator of - mode transforms as Eq. (4.30). These parameters are uniquely determined by a real scalar auxiliary function b(t) that satisfies Eq. (4.14), i.e., ν2 ¨b(t)+ ν2 (t)b(t) 0 =0 , (8.14) − − b3(t)

with b(t < 0) = 1. Parametric excitation is absent if η∗(t > T ) 1 and ζ(t > T ) → → 0. This imposes a boundary condition on the auxiliary function: b(t > T ) 1. An → additional boundary condition is required on b(t) that yields the desired ﬁnal phase

θ (T )= θ+(T )+ π = ν0T + π. −

132 Chapter 8. Rapid ion re-cooling by swapping beam splitter

In order to obtain the local trap parameters that realise a SBS, let us adopt an ansatz for b(t) to obtain an appropriate ν2 (t). The ion separation, r(t), is then determined by − 2 2 Constraint 3. ξ1 (t) and ξ2 (t) are obtained by Constraint 2 and 1 respectively. For

R1(t) and R2(t), one constraint is already imposed by the desired r(t) and the classical equation of motion, Eq. (8.2); an additional constraint on the classical centre-of-mass motion is needed to ﬁx the parameters. A possible choice of this Constraint 4 is the symmetric ion motion: x (t)= x (t)= r(t)/2. 1 − 2

8.4.1 Ansatz

I here suggest a class of ansatz of b(t) that all the boundary conditions are satisﬁed at construction: 1/2 √π (t 0.5T )2/σ2 − b(t)= e− − +1 . (8.15) ν σ 0 2 2 The speed of the SBS is determined by the parameter σ. For ν0 σ = 2 and 3, the SBS process time, T , is 8.3 and 10.2 trap oscillation periods respectively 1. I note that the scaled process time, ν0T , is independent of the ions’ mass and ν0, and does not aﬀect

the quality of cooling within the quadratic approximation of Hˆ2. Therefore, the qubit cooling time is generally in the µs range if the trap frequency is a few MHz. As an example, a controlled collision between a 40Ca+ qubit and a 24Mg+ coolant 2 2 with ν0 = 2π MHz was simulated. The cooling time is T = 1.3µs for ν0 σ = 2. Time variation of mean phonon number, ion separation, and local trap parameters are shown in Fig. 8.2.

8.5 Ground state qubit pair

Now I discuss how to use the above cooling scheme to prepare a ground state qubit pair for high ﬁdelity quantum logic operation. Here I speciﬁcally consider the KMW quantum computer architecture that is constituted by numerous interconnected traps (c.f. Sec. 4.3), though the method is also applicable to other architectures that ions are movable in linear traps, e.g. Ref. [135]. Let us consider two qubits are transported to the linear trap that contains an array of individually trapped coolants (Fig. 8.3a). The qubits can be individually cooled by three rounds of SBS (Fig. 8.3b I-III). Between each round, 2 the qubits (coolants) are transported by moving harmonic wells with strength m1ν0

1For numerical reason, T is defined at when the total phase difference is at least 10−4 deviated from −4 −6 π, i.e., θ−(T ) θ+(T ) π 10 . With this setting, the qubit possesses no more than 10 final motional| excitation− at t−= T|. ≤

133 Chapter 8. Rapid ion re-cooling by swapping beam splitter

(a) (b) 6 10

8 4 6

4 2 2

0 0 - 4 - 2 0 2 4 - 4 - 2 0 2 4

5 1. 0 0.5 - 5

- 10 0. - 4 - 2 0 2 4 - 4 - 2 0 2 4

Figure 8.2: (a) Let the qubit be initially in a thermal state with aˆ1†aˆ1 = 5. The mean phonon number of qubit (solid) and coolant (dashed) is swappedh afteri the SBS. Time 2 2 variation of (b) ion separation, (c) R1 (solid) and R2 (dashed) , (d) ξ1 (solid line) and ξ2 2 2 (dashed line), for the SBS following the ansatz in Eq. (8.15) with ν0 σ = 2. All the length 3 2 2 in the ﬁgure is expressed in the unit of the characteristic length, l0 = 2e /4πǫ0m1ν0 5.61µm, which is the separation of two 40Ca+ ions in a single harmonic well with ν =2≈π p 0 MHz.

134 Chapter 8. Rapid ion re-cooling by swapping beam splitter

(a)

(b) Q1 Q2 C1 C2 I II III IV

Figure 8.3: (a) Heated qubits are transported to the linear trap containing individually trapped coolants. (b) Sequence of constructing ground state qubit pair: (I) Motional excitation is transferred from qubit Q2 to coolant C1 through a SBS. (II) Two SBS are simultaneously conducted to swap the motion between qubit Q1 and Q2, and between coolant C1 and C2. (III) Repeat the procedure in I. Both Q1 and Q2 are cooled after this step. (IV) The qubits are combined in a single harmonic well through the heatingless ion combination process.

2 (m2ν0 ). As discussed in Sec. 4.3, the ions’ classical motion can be freely manipulated by precisely controlling the harmonic well, while the quantum fluctuation is unaffected [168, 106, 46, 175, 33]. After two qubits are individually cooled, they have to be combined in a single harmonic well (see Fig. 8.3b IV) for entanglement operation. The ion combination can be rapid and causing minimal excitation. This heatingless ion combination can be viewed as half of a SBS, which the double well potential stops varying when it converges to a single harmonic well. Time variations of trap parameters in this process can be obtained by a similar procedure as in SBS. Here, Constraint 1 is not necessary as the + and - modes are decoupled when both qubits have the same mass. This constraint can be 2 2 modified to require symmetric local trap strength, i.e., ξ1 (t) = ξ2(t). Constraint 2, which requires constant + mode frequency, can be retained so that the + mode is not parametric excited during the combination. Constraint 3 is still required to yield the desired - mode frequency by inverse engineering. Constraint 4 can still be imposed to require symmetric motion. For the ansatz of b(t), b(t< 0) 1 is remained because the initially separated ions are → not excited. The crucial difference here is the boundary condition of b(t > T ). According 2 2 to Eq. (8.14) and the fact that ν =3ν0 when two ions are trapped in a single harmonic − well [90, 107], the steady value of the auxiliary function after the combination should be 1/4 b(t > T ) 3− . The - mode will not be parametric excited if b(t) remains constant for →

135 Chapter 8. Rapid ion re-cooling by swapping beam splitter

III

Figure 8.4: Variations of potentials during diabatic ion separation. Step I, a quartic potential is added to the common trap to form a double well potential. Step II, the double well potential expands to separate the ions, until they reach the pick-up positions. Step III, harmonic wells pick-up the ions and bring them to other traps.

t > T , which is a necessary condition that the ion combination process does not cause any motional excitation [113]. An example of ansatz of such b(t) is

1 (t 0.5T )3/σ3 b(t)= 1 e − +1 , (8.16) √4 3 −

40 + As an illustration, if the initial separation between two Ca ions is r(0) = 100l0, they can be combined to a single well, i.e., r(T ) = l , in T 5/ν 0.8µs for ν = 2π MHz 0 ≈ 0 ≈ 0 and an ansatz with ν0σ = 2. I note that the above ion combination process can be reversed to diabatically separate an ion pair. While all the constraints are the same (with the same flexibility that they can be chosen according to practical needs), the only difference is that the boundary 1/4 conditions of b(t) are reversed, i.e., b(t 0) = 3− and b(t T ) = 1. Any ansatz ≤ ≥ that implements a diabatic combination can be modified for the separation process by setting t T T t. The diabatic ion separation process is applicable after the ions − → − have conducted entanglement operation in a single trap. Since the motional excitation is reduced, subsequent cooling time is shortened and the speed of quantum computation is hence improved. The whole process is summarized in Fig. 8.4.

8.6 Transport between traps

As shown as Step I (III) in Fig. 8.1, the ions are picked-up by (from) the double well before (after) the SBS. The classical motion of the ions during the SBS is speciﬁed by the ansatz of b(t). Before the SBS, the ions have to be transported to the speciﬁed pick-up

positions, xi(0), and accelerated to the speciﬁed pick-up velocities,x ˙ i(0). Similarly after

the SBS, the ions have to be transported away from xi(T ) and decelerated fromx ˙ i(T ). As discussed in Sec. 7.4.1, the ions can be transported by moving harmonic potentials

136 Chapter 8. Rapid ion re-cooling by swapping beam splitter

with fixed strength to avoid parametric excitation [168, 106, 46]. For a smooth transition between the double well and the moving harmonic potential, the local trap parameters should be continuous at t = 0 and T . This requires the trap strength of the harmonic wells to be ξ2(t)= ξ2 and ξ2(t)= m2 ξ2. 1 0 2 m1 0 If the trap potential can be changed sufficiently rapidly and accurately, the speed of ion transportation is not limited at the range of the harmonic oscillation period [106]. As demonstrated in recent experiments, ions in a MHz trap can be transported in a few µs without significant heating [175, 33]. The error of the transportation stage is thus not likely to reduce the performance of the cooling process at the current level of technology.

8.7 Implementation of potential

In the above discussion, the trap potential is represented by two displaced harmonic wells, i.e., Eq. (8.1). This representation is an approximation of the global trap potential around the classical position of the ions. In practice, such local potential can be implemented by various kinds of global potential, the choice of which depends on the experimental convenience. Here I give an example of a feasible implementation

V (x)= E (t)x + α(t)x2 + β (t)(x x )4 + β (t)(x + x )4 . (8.17) 0 1 − 0 2 0

The first two terms denote a harmonic potential with variable trap strength and centre, which can be realised by segmented traps (e.g. [33]); the last two terms are two quartic wells with fixed centre at x and x respectively, which can be realised by two sets 0 − 0 of octupole potentials [85]. By Taylor-expanding V (x) around the vicinity of the ions’ classical position, x1 and x2, and collecting the first and second order terms, at any time t the local and global trap parameters are related as

3 3 2 1 2x1 4x1 4x1+ E0 (x1 R1)ξ1 − − 3 3 2  1 2x2 4x2 4x2+   α   (x2 R2)ξ2  − = − , 2 2 2 0 2 12x1 12x1+ · β1 ξ1  −       2 2     2   0 2 12xx 12x2+   β2   ξ2   −            2 where xi xi x0. If the desired time variation of the local trap parameters, ξ1 (t), ± ≡ ± 2 ξ2 (t), R1(t), and R2(t), are known, then the time variation of the global trap parameters,

E0(t), α(t), β1(t), and β2(t) can be obtained by inverting the above equation.

137 Chapter 8. Rapid ion re-cooling by swapping beam splitter

8.8 Anharmonicity

In the above discussion, I have shown that a SBS can perfectly cool a qubit in case of the quadratic approximation, i.e., the model of coupled harmonic oscillators in Eq. (8.3) is a good approximation to the full Hamiltonian. In practice, the local potential experienced by the ions is not purely harmonic; the anharmonicity would induce motional excitation during the SBS. The anharmonicity comes from both the global trap potential and the Coulomb interaction. The global trap anharmonicity highly depends on the configuration of the experiment. For example, if the ion motion is symmetric and the applied global poten- 4 j tial is a fourth order polynomial of position, i.e. V (x) = j=1 Vjx , where Vj are real parameters related to ξ’s and R’s, then the leading order anharmonicP terms in Eq. (8.3) are at the third order ofq ˆ’s, viz. [(ξ2 ξ2)/6r +(ξ2R ξ2R )/r2]ˆq3 + [(ξ2 ξ2)/6r 1 − 2 1 1 − 2 2 1 1 − 2 − (ξ2R ξ2R )/r2]ˆq3. According to my simulation, the motional excitation caused by 1 1 − 2 2 2 these terms can be a few times higher than that caused by the Coulomb anharmonicity. However, these third order terms can be suppressed if the applied potential is a higher order polynomial of x. In practice, the global trap anharmonicity can be suppressed by optimising the geometry and the potentials of the electrodes [85, 19]. On the other hand, the anharmonicity of the Coulomb potential are the higher order terms in the Taylor expansion in Eq. (8.3). Such terms represent the nonlocal interaction between ions that could not be fully suppressed by adjusting the trap potential. Each order of the anharmonic terms roughly scales as qˆ2 /r, so the anharmonicity becomes h i significant if the mean initial excitation nˆ is higher or the ion separation is shorter. h 1iin p As a brief estimation, the minimum ion separation is about l0, so the scale factor is ~ roughly nˆ1 in /l0, which is at the range of 0.01 if the mean phonon number is h i 2mν0 about orp less thanq 10. The effect of Coulomb anharmonicity on the cooling performance was assessed in more details. Eq. (8.1) is numerically integrated when the lowest order term in O(ˆq3) in e2 3 Eq. (8.3), 4 (ˆq1 qˆ2) , is included. Although the anharmonic heating is expected to 4πǫ0r − be more serious in a faster SBS, because the ions should be brought closer for a stronger Coulomb interaction, my numerical results show that the heating effect is more sensitive to the ansatz of b(t) rather than only the speed. Fig. 8.5 shows the simulation result for three cooling process with different ansatz. 2 2 Case I and Case II correspond to the ansatz in Eq. (8.15) with ν0 σ = 2 and 3. The ansatz 4 4 of Case III is chosen as b (t)=1/ γ exp( t /k′) + 1 with k′ =8/ν0 and γ 3.048. The − − ≈ process time of the three cases are respectively T 8.3/ν , 10.2/ν , 6.3/ν . The numerical p ≈ 0 0 0

138 Chapter 8. Rapid ion re-cooling by swapping beam splitter

Figure 8.5: Final motional excitation of the qubit, nˆ , induced by the Coulomb an- h 1if harmonicity when the two ions are initially prepared in a Fock state, nin 1 0 2, and then undergo the SBS process speciﬁed by Case I (blue), II (red), III (brown).| i | i

results show that higher excitation is induced in Case I than in Case III although the former one is slower. This result could be understood from the fact that the minimum ion separation in case I is shorter than that of case III. In either case, our result shows that even if the qubit initially has 40 phonons, the Coulomb anharmonicity induces no 3 more than 10− phonon on the qubit. Therefore the anharmonicity of potential is not deemed a serious threat to the performance of the SBS cooling process. Similar to the phonon beam splitter, the error of the diabatic ion combination method is produced by the anharmonicity of the global trap potential and the Coulomb interaction. To demonstrate the feasibility of the scheme, the evolution equation Eq. (8.1) is again numerically integrated when the lowest order term in O(ˆq3) included. The ansatz + of b(t) is taken as Eq. (8.16). I consider the ions are Ca and ν0 = 2π MHz, where the

ﬁnal separation r(0) = l0. After combining the ions from about 80 l0 to l0, which the duration is about 5 1/ν 0.8µs, less than 0.001 quanta is excited for both the + mode × 0 ≈ and the - mode.

8.9 Fluctuation

Practical implementation of the trap potential may involve imperfection. Here I numerically assess how the cooling performance would be aﬀected by the inaccuracy of the

139 Chapter 8. Rapid ion re-cooling by swapping beam splitter

global trap potential. I employ the implementation of potential in Sec. 8.7. For each of the global trap parameters, a random Gaussian error is added. The characteristic width and time of the Gaussian error are denoted by κ and τ respectively 2. The effect of the error is two-fold. Firstly, the constraints will be violated and so the energy swap is not complete. In general, the quantum fluctuation will experience a two- mode squeezing, i.e., the annihilation operator transforms asa ˆ A aˆ + A aˆ† + B aˆ + 1 → 1 1 2 1 1 2 3 B2aˆ2† . Let us assume that the qubit is initially a thermal state with motional excitation n , the erroneous SBS will yield a final excitation of quantum fluctuation as n = h 1iin h 1iq ( A 2 + A 2) n + A 2 + B 2. Secondly, the error potential will cause unexpected | 1| | 2| h 1iin | 2| | 2| acceleration on the ions, so the final classical position and momentum will be uncertain, and hence will lead to motional heating. A position discrepancy δx1 and a momentum discrepancy δp will give an excitation of classical motion as n =(δx )2(m /ν /2~)+ 1 h 1ic 1 1 0 (δp )2/(2~m ν ). The total motional excitation after the process will be n + n . 1 1 0 h 1ic h 1iq I have numerically assess the magnitude of motional heating caused by such Gaussian potential fluctuation with different κ and τ. The result is shown in Fig. 8.6. In all cases investigated, the motional excitation due to incomplete energy swap is insignificant when comparing to that due to the uncertainty of classical displacement. Nevertheless, precise manipulation of ions’ classical position and momentum has been recently demonstrated in ion shuttling and splitting experiments [175, 33]. So I claim that, by using such experimental techniques, the SBS cooling scheme can be realised without causing serious motional heating. I further note that the actual motional excitation due to the random potential error is highly dependent on the experimental implementation of the global potential. The model of global potential in Eq. (8.17) is far from optimal, because a strong potential has to be applied (applying high voltage to the electrodes) to implement the displaced harmonic well when the ions are far separated. The same situation can be implemented with much weaker potential by using segmented traps, which can reduce the magnitude of the random error.

8.10 Conclusion

In this chapter, I propose that the axial motional excitation of an ion qubit can be removed by a swapping beam splitter. The SBS can be implemented by a controlled

2In the numerical simulation, I added a random error, which follows a Gaussian distribution with width κ, to the potential at every time interval τ. The potential error between two time intervals is interpolated with respect to time. 3 In the ideal case that energy transfer is complete, the parameters are A1 = A2 = B2 = 0 and iθ B1 = e for some real angle θ.

140 Chapter 8. Rapid ion re-cooling by swapping beam splitter

Figure 8.6: (Left) Motional excitation caused by the uncertainty of classical displacement, 3 4 n1 c. The spread of the Gaussian error is κ = 10− (solid) and κ = 10− (dashed). (Right)h i Parameters that determines the motional excitation due to incomplete phonon 2 2 2 2 3 swap. A1 + A2 (red) and A2 + B2 (green) is plotted for the case with κ = 10− | | | | 4 | | | | (solid) and κ = 10− (dashed). These results show that n1 q is comparable to n1 c only if n is at the order of 105, which is unconventionallyh highi in trapped ionh quantumi h iin logic experiments.

collision between the qubit and a ground state coolant ion, i.e., the ion separation is precisely controlled that the mutual Coulomb interaction swaps the motional state of the ions. The whole process can take less than ten oscillation periods of the trap, which is at µs range for current state MHz traps. The cooled individual ions can then be diabatically combined into a single well for high ﬁdelity quantum logic operation. I have outlined a systematic procedure to obtain the time variation of the trap parameters for both the SBS and the ion combination processes. If excessive coolants are prepared before the quantum computation, a new coolant can be employed in each round of cooling, thus lengthy laser cooling is not necessary during the computation. Therefore, the cooling scheme can improve the operational speed of an ion trap QC. I note that the core of my SBS cooling scheme is the tuneable trap frequency and quadratic interaction of two coupled harmonic oscillators. The idea could also be applied to cool systems with similar behaviours, such as polar molecules [87] and nanomechanical oscillators [82, 166].

141 Chapter 9

Summary

Aiming to improve the practicality of quantum information processing (QIP), in my PhD period I have studied various applications in quantum cryptography and ion trap quantum computation. In the work presented in Ch. 2, we have examined the assumptions behind a new quantum cryptography scheme called position-based quantum cryptography (PBQC). We discovered that an incorrect assumption was made in the previous literature that the cheaters do not share entangled resources. In fact, we showed that all known PBQC protocols could be deterministically cheated if the cheaters share entanglement. By generalising the entanglement attack in our work, the most general PBQC protocol was shown to be insecure if the cheaters employ sufficient entangled resources to conduct nonlocal measurement. Therefore PBQC is not unconditionally secure, even when all apparatus is perfect. Although PBQC is still claimed to be secure in the condition that the cheaters share only bounded amount of entanglement [40], conducting a useful PBQC protocol requires demanding infrastructure, such as straight and concrete transmission channels, and extremely fast detectors and transmitters. Based on these issues, PBQC is unlikely a practical application of QIP. Nevertheless, there are still academically interesting problems inspired by PBQC. In all known cheating strategies today, the success rate scales as a polynomial of the dimension of the encrypting quantum system. In principle, if the message is encrypted in an infinite-level quantum system, successful cheating would require unrealistically large amount of entangled resources. Such an infinite-level system can be found in continuous- variable (CV) systems, which the information can be an arbitrary complex number. However, according to an ongoing work that I am studying, if the message is encrypted as a Gaussian state, then the cheaters can reproduce the correct response if they share a EPR state and use it to conduct a CV version of the teleportation attack presented

142 Chapter 9. Summary

in Ch. 2. Therefore, employing an infinite-level system does not necessarily improve the security of PBQC. A possibly more secure scheme can be a variation of Protocol A that the message is encrypted as either a coherent state or a Fock state. Since the Fock basis is not Gaussian, a more complicated entanglement attack is deemed required. I believe that both the protocol and the corresponding cheating strategy of CV PBQC are interesting directions for further investigations. In the work presented in Ch. 3, we have analysed the security of quantum secret sharing (QSS) when CV cluster states are employed as the resources. We proposed a procedure to transform a multipartite QSS scheme to a bipartite quantum key distribution (QKD) scheme, so the security of QSS can be analysed by using well-developed techniques in CV QKD. While in the literature the security of CV QSS is guaranteed only when the resources state is infinitely squeezed, our studies show that a finitely but sufficiently squeezed CV cluster state can also produce a nonzero secret sharing rate. Our results relax the stringent requirement of the resources that could be used in secure CV QSS, thus the practicality of QSS is improved. However, there are more problems have to be solved before QSS is practically useful. One problem is that realistic apparatus is imperfect, for example the transmission channels are lossy, and the state initialisation and detection are not 100% accurate. These imperfections may weaken the correlation between the dealer and the access structure, thus reduce the secret sharing rate. Besides, our calculations are conducted in the asymptotic limit that infinite rounds of state distribution have been conducted; in practice secret sharing rate has to be calculated with the finite-key effect considered. Nevertheless, by using the transformation procedure in our work, the imperfection analysis in CV QKD could be borrowed to study the security of QSS under practical situations. In the work presented in Ch. 5, we studied how the information encoded in the ion electronic states would be influenced by the electric field for transporting ion qubits. We found that the electric field would produce a dc Stark shift on the energy levels, which induces a phase shift on the quantum information. We obtained the optimal transportation trajectory that minimises the accumulative dc Stark shift, and deduced a threshold speed above which the phase shift would become a significant error on the quantum information. Our calculation shows that the threshold speed is at least two orders of magnitude faster than the transportation speed in state-of-the-art ion trap experiments. Our work verifies previous claim that dc Stark effect is not a major source of error in ion trap quantum computation. As a by-product of this work, we have derived a formula, Eq. (5.3), to describe the motional state transformation of an ion after being transported by a harmonic well. This

143 Chapter 9. Summary

formula is useful for deducing the optimal well trajectory that causes negligible motional excitation after the transportation; such optimal transportation has been realised in experiments [33]. A generalisation of this formula is to consider the transportation of a chain of ions. According to our preliminary work shown in Appendix A.6, if all ions have the same mass, Eq. (5.3) is still applicable to describe the centre-of-mass mode, while other motional modes will not be excited if the harmonic well strength is kept constant. However, the situation becomes much complicated if the ions have unequal mass. This is because the motional modes are coupled, which the general analytic solution is difficult to obtain. Nevertheless, it is still of high interest to develop a numerical procedure to deduce the optimal trajectory for inhomogeneous ion chain transportation. Because each harmonic well in recent experiments usually involves one qubit and one coolant ion, a transportation process with lower motional excitation rate would reduce subsequent cooling time. In the work presented in Chs. 6 and 7, we proposed a new application of ion trap systems: universal bosonic simulator (UBS). Being trapped in a harmonic potential, the quantised ion motion exhibits bosonic behaviours, so each degree of freedom of ion motion can simulate a bosonic mode. In the architecture in Ch. 6, which involves multiple ions being trapped in a single harmonic well, boson initialisation and transformation can be conducted by applying radiation fields with precisely tuned frequency. Although the speed and the quality of the operations would be reduced when much more than four modes are simulated, this architecture is implementable with today’s technology and is useful to simulate small-scale bosonic phenomena, such as the Hong-Ou-Mandel effect. In the UBS architecture in Ch. 7, which involves separately trapped ions, boson initialisation and transformation can be conducted by varying the trap potential. This architecture has the advantage that the operation applied on one mode would barely influence the uninvolved modes, and both the speed and the quality of each operation are independent of the total number of modes in the simulation. Implementing this architecture requires the trap potential to be controlled with high speed and accuracy; such level of control has been demonstrated in recent experiments [33, 175]. When comparing to conducting optical experiments, the ion trap UBS has the advantage that a wider range of state can be deterministically initialised, stronger nonlinear interaction can be applied, and higher efficiency measurements can be conducted. However, the practicality of the UBS depends on the level of imperfections, such as the fluctuation of trap potential that excites the motional states, the higher order non-harmonic potential that induces unwanted coupling of modes, and the infidelity of bosonic operations that is due to the imprecision of the trap potential and of the applied radiation field.

144 Chapter 9. Summary

The magnitude of these imperfections has to be verified in experiments. Nevertheless, the detrimental effects of the imperfections can be relieved by imposing rectifying methods. For example, the speed of Fock state initialisation and measurement can be improved by using composite pulses of radiation field [78], so the motional heating due to potential fluctuation can be reduced. Another method is to employ dynamical decoupling to remove undesired interaction. Furthermore, when comparing to quantum computation, less accuracy is generally required to produce meaningful results in quantum simulations [41]. In summary, we believe UBS is a promising application of ion trap systems that deserves further investigations. In the work presented in Ch. 8, I proposed a new method to re-cool ion qubits during quantum computation. The principle is to apply a swapping beam splitter to transfer the motional excitation of the qubit to a coolant ion that was prepared in the motional ground state. I showed that by using this method, ions can be re-cooled for over ten times faster than by using laser cooling. Since ion qubits have to be frequently re-cooled for implementing high fidelity logical operations, my method can improve the operational speed of ion trap quantum computers. One possible threat to the cooling scheme is the fluctuation of trap potential. My investigation shows that the influence of fluctuation is sensitive to how the potential is implemented, e.g. the distribution of the electrodes and the ansatz of potential variation; the actual performance of the cooling scheme has to be verified in experiments. Never- theless, in the inverse-engineering procedure discussed in Sec. 8.4, the ansatz that I have discussed yields a diabatic process, i.e., a shortcut to adiabaticity, which is generally believed to be robust against parameter fluctuations. Additionally, the constraints that I have proposed in Sec. 8.4 are sufficient but not necessary conditions for implementing a swapping beam splitter; in practice the constraints can be optimised for experimental convenience. In summary, in my PhD research I have resolved various hardware and software issues in quantum cryptography and ion trap quantum computation. I believe my works have brought quantum information processing a tiny step closer to practical application.

9.1 Prospects

I believe the practicality of quantum information processing would be continually improved by the advancement of apparatus and theoretical studies; there are several directions that the development is particularly promising. For example, room temperature quantum storage has been realised in a single defect centre of a semiconductor crystal for

145 Chapter 9. Summary

over 30 minutes [157]. It is anticipated that if the quantum information is encoded in the decoherence-free subspace of multiple defect centres, the coherent storage time can be further extended. Such techniques might be useful for implanting quantum certification in valuable crystals, such as diamonds. Because only well-trained experts could determine the grade of a diamond (the 4C’s), ordinary customers and jewellers rate a diamond solely by trusting the certification report, which can be easily faked. By implementing the idea of quantum money on diamonds, i.e., the certifying information is encoded in non-orthogonal states of multiple defect centres (see e.g. Refs. [179] and [16]), the buyers and sellers could have more reliable information about the genuineness of the diamond. Another promising direction of development is building hybrid quantum devices. Dif- ferent implementations of QIP, such as photons, ion traps, and superconductors, have respective strengths and weaknesses. For instance, the measurement of trapped ion qubits is high in fidelity but low in speed; on the other hand, the quantum logical operations of superconducting circuits are fast but the readout of information suffers from certain limitations. Quantum information processing is potentially more practical by building a hybrid quantum device that is composed of and could take advantage of multiple physical systems [186]. Nevertheless, new problem is encountered that the interaction between quantum systems, such as between trapped atoms and superconducting circuits, is weak and is volatile against environmental noise. Such problem could be relievable by extending a recent proposal that creates entanglement by dissipation [94]. Analysis showed that such entanglement creation process is more resistant to noise. Designing a hybrid device to realise the dissipation-mediated entanglement creation would be a promising direction of research. In addition to constructing a quantum information processor that implements the circuit elements of universal quantum computation [143], an alternative processor implementation is to create a large entanglement state for conducting measurement-based quantum computation (MBQC). In today’s MBQC model, the resources state is assumed to be created by appropriately controlling the interaction between particles. However, in some quantum systems, such as the electron spins in condensed matter systems, the particle-particle interaction is barely controllable. Nevertheless, it is easy to see that if the particle interaction is of the Ising type, a cluster state will be created occasion- ally. To the best of my knowledge, there are only a few studies about the application of such persistently and uncontrollably interacting system in MBQC. I anticipate that further theoretical studies of the computational model and the error tolerance would allow practical MBQC to be implemented in a wider range of quantum system. Just like classical communication systems, while long (kilometer) range transmission

146 Chapter 9. Summary facilities could be installed in foreseeable future, a major obstacle to the practicality of quantum communication is the ‘last mile problem’, i.e., it is not easy to install quantum channels to deliver secure quantum signals to every household. This last mile problem can be solved if the security of the quantum signal is preserved after passing through existing infrastructure. One idea is to transmit quantum signals through copper wires, such as telephone wires and television cables, that have been installed in every modern building. Unfortunately, according to my preliminary investigation, a domestic copper wire with centimetre length would expose to environmental noise that is serious enough to destroy the security of the QKD scheme, if the signal’s frequency is below 100 MHz. Nevertheless, in a recent experiment entanglement distribution through a metre-long coaxial cable has been demonstrated by using microwave frequency signal [152]. It would be promising to develop quantum communication protocols that could be implemented by such techniques, so that secure quantum signals could be transmitted through existing household copper wire networks.

147 Appendix A

Appendix

A.1 Security of Modiﬁed Protocol

In Sec. 2.7, I have demonstrated that Protocol A′ cannot be cheated by the teleportation- based strategy. Recently, all PBQC protocols are shown to be insecure if the cheaters possess entanglement resources that scales polynomially as the accuracy, 1/ǫ, where ǫ is the failure rate of cheating. However, it is still an open question that what is the minimum amount of entanglement resources required to cheat a general PBQC protocol. In the following, I will prove that Protocol A’ cannot be 100% successfully cheated if the cheaters share only a pair of entangled qubit or qutrit. Therefore the cheaters must possess at least two Bell pairs for successful cheating.

A.1.1 Security Against Attacks with One Entangled Qubit

Let us ﬁrst consider the two-veriﬁer case. Since any two qubit entangled state can be created by teleportation through a Bell state, without loss of generality I consider the cheaters share a Bell state in Eq. (2.3). It is easy to observe that the state has to be measured at t =(d l)/c, otherwise any measurement outcome obtained after this time − cannot help making the correct response. As the B2 cheater possesses only the qubit from the Bell state, he can measure the qubit according to the basis information from

V2 veriﬁer. This action is actually a remote state preparation [28], which leaves the Bell

state qubit of B1 cheater in either

= g(θ,φ) 0 + h(θ,φ) 1 or = h∗(θ,φ) 0 g∗(θ,φ) 1 , (A.1) | ↑i | i | i | ↓i | i − | i

148 Chapter A. Appendix

where g, h depend on the basis information received by B2 cheater. The appearance of and , with the probability 50% each, depends on the measurement outcomes of | ↑i | ↓i B2 cheater. This information will immediately be sent to B1 cheater together with the encryption basis information and the measurement basis of the Bell state qubit.

Suppose the encryption is in the Z-basis, i.e., θ = 0. Once received the encrypted

qubit, B1 cheater conducts a von Neumann measurement with basis

M = α 0 0 + β 1 1 , M = β∗ 0 0 α∗ 1 1 | 1i | i| i | i| i | 2i | i| i − | i| i M = γ 0 1 + δ 1 0 , M = δ∗ 0 1 γ∗ 1 0 , (A.2) | 3i | i| i | i| i | 4i | i| i − | i| i

for some complex coeﬃcients α,β,γ,δ. The ﬁrst bracket belongs to the encrypted qubit and the second one belongs to the teleported qubit. The above measurement basis states are chosen that each component of 0 and 1 of the teleported qubit is not associated | i | i with the superposition 0 or 1 of the encrypted qubit. Otherwise if a measurement | i | i basis state, say M , contains terms like ( 0 + 1 ) 1˜ , the von Neumann operator of | 1i | i | i | i M will project the encrypted qubit to a superposition state that the cheaters cannot | 1i distinguish its original identity.

Since B1 cheater knows nothing about the encryption basis, the measurement con-

ducted always has the same measurement basis as in Eq. (A.2). For general θ and φ, B1 cheater gets one of the four states before the measurement:

ψ , ψ , ψ , ψ . (A.3) {| 0i| ↑i | 0i| ↓i | 1i| ↑i | 1i| ↓i}

An important observation here is that the cheaters are able to distinguish the encoded qubit, only if each measurement basis state M that contains the components of either | ii only ψ and ψ , or only ψ and ψ . This statement can be refor- | 0i| ↑i | 1i| ↓i | 0i| ↓i | 1i| ↑i mulated to say that each state in Eq. (A.3) is a superposition of at most two states in Eq. (A.2).

When ψ is expanded, we have | i| ↑i θ θ ψ = cos g(θ,φ) 0 0 + cos h(θ,φ) 0 1 | 0i| ↑i 2 | i| i 2 | i| i θ θ + sin eiφg(θ,φ) 1 0 + sin eiφh(θ,φ) 1 1 . (A.4) 2 | i| i 2 | i| i

Without loss of generality, I assume it is a superposition of M and M . By comparing | 1i | 3i

149 Chapter A. Appendix

the component coeﬃcients in Eqs. (A.2) and (A.4), the following relations are imposed:

θ iφ g α θ iφ h γ cot e− = and cot e− = . (A.5) 2 h β 2 g δ

Similarly, ψ can be expanded as | 1i| ↓i θ θ ψ = sin h∗(θ,φ) 0 0 sin g∗(θ,φ) 0 1 | 1i| ↓i 2 | i| i − 2 | i| i θ iφ θ iφ cos e h∗(θ,φ) 1 0 + cos e g∗(θ,φ) 1 1 . (A.6) − 2 | i| i 2 | i| i

This state has to be a superposition of either M and M or M and M , in order | 1i | 3i | 2i | 4i to avoid unphysical result ψ ψ = 0. h 0 ↑| 1 ↓i 6 I first consider ψ is a superposition of M and M . By comparing the compo- | 1 ↓i | 2i | 4i nent coefficients in Eq. (A.2) and (A.6), the following relations have to be satisfied:

θ iφ g α θ iφ h γ cot e− = and cot e− = . (A.7) 2 h −β 2 g − δ

Together with Eq. (A.5), either α = δ = 0 and g(θ,φ)=0,or β = γ = 0 and h(θ,φ) = 0, for any (θ,φ). These relations imply that B2 cheater always measures the Bell state qubit in the Z basis, and B1 cheater measures both encrypted qubit and Bell state qubit individually in Z basis. As a result, the measurement of B1 cheater can project, for example, both ψ 0 and ψ 0 to M . Therefore, the cheaters cannot always distinguish | 0 i | 1 i | 2i ψ and ψ from the measurement outcomes. | 0i | 1i Now I consider another case that ψ is a superposition of M and M . By | 1 ↓i | 1i | 3i considering the coeﬃcients of components in Eq. (A.2) and (A.6), the following relations are imposed θ iφ h∗ α θ iφ g∗ γ tan e− = , tan e− = . (A.8) 2 g∗ β 2 h∗ δ Together with Eq. (A.5), we get the following relations,

g 2 θ h 2 θ | | = tan2 , | | = tan2 . (A.9) h 2 2 g 2 2 | | | | These two relations can be satisﬁed at the same time only when θ = π/2. I note that according to Eq. (2.23), θ = π/2 implies that cheating is possible if the encoding qubit is encrypted in the X basis. This explains why the cheating strategy in Sec. 2.4 is possible.

By the above arguments, I can claim that in the two-veriﬁer case, Protocol A′ cannot be cheated with 100% successful probability if the cheaters share only a pair of entangled

150 Chapter A. Appendix

qubit. In the case with N > 2 verifiers, I can prove by contradiction that Protocol A′ is secure if the cheaters share less than or equal to N 1 Bell pairs. In this case, there − must be a cheater possesses only one Bell state qubit, say the Bn cheater possess a qubit that entangles with Bm cheater. Since the encrypted qubit can be sent from any verifier, let us consider the case that Vn verifier sends the qubit. We can further assume that Ui’s are identity except Um. This case reduces to the two-verifier case that have discussed in previous paragraph, which cannot be cheated with 100% successful probability.

A.1.2 Security Against Attacks with One Entangled Qutrit

Here I outline the proof of security of Protocol A′ against arbitrary attacks if the two cheaters possess only a pair of maximally entangled qutrit in the two-veriﬁer case. First of all, I discuss the necessary properties of a maximally entangled d-level system (qudit) if the cheating can succeed with certainty. Let us consider B2 cheater conducts a remote state preparation by measuring the qudit pair in a d-level basis, then the qudit of B1 cheater will become, with equal probability, one of the eigenstates of the d-level basis. The choice of the d-level basis depends on the information of the encryption. If the encrypted qubit is an eigenstate of Z, I assume B1 cheater will receive a state in the set φ ,..., φ . Let us deﬁne a vector Φ~ ( φ ... φ )T . If the encrypted qubit is an {| 1i | di} | i ≡ | 1i | ni eigenstate ofn ˆ(θ,φ) ~σ, B cheater’s qudit will be an element of the vector Tˆ(θ,φ) Φ~ , · 1 · | i where Tˆ(θ,φ) is a d d unitary matrix function of polar angles that is known to both × cheaters.

Let B1 cheater measures the encrypted qubit and the entangled qudit in a basis with eigenstates M ,..., M . In order to distinguish the identity of the encrypted qubit {| 1i | di} after obtaining information from B cheater, each M should not contain components of 2 | ii both φ 0 and φ 1 for any j. Let us deﬁne a selection matrix S(i) that is a diagonal | ji| i | ji| i matrix, where S(i) = 1 if M contains the component of φ 0 ; and S(i) = 0 if M jj | ii | ji| i jj | ii contains the component of φ 1 . In terms of S(i), M can be written as | ji| i | ii

d (i) (i) Mi = αij′ Sj′j φj 0 + αij′ (I Sj′j) φj 1 , (A.10) | i ′ | i| i − | i| i j,jX=1 where α 2 + ... + α 2 = 1; I is the d d identity matrix. Since B knows nothing | i1| | id| × 1 about the basis, his measurement is always the same. Similar to the argument above, if B cheater is able to distinguish between ψ and 1 | i

151 Chapter A. Appendix

ψ for general θ and φ, M has to be | 1i | ii

d ˜(i) ˆ ˜(i) ˆ Mi = βikSkk′ Tk′j φj ψ0 + βik(I Skk′ )Tk′j φj ψ1 , (A.11) | i ′ | i| i − | i| i k,kX,j=1 where β 2 + ... + β 2 = 1; S˜(i) is the selection matrix for the speciﬁc θ and φ. By | i1| | in| comparing Eqs. (A.10) and (A.11), we have

θ θ α [cos S(i) + sin eiφ(I S(i))] = β S˜(i)Tˆ , (A.12) ij 2 jj 2 − jj ik kk kj θ θ α [sin S(i) cos eiφ(I S(i))] = β (I S˜(i))Tˆ . (A.13) ij 2 jj − 2 − jj ik − kk kj

After summing the above two relations, and taking the scalar product of themselves, we have

d (i) (i) α [(1 + sin θ)S + (1 sin θ)(I S )]α∗ = β β∗ =1 , (A.14) ij jj − − jj ij ik ik j X Xk where the identities S2 = S, (I S)2 = (I S), and S(I S) = 0 are employed; the − − − relation on the right hand side is the normalisation condition of Eq. (A.11). The above equation is true for any θ except when sin θ = 0, so the following relation can be deduced: 6 d (i) (i) 1 α S α∗ = α (I S )α∗ = , (A.15) ij jj ij ij − jj ij 2 j j X X where I have implicitly employed the normalization condition of M in Eq. (A.10). This | ii relation restricts the kind of measurement that should be conducted by B1 cheater, if the cheating is successful with 100% probability.

In the following, I will show that there does not exist a set of M for d = 3 {| ii} that Eq. (A.15) is satisfied. Let us assume that at least one measurement basis state, say M , consists of three components, say φ 0 , φ 0 , φ 1 , with nonzero coefficient. | 1i {| 1i| i | 2i| i | 3i| i} Then M ,..., M should consist of more than one component that is the same as M . | 2i | 6i | 1i Let us consider M shares two common components as M ; for example M consists | 2i | 1i | 2i of φ 0 , φ 1 , φ 1 . By the completeness of the measurement basis states, there {| 1i| i | 2i| i | 3i| i} must be a state, say M , that consists of φ 1 . However, such a state cannot be | 3i | 1i| i orthogonal to both M and M while satisfying Eq. (A.10). This is because the φ | 1i | 2i | 3i component of M should have either finite or no overlap with both M and M , | 3i | 1i | 2i while the φ component of M should overlap only with either M and M . As | 2i | 3i | 1i | 2i

152 Chapter A. Appendix

a consequence M must contain 1 φ , 1 φ , 0 φ . By applying the same idea | 2i {| i| 1i | i| 2i | i| 3i} to M ,..., M , at least three of the six measurement basis states must consist of the | 3i | 6i same set of components. Let us consider M , M , M consist of 0 φ , 0 φ , 1 φ . With Eq. (A.15) | 1i | 3i | 5i {| i| 1i | i| 2i | i| 3i} respected, the states should be expressed as

1 iµi iνi Mi = cos θi 0 φ1 + sin θie 0 φ2 + e 1 φ3 . (A.16) | i √2 | i| i | i| i | i| i Since the three states are orthogonal, we require

i(µ µ ) i(ν ν ) cos θ cos θ + sin θ sin θ e i− j = e i− j , (A.17) i j i j − for i = j. The norm of the term on the right hand side is equal to 1; the norm of the left 6 hand side term is equal to 1 if and only if θi = θj, µ1 = µj, and νi = νj + π. In terms of the parameters of M , M can be expressed as | 1i | 3i

1 iµ1 iν1 M3 = cos θ1 0 φ1 + sin θ1e 0 φ2 e 1 φ3 . (A.18) | i √2 | i| i | i| i − | i| i However, there does not exists a set of (θ5,µ5, ν5) that both satisﬁes Eq. (A.17) and makes M orthogonal to M and M . Therefore, there does not exist three-component | 5i | 1i | 3i measurement basis states that all the above criteria are satisﬁed. In other words, Protocol

A′ cannot be cheated if B1 cheater measures the encrypted qubit and the qutrit in a basis with three-component states. I have assumed above that at least one measurement basis state is a superposition of three components. I now consider all states contain only two components. With Eq. (A.15) satisﬁed, the states are given by

1 iµ1 M1,2 = 0 φ1 e 1 φ2 (A.19) | i √2 | i| i ± | i| i 1 θ θ i(µ1 φ) 1 θ θ i(µ1 φ) = ψ0 cos φ1 sin e − φ2 + ψ1 sin φ1 cos e − φ2 , √2| i 2| i ± 2 | i √2| i 2| i ∓ 2 | i

1 iµ2 M3,4 = 0 φ2 e 1 φ3 (A.20) | i √2 | i| i ± | i| i 1 θ θ i(µ2 φ) 1 θ θ i(µ2 φ) = ψ0 cos φ2 sin e − φ3 + ψ1 sin φ2 cos e − φ3 , √2| i 2| i ± 2 | i √2| i 2| i ∓ 2 | i

1 iµ3 M5,6 = 0 φ3 e 1 φ1 (A.21) | i √2 | i| i ± | i| i 1 θ θ i(µ3 φ) 1 θ θ i(µ3 φ) = ψ0 cos φ3 sin e − φ1 + ψ1 sin φ3 cos e − φ1 , √2| i 2| i ± 2 | i √2| i 2| i ∓ 2 | i 153 Chapter A. Appendix

or some cyclic permutation of φ ’s. On the other hand, because I have proved every | ii M cannot contain three components, they should be written as, | ii

1 iν1 M1,2 = ψ0 Tˆ1i φi e ψ1 Tˆ2i φi , (A.22) | i √2 | i | i ± | i | i 1 iν1 M3,4 = ψ0 Tˆ2i φi e ψ1 Tˆ3i φi ) , (A.23) | i √2 | i | i ± | i | i 1 iν1 M5,6 = ψ0 Tˆ3i φi e ψ1 Tˆ1i φi . (A.24) | i √2 | i | i ± | i | i Let us consider the qutrit state associated with ψ in M , and that associated with | 1i | 1i ψ M . Although both states should be Tˆ φ in Eq. (A.22) and Eq. (A.23), but | 0i | 3i 2i| ii they are unequal in Eq. (A.19) and Eq. (A.20). Therefore, there does not exist two- component measurement basis states that all the above criteria are satisﬁed. As a result,

Protocol A′ cannot be cheated with 100% successful probability in the two-veriﬁer case if the cheaters only share one pair of maximally entangled qutrit. I note that, by using

similar argument as in Sec. A.1.1, N-veriﬁer Protocol A′ cannot be cheated with 100% successful probability if the cheaters share less than N qutrits.

A.2 Example of CC QSS

A.2.1 Example 1: (2,3)-CC protocol

In a (2,3)-CC protocol, the access structure is any two of the three parties collaborating, while the adversary structure is any collaboration with only one party. The (2,3)-CC protocol can be implemented by a linear three-mode cluster, as shown in Fig. A.1. I assume the dealer picks the secret classical value s according to a Gaussian probability distribution with a width Σ, i.e.,

1 s2/Σ2 (s)= e− . (A.25) PD √πΣ

The state is encoded by displacing mode 2 by is/√2 and mode 3 by is/√2. In the − inﬁnitely squeezed case, the nulliﬁers of the cluster state are

Nˆ =p ˆ qˆ qˆ ; Nˆ =p ˆ qˆ + s ; Nˆ =p ˆ qˆ s . (A.26) 1 1 − 2 − 3 2 2 − 1 3 3 − 1 −

154 Chapter A. Appendix

1 1 1 1 1 1 5 2 2 3 1 1 4 1 3

Figure A.1: Schematic representation of the cluster states for the (2,3)-protocol (left), and the (3,5)-protocol (right). Each oval represents a squeezed mode, which will be distributed to the party denoted by the label inside. The subscript of each mode denotes the squeezing parameter and the displacement before the CPHASE operation (for clarity of the graph, each displacement is divided by i/√2). The edges joining the modes represent CPHASE operations, of which the strength is denoted by the edge’s label.

A.2.1.1 Parties {1,2} collaboration

Let us consider parties 1 and 2 to be the access structure. If the cluster state is inﬁnitely squeezed, the shared secret is the diﬀerence betweenp ˆ measurement outcome of party 1 andq ˆ measurement outcome of party 2, i.e., s = q p according to Nˆ . Derived from 1 − 2 2 Eq. (3.8), the reduced Wigner function of party 3 is a constant function independent of s, thus the protocol is secure.

In the ﬁnitely squeezed case, the Wigner function of the cluster state is given by Eq. (3.7) with the nulliﬁers in Eq. (A.26). For simplicity, I assume the modes are equally

squeezed, i.e., σi = σ for all i, but my analysis is applicable to the states with inhomoge-

neous σi. The measurement basis of the parties is the same as in the inﬁnitely squeezing case. The classical probability of having the measurement outcomes p2 and q1 is

(s) 1 2 2 2 2 (p2 q1 s) /σ σ q1 A D; 1,2 (q1,p2)= e− − − e− , (A.27) P | { } π

which is obtained by tracing out q1, p2, and the contributions of party 3 in the Wigner

function. The probability distribution of the diﬀerence of the outcomes, s′ = q p , can 1 − 2 be obtained by tracing out an orthogonal quantity, e.g. (p1 + q2)/2. Then we get

2 1 (s s′) A D; 1,2 (s,s′)= exp − , (A.28) P | { } √πσ − σ2 155 Chapter A. Appendix

and thus according to Eq. (3.12),

1 s′2/(σ2+Σ2) A; 1,2 (s′)= e− . (A.29) P { } √π√σ2 + Σ2

The mutual information between the dealer and the access structure can then be calculated using Eq. (3.13).

I now consider the adversary structure. The reduced Wigner function ofρ ˆE and

ρˆE D(s) are |

2 2 2 2 4 σ ((p3−s) +q3(1+σ )) σ 4 WE D; 3 = e− 1+σ ; (A.30) | { } π√1+ σ4 p2 2 σ2 q2+ 3 σ − 3 1+σ4+σ2Σ2 WE; 3 = WE D; 3 ds = e . (A.31) { } | { } π√1+ σ4 + σ2Σ2 Z The covariance matrices of these states are given by

1 1 2σ2 0 2σ2 0 VE D; 3 = 4 , VE; 3 = 4 2 2 , (A.32) | { } 1+σ { } 1+σ +σ Σ 0 2σ2 ! 0 2σ2 !

4 2 4 2 2 2 where the symplectic eigenvalues are νE D; 3 = √1+ σ 2σ and νE; 3 = √1+ σ + σ Σ /2σ , | { } { } respectively. The Holevo bound can then be calculated using Eq. (3.20) and (3.22), and hence the secret sharing rate can be obtained from Eq. (3.23). Because both parties 2 and 3 hold the end mode of the cluster state, their states are local-unitarily equivalent, i.e., all Wigner functions will be the same as above except replacing the subscript 2 by 3 and every value s by s. The security for party 1, 3 − { } collaboration can be analysed by the same procedure as the 1, 2 collaboration, and the { } secret sharing rate of both collaborations will be the same.

A.2.1.2 Parties {2,3} collaboration

Let us consider now that parties 2 and 3 are the access structure. In the inﬁnitely squeezed case, because the operator Nˆ Nˆ =p ˆ pˆ +2s is also a nulliﬁer, the secret s can be 2 − 3 2 − 3 obtained if both parties conductp ˆ measurement, i.e., s =( p + p )/2. The protocol is − 2 3 secure because the reduced Wigner function of party 1 is a constant independent of s.

In the ﬁnitely squeezed case, the measurement outcomes ofp ˆ1 andp ˆ2 follow the probability distribution

(p −p +2s)2 (s) 1 2 3 2 2 2 4 − σ2(2+σ4) σ ((p2+s) +(p3 s) )/(2+σ ) A D; 2,3 (p1,p2)= e e− − . (A.33) P | { } π√2+ σ4

156 Chapter A. Appendix

The ﬁrst exponent accounts for the strong correlations while the last exponent is re-

sponsible for higher order weak correlations. The quantity s′ =( p + p )/2 follows the − 2 3 probability distribution

2 1 2(s s′) A D; 2,3 (s,s′)= exp − , (A.34) P | { } √2πσ − σ2 and thus 2 2 2s′ A; 2,3 (s′)= exp . (A.35) P { } sπ(σ2 + 2Σ2) − σ2 + 2Σ2

For the adversary structure, party 1, the Wigner function of the reduced stateρ Ê D; 1 | { } is given by 2 2 σ 2 2 p1 WE D; 1 = exp σ q1 + . (A.36) | { } π√2+ σ4 − 2+ σ4 Because Eq. (A.36) is independent of s, the Wigner function ofρ Ê; 1 andρ Ê D; 1 would { } | { } be the same, i.e., WE D; 1 = WE; 1 . Therefore the Holevo bound vanishes, i.e., party 1 | { } { } cannot get any information, and hence the secret sharing rate is simply I(D : A).

A.2.2 Example 2: (3,5)-CC protocol

In a (3,5)-CC protocol, the access structure is any three of the five parties collaborating, while the adversary structure is any collaboration with less than three parties. The (3,5)-CC protocol can be implemented by a star-shaped five mode cluster, as shown in Fig. A.1. All five modes of the cluster state are displaced by is/√2, where the classical − secret s is assumed to be chosen according to the probability distribution in Eq. (A.25). In the infinitely squeezed case, the nullifiers of the cluster state are given by

Nî =p î qî+1 qî 1 s , (A.37) − − − − where i + 1 = 1 when i = 5; i 1 = 5 when i = 1. − Ten different combinations of access structure can be formed in this protocol, but they can be categorised into two classes of collaborations: three neighbouring parties, and two neighbours with one disjoint party. Without loss of generality, I consider parties 1,2,3 { } as an example of the three-neighbour collaboration, and parties 1,3,4 for two-neighbour { } collaboration. The security analysis and secret sharing rate of these two examples can be adapted to other collaborations after indices changing.

157 Chapter A. Appendix

A.2.2.1 Parties {1,2,3} collaboration

Let us ﬁrst consider parties 1, 2, and 3 are the access structure. In the inﬁnitely squeezed case, the secret s can be obtained if both parties 1 and 3 measureq ˆ and party 2 measures pˆ, i.e., s = q + p q according to Nˆ in Eq. (A.37). The reduced Wigner function of − 1 2 − 3 2 parties 4, 5 collaboration is a constant function after tracing out the contributions of { } the access structure in Eq. (3.8).

In the ﬁnitely squeezed case, the outcome probability of the measurement by parties

1,2,3 is determined by the reduced Wigner function WA D; 1,2,3 (q1,p1, q2,p2, q3,p3), { } | { } which is obtained by tracing out the contributions of parties 4 and 5 in the full Wigner function (Eq. (3.7) with the nulliﬁers in Eq. (A.37)). The measurement bases areq ˆ2,p ˆ1, andq ˆ3; the outcome probability is obtained by tracing out the dependence of p1, q2, and p3 from the reduced Wigner function. The probability distribution of the received secret, s′ = p q q , can be obtained by ﬁrst substituting the set of variables (q ,p , q ) by 2 − 1 − 3 1 2 3 another linearly independent set of variables, e.g., (q1,s′, q3), and then tracing out the independent variables, i.e., q1 and q3. The Jacobian matrix of this variable transformation is 1, so the form of probability distribution remains the same [13].

Each process of trace-out described above involves a speciﬁc physical meanings, but the end result is that all contributions except s′ are traced out. So the probability distribution of s′ can be obtained in only two steps: ﬁrst, substituting one variable with s′, e.g. p2 = s′ + q1 + q3, in the full Wigner function, and then tracing out all variables except s′. In this case, we get

2 1 (s s′) A D; 1,2,3 (s,s′)= exp − , (A.38) P | { } √πσ − σ2 and thus 1 s′2/(σ2+Σ2) A; 1,2,3 (s′)= e− . (A.39) P { } √π√σ2 + Σ2

For the adversary structure parties 4 and 5, the reduced Wigner function WE D; 4,5 | { } and WE; 4,5 can be obtained by tracing out the contribution of parties 1,2,3. The co- { }

158 Chapter A. Appendix

variance matrices of these states are

1 1 2σ2 0 0 2σ2 1 σ2 1  0 σ2 + 2 2σ2 0  VE D; 4,5 = , (A.40) | { } 0 1 1 0  2σ2 2σ2   1 1 σ2   2 0 0 2 +   2σ σ 2  1  1  2σ2 0 0 2σ2 1 σ2+Σ2 1  0 σ2 + 2 2σ2 0  VE; 4,5 = , (A.41) { } 0 1 1 0  2σ2 2σ2   1 1 σ2+Σ2   2 0 0 2 +   2σ σ 2    4 2 4 2 where the symplectic spectrum are νE D; 4,5 = √1+ σ /2σ , √1+ σ /2σ and νE; 4,5 = | { } { } { } √1+ σ4/2σ2, √1+ σ4 +2σ2Σ2/2σ2 . The von Neumann entropy can then be calculated { } by using Eq. (3.20), and the secret sharing rate is calculated from Eq. (3.23).

A.2.2.2 Parties {1,3,4} collaboration

Let us consider parties 1, 3, and 4 are the access structure. In the inﬁnitely squeezed case, the secret s can be obtained when party 1 measuresp ˆ, party 3 and 4 measures

pˆ′ = (ˆp qˆ)/√2. Because Nˆ Nˆ Nˆ is a nullifier, their measurement results are − 1 − 3 − 4 correlated as p + √2p′ + √2p′ = s. The reduced Wigner function of parties 2,5 − 1 3 4 { } collaboration is a constant function, so the secret sharing is secure. In the finitely squeezed case, similar to the case of 1,2,3 collaboration, the prob- { } ability distribution of the quantity s′ = p + p q + p q can be obtained by − 1 3 − 3 4 − 4 first substituting the set of variables (q1,p1, q3,p3, q4,p4) with the new set of variables

(q1,s′, q3,p3, q4,p4) in the Wigner function. The determinant of the Jacobian matrix of this transformation is 1. All variables except s′ are traced out from the full Wigner function, then we get

2 1 (s s′) A D; 1,3,4 (s,s′)= exp − , (A.42) P | { } √3πσ − 3σ2 and thus 1 s′2/(3σ2 +Σ2) A; 1,3,4 (s′)= e− . (A.43) P { } √π√3σ2 + Σ2

For the adversary structure parties 2 and 5, the covariance matrices of the states

159 Chapter A. Appendix

Figure A.2: Secret sharing rate of CC QSS protocols using CV cluster states with diﬀerent squeezing parameters σ. The variance of the classical secret probability is chosen as Σ = 1. Left panel: (2,3)-protocol with 2,3 collaboration (solid line) and 1,3 collaboration (dashed line) as the access structure.{ } Left panel: (3,5)-protocol with 1,2,3{ } collaboration (solid line) and 1,3,4 collaboration (dashed line) as the access structure.{ } { }

ρˆE D; 2,5 andρ ˆE; 2,5 are | { } { }

1 2σ2 0 0 0 1 σ2 1  0 σ2 + 2 0 2σ2  VE D; 2,5 = , (A.44) | { } 0 0 1 0  2σ2   1 1 σ2   0 2 0 2 +   2σ σ 2  1   2σ2 0 0 0 1 σ2 Σ2 1 Σ2  0 σ2 + 2 + 2 0 2σ2 + 2  VE; 2,5 = . (A.45) { } 0 0 1 0  2σ2   1 Σ2 1 σ2 Σ2   0 2 + 0 2 + +   2σ 2 σ 2 2    4 2 4 2 The symplectic spectrum are νE D; 2,5 = √1+ σ /2σ , √3+ σ /2σ and νE; 2,5 = | { } { } { } √1+ σ4/2σ2, √3+ σ4 +2σ2Σ2/2σ2 , respectively. The von Neumann entropy can then { } be calculated by Eq. (3.20), and hence the secret sharing rate by Eq. (3.23).

The secret sharing rate for the (2,3)- and (3,5)-protocols is plotted against σ in Fig A.2. Apart from the 2, 3 collaboration in (2,3)-protocol that the correlation can be { } completely removed from the adversary structure, CC QSS is secure unless the squeezing parameter is larger than some threshold limit.

160 Chapter A. Appendix

1 σ 1 σ

2 3 σ σ 5 2 σ D σ σd -1 D 4 3 σd σ σ

Figure A.3: Schematic representation of the cluster state for the (2,3)-CQ protocol (left), and the (3,5)-CQ protocol (right). All of the modes have zero displacement before the CPHASE operation. The strength of unlabelled edges is A = 1.

A.3 Examples of CQ QSS

A.3.1 Example 1: (2,3)-CQ protocol

In a (2,3)-CQ protocol, any collaboration involving two of the three parties can form a strong correlation with the dealer, while any one party alone is only weakly correlated with the dealer. The protocol can be implemented by a diamond-shaped CV cluster state with A = A = A = 1 and A = 1, as shown in Fig. A.3. 1 In the inﬁnitely D3 13 12 D2 − squeezed case, the nulliﬁers are

Nˆ =p ˆ +ˆq qˆ ; Nˆ =p ˆ qˆ qˆ D D 2 − 3 1 1 − 2 − 3 Nˆ =p ˆ +ˆq qˆ ; Nˆ =p ˆ qˆ qˆ . (A.46) 2 2 D − 1 3 3 − D − 1

The ﬁnitely squeezed state is described by Eq. (3.7) with the above nulliﬁers.

The access structure can be composed by parties 1, 2 , 1, 3 , and 2, 3 . The state { } { } { } possessed by collaborations 1, 2 and 1, 3 are equivalent up to a local unitary, because { } { } the nulliﬁers of 1, 2 will be the same as that of 1, 3 if the dealer applies a π-phase { } { } operation, Fˆ(π), to his mode. On the other hand, the collaboration 2, 3 possesses a { } diﬀerent state.

1The diamond-shaped CV cluster state is the same as the error ﬁltration code in cluster state quantum computation [173].

161 Chapter A. Appendix

A.3.1.1 Parties {1,2} collaboration

If parties 1, 2 are the access structure, the strong correlations are specified by the { } nullifiers Nˆ Nˆ =p ˆ pˆ +2ˆq and Nˆ =q ˆ qˆ +ˆp . (A.47) D − 1 D − 1 2 2 D − 1 2 A global operation is applied on the access structure’s modes to transfer the strong correlation to mode 2, i.e., mode 2 is treated as mode h. The transformation UÂ can be implemented by various sequence of operations, but the final measurement results and the covariance matrix are not affected. One possible choice is the 1,2 Decoding { } Sequence: (i) apply exp( iqˆ qˆ ); (ii) then exp(ipˆ pˆ ); (iii) finally Fˆ (π). − 1 2 1 2 2 After tracing out the modes other than mode D and mode 2, the covariance matrix of the resultant stateρ ˆDA is given by

1 1 2 0 0 2 2σD 2σD 2 1 σD 1  0 σ2 + 2 σ2 0  VDA; 1,2 = 2 . (A.48) { } 0 1 1 + σ 0  σ2 σ2 2   1 σ2 1   2σ2 0 0 2 + 2σ2   D D    The covariance matrix will be revealed in the parameter-estimation stage when half of the states are measured.

As VDA; 1,2 is not in the standard form, i.e., Eq. (3.25), rectifying quantum operations { } are applied onto the residual states. First of all, the variance ofq ˆD andp ˆD are balanced by squeezing mode D with the squeezing parameter

σ γ = D 2+ σ2σ2 . (A.49) D σ D r q Next, mode 2 is squeezed to balance the coherent (oﬀ-diagonal) terms, i.e., ∆ˆq ∆ˆp h D 2i and ∆ˆp ∆ˆq . The squeezing parameter is given by h D 2i

2σD γ2 = . (A.50) 2 2 sσ 2+ σ σD p In practice, both γD and γ2 can be obtained empirically from the results in parameter- estimation stage, i.e., without knowing the squeezing parameter of the initial cluster

state. This squeezing stage will transform the stateρ ˆDA as

ρˆ ρˆ′ = Sˆ (γ )Sˆ (γ )ˆρ Sˆ†(γ )Sˆ† (γ ) , (A.51) DA → DA D D 2 2 DA 2 2 D D

162 Chapter A. Appendix

where the covariance matrix becomes

V(2,3) 0 0 c

 0 V(2,3) c 0  VDA′ ; 1,2 = . (A.52) { }  0 c Vq′ 0     c 0 0 V ′   p    2 2 V(2,3) = 2+ σ σD/2σσD ; c =1/√2σσD; the variances of mode 2 are p 2 2 4 2 2 1+ σ σD (2 + σ ) 2+ σ σD V ′ = ; V ′ = . (A.53) q 2 2 p 4σσ σσD 2+ σ σD p D p I note that γD and V(2,3) are the same for any collaboration in the (2,3)-protocol, because mode D is kept with the dealer that is not aﬀected by operations on delivered

modes. The disparity between Vq′ and Vp′ implies imbalanced noise for the quadraturesq ˆ2

andp ˆ2, which has to be rectiﬁed by state-averaging. The state will then be transformed as

1 1 ρˆ′ ρˆ′′ = Fˆ ( π/2)ˆρ′ Fˆ† ( π/2)+ Fˆ ( π/2)ˆρ′ Fˆ†( π/2) . (A.54) DA → DA 2 D − DA D − 2 2 − DA 2 −

The covariance matrix ofρ ˆDA′′ is given by

V(2,3) 0 c 0

 0 V(2,3) 0 c  VDA′′ ; 1,2 = − , (A.55) { }  c 0 VA; 1,2 0   { }   0 c 0 VA; 1,2   − { }    where VA; 1,2 =(Vq′ + Vp′)/2. Finally, the modes are measured by the dealer and party 2 { } in either theq ˆ orp ˆ basis. The variances of the measurement outcomes will be given by Eq. (A.55).

I note thatρ ˆDA′′ is not a Gaussian state because the stage-averaging process in Eq. (A.54) is not a Gaussian operation. To calculate the secret sharing rate by using the techniques in Sec. 3.4.2, I can pretend the measurement results originates from a

Gaussian stateρ ˆG where its covariance matrix is VDA′′ ; 1,2 . According to Refs. [184, 68], { } Gaussian states minimise the secure key rate for all states with the same covariance matrix. Therefore state-averaging maximises the power of the unauthorised parties and lower-bounds the secret sharing rate.

By comparing Eqs. (A.55) and (3.25), the variance of the dealer’s mode is V = V(2,3),

163 Chapter A. Appendix

and the analogous channel parameters can be deduced as τ = c2/(V 2 1/4) = 1 (2,3) − and χ = VA; 1,2 V(2,3). The minimal secret sharing rate can then be calculated by { } − Eqs. (3.31)-(3.34) and (3.36).

A.3.1.2 Parties {2,3} collaboration

If parties 2, 3 are the access structure, the strong correlations are speciﬁed by the { } nulliﬁers Nˆ Nˆ pˆ pˆ Nˆ =p ˆ +ˆq qˆ and 2 − 3 =q ˆ + 2 3 . (A.56) D D 2 − 3 2 D 2 − 2 The quantum correlations can be transferred to mode 2 by the 2,3 Decoding Sequence, { } which is simply a 50:50 beam splitter that transformsq ˆ (ˆq +ˆq )/√2,p ˆ (ˆp + 2 → − 2 3 2 → − 2 pˆ )/√2,q ˆ (ˆq qˆ )/√2, andp ˆ (ˆp pˆ )/√2. The resultant covariance matrix of 3 3 → 2 − 3 3 → 2 − 3 mode D and mode 2 becomes

1 1 2σ2 0 0 √ 2 D 2σD σ2  0 1 + D 1 0  σ2 2 √2σ2 VDA; 2,3 = . (A.57) { } 0 1 1 0  √2σ2 2σ2   2   1 0 0 σ + 1   √2σ2 2 σ2   D D    The secret sharing rate can be deduced by similar processes as in the 1, 2 collab- { } oration: squeezing and transforming local modes to construct a state with standardised covariance matrix, and then measuring the states to obtain the analogous channel parameters for computing the information of diﬀerent parties. However, the 2, 3 collaboration { } is special as the dealer and party 2 are actually holding a pure state, i.e., the beam splitter has removed all entanglement from the unauthorised parties. This can be seen from the symplectic spectrum of Eq. (A.57), ν 2,3 = 1/2, 1/2 , so the entropy of the system { } { } DA vanishes, i.e., S(DA) = 0. Therefore the unauthorised parties cannot obtain any information about the secret by entangling their modes to the dealer’s mode. The secret sharing rate is hence the same as the mutual information between the dealer and the access structure, which is given by

I(D : A) = log(2V(2,3)) . (A.58)

A.3.2 Example 2: (3,5)-CQ protocol

In a (3,5)-CQ protocol, any collaboration with three of the ﬁve parties can form a strong correlation with the dealer, while any collaboration with less than two parties is only

164 Chapter A. Appendix

weakly correlated with the dealer. The protocol can be implemented by a pentagonal CV cluster state, as shown in Fig. A.3, where each connected vertex is entangled by a CPHASE operation with = 1. In the inﬁnitely squeezed case, the nulliﬁers are Aij

NˆD =p ˆD qî ; Nî =p î qî+1 qî 1 qˆD , (A.59) − − − − − i=1 X where i+1 = 1 when i = 5; i 1 = 5 when i = 1. The finitely squeezed state is described − by the Wigner function in Eq. (3.7) with the above nullifiers. The access structure can be composed by two categories of collaboration: three neighbouring parties, e.g. parties 1, 2, 3 , and two neighbours with one disjoint party, e.g. { } 1, 3, 4 . The collaborations in each category hold nullifiers with the same form, so the { } decoding sequence will be the same. If the squeezing parameter is identical for all five modes, the secret sharing rate of the collaborations in each category will also be the same.

A.3.2.1 Parties {1,2,3} collaboration

If parties 1, 2, 3 are the access structure, the strong correlations are speciﬁed by the { } nulliﬁers

Nˆ Nˆ +2Nˆ Nˆ =p ˆ (ˆp +3ˆq )+(2ˆp +ˆq ) (ˆp +3ˆq ) D − 1 2 − 3 D − 1 1 2 2 − 3 3 and Nˆ =q ˆ pˆ +ˆq +ˆq . (A.60) − 2 D − 2 1 3

The quantum correlations can be transferred to party 2 by the 1,2,3 Decoding Sequence: { } (i) apply exp( iqˆ qˆ ) and exp( iqˆ qˆ ); (ii) then exp(ipˆ pˆ ) and exp(ipˆ pˆ ); (iii) ﬁnally − 1 2 − 2 3 1 2 2 3 exp(ipˆ2(2ˆp1 +ˆq1)) and exp(ipˆ2(2ˆp3 +ˆq3)).

The covariance matrix of the stateρ ˆDA of mode D and mode 2 is given by

1 1 2 0 0 2 2σD 2σD 2 2 5+σ σD 5  0 2σ2 2σ2 0  VDA; 1,2,3 = 4 . (A.61) { } 5 5+6σ 2 0 2σ2 2σ2 σ  − 2 2   1 2 1+σ σD   2σ2 0 σ 2σ2   D − D    All terms in VDA; 1,2,3 can be revealed byx ˆ andp ˆ measurements in the parameter- { } estimation stage except for the local coherent terms (∆ˆq ∆ˆp +∆ˆp ∆ˆq )/2 . However h 2 2 2 2 i these terms do not aﬀect the parameters in the squeezing stage, and will be eventually cancelled during state-averaging.

165 Chapter A. Appendix

The unmeasured states are locally squeezed to balance the variances ofq ˆD andp ˆD, as well as the coherent terms. The state is transformed as in Eq. (A.51), where the parameters for the 1, 2, 3 collaborations are { }

σD 2 2 σ 2 2 γD = 5+ σ σD ; γ2 = 5+ σ σD . (A.62) σ 5σD r q r q The covariance matrix of the transformed state is given by

V(3,5) 0 0 c

 0 V(3,5) c 0  VDA′ ; 1,2,3 = (A.63) { }  0 c Vq′ 0     c 0 0 V ′   p    2 2 where V(3,5) = 5+ σ σD/2σσD ; c = √5/2σσD, and the variance of mode 2 is given by p 4 2 2 2 2 (5+6σ ) 5+ σ σD 5(1 + σ σD) V ′ = ; V ′ = . (A.64) q 10σσ q 2 2 p D 2σσD 5+ σ σD p The value of γD and V(3,5) are the same for any collaboration in the (3,5)-protocol.

state-averaging ensues to balance the correlations of p q and q p . Half of the D − 2 D − 2 unmeasured states are transformed by Fˆ ( π/2), while the other half are transformed D − by Fˆ ( π/2). After discarding the choice of division, the state transforms as Eq. (A.54), 2 − and the covariance matrix becomes

V(3,5) 0 c 0

 0 V(3,5) 0 c  VDA′′ ; 1,2,3 = − , (A.65) { }  c 0 VA; 1,2,3 0   { }   0 c 0 VA; 1,2,3   − { }    for VA; 1,2,3 = (Vq′ + Vp′)/2 with the deﬁnition in Eq. (A.64). I note that the local { } coherent terms vanish after state-averaging because their sign in Fˆ ( π/2)ˆρ′ Fˆ† ( π/2) D − DA D − and Fˆ ( π/2)ˆρ′ Fˆ†( π/2) are opposite. 2 − DA 2 − As discussed before, the measurement results can be pretended as coming from a

Gaussian state with the same covariance matrix VDA′′ ; 1,2,3 . The variance of the dealer’s { } mode is recognised as V = V(3,5), and the analogous channel parameters can be deduced 2 2 as τ =c ¯ /(V 1/4) = 1 and χ = VA; 1,2,3 V(3,5). The minimal secret sharing rate (3,5) − { } − can then be calculated by Eqs. (3.31)-(3.34) and (3.36).

166 Chapter A. Appendix

A.3.2.2 Parties {1,3,4} collaboration

If parties 1, 3, 4 are the access structure, the strong correlations are speciﬁed by the { } nulliﬁers

Nˆ 2Nˆ + Nˆ + Nˆ =p ˆ (2ˆp +ˆq )+(ˆp 2ˆq )+(ˆp 2ˆq ) D − 1 3 4 D − 1 1 3 − 3 4 − 4 and Nˆ Nˆ Nˆ =q ˆ +ˆp (ˆp qˆ ) (ˆp qˆ ) . (A.66) 1 − 3 − 4 D 1 − 3 − 3 − 4 − 4

The quantum correlations can be transferred to mode 1 by the 1,3,4 Decoding Sequence: { } (i) apply exp( i(ˆq +ˆp qˆ )) and exp( i(ˆq +ˆp qˆ )); (ii) followed by exp(ipˆ pˆ ) and − 1 3 − 3 − 1 4 − 4 1 3 2 ˆ exp(ipˆ1pˆ4); (iii) then exp(i2ˆp1); (iv) ﬁnally F (π).

The covariance matrix of the stateρ ˆDA between mode D and mode 1 is given by

1 1 2 0 0 2 2σD 2σD 2 2 5+σ σD 5  0 2σ2 2σ2 0  VDA; 1,3,4 = 4 . (A.67) { } 5 5+6σ 2 0 2σ2 2σ2 2σ  − 2 2   1 2 1+3σ σD   2σ2 0 2σ 2σ2   D − D   

After the parameter-estimation stage, local-squeezing is applied as in Eq. (A.51) except mode h is now mode 1. The squeezing parameters in the current collaboration are

σD 2 2 σ 2 2 γD = 5+ σ σD ; γ1 = 5+ σ σD . (A.68) σ 5σD r q r q The covariance matrix of the unmeasured states then becomes

V(3,5) 0 0 c

 0 V(3,5) c 0  VDA′ ; 1,3,4 = (A.69) { }  0 c Vq′ 0     c 0 0 Vp′      where c = √5/2σσD, and the variances of mode 1 are given by

4 2 2 2 2 (5+6σ ) 5+ σ σD 5(1+3σ σD) V ′ = ; V ′ = . (A.70) q 10σσ p 2 2 p D 2σσD 5+ σ σD p state-averaging ensues to balance the correlations of p q and q p . The covari- D − 1 D − 1

167 Chapter A. Appendix

Figure A.4: Secret sharing rate of CQ QSS protocols using CV cluster states with diﬀerent squeezing parameters σ. The squeezing parameter of dealer’s mode is set as σD = σ. Left panel: (2,3)-protocol for 2,3 collaboration (solid line) and 1,3 collaboration (dashed line). Right panel: (3,5)-protocol{ } for 1,2,3 collaboration (solid{ line)} and 1,3,4 { } { } collaboration (dashed line).

ance matrix becomes

V(3,5) 0 c 0

 0 V(3,5) 0 c  VDA′′ ; 1,3,4 = − , (A.71) { }  c 0 VA; 1,3,4 0   { }   0 c 0 VA; 1,3,4   − { }    where VA; 1,3,4 = (Vq′ + Vp′)/2 with the deﬁnition in Eq. (A.70). Similar to the 1, 2, 3 { } { } collaboration, the local coherent terms are eliminated by state-averaging.

After localq ˆ andp ˆ measurements, the results are treated as coming from a Gaussian state. The variance of the dealer’s mode is recognized as V = V(3,5), and the analogous 2 2 channel parameters can be deduced as τ =¯c /(V 1/4) = 1 and χ = VA; 1,3,4 V(3,5). (3,5) − { } − The minimal secret sharing rate can then be calculated by Eqs. (3.31)-(3.34) and (3.36).

The secret sharing rates of the (2,3)- and (3,5)-CQ protocol are plotted in Fig. A.4 for diﬀerent σ.

168 Chapter A. Appendix

A.4 Example of QQ QSS

A.4.1 Example 1: (2,3)-QQ protocol

In the (2,3)-QQ protocol, the collaboration of any two out of the three parties can recover the shared secret state with high fidelity, while any one party alone achieves much less information about the secret. This protocol can be implemented by the same diamond- shaped CV cluster state as that for the (2,3)-CQ protocol in Sec. A.3.1. In the infinitely squeezed case, the nullifiers are given by Eq. (A.46), and these nullifiers with Eq. (3.7) characterise the finitely squeezed cluster state. Three different collaborations can be formed: parties 1 and 2, parties 1 and 3, and parties 2 and 3. The state of the 1, 2 collaboration is local-unitarily equivalent to that { } of the 1, 3 collaboration. As a result, the entanglement extracted between the dealer { } and parties 1, 2 is the same as that between the dealer and parties 1, 3 . { } { } For the 1, 2 collaboration, the quantum correlation can be transferred to mode 2 by { } the 1,2 Decoding Sequence in Sec A.3.1.1. In the infinitely squeezing case, the nullifiers { } in Eq. (A.47), which specifies the strong correlation, is transformedtop ˆ qˆ andq ˆ pˆ . D − 2 D − 2 An infinitely squeezed two-mode cluster is hence extracted for teleportation. In the finitely squeezing case, the 1,2 Decoding Sequence also transfers the strong correlation { } to party 2. The covariance matrix of the extracted state between mode D and mode 2 is given by VDA; 1,2 in Eq. (A.48). { } For the 2, 3 collaboration, the quantum correlation, which is specified by the nul- { } lifiers in Eq. (A.56), can be transferred to party 2 by applying a 50:50 beam splitter between mode 2 and mode 3. An infinitely squeezed two-mode cluster state between mode D and mode 2 is extracted in the infinitely squeezing case. While in the finitely squeezing case, a strongly entangled state is extracted, of which the covariance matrix

VDA; 2,3 is given in Eq. (A.48). { }

A.4.2 Example 2: (3,5)-QQ protocol

In the (3,5)-QQ protocol, any collaboration with three out of five parties can recover the shared secret state with high fidelity, while fewer than three parties achieve much less information about the secret. This protocol can be implemented by the same pentagonal CV cluster state as used for the (3,5)-CQ protocol in Sec. A.3.2. In the infinitely squeezed case, the nullifiers are given by Eq. (A.59), and these nullifiers with Eq. (3.7) characterise the finitely squeezed cluster state. Two categories of access structure can be formed: three neighbouring parties, and

169 Chapter A. Appendix

Figure A.5: Logarithmic negativity of the state extracted from a CV cluster state in QQ QSS. Left panel: (2,3)-protocol for 2,3 collaboration (solid line) and 1,3 collaboration { } { } (dashed line). Right panel: (3,5)-protocol for 1,2,3 collaboration (solid line) and 1,3,4 collaboration (dashed line). { } { } two neighbours with one disjoint party. Within each category, the procedure of decoding and the final entanglement extracted are the same for each collaboration. Let us take the parties 1, 2, 3 as an example of the three neighbouring parties collab- { } oration. The quantum correlation can be transferred to party 2 by the 1,2,3 Decoding { } Sequence in Sec. A.3.2.1. In the infinitely squeezed case, the nullifiers in Eq. (A.60) that specify the strong correlation are transformed top ˆ qˆ andq ˆ pˆ . This indicates D − 2 D − 2 that an infinitely squeezed two-mode cluster is extracted in mode D and mode 2. While in the finitely squeezed case, the decoding operation extracts a strongly entangled state with the covariance matrix VDA; 1,2,3 in Eq. (A.61). { } On the other hand, parties 1, 3, 4 is an example of the two neighbours with one { } disjoint party collaboration. The quantum correlations can be transferred to party 1 by the 1,3,4 Decoding Sequence in Sec A.3.2.2. In the infinitely squeezed case, an { } infinitely squeezed two-mode cluster is extracted in mode D and mode 1, because the nullifiers in Eq. (A.66) that specify the strong correlation is transformed top ˆ qˆ and D − 1 qˆ pˆ . While in the finitely squeezing case, a strongly entangled state is extracted, and D − 1 its covariance matrix is given by VDA; 1,3,4 in Eq. (A.67). { } The logarithmic negativity of the extracted state for different collaborations in the (2,3)- and the (3,5)-protocols is calculated by using Eq. (3.57) with the corresponding covariance matrices. The result is plotted in Fig. A.5 against different squeezing parameters.

170 Chapter A. Appendix

A.5 Example of Application: Demonstration of Hong- Ou-Mandel Eﬀect

With the beam splitter and readout processes described in Ch. 6, numerous bosonic effects can be simulated by the ion trap UBS. One of the particular important bosonic phenomena is the HOM effect (see, e.g., ref. [14] for details), which shows the quantum nature of bosons and becomes the foundation of some proposals of linear optics quantum computation [101]. The HOM effect happens when two identical photons in different input modes hit a 50:50 beam splitter at the same time. The resultant state will be a superposition of two-photon state in either one of the output mode, i.e.

50:50 beam splitter 1 2 1 2 1 1 1, 1 = a†a† vac a† a† vac = 2, 0 0, 2 . (A.72) | i 1 2| i −−−−−−−−−−−→ 2 1 − 2 2 | i √2| i − √2| i On the contrary to classical prediction that the photons will randomly distribute to the output modes with equal probability, the above quantum state predicts that no coincidence of photon can be measured in the output modes. The reason of this effect is originated from the interference of photon state, which is related to the quantum nature of bosons. The HOM effect was first demonstrated experimentally in 1987 [86] by using photons generated from spontaneous downward conversion. Various subsequent experiments have been conducted by using photons from synchronized but less dependent sources [29], but HOM effect is not demonstrated on systems other than optics. Thus, it is of great interest to realise the HOM effect in other bosonic system, in order to verify if the HOM effect is a generic bosonic behaviour, or it is simply an optical phenomenon that is not fully understood. Furthermore, observing HOM effect in trapped ions is a strong evidence of the motional states’ bosonic nature, which is still a theoretical prediction. In this section, I outline the steps required to demonstrate HOM effect on a trapped-ion bosonic simulator. I consider a simulator that involves only two ions. Two phonon modes are available for simulation: the centre-of-mass (CM) mode and the stretching mode. The circuit diagram of the demonstration is shown in Fig. A.6. The ions are initially cooled to (or very close to) the ground state. The simulation starts by initializing one phonon in each mode by the following procedure: a blue sideband pulse is first applied to transfer the state from gg 0, 0 to ge 1, 0 , and then a carrier transition restores the electronic state | i| i | i| i to g , so the final state becomes gg 1, 0 [127]. By repeating the above procedure for | i | i| i the other mode, the total motional state becomes gg 1, 1 . | i| i

171 Chapter A. Appendix

B g,e

g,e Preparation

B g,e

g,e

BS BS Phonon g splitting

R g,e

R e,r

R g,e Measurement R g,e

R e,r

R g,e

g,e,r g,e,r

Figure A.6: Circuit diagram of the Hong-Ou-Mandel effect demonstration. Both the centre-of-mass mode (left dashed line) and the stretching mode (right dashed line) are involved. The state is first prepared as gg 1, 1 after the preparation stage, and then | i| i transforms to gg 1 ( 0, 2 2, 0 ) by the beam splitter. The motional state of each √2 mode is transferred| i to| thei−| internali state of different ions, so both of the modes can be measured. If HOM effect is realized, final motional excitation will not be observed in both the CM and stretching modes. 172 Chapter A. Appendix

Beam splitter operation is conducted by switching on the Hamiltonian (6.23) for t = π/(2η η Ω), which changes the state to ψ = gg ( 2, 0 0, 2 )/√2 2. The 1 2 | i | i | i−| i next step is to measure the coincidence of phonon detection in the modes. If adiabatic passage is employed, Raman field is applied to the first ion which the frequency is tuned around the red sideband of the CM mode. The total motional state becomes ( eg 1, 0 gg 0, 2 )/√2. Similarly, the Raman field of the second ion is tuned around | i| i−| i| i the red sideband of the stretching mode, and the state becomes ( eg 1, 0 ge 0, 1 )/√2. | i| i−| i| i Internal states of the ions are then measured by fluorescence measurement. If HOM effect is realised, one and only one of the ions is in the state e . On the other hand, | i if the final state contains a component of gg 1, 1 , it will be transformed to ee 0, 0 , | i| i | i| i which both ions are in the e . Because the adiabatic transfer measurement method is | i equivalent to a phonon number non-resolving detection, the coincidence of the phonon is thus characterized by the rate of getting both ions in e . | i Apart from using adiabatic transfer, phonon coincidence can also be measured by the resonant pulses method. If the total phonon number in the system remains 2, and if the beam splitter operation does not cause additional motional excitation, phonon coincidence is solely contributed by the component 1, 1 . The post-selection sequence | i for measuring 1, 1 is as follow. Firstly, a red sideband pulse of CM mode is applied to | i transfer g 1 to e 0 . Then g is measured by fluorescence measurement. The exper- | i | i | i iment is terminated when positive outcome is obtained (scattered photon is detected). Otherwise, the remaining state is e associated with some phonon state. Particularly, | i e 0 comes from the original g 1 , which is the component related to the HOM effect. | i | i A blue sideband pulse is then applied to transfer the e 1 to g 0 while leaving the | i | i e 0 unchanged. g is then measured again. If the system is assumed to involve only | i | i 2 phonons, then the measurement scheme can stop at this point, otherwise subsequent blue sideband pulses are applied to clear the higher number Fock states associated to e . | i The above procedure is repeated for the stretching mode with another ion. If the bosonic simulator could realise the HOM effect, the probability of getting all negative results (no photons detected in all measurement), which denotes the presence of 1, 1 , would be very | i small. In other words, experiments will be terminated with high probability due to the measurement of g . | i If there is one more metastable state available in each ion and if there are only two phonons involved in the system, then post-selection is not required for the readout. The operation described in section 6.5.2.2 can be conducted on both ions, which each ion

2Before the simulation, the action of beam splitter should be calibrated, for example by considering its action on some single mode states.

173 Chapter A. Appendix

is responsible for one phonon mode. The operation transforms gg ( 02 20 )/√2 to | i | i−| i ( gr rg )/√2 0, 0 , while the coincident state gg 1, 1 will become ee 0, 0 . The | i−| i | i | i| i | i| i electronic states g , e , r of each ion are then measured. This process is equivalent to | i | i | i measuring the PVM 0 0 , 1 1 , 2 2 on the motional state. The occupation of the {| ih | | ih | | ih |} phonon states can then be deduced by the statistics of the measurement results. Phonon coincidence will be recognised if both ions are measured as e or r . In other words, if | i | i HOM eﬀect exists, either one of the ion must be revealed to be in g . | i

A.6 Moving multiple ions by harmonic trap

If N ions are trapped in a single harmonic potential with the trapping strength Ω2(t) and centre R(t), the Hamiltonian of the system is

N Pˆ2 1 N e2 Hˆ = i + Ω2(t)(ˆx R(t))2 + , (A.73) 2m 2 i − 4πǫ (ˆx xˆ ) i i ! i>j 0 i j X X − where an ion with a larger index should have a larger displacement, i.e. xˆ > xˆ for h ii h ji i > j. Let the total state of the ions be Ψ . The classical and quantum contribution of | i the state can be decoupled by deﬁning

Ψ Dˆ (x ,p ) ψ , (A.74) | i ≡ i i i | i i Y where Dˆ is the displacement operator of the ith ion; ψ contains only the quantum i | i ﬂuctuation; xi and pi are the classical position and momentum of the ith ions following the classical equation of motion

p e2 e2 x˙ = i ;p ˙ = Ω2(t)(x R)+ where r = x x . (A.75) i m i − i − 4πǫ r2 − 4πǫ r2 ij i − j i i>j 0 ij i

butions by the expansionx î = xi +ˆqi ; Pî = pi +ˆpi, whereq î andp î are the position and momentum operators of the quantum fluctuation only. The evolution of the quantum fluctuation follows

i∂ ψ = Hˆ ψ where (A.76) t| i Q| i pˆ2 1 e2 Hˆ i + Ω2(t)ˆq2 + (ˆq qˆ )2 . (A.77) Q ≈ 2m 2 i 4πǫ r3 (t) i − j i i i>j 0 ij X X

174 Chapter A. Appendix

Here we have made an expansion of the Coulomb potential and collect only the second order terms of position operators, i.e. the third or higher order terms ofq ˆ qˆ are i − j neglected. This approximation is valid if the quantum fluctuation of each ion is much narrower than the distance between ions, i.e. xˆ xˆ ∆ˆx2 for any i, j. In the |h ii−h ji| ≫ h i i Heisenberg picture, a quadratic Hamiltonian evolvesq î andp î only to combinations of qˆ’s andp ˆ’s [117], i.e.,

qˆ Uˆ † qˆ Uˆ = A (t)ˆq +B (t)ˆp ;p ˆ Uˆ † pˆ Uˆ = C (t)ˆq +D (t)ˆp , (A.78) i → Q i Q ij j ij j i → Q i Q ij j ij j j j X X where UˆQ is the corresponding evolution operator of HˆQ. Aij(t), Bij(t),Cij(t),Dij(t) are real functions of time determining the multi-mode squeezing of the ions’ motion that corresponds to the parametric excitation. Using the identities

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ i∂t(UQ† qˆiUQ)= UQ† [ˆqi, HQ]UQ and i∂t(UQ† qˆiUQ)= UQ† [ˆpi, HQ]UQ , (A.79)

we can get the equations for A, B, C, D :

miA˙ ij = Cij ; (A.80) 2 i N 2 i N ˙ 2 2e 1 1 2e 1 1 Cij = Ω 3 + 3 Aij + 3 Akj + 3 Akj ; − − 4πǫ0 rik rki 4πǫ0 rik rki h Xk=1 Xk=i i Xk=1 Xk=i miB˙ ij = Dij ; 2 i N 2 i N ˙ 2 2e 1 1 2e 1 1 Dij = Ω 3 + 3 Bij + 3 Bkj + 3 Bkj . − − 4πǫ0 rik rki 4πǫ0 rik rki h Xk=1 Xk=i i Xk=1 Xk=i Therefore under the quadratic approximation, the evolution operator, Uˆ, of Hˆ in Eq. (A.73) transforms the state, the position and momentum of the ions as

Ψ(t) = Dˆ (x ,p ) Uˆ ψ(0) . (A.81) | i i i i Q| i i ! Y A.6.1 Motional Excitation

Assume the initial and ﬁnal trapping frequency of the well are the same, i.e. Ω(0) =

Ω(T ) = Ω0, and the centre is moved from R(0) = 0 to R(T ). If the transportation is ideal, the state of the ions, Ψ(T ) , is assumed to be the same as the state at t = 0 except | i all the ions’ position are shifted by R(T ). In other words if we start from a ground

175 Chapter A. Appendix

state, we expect the final quantum fluctuation is in the ground state, the final classical

momentum is zero, and the ﬁnal classical position of the ions are xi(0)+ R(T ), satisfying

e2 e2 Ω2x (0) = , where r (0) = x (0) x (0) . (A.82) 0 i 4πǫ r2 (0) − 4πǫ r2 (0) ij i − j i>j 0 ij i

2 2 2 pˆi 1 2 2 e 2 pˆµ 1 2 2 Ωµ 1 + Ω0qˆi + 3 (ˆqi qˆj ) + Ωµqˆµ aˆµ† aˆµ+ , 2mi 2 4πǫ0rij(0) − ≡ 2mµ 2 ≡ √mµ 2 i i>j µ µ X X X X (A.83) which is the quadratic approximated HˆQ in Eq. (A.77) at t = 0. This quadratic Hamil- tonian can be diagonalised with respect to the eigenmodes. We hereafter denote ions with English alphabet indices (i, j, k, . . .) and eigenmodes with Greek alphabet indices

(µ, ν, . . .). Ωµ and mµ are the eﬀective trapping frequency and the eﬀective mass of the mode µ. The position and momentum operator of the modes and ions are related by linear combinations

qˆµ = Eµiqî ;p ˆµ = Fµipî , (A.84) i i X X where Eµi and Fµi are real constants determined by the eigenvectors of the diagonaliza- tion. The final excitation of the quantum fluctuation, represented by the state ψ˜(T ) , | i is defined by subtracting the expected classical displacement from the ions’ final state, Ψ(T ) , i.e., | i

ψ˜(T ) Dˆ † x (0) + R(T ), 0 Ψ(T ) | i ≡ i i | i i Y = Dˆ † x (0) + R(T ), 0 Dˆ x (T ),p (T ) Uˆ ψ(0) , (A.85) i i i i i Q| i i Y where the last identity comes from Eq. (A.81). Therefore the ﬁnal excitation of the mode µ is given by

n¯ ψ˜(T ) aˆ† aˆ ψ˜(T ) (A.86) µ ≡ h | µ µ| i 2 pˆµ 1 2 1 = ψ(0) Uˆ † Dˆ †(∆x ,p ) + √m Ω qˆ Dˆ (∆x ,p ) Uˆ ψ(0) , h | Q i i i 2√m Ω 2 µ µ µ − 2 i i i Q| i i µ µ i Y Y where ∆x x (T ) x (0) R(T ) is the position displacement mismatch. i ≡ i − i −

176 Chapter A. Appendix

We tackle the terms in Eq. (A.87) one by one. The displacement operator Dˆ i transforms the ions’ position and momentum operators as

ˆ ˆ ˆ ˆ Di†(∆xi,pi)ˆqiDi(∆xi,pi)=ˆqi + ∆xi ; Di†(∆xi,pi)ˆpiDi(∆xi,pi)=ˆpi + pi , (A.87)

and the modes’ operators as

Dˆ †(∆x ,p ) qˆ Dˆ (∆x ,p ) = E qˆ + E ∆x qˆ + x i i i µ i i i µi i µi i ≡ µ µ i i i i Y Y X X Dˆ †(∆x ,p ) pˆ Dˆ (∆x ,p ) = E pˆ + E p pˆ + p . (A.88) i i i µ i i i µi i µi i ≡ µ µ i i i i Y Y X X Consider the initial ground state satisﬁes qˆ = pˆ = 0, where we deﬁne for any h µi0 h µi0 operator Oˆ, Oˆ ψ(0) Oˆ ψ(0) . The motional excitation can be separated asn ¯ = h i≡h | | i µ coherent parametric n¯µ +¯nµ , where

2 coherent pµ 1 2 n¯µ = + √mµΩµxµ ; (A.89) 2√mµΩµ 2 pˆ2 parametric ˆ µ 1 2 ˆ 1 n¯µ = ψ(0) UQ† + √mµΩµqˆµ UQ ψ(0) . (A.90) h | 2√mµΩµ 2 | i − 2 The ﬁrst term is the coherent excitation, and the second term is the parametric excitation.

Combining Eq. (A.81) and (A.84), the mode operators transform under UˆQ as

1 1 Uˆ † qˆ Uˆ = E A E− qˆ + E B F − pˆ A′ qˆ + B′ pˆ Q µ Q µi ij jν ν µi ij jν ν ≡ µν ν µν ν ijν ν X X 1 1 Uˆ † pˆ Uˆ = F C E− qˆ + F D F − pˆ C′ qˆ + D′ pˆ , (A.91) Q µ Q µi ij jν ν µi ij jν ν ≡ µν ν µν ν ijν ν X X 1 1 where E− and F − are the inverse of E and F respectively. Putting into Eq. (A.89), the parametric excitation is given by

2 2 2 2 parametric (Cµν′ ) qˆν 0 +(Dµν′ ) pˆν 0 1 2 2 2 2 1 n¯ = h i h i + √mµΩµ (A′ ) qˆ 0 +(B′ ) pˆ 0 µ 2 m Ω 2 µν h νi µν h νi − 2 ν √ µ µ X (A.92) where we have used the fact that qˆ pˆ +p ˆ qˆ = 0 for any µ and ν. We note that h µ ν µ νi0 although the initial state is assumed to be the ground state, but our calculation can be easily extended to general mixed initial state. Our method has an advantage that the evolution of N ions can be obtained by numerically solving 2N 2 diﬀerential equation of real functions. In conventional method,

177 Chapter A. Appendix

the final excitation of the transportation has to be calculated by solving the coupled evolution of the states of N phononic modes, which the Hilbert space will be large if the modes can be significantly excited during the diabatic transportation. We note that the effect of anharmonic terms can be calculated perturbatively using the squeezing parameters we have calculated [18].

A.6.2 Example: Two ions in a trap

If two diﬀerent ions trapping in a single well is transported, the evolution is governed by the Hamiltonian

Pˆ2 Pˆ2 1 1 e2 Hˆ = 1 + 2 + Ω2(t)(ˆx R)2 + Ω2(t)(ˆx R)2 + . (A.93) 2m 2m 2 1 − 2 2 − 4πǫ (ˆx xˆ ) 1 2 0 2 − 1 The classical position and momentum can be extracted, and they satisfy the classical equation of motion

p e2 x˙ = 1 ;p ˙ = Ω2(t) x R(t) ; 1 m 1 − 1 − − 4πǫ (x x )2 1 0 2 − 1 p e2 x˙ = 2 ;p ˙ = Ω2(t) x R(t) + . (A.94) 2 m 2 − 2 − 4πǫ (x x )2 2 0 2 − 1

The quantum contribution satisﬁes Eq. (A.76), where the quadratic approximated HˆQ is

pˆ2 pˆ2 1 1 e2 Hˆ 1 + 2 + Ω2(t)ˆq2 + Ω2(t)ˆq2 + (ˆq qˆ )2 . (A.95) Q ≈ 2m 2m 2 1 2 2 4πǫ (x x )3 1 − 2 1 2 0 2 − 1

In the Heisenberg picture, the evolution operator of HˆQ transforms the position and momentum operator as

ˆ ˆ UQ† qˆ1UQ = A11qˆ1 + A12qˆ2 + B11pˆ1 + B12pˆ2 ; ˆ ˆ UQ† qˆ2UQ = A21qˆ1 + A22qˆ2 + B21pˆ1 + B22pˆ2 ; ˆ ˆ UQ† pˆ1UQ = C11qˆ1 + C12qˆ2 + D11pˆ1 + D12pˆ2 ; ˆ ˆ UQ† pˆ2UQ = C21qˆ1 + C22qˆ2 + D21pˆ1 + D22pˆ2 , (A.96)

178 Chapter A. Appendix

where the variation of the squeezing parameters satisfy

miA˙ ij = Cij ; miB˙ ij = Dij 2 2 ˙ 2 2e 2e C11 = Ω (t) 3 A11 + 3 A21 ; − − 4πǫ0r (t) 4πǫ0r (t) 2 2 ˙ 2 2e 2e D11 = Ω (t) 3 B11 + 3 B21 ; − − 4πǫ0r (t) 4πǫ0r (t) 2 2 ˙ 2 2e 2e C21 = Ω (t) 3 A21 + 3 A11 ; − − 4πǫ0r (t) 4πǫ0r (t) 2 2 ˙ 2 2e 2e D21 = Ω (t) 3 B21 + 3 B11 ; − − 4πǫ0r (t) 4πǫ0r (t) 2 2 ˙ 2 2e 2e C12 = Ω (t) 3 A12 + 3 A22 ; − − 4πǫ0r (t) 4πǫ0r (t) 2 2 ˙ 2 2e 2e D12 = Ω (t) 3 B12 + 3 B22 ; − − 4πǫ0r (t) 4πǫ0r (t) 2 2 ˙ 2 2e 2e C22 = Ω (t) 3 A22 + 3 A12 ; − − 4πǫ0r (t) 4πǫ0r (t) 2 2 ˙ 2 2e 2e D22 = Ω (t) 3 B22 + 3 B12 . (A.97) − − 4πǫ0r (t) 4πǫ0r (t)

At t = 0, the classical separation between the ions is

e2 e2 Ω2(0)x (0) = ; Ω2(0)x (0) = . (A.98) 1 −4πǫ (x (0) x (0))2 2 4πǫ (x (0) x (0))2 0 2 − 1 0 2 − 1

The vibrational modes α and β can be deﬁned by diagonalising the HˆQ in Eq. (A.95) at 2 pˆ2 t = 0, viz. pˆα + β + 1 Ω2 qˆ2 + 1 Ω2 qˆ2 , where the mode operators are deﬁned as: 2mα 2mβ 2 α α 2 β β

qˆα = Eα1qˆ1 + Eα2qˆ2 ;q ˆβ = Eβ1qˆ1 + Eβ2qˆ2 ;p ˆα = Fα1pˆ1 + Fα2pˆ2 ;p ˆβ = Fβ1pˆ1 + Eβ2pˆ2 , (A.99)

179 Chapter A. Appendix

where

2 2 m1 m2 + m1 m1m2 + m2 Eα1 = − − ; Eα2 = 1 ; pm2 2 2 m1 m2 m1 m1m2 + m2 Eβ1 = − − − ; Eβ2 = 1 ; pm2 2 2 2 2 Eα1 Eα2 Eβ1 Eβ2 mα = 1/ + ; mβ =1/ + ; m1 m2 m1 m2 2 2 2 mα(m1 + m2 m1 m1m2 + m2) 2 Ωα = − − Ω0 ; mp1m2 2 2 2 mβ(m1 + m2 + m1 m1m2 + m2) 2 Ωβ = − Ω0 ; mp1m2 mα mα Fα1 = Eα1 ; Fα2 = Eα2 ; m1 m2 mβ mβ Fβ1 = Eβ1 ; Fβ2 = Eβ2 . m1 m2

We note that both ions move in phase for mode α, while they move out of phase for mode β. Using the ions’ ﬁnal classical position and momentum at t = T , obtained by solving Eq. (A.94), and the relations to the modes’ classical position and momentum in Eq. (A.88) and (A.99), the coherent excitation of any mode can be calculated by Eq. (A.89). The parametric excitation of any mode can be calculated by Eq. (A.91), (A.92), (A.99) and the squeezing parameters A, B, C, D at t = T , obtained by solving Eq. (A.97) with the initial conditions

A11(0) = A22(0) = D11(0) = D22(0)=1 ; others =0 at t =0 . (A.100)

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