Quantum Engineering of Continuous Variable Quantum States

Gezielte Beeinﬂussung von Quantenzust¨anden mit kontinuierlichen Variablen

Der Technischen Fakultät der Universität Erlangen-Nürnberg zur Erlangung des Grades

DOKTOR-INGENIEUR

vorgelegt von Metin Sabuncu

Erlangen, 2010 Als Dissertation genehmigt von der Technischen Fakultät der Universität Erlangen-Nürnberg

Tag der Einreichung: 08-06-2009 Tag der Promotion: 29-10-2009 Dekan: Prof. Dr.-Ing. habil. Reinhard German Berichterstatter: Prof. Dr.-Ing. Bernhard Schmauss Prof. Dr. Gerd Leuchs Abstract

Quantum information with continuous variables is a field attracting increasing attention recently. In continuous variable quantum information one makes use of the continuous information encoded into the quadrature of a quantized light field instead of binary quantities such as the polarization state of a single photon. This brand new research area is witnessing exciting theoretical and experimental achievements such as teleportation, quantum computation and quantum error correction. The rapid development of the field is mainly due higher optical data rates and the avail- ability of simple and efficient manipulation tools in continuous-variable quantum information processing. We in this thesis extend the work in continuous variable quantum information processing and report on novel experiments on amplification, cloning, minimal disturbance and noise erasure protocols. The promising results we obtain in these pioneering experiments indicate that the future of continuous variable quantum information is bright and many advances can be foreseen.

Zusammenfassung

Quanteninformation mit kontinuierlichen Variablen ist ein Gebiet, das in der let- zten Zeiten grosses Interesse auf sich zieht. Auf dem Gebiet der Quanteninforma- tion mit kontinuierlichen Variablen nutzt man die kontinuierliche Information, die in der Quadratur eines quantisierten Licht Feldes kodiert ist anstelle von binaeren Parametern wie z.B. den Polarisationszustand eines einzelnen Photons. Dieses neue Forschungsgebiet erlebt derzeit interessante theoretische und experimentelle Fortschritte wie Teleportation, Quanten-Berechnung und Quanten-Fehlerkorrektur. Die rasche Entwicklung auf diesem Gebiet ist in erster Linie bedingt durch die Moeglichkeit hohe optische Datenraten zu Verwenden sowie durch die Verfueg- barkeit einfacher und eﬃzienter Werkzeuge zur Quanten-Informationsverarbeitung. In dieser Arbeit wird die gezielte Beeinﬂussung der Quantenzustaende mit kontinuierlichen Variablen untersucht und ueber neuartige Experimente zur Verstaerkung, zum Klonen, zur Messung mit minimalen Stoerungen und zur Loeschung des Rau- schens von Quantenzustaenden berichtet. Dazu wurde insbesondere die sog. feed forward Methode eingesetzt. Die vielversprechenden Ergebnisse der beschriebenen Experimente weisen auf eine bedeutende zukuenftige Entwicklung dieses Forschungs- gebiets hin.

Acknowledgments

This work is maybe a small step (not even a step actually) for mankind though has been a giant leap for me. I am grateful to many people who helped me in this hard and challenging path, without their help it would have been absolutely impossible to accomplish any of the work carried out in my PhD. I would like to express my sincere thanks to Professor Bernhard Schmauss, who has always helped me and given me continuous support throughout my PhD thesis. I owe a great debt of thanks to Professor Gerd Leuchs for giving me the opportunity to complete my thesis at the Institute of Optics, Information and Photonics, the brand new Max Planck Institute for the Science of Light. Special thanks go to Professor Ulrik L. Andersen, the head of the Quantum Infor- mation Processing Group for being an excellent group leader, and being very hospitable throughout my stay in Denmark. I would like to express my great gratitude to Ulrik for his excellent supervision of my work and for his enthusiasm for the subject. I am particularly grateful to Vincent Josse who introduced me to the cloning machine and taught me all the tricks with the optic and electronic components. Merci beaucoup Vincent, without your help I wouldn’t have been able to achieve any of the results I got in my PhD. I must not forget to thank Dr. Radim Filip who was always very friendly to me and helped me with the theoretical aspects of quantum optics and for inviting me to Olomouc. I am most grateful to my roommates Jessica, Josip and later on Alex who apart from the fruitful scientific discussions always created a friendly atmosphere in the office. I enjoyed discussing over a Kebab and tea in Norrebronx with you Alex! I owe thanks to Dr. Haim Abitan who always had something interesting to talk about during thetimewewereinthesameofficeinLyngby.Emir,Darko,StefanandMartinmade DTU a better place to be at. Leonid Krivitsky, Jiri Janousek and Martin Andersen, thanks for the fun times at the Physics Department at DTU. I thank Julien Niset and Ladislav Mista for providing excellent theoretical proposals which resulted in fruitful collaborations. I would also like to thank the other members of the QIP Group: Oliver Gloeckl, Christoph Marquardt, Carlos Wiechers, Mario A. Usuga, Christopher Wittmann for sharing their experience in Quantum Optics with me and always being friendly. Spasibbo to Denis Sych, I will never forget the conference in Calgary Mr. Einstein! I enjoyed working together in the lab with Mikael Lassen and playing pool in building 306 or Stefanshus. Pavel Marchenko, spassibo, thanks to you I am now a famous film star! Pasha I will never forget the time we spent together and the deep discussions we had. Xie xie to Ruifang Dong for always being friendly to me and allowing me to borrow any kind of equipment that I needed, even when sometimes I was taking the equipment more than 1000 km away! Alessandro Villar and Katiuscia Cassemiro: Obrigado, it is always a pleasure to spend timewithyouguys.FilouRaynal,JanSchaefer, Peter Panzer (one of the most helpful and positive persons I have met), Jochen Mueller, Dominique Elser, and Wenjia Zhong made the insititute and Erlangen a place I always would like to be at. Dominique and Wenjia always remember the DPG conference in Frankfurt ok? Many thanks to Vishal Vaibhav and Rashit Sharma who were always smiling and emiting positive energy. I owe many thanks to Eva who has always been willing to help with any kind of beau- rocratic problem I had and made life easier. I faced administrative hurdles when I started my thesis in Erlangen and thanks to secretaries Frau Dollinger and Frau Schwender, it was easy to overcome problems related to paperwork. Thanks to the other members of the Institute and our Project Coordinator Dr. Sabine Koenig who have always been friendly and supportive. Without Bruno and Adam from the electronics workshop I couldn’t have performed any of the measurements. To Seckin Sefi my fellow country- man: Yolun acik olsun dostum! Nebi Unlenen the computer specialist who brought my laptop back to life again deserves a big thank you. Cagdas followed me around from Copenhagen to Nuremberg, every now and then it was great to go out and have fun together. My Somalian students/friends in Copenhagen helped me figure out who I am in life: Waad mahadsan tahay Said, Ahmed, Edmond, Abdifatah, Gulled, Aydarus and Ismail. Erdem my special friend from Vesterbro, I learnt a lot from you, and am looking forward to our next encounter already! Millions of aciu go to Milda who diverted me from my daytodayproblemsandtookmymindawayfromwork,over! My true friends who were always with me on this journey Savaskan Bulek and Murat Isikhan, I can‘t thank you enough for your support: Kartallar sonsuz tesekkurler! There are surely many other people that I should have thanked and maybe forgot, thank you all! Lastly, and most importantly, I would like to thank my parents and brother for their years of love and devotion, I dedicate this work to them. Contents

1 Introduction 1

2 Quantum Optics Formalism 7 2.1TheQuantizedElectromagneticField...... 7 2.2QuantumStatesoftheField...... 9 2.2.1 Thephotonnumberstate...... 9 2.2.2 Thecoherentstate...... 10 2.2.3 Squeezed states ...... 13 2.3GaussianStates...... 13 2.4ExperimentalConsiderations...... 17 2.4.1 Experimentalgenerationofcoherentstates...... 18 2.4.2 LinearOpticsComponents...... 20 2.4.3 Stablemeasurements...... 24 2.5Detectionschemes...... 29 2.5.1 SinglePhotonDetection...... 29 2.5.2 DirectDetection...... 30 2.5.3 HomodyneDetection...... 31 2.5.4 HeterodyneDetection...... 32 2.6Summary...... 35

3 The Quantum Ampliﬁer 37 3.1TheBasics...... 37 3.2TheScheme...... 40 3.3TheExperimentalDemonstrationandResults...... 43 3.4Conclusion...... 48

4 The Quantum Cloner 49 4.1Introduction...... 49 4.2AuniversalCVcloner...... 51 4.3Towardsasuperiorcloningmachine...... 55 4.4Asuperiorcloningmachine...... 60 4.5Cloningofpartialquantuminformation...... 68 4.6Conclusion...... 77

5 Minimal Disturbance Measurements 79 5.1Introduction...... 79 5.2Gaussianminimaldisturbancemeasurements...... 82 5.3Quantumteleportationasaminimaldisturbancemeasurement.... 85 5.4Proofsofoptimality...... 88 5.4.1 ProofI...... 90 5.4.2 ProofII...... 94 5.5Linearopticsscheme...... 97 5.6Experiment...... 99 5.7Discussionandconclusions...... 104 5.8UnityGainMDMwithanapplication...... 105 5.9Conclusion...... 112

6 Noise Erasure 113 6.1Introduction...... 113 6.2Theory...... 115 6.3Experiment...... 119 6.4Results...... 121 6.5Conclusion...... 126

7 Conclusion and Outlook 127 List of Figures

2.1Coherentstate...... 15 2.2 Squeezed state ...... 16 2.3Thermalstate...... 17 2.4Modulators...... 21 2.5TheBeamSplitter...... 22 2.6Modelinglinearlosses...... 23 2.7Displacement...... 24 2.8Lockingscheme...... 25 2.9Lockscheme...... 27 2.10SinglePhotonDetector...... 30 2.11DirectDetection...... 31 2.12Homodynedetectionscheme...... 32 2.13Heterodynedetectionscheme...... 33 2.14Equivalentheterodynedetectionscheme...... 34 3.1Phaseinsensitiveamplification...... 38 3.2Theschemeforthequantumlimitedamplifier...... 40 3.3AmplifierperformanceforG=1.5...... 44 3.4Amplifierperformancefordifferentgainfactors...... 45 3.5ThePhaseConjugatingAmplifier...... 46 4.1Cloningisforbidden...... 50 4.2AGaussiancloner...... 52 4.3Theuniversalcloner...... 54 4.4Phaseconjugation...... 55 4.5Superiormeasurement...... 57 4.6ResultsofaSuperiorMeasurement...... 58 4.7Fidelity...... 61 4.8GeneralPhaseConjugateCloner...... 62 4.9 Experimental setup for cloning with phase conjugate inputs ..... 64 4.10 2 → 2phaseconjugatecloning...... 65 4.11 2 → 3phaseconjugatecloning...... 66 4.12ClonerwithPartialInformationProposal...... 70 4.13Experimentalsetupforpartialinformationcloning...... 71 4.14Cloningofpartialquantuminformationresults...... 72 4.15Phasecovariantcloning...... 76 5.1TheMDMconcept...... 81 5.2Optimaltrade-off...... 88 5.3MDMexperimentalscheme...... 97 5.4MDMresultsforoptimizedgain...... 102 5.5MDMresultsforfixedgain...... 103 5.6DifferentMDMstrategies...... 108 5.7Statefidelityvs.estimationfidelity...... 109 5.8AnMDMappication...... 111 6.1Thedecoherenceprocess...... 116 6.2Schematicoftheprotocol...... 120 6.3Deterministicapproachresults...... 121 6.4Optimizedgainresults...... 123 6.5 Probabilistic results for thex ˆ quadrature...... 124 6.6 Probabilistic results for thep ˆ quadrature...... 125 Chapter 1

Introduction

Moore’s law holds an important place for the current technology and basically states that the number of transistors that can be placed inexpensively on an integrated circuit doubles approximately every two years [97]. This trend has continued for almost half a century and is expected to continue until the year 2015. The transistors used in electronic devices are getting smaller and faster every day. As the electronic components are getting smaller each day soon we will be reaching a point where the components will be consisting of a few atoms resulting in quantum mechanical effects. This means that some day in the future quantum technology will be replacing the present technology. Quantum mechanical effects provide results which allow more effective computation and communication than classical ones [101]. It was Zoller and Cirac who proposed to use the ion trap technology of quantum optics which assisted the birth of quantum information processing [38]. In their proposal they suggested to utilize an external laser to process multiple quantum information stored in the internal electronic state of single trapped ions, in order to correlate their internal states using collective vibrational degrees of freedom. To- day ion trap quantum information processing is progressing quickly and with the present technology demonstration of entanglement with more than eight qubits was performed in the groups of Wineland at NIST in Colorado and Blatt in Innsbruck. Recent advances in quantum technology has enabled the field of quantum information to flourish. It is now possible to encode information in single quantum particles and then manipulate them successfully and effectively. The thought experiments which gained popularity in the early stages of the birth of quantum mechanics known as ”Gedanken Experiments” have also been realized in many quantum labs 2 Introduction around the world. Quantum information is basically categorized in two subgroups; the discrete variable (corresponding to qubits such as for example the spin of an electron or polarization of a photon) or the continuous variable groups. The first realizations of quantum information protocols involved qubits because they were thought as the quantum counterpart of the standard classical bits. Many protocols could be realized making use of the qubits some examples being: teleportation, quantum key distribution, entanglement generation to name a few. The qubits are very attractive in theory, because they enable an easy framework to work with, though on the experimental side they have some major drawbacks. Experimentally it is extremely challenging to generate and manipulate or effectively control single photons. Gen- erating resources for quantum information tasks such as entanglement production is also very demanding in the qubit regime. In this regime the generation of entangled states is unconditional. These experimental drawbacks make discrete variable quantum information less realizable. There is a promising new alternative way to encode quantum information in the quadratures of the electromagnetic field. This new area relies on canonical observables with continuous spectra hence it’s known as the continuous variable (CV) regime [19]. The experimental demonstration of unconditional quantum teleportation in 1998 triggered an explosion of research in the continuous variable regime of quantum information [60]. The CV approach holds many advantages on the experimental side. One of the major advantages is the effective state characterization method known as homodyne detection which can be very efficient and fast. Also the fact that entangled state generation and control is efficient and easy makes CV quantum information processing highly attractive on the practical side. In addition to the advantages on the experimental side the theoretical side also holds a major advantage which is the simplicity to deal with CV states belonging to Gaussian states. Since the first demonstration of CV teleportation there has been much advance- ment in several directions in CV Quantum Information. Very recently a sequence of two quantum teleportations of optical coherent states was accomplished by combining two high-fidelity continuous variables teleporters [140]. For the teleportation protocol entanglement is a crucial ingredient. In the field of entanglement for continuous variables many theoretical and experimental studies has been conducted [125, 46, 80, 128, 15, 58]. Recently E. Shchukin and W. Vogel developed a the- 3 oretical framework which gave the conditions for multipartite continuous-variable entanglement [123]. They developed methods which could certify entanglement for a manifold of multimode quantum states, by making use of a numerical program which provides a systematic test of the hierarchy of conditions. Hyllus and Eisert also contributed to the theoretical advances of the field by analyzing all tests for continuous-variable entanglement that arise from linear combinations of the variances of canonical coordinates [74]. This theoretical analysis is relevant from an experimental point of view since variances of canonical coordinates are commonly used to verify entanglement in experiments. Of course the developments were not only limited to theory and many experiments in entanglement generation have been performed. In 2005 Laurat et al. designed an experiment based on a type-II OPO (optical parametric oscillator) enabling a clear proof of many theoretical predictions and providing a better understanding of the general properties of two-mode Gaussian states and entanglement resource manipulation [86]. It was shown by Bradley et el., that when operated above threshold an optical parametric oscillator based on three concurrent χ2 nonlinearities can produce bright output beams of macroscopic intensities which exhibit strong tripartite continuous-variable entanglement [18]. Villar et al. considered again an optical parametric oscillator operating above threshold and showed the first measurement of squeezed-state entanglement between the generated twin beams [132]. The squeezing was measured in the sum of phase quadratures of the twin beams. This scheme has potential applications such as the transfer of quantum information between different parts of the electromagnetic spectrum since it enables the measurement of phase anticorrelations between fields of different frequencies. Gloeckl et al. studied the verification of continuous variable entanglement of intense light beams generated by the Kerr effect in optical fibres by making use of large interferometers to access the two conjugate variables and show the presence of entanglement [67]. One of the most promising fields of CV quantum information is Quantum Key Distribution. By encoding Gaussian modulations onto the amplitude and phase quadratures of light beams quantum key distribution protocol using continuous- wave coherent light was recently achieved [84]. The protocol used information encoded into conjugate quadratures respectively with postselection achieving high secret rates. Analysis of an optimal eavesdropping attack against this protocol was performed for the case of individual Gaussian attacks. A comparison of this ap- 4 Introduction proach was made with the original BB84 protocol. Recently a new form of quantum computation known as cluster state quantum computation was developed. Menicucci et al. created the continuous variable quantum state cluster state computation scheme which involved the usage of squeezed light and linear optics combined with homodyne detection [95]. They showed that while performing Gaussian operations one can exhibit cluster based error reduction. Later Loock et al. developed an efficient way which required the use of non-Gaussian measurements to generate linear optical Gaussian cluster states that can be used as resources for universal quantum computation [129]. Purification of coherent states was achieved by Andersen et al. making use of two imperfect copies of coherent states which had travelled through independent noisy channels [1]. This method due to its simplicity was said that could become an alternative to the more complex protocols of entanglement distillation and quantum error correction. Distillation of squeezing from continuous variable non-Gaussian states was achieved using linear optics and conditional homodyne detection [58, 72] which was recently followed by the distillation of entanglement [43, 70]. Lassen et al. introduced the methods of modulation, detection and spatial quantum correlations for CV multimode quantum information which are decisive for practical multimode optical quantum information systems [85]. Experimental demonstration of quantum memory for light was achieved by Julsgaard and colleagues [108]. The team successfully managed to record an externally provided quantum state of light onto the atomic quantum memory with a fidelity of 70% and a lifetime of 4 ms, which is higher than that for the classical recording. Andersen et al. proposed a Gaussian quantum cloner for coherent states which relied entirely on simple linear optical elements and homodyne detection [3]. Ex- perimentally, clones were generated with a cloning fidelity which almost touched the optimal value of 2/3. Using a novel encoding strategy by making use of phase conjugated inputs a superior cloner for coherent states was also built recently [111]. Later on Koike et al. demonstrated for the first time quantum telecloning of optical coherent states, where coherent states were symmetrically teleconed from one sender to two receivers in an unconditional manner with fidelities exceeding the classical limits [79]. Yoshikawa and coworkers demonstrated a sum gate for continuous variable universal quantum computation, which can be seen as the analogue of a C-NOT gate for qubits [141]. 5

In this thesis we will introduce the basics of continuous variable quantum optics tools and report on the most recent demonstration of CV QI protocols. Chapter 2 will give the backround which is necessary to understand the quantum information protocols demonstrated in this thesis. Chapter 3 is on the demonstration of the quantum limited ampliﬁer, which is crucial for the construction of the quantum cloner which is described in Chapter 4. Minimal disturbance measurements of coherent states are analysed in Chapter 5. Chapter 6 is devoted to Environmental Assisted Noise Erasure. Finally the thesis ends with Chapter 7 with a conclusion and outlook. 6 Introduction Chapter 2

Quantum Optics Formalism

We will start introducing the basic concepts of Quantum Optics in this chapter. We will present a discussion on the quantization of the electromagnetic field and show its basic properties. Then the quantum states of the light field will be explained. Common experimental components used and methods in order to measure the quantum states will be investigated. Various detection methods for the light field in quantum optics will also be discussed.

2.1 The Quantized Electromagnetic Field

The quantization of the electromagnetic field in free space starts with a classical description established upon the well known Maxwell equations. Maxwell in his work Dynamical Theory of the Electromagnetic Field created a unified model of electromagnetism by showing that light also had wave nature similar to electricity and magnetism and resulted from an electromagnetic field. The Maxwell Equations build a link between the electric field E and magnetic field B. In vacuum there are no charges (current or electric) and the equations take the following form:

∇·E =0, (2.1)

∇·B =0, (2.2) ∂B ∇×E + =0, (2.3) ∂t ∂E ∇×B − μ00 =0. (2.4) ∂t 8 Quantum Optics Formalism

In these equations μ0 and 0 are the magnetic permeability and electric permittivity of free space respectively and are related to c the free space speed of light via the relation: √ μ00 = c. (2.5)

It can be shown that spatial and time dependent electric ﬁeld E(r, t)isthesolution of the wave equation: ∂2 ∇2 − E =0. (2.6) ∂t2 The solution of this equation consists of the positive and negative components giving the total electric ﬁeld as follows:

hω k 1/2 −iωkt − ∗ ∗ iωkt E(r, t)= ( ) [αkuk(r)e αkuk(r)e ] (2.7) 20 k here is the reduced planck constant, k is the index of the mode, ωk is the angular frequency of the mode k, α and α∗ are dimensionless complex Fourier amplitudes and u is a mode function which includes also the polarization mode. We perform the quantization of the electromagnetic ﬁeld by replacing α and α∗ by the annihilation † and creation operators bya ˆk anda ˆk, respectively which are mutually adjoint operators. We use the fact that photons are bosons to choose appropriate commutations † for the operatorsa ˆk anda ˆk the following bosonic commutator relations:

† † † [âk, aˆk]=[âk, aˆk]=0, [âk, aˆk]=δkk. (2.8)

The quantized electric ﬁeld then becomes: hω † k 1/2 −iωkt − ∗ ∗ iωkt E(r, t)= ( ) [ˆakuk(r)e aˆk uk(r)e ] (2.9) 20

The Hamiltonian for the electromagnetic ﬁeld is: † 1 H = ω (ˆa aˆ + ). (2.10) k k k 2 k 2.2 Quantum States of the Field 9

2.2 Quantum States of the Field

2.2.1 The photon number state

1 | The eigenvalues of the Hamiltonian H are ωk(nk + 2 ). nk are known as number or Fock states. The eigenstates |n of the number operator Nˆ =ˆa†aˆ with eigenvalue n,ˆa†aˆ |n = n |n are called the photon number states1. The vacuum state of the

field mode is defined bya ˆk |0 = 0 Using the Hamiltonian expression: ˆ † H = ωk aˆkaˆ +1/2 , (2.11) k one can show that the energy of the vacuum state is nonzero. The vacuum state † has zero mean photonsn ˆ =âkaˆ = 0 such that the Hamiltonian becomes:

1 Hˆ = ω . (2.12) 2 k

The photon annihilation and creation operators,a ˆ anda ˆ†, act on photon number states |n (or also known as Fock states) as follows: √ aˆ† |n = n +1|n +1 (2.13) and √ aˆ |n = n |n − 1 . (2.14)

The effect ofa ˆ† (creation operator), anda ˆ (annihilation) on the state are raising and lowering the photon number respectively as their name implies. One can generate any desired number state by exciting sufficient number of photons successively from the vacuum as: (â†)n |n = 1 |0 (2.15) 2 n (n!) The Fock states are orthogonal and complete:

n|m = δmn, (2.16)

1A photon number state(or Fock state) has a definite number of photons in the mode (this means that each mode has a definite average power) but completely random phase. Fock states have zero average electric field because of the random phase. 10 Quantum Optics Formalism n|n =1. (2.17) n Photon number states are nice to deal with theoretically though they are extremely challenging to generate experimentally. Only photon numbers with 0,1 or 2 photons can be produced efficiently. States with higher photon numbers cannot yet be generated easily, effectively and most importantly in a controlled fashion.

2.2.2 The coherent state

The coherent states are a much more convenient way to represent optical fields defined as being the eigenstate of the annihilation operatora ˆ. The coherent state is the best suitable model to correctly describe the radiation coming from an ideal laser [122]. Also a monochromatic dipole with an oscillating current having a frequency ω will emit a coherent light field. If |α is representing a coherent state, we may write the following : aˆ |α = α |α . (2.18)

Here α is the resulting eigenvalue of the operator acting on the coherent state. Co- herent states can also be expressed in terms of number states. Taking the conjugate of equation 2.13 we obtain: √ n| aˆ = n +1n +1| . (2.19)

Now operating this on a coherent state |α and using equation 2.18 we get, √ n| aˆ |α = n +1n +1|α = α n|α , (2.20) or replacing n by n − 1 in equation and rearranging the terms,

α αn n|α = √ n − 1|α = √ 0|α . (2.21) n n!

Using the following identity: |nn| =I, (2.22) n we can write: αn |nn|α = 0|α √ |n. (2.23) n n n! 2.2 Quantum States of the Field 11

If we normalise this eigenket we obtain

− 1 | |2 0|α =e 2 α . (2.24)

Using Eq. 2.23 and Eq. 2.24 we can write a coherent state in terms of photon number states as follows: n − 1 | |2 α |α =e 2 α √ |n . (2.25) n n! Since the photon number in a coherent state is not certain, an important question we can raise is what the probability of having a certain number of photons in the state is. The probability of ﬁnding ’n’ photons in a coherent state can be calculated using the following formula:

2n 2 α P (n)=|n|α|2 =e−|α| . (2.26) n!

Equation 2.26 indicates that the photon number distribution for a coherent state is poissonian. One can alternatively derive a formalism treating the coherent state as a displaced vacuum state. We can iteratively derive any photon number state we want from vacuum by applying the photon creation operator on vacuum:

(ˆa†) n |n = √ |0. (2.27) n!

One can also write the coherent state as:

† n − 1 | |2 (αaˆ ) ( ˆ†− 1 | |2) |α =e 2 α √ |0 =eαa 2 α |0. (2.28) n n!

It can be shown that [77]:

( ˆ†− 1 | |2) ˆ†− ∗ˆ e αa 2 α |0 = eαa α a|0 (2.29)

If we deﬁne the unitary displacement operator D(α)suchas:

† ∗ D(α)=eαaˆ −α aˆ (2.30) then we obtain |α = D(α)|0. (2.31) 12 Quantum Optics Formalism

This shows that the coherent state is obtained by displacing vacuum. The displacement operator when acted on the annihilation and creation operators displaces them by an amount α and α∗ respectively:

D†(α)ˆaD(α)=ˆa + α, (2.32)

D†(α)ˆa†D(α)=ˆa† + α∗. (2.33)

The photon creation and annihilation operators do not correspond to any physi- cally measurable quantity and it is therefore more convenient to define the measurable operators: xˆ =â +â†, (2.34) and pˆ = i(â† − aˆ), (2.35) wherex ˆ andp ˆ are the ‘in phase’ and ‘out of phase’ quadratures of the electromagnetic field generally referred to as the amplitude and phase quadratures respectively.. These operators do not commute [ˆx, pˆ]=2i and the normalised variances obey the Heisenberg Uncertainty Principle:

Δ2xˆΔ2pˆ ≥ 1, (2.36) where the variances for the quadratures are deﬁned as:

2 Δ2xˆ = xˆ2 −xˆ , (2.37) and 2 Δ2pˆ = pˆ2 −pˆ . (2.38)

For coherent states we have the Heisenberg relation saturated and therefore refer to coherent states as the minimum uncertainty states:

Δ2xˆΔ2pˆ =1. (2.39)

The coherent states are nonorthogonal and the inner product of two coherent states α and β is equal to: 2 β|α = e−|(α−β) |/2. (2.40) 2.3 Gaussian States 13

In principle the above expression will always be nonzero for two coherent states, a consequence of the nonorthogonality of the states. However coherent states will become approximately orthogonal as the distance between α and β becomes large. Mathematically speaking the distance between two coherent states can be represented by x and the bigger the distance is the more orthogonal the states will be to each other: 2 lim e−|(x) |/2 =0. (2.41) x→∞ Coherent states are referred to as the closest quantum mechanical states to classical states.

2.2.3 Squeezed states

The squeezed state of light is a nonclassical state one for which the variance of the quadratures is no longer equal to each other. One of the variances falls below the limit of a coherent state. For example, the amplitude squeezed state would have the property that the variance of the amplitude quadrature would be less than 1. The Heisenberg Uncertainty Principle is not violated since the variance in the conjugate phase quadrature will be bigger than unity making the product of the variances still greater or equal to unity. In this thesis we will report on experiments performed using coherent states.

2.3 Gaussian States

States which have a Gaussian phase space representation such as characteristic function etc. are known as Gaussian states. Gaussian states are an exception of continuous variable states where knowing their first two moments is sufficient to fully describe them. They hold an especially important place for experimental quantum optics. These states are easy to generate and manipulate in the lab by using linear optics components such as beam splitters, mirrors and phase shifters. Some examples of Gaussian states are the quantum states we already introduced such as the vacuum state, coherent states, squeezed states and thermal states. All the states which have been experimented with in this thesis belong to the class of Gaussian states. A Gaussian states’ quadrature’s first moment is described by the displace- 14 Quantum Optics Formalism ment vector d , and the second moments of the quadrature by the covariance matrix:

γij = Tr[ρ(ˆri − di), (ˆrj − dj)]. (2.42)

The curly brackets refer to the anticommutator. For two operators Aˆ and Bˆ the anticommutator is deﬁned as:

{A,ˆ Bˆ}≡AˆBˆ + BˆA.ˆ (2.43)

ρ is the density matrix, r is a vector where the quadratures are grouped as:

T T rˆ =(ˆr1, ..., rˆ2N ) =(ˆx1, pˆ1, xˆ2, pˆ2, ..., xˆN , pˆN ) . (2.44)

N represents the number of modes of the electromagnetic ﬁeld which can correspond to diﬀerent frequencies, spatial modes or even polarization. The Wigner function is given by the Gaussian:

1 T −1 W (r)= √ e−(r−d) γ (r−d). (2.45) π2N detγ

The matrix equivalence of the Heisenberg Uncertainty Principle is as follows:

γ + iΩ ≥ 0. (2.46)

Ω is the symplectic matrix deﬁned as: N 01 Ω= (2.47) − i=1 10

The conjugate operators obey the following canonical commutation relations:

[ˆrj, rˆk]=2iΩjk. (2.48)

A vacuum state is located around origin in phase space since it has no displacement, actually it is a coherent state with no excitation (no displacement=d(0,0)) 2.3 Gaussian States 15 and its covariance matrix is the identity matrix. γ = I : 10 γ = (2.49) 01

The coherent state is obtained by displacing a vacuum state in phase space, hence it has the same covariance matrix in addition to displacements in the two conjugate quadrature directionsx ˆ andp ˆ. The displacement can be represented by d =(dx,dp) and the covariance matrix is the same as the vacuum state γ = I: The squeezed state

d p {

{ X dx

Figure 2.1: The coherent state in phase space. The state is displaced from the origin depending on its displacement vector d =(dx,dp). The coherent state has an uncertainty circle equal to the vacuum state ie the standard quantum limit. may be situated around vacuum without having an excitation or have an excitation and be displaced from the origin in phase space. In any case its covariance matrix willhavethesameformas: e−2s 0 γ = (2.50) 0 e2s s is the squeezing parameter. If the state is a vacuum squeezed state then the state will possess a null displacement vector and a bright squeezed state will possess a nonzero displacement vector such as: d =(dx,dp). The thermal state is centered around the origin in phase space, and has a covariance matrix which has equal diagonal members which are greater than unity. The entry of the matrix depends 16 Quantum Optics Formalism

d p {

^ { X dx

Figure 2.2: The vacuum and bright squeezed states in phase space. The vacuum squeezed state is situated around the origin. And the bright squeezed state is displaced from the origin depending on its displacement vector d. The squeezed state has an uncertainty ellipse where in the squeezing direction the uncertainty is less than the standard quantum limit and in the conjugate direction larger, as the Heisen- berg Uncertainty Principle implies.

on the average number of photons (n) in the state as:

V =2n + 1 (2.51) V 0 γ = (2.52) 0 V The thermal state’s displacement vector is null and V is greater than unity. The two mode squeezed state is a necessary resource for quantum information protocols such as teleportation. The covariance matrix is given as: cosh rI sinh rσ z (2.53) sinh rσz cosh rI

I is the 2x2 identity matrix: 10 I = (2.54) 01 2.4 Experimental Considerations 17

Figure 2.3: The thermal state in phase space. The state is centered around origin because it’s displacement vector is null, and it has a higher uncertainty circle than vacuum which is shown as the smaller circle.

And σz is the 2x2 Hermitian Pauli matrix deﬁned as: 10 σz = (2.55) 0 −1

This state may have a nonzero displacement vector. The two mode squeezed vacuum states are a vital ingredient of the most important CV quantum information protocol: teleportation. A typical way of generating two mode squeezed vacuum states experimentally is to ﬁrst generate two squeezed states having orthogonal squeezing angles and interfere them on a 50/50 beam splitter.

2.4 Experimental Considerations

In order to analyse quantum optics experiments we will make use of the linearization of the operator fields. The photon creation and annihilation operators will be defined as having a constant mean value (represented by α) and fluctuations (represented by δ) around this value: aˆ = α + δa,ˆ (2.56)

aˆ† = α∗ + δaˆ†. (2.57) 18 Quantum Optics Formalism

In general all operators will be considered to have a mean value and a second order term for the fluctuations. Any higher order terms arising from interactions will be neglected in the linearization procedure. An experimentally practical way to treat the quantum states is in the sideband picture. This picture becomes useful when we are using modulators in the light path in order to generate coherent states by exciting sidebands. This way information can be encoded at a frequency sideband with respect to the carrier frequency. This is typically achieved by modulating the phase or amplitude of the light at some frequency Ω which is much less than the optical frequency ω of the light. Such modulation may be visualised by the addition of modulation sidebands to the optical carrier at frequencies Ω relative to the carrier frequency ω [17]. 1 aˆsb(Ω) = √ [âω−Ω +âω+Ω]. (2.58) 2

2.4.1 Experimental generation of coherent states

The displacement operator on vacuum in order to generate coherent states D(α)|0 = |α in the lab is realized by using electrooptic modulators (EOMs) in the beam path. To see how this works we will analyse the amplitude and phase modulators. When an oscillating electric field at frequency Ω is applied to an EOM, photons from the carrier field are removed and amplitude or phase sidebands at ±Ωare excited. Nonlinear crystals whose refractive index changes with the strength of the local electric field are placed in EOMs. A typical crystal which is used can be Lithium niobate. Light travels slower in a medium with high refractive indices, meaning that the phase of the light leaving the crystal is related to the refractive index it passes through. By changing the electric field in the crystal one can directly control/modulate the phase of the light passing through it. A phase modulated beam can easily be converted into an ampitude modulated light by using interferometric techniques such as a Mach-Zehnder interferometer or operating the device as a polarization modulator followed by a polarizing beam splitter. The effect of a phase modulator operating at a modulation frequency Ω on the fields annihilation operator having a carrier frequency ω is as follows [17]:

(iωt+iβ cos Ωt) aˆpm =ˆa(t)e . (2.59)

β is the modulation strength (note that no modulation corresponds to β =0and 2.4 Experimental Considerations 19 full modulation means β = 1) and one can make use of the Taylor series expansion and considering weak modulation (β 1) the equation 2.59 becomes

iωt aˆpm ≈ aˆ(t)e (1 + iβ cos Ωt), (2.60) which can be rewritten by making use of trigonometric identities iβ iβ aˆ(t) eiωt + ei(ω+Ω)t + ei(ω−Ω)t . (2.61) 2 2

Actually in the Heisenberg picture of quantum mechanics an electromagnetic oscillation at frequency ω is representable by the ﬁeld annihilation operatora ˆ(ω), which together with its conjugate, the ﬁeld creation operatora ˆ†(ω), satisfy the boson commutation relations. The frequency domain operators are related through the Fourier transform operation to the time domain operators:

aˆ(ω)=F(ˆa(t)), (2.62)

aˆ†(ω)=F(ˆa†(t)), (2.63)

xˆ(ω)=F(x(ˆt)), (2.64)

pˆ(ω)=F(p(ˆt)). (2.65)

In order to switch to the frequency domain we will then simply take the Fourier transform of this expression:

β β aˆ (ω)=ˆa(ω)+i aˆ(ω − Ω) + i aˆ(ω +Ω). (2.66) pm 2 2

This expression for β<<1 can be approximated to:

β β aˆ (ω) ≈ aˆ(ω)+i δ(Ω)α(t)+i δ(−Ω)α(t). (2.67) pm 2 2

This expression means that the phase modulation (PM) has created 2 sidebands proportional to the modulation strength β separated by the modulation frequency ω from the carrier frequency Ω. Similarly, amplitude modulation (AM) can be 20 Quantum Optics Formalism expressed as:

β aˆ =(1+β cos Ωt)â(t)eiωt =â(t)eiωt +â(t) ei(ω+Ωt) + ei(ω−Ωt) . (2.68) am 2

Applying a Fourier transformation in the frequency domain we get:

β β aˆ (ω)=ˆa(ω)+ aˆ(ω − Ω) + aˆ(ω +Ω), (2.69) am 2 2 and when β 1:

β β aˆ (ω) ≈ aˆ(ω)+ δ(Ω)α(t)+ δ(−Ω)α(t). (2.70) am 2 2

In the sideband picture we can write the sideband operators corresponding to phase and amplitude modulation as follows:

iβ aˆPM,sb(Ω) =a ˆsb(Ω) + √ α, (2.71) 2

β aˆAM,sb(Ω) =a ˆsb(Ω) + √ α. (2.72) 2 These two expressions indicate that a phase modulation with weak excitation (β 1) will perform a displacement in the phase quadrature (ˆp), whereas an amplitude modulation will make a displacement in the π/2 rotated conjugate quadrature which is the amplitude quadrature (ˆx). Hence by placing an amplitude and phase modulator in a laser beam one can perform the displacement operator D(α) and generate coherent states.

2.4.2 Linear Optics Components

There are some crucial linear optical operations which are performed on an optical beam such as phase shifting, displacement and beam splitting. These operations preserve the Gaussian statistics of the inputs and they are building blocks of the quantum information protocols performed in this thesis. Beam splitter: The most important and common component is the beam splitter, which actually is a semitransparent mirror transmitting a portion of the input signal and reﬂecting the rest. The input output relations for a beam splitter are given as follows (refer 2.4 Experimental Considerations 21

Figure 2.4: Two concatenated electrooptical phase and amplitude modulators were placed in the laser beam to generate the coherent beams. The coaxial cables attached to the modulators carry radio frequency signals in order to impose the modulations. This way the displacement operation was performed experimentally.

to Fig.2.5): √ √ aˆ1,out = Taˆ1,in + Raˆ2,in, (2.73) √ √ aˆ2,out = Raˆ1,in − T aˆ2,in, (2.74) √ √ rˆ1,out = T rˆ1,in + Rrˆ2,in, (2.75) √ √ rˆ2,out = − Rrˆ1,in + T rˆ2,in. (2.76)

Here the vectorr ˆ represents the quadrature operators such thatr ˆ =(ˆx, pˆ)T .The beam splitter is frequently used in quantum optics experiments. It is also very important in the theoretical analysis of quantum optical experiments as the beam splitter is used to model losses and nonideal optical components. If a quantum state propagates through a lossy environment or for example is detected by a nonunity efficient detector this can be modeled by a linear beam splitter as in Fig. 2.6. The efficiency of the process is equal to the transmission (η) of the beam splitter. There will be vacuum noise entering through the empty port of the beam splitter and inflicting the quantum state. Using the beam splitters input output relations we can write the output field operator as: 22 Quantum Optics Formalism

2,in

1,out 1,in BS

2, out

Figure 2.5: The linear beam splitter is a unitary transformer with two input and two output modes. It is frequently used in quantum optics experiments, and in theory it is used to model linear losses.

√ aôut = ηaîn + 1 − ηaˆv (2.77) we can write the same relation for the amplitude quadrature as: √ xôut = ηxîn + 1 − ηxˆv (2.78)

Now we can write the relation for the variance between the input and output fields as: 2 2 2 Δ xôut = ηΔ xîn +(1− η)Δ xˆv. (2.79)

Since vacuum is a coherent state having unity variance we can write:

2 2 Δ xˆout = ηΔ xˆin +(1− η) (2.80)

The formula given by Equation. 2.80 is used when one wants to infer back a nonideal quantum measurement having efficiency given by η. Representing the amplitude quadrature of the inferred field and measured field byx înf andx ˆmes respectively one can write the following extremely useful relation to switch from the measured into the inferred variances: 2 2 Δ xînf =(Δxˆmes + η − 1)/η (2.81) 2.4 Experimental Considerations 23

v,in

1,out 1,in h

Figure 2.6: The linear beam splitter used to model linear losses. The beam splitter transmission (η) is set to be the same as the eﬃciency of the measurement.

The above equation was frequently used in order to infer variances, which was often also referred to as correcting for the losses in the system. Phase shift: A phase shift arises directly from the wave nature of the electric ﬁeld. In the lab a phase shift is applied on an optical beam by changing its path length with respect to other beams. This is achieved typically by using piezoelectric transducers attached on mirrors in order to move them very accurately. The input-output relations of a phase shifter for the quadratures read:

xôut =cosθxîn +sinθpîn, (2.82)

pôut = − sin θxîn +cosθpîn. (2.83)

The resultant eﬀect of the phase shifter is just to rotate the quadrature by an angle θ without eﬀecting its magnitude. Displacement: The theoretical Displacement Operator acting on a coherent state is represented by the operator Dˆ(α). In a quantum optics experiment the way to realize this is to make use of a high transmitting beam splitter (T ≈ 1,R ≈ 0) and interfere a bright beam on the beam splitter with the coherent state |α we want to displace as shown in Fig 2.7. Let’s do the analysis for a single quadrature;x ˆ quadrature, using symmetry one can do the same for the conjugate√ p ˆ quadrature. Choosing the auxiliary beams quadrature magnitude xaux = xdisp/ 1 − T and making use of the 24 Quantum Optics Formalism

Xaux

Beam X Xin T~1 Splitter disp

Figure 2.7: The displacement operation performed experimentally. beam splitter relations we can write the output quadrature as: √ √ xôut = T xîn + 1 − Txâux (2.84) √ xôut = Txîn +ˆxdisp (2.85) in terms of the quadrature variances. As T approaches 1 (perfect transmission) the displacement process becomes purer and the output quadrature is exactly the input quadrature displaced. Of course perfect displacement operation is not possible since it would require infinite energy in the auxiliary beam. The net effect of the displacement operator on any Gaussian state is therefore to translate its mean value. Experimentally we implemented a 99/1 beam splitter to accomplish high purity displacement operations.

2.4.3 Stable measurements

Another necessary point in optics experiments is to measure signals stably in order to characterize quantum states accurately. This is achieved by using electronic feedback techniques. Typically one ﬁrst will generate error signals which will then be used to control some component in the experiment which will stabilize the measurements. This controlled component is generally the position of a mirror, which will move in a way to keep optical path lengths uninﬂuenced by environmental disturbances. First we will introduce a method which enables a stable relative phase of π/2 between two optical signals. The scheme is shown in Fig. 2.8. We write the currents generated at the detectors:

∗ 1 2 1 2 i (t) ∝ α α3 = α + α + α1α2 cos θ, (2.86) DetectorA 3 2 1 2 2 2.4 Experimental Considerations 25

LO q=p/2 |b>

Detector B

|a> Beam Splitter

Detector A - Error Signal

Figure 2.8: Scheme to lock a stable relative phase of π/2 between two beams, can be used in the phase quadrature measurement or also the heterodyne measurement strategy employing the auxiliary beam.

∗ 1 2 1 2 i (t) ∝ α α4 = α + α − α1α2 cos θ. (2.87) DetectorB 4 2 1 2 2 Taking the diﬀerence of the photocurrents generated at the detectors results in an error signal E;

E ∝ α1α2 cos θ. (2.88) using the trigonometric relations:

π cos θ =sin( − θ)andsin(θ) ≈ θ, (2.89) 2 for small θ ≈ 0 we obtain: π E ≈ α1α2( − θ). (2.90) 2 Clearly the error signal will vanish around θ = π/2. By feeding this signal to a piezo mounted mirror in one of the beam paths we can ﬁx the relative phase between the two beams to a stable level of π/2. This phase relation is necessary when performing heterodyne measurement of a coherent state, or when measuring the phase quadrature of a coherent state. If we want to measure the amplitude 26 Quantum Optics Formalism quadrature of a coherent state then the relative phase between the local oscillator and the signal should be 0. We will use a special trick in order to obtain an error signal which will enable locking the phase between two beams to 0. We will phase modulate one of the beams at a sideband frequency such that now:

αPM1 = α1 (1 + iβ sin(ωt)) , (2.91)

1 2 1 2 i (t) ∝ α1(t) + α2(t) + α1(t)α2(t)cos(θ)+βα1(t)α2(t)sin(θ)sin(ωt). DetectorB 2 2 (2.92) Again the ﬂuctuating terms were neglected. Now we can mix this photocurrent down with an electronic local oscillator to extract the error signal from the oscillating term. We choose an electronic local oscillator such as sin(ωt). The mixed down signal will read: 1 2 1 2 iDetectorB(t) × sin(ωt) ∝ α1(t) + α2(t) + α1(t)α2(t)cos(θ) sin(ωt) 2 2 (2.93) 2 +α1(t)α2(t)sin(θ)sin (ωt).

Now making use of the trigonometric identity:

1 sin2(x)= (1 − cos(2x)), (2.94) 2 we obtain 1 2 1 2 iDetectorB(t) × sin(ωt) ∝ α1(t) + α2(t) + α1(t)α2(t)cos(θ) sin(ωt) 2 2 (2.95) +α1(t)α2(t)sin(θ)(1 − cos(2ωt)).

Sending this signal through an analog low pass ﬁlter, which rejects the high frequency components we ﬁnally get the error signal;

E ∝ βα1(t)α2(t)sin(θ), (2.96)

For θ ≈ 0 we can once again use the trigonometric identity sin(θ) ≈ θ, The error signal can now be approximated to be proportional to:

E ∝ βα1(t)α2(t)θ. (2.97) 2.4 Experimental Considerations 27

The error signal will be zero when θ is zero. So by feeding this signal to a piezo

LO q=0 PM |b>

Detector B

|a> Beam Splitter

Signal Generator X Lowpass Filter

Figure 2.9: Scheme to lock a stable relative phase of 0 between two beams, can be used in the amplitude quadrature measurement. mounted mirror in one of the beam paths we can ﬁx the relative phase between the two beams to a stable level of 0. When performing theoretical analysis of quantum optical devices and hence experiments we use the following version of the operators:

a(ˆt)=α(t)+δa(ˆt) (2.98)

where α is the (complex) expectation value and δa(ˆt) is the operator for the ﬂuctuations. The variances of operators arise from the ﬂuctuating terms as: V (ˆx)=Δ2xˆ = δxˆ2 , (2.99)

V (ˆp)=Δ2pˆ = δpˆ2 . (2.100)

Now assume we put a photodetector in front of this ﬁeld and measure its properties. A photodetector will give an output proportional to the photon number n as: nˆ =ˆa(t)†aˆ(t), (2.101) 28 Quantum Optics Formalism performing the linearization by neglecting higher order terms one obtains:

nˆ = α(t)2 + α(t)δxˆ(t), (2.102) so that the photodetectors output directly is probing the time domain amplitude quadrature operatorx ˆ of the ﬁeld. One usually is interested in the frequency domain operators high frequency components in quantum optics because of the presence of low frequency noise sources destroying the signal quality. There are many noise sources in the lab arising for example from mechanical vibrations, noise, the lasers internal relaxation oscillation and thermal noise. One can easily switch the analysis to the frequency domain by a Fourier transformation. We will avert the noise sources by looking at a speciﬁc sideband signal by just a Fourier transform yielding the frequency domain amplitude quadrature operator of the detector photocurrent:

i(ω) ∝ α(t)ˆxΩ+ω(ω) (2.103) now when we analyse the variance of the expression we obtain

V (n(t)) = α(t)2V (ˆx) (2.104)

Spectrum analysers measures the power spectrum S(ω) of a signal enabling us to gather information about different frequency components of the signal, in our case the optical signal. Their working principle is similar to the Heterodyne Detection System in electronics. There is an internal local oscillator, whose frequency is ad- justable, according to which frequency component of the signal we are interested in. The fluctuations of the field in a certain bandwidth(measurement window) about the optical carrier frequency can be resolved. The spectral power density S(ω)for a stochastic variable δx(t) having zero mean is calculated via the Fourier transform of the autocorrelation function G(τ) [134]: S(ω)= G(τ)exp−iωτ dτ. (2.105) T

In an experiment, the value of the power spectrum at some frequency ω is measured over some bandwidth B about the frequency ω. This bandwidth is the resolution bandwidth (RBW) of the spectrum analyser. To accurately measure the spectrum, B is chosen so that the value of the spectral density is approximately 2.5 Detection schemes 29 constant over the range of B. This ensures that the details of the spectrum are not blurred in the measurement. Assuming that the spectrum doesn’t change in this window we deﬁne the bandwidth limited spectrum SB(ω) as [25]: 2 SB(ω)=S(ω) × B = |δxˆ(ω)| (2.106)

One can deﬁne a normalized spectrum V (ω):

S (ω) V (ω)= B = |δxˆ(ω)|2 , (2.107) B where we have normalised out the measurement bandwidth. Note that V (ω)isthe frequency domain equivalent to the variance, Δ2, introduced in the time domain. For a vacuum(coherent) light beam we will have V (ω) = 1. The phase and amplitude quadratures power spectra at each frequency will still comply with the Heisenberg Uncertainty Principle such that:

Vx(ω)Vp(ω) ≥ 1. (2.108)

2.5 Detection schemes

2.5.1 Single Photon Detection

When one is dealing with single photon quantum experiments single photon detectors are vital. These devices are capable of detecting signals as low as single photons. A reversed biased p-n junction diode is excited when a photon hits it and an avalanche eﬀect causes an electrical current. The measurement is naturally in the photon number mode or Fock basis. The detection will in general be a discrimination between no photon, and one or more photons, usually referred to as no click or click. The experiments in this thesis were performed in a regime where coherent states composed of many photons were detected, and single photon detectors were not used. However there is recent interest in combining CV with DV quantum information in order to generate special states. For example Grangiers group in Paris, France included single photon detectors in CV experiments in order to generate Schroedinger kitten states [106]. 30 Quantum Optics Formalism

n(t) . . 010010 ......

Figure 2.10: The single photon detector. In general if photons hit on the detector surface a signal will be generated, and if no photons hit then there is no output.

2.5.2 Direct Detection

By placing a photodiode in the beam, we can measure it directly. A photocurrent which is linearly proportional to the number of photons hitting the surface of the photodiode will be produced. The number of photons can be calculated from the creation and annihilation operators as follows:

i(t) ∝ nˆ(t)=ˆa†a.ˆ (2.109)

We can once again make use of the operatorsa ˆ = α + δaˆ and ﬁnd the current generated at the detector to be:

i ∝ n ≈|α|2 + αδxˆ(t). (2.110)

The ﬁrst term is proportional to the optical intensity whereas the second term containing the ﬂuctuations corresponds to the amplitude quadrature. One can write the frequency domain equivalence of 2.110 as:

n(ω) ≈|α|2δ(0) + αδxˆ(ω) (2.111)

The first term is a constant term referred to as the DC term, and the second term is the oscillating term generally called the radio frequency (RF) component. So by direct detection we can measure one of the quadratures, namely the amplitude (ˆx) quadrature. The direct detection scheme cannot therefore be used to characterize quantum states, or measure different quadratures. One requires a phase dependent measurement scheme in order to switch between different measurement bases, and that can be achieved by homodyne detectors. 2.5 Detection schemes 31

n(t) i(t) ......

Figure 2.11: The direct detection scheme. The current generated by the detector will be proportional to the number of photons arriving at the detector.

2.5.3 Homodyne Detection

Homodyne detection is one of the most important light measurement techniques used in quantum optics. A bright local oscillator (|β) beam which is much more intense than the input (|α) is used to probe and analyse the input reference beam. a = α+δa is the input signal beams ﬁeld operator and b = β+δb is the local oscillator (LO) beam, where θ is the relative phase between the input signal and local oscillator beam. For homodyne detection we have the requirement that the signal must be dim compared to the LO so that |α|2 |β|2. The homodyne detection scheme is shown in Fig. 2.12. We can write the ﬁelds in the paths 1 and 2 by making use of the beam splitter transformations and the linearized operator formalism:

∝ † i1,2 nˆ1,2 =ˆa1,2aˆ1,2. (2.112)

The ﬁelds general quadrature amplitude is:

θ −iθ † −iθ δxa = δae + δa e . (2.113)

The photocurrents generated at the detectors is proportional to the photon numbers. In an actual experiment one subtracts the two currents from the detectors which corresponds to amplifying the electronic signal using a low noise ampliﬁer.

( + π ) θ 2 i− ∝ αβ cos θ + βδxa (2.114)

This enables us to measure directly thex ˆ amplitude quadrature of the input mode. Making a phase shift of the local oscillator one can measure any quadrature combination one desires. Actually in order to switch the measurement base from the amplitude into the phase quadrature one will need to introduce a π/2 phase shift to the LO. Any 32 Quantum Optics Formalism

LO q |b> 1 Detector B

|a> Beam Splitter 2

Detector A homodyne - signal

Figure 2.12: Homodyne detection scheme. quadrature combination in between can be easily measured by tuning the phase of the LO. The eﬃciency of the homodyne measurement is calculated through the mode matching between the LO and the input signal. One ﬁrst calculates the visibility(V) of the interference of the two beams:

I − I V = max min . (2.115) Imax + Imin

Here Imax and Imin represent the maximum and minimum power of the interference fringes respectively. The homodyne detection eﬃciency(ηhom)isthencalculatedas 2 ηhom = V .

2.5.4 Heterodyne Detection

We saw that a homodyne detector enables us to measure any quadrature of the input signal. When we want to measure not only one quadrature but two conjugate quadratures simultaneously we hit a problem. It is impossible to measure two noncommuting variables precisely simultaneously, due to the Heisenberg Uncertainty Principle (HUP). If we are interested in measuring two conjugate variables (ˆx and 2.5 Detection schemes 33 pˆ) of a coherent state the best strategy is to split the state by a 50/50 beam splitter and perform homodyne measurements on each quadrature separately as shown in Fig. 2.13. This measurement process will add vacuum noise to the process at

LO q=0

Beam Splitter

|a> 50:50

q=90

x - - LO

Figure 2.13: Heterodyne detection scheme.

the beam splitter in accordance with HUP, preventing accurate knowledge about the conjugate variables. An equivalent scheme to measure simultaneously conjugate quadratures of coherent states is to mix the state with an equally bright auxiliary beam and measure the output beams of the beam splitter using photodetectors as in Fig 2.14. When we write the operators as a = α + δa for the input signal beam and b = β + δb for the auxiliary beam (AUX) and make use of the beam splitter transformation we can derive the photon number ﬂuctuations at the two detectors as [87]: 1 † i i † δnˆB = (δa + δa )α +( δa − δa )β 2 2 2 (2.116) i i 1 +(− δb + δb†)α + (δb + δb†)β. 2 2 2 34 Quantum Optics Formalism

AUX q=p/2

Detector B

|a> Beam Splitter

Detector A - phase + amplitude Figure 2.14: The equivalent alternative heterodyne detection scheme works for bright beams.

1 † i i † δnˆA = (δa + δa )α +(− δa + δa )β 2 2 2 (2.117) i i 1 +( δb − δb†)α + (δb + δb†)β. 2 2 2 We reformulate the photocurrents ﬂuctuations at the detectors in terms of the quadrature operators variances: 2 2 2 2 2 Δ iB = Δ xˆ1 +Δ xˆ2 +Δ pˆ1 − Δ pˆ2 /2 (2.118) 2 2 2 2 2 Δ iA = Δ xˆ1 +Δ xˆ2 − Δ pˆ1 +Δ pˆ2 /2 (2.119)

The terms having subscript 1 and 2 represent the input signal and the auxiliary beam respectively. When the two signals have equal brightness (ie. α = β)then taking the diﬀerence of the photocurrents will give us the phase quadrature (p) of the input signal with a shot noise contribution from the auxiliary beam. The summation of the photocurrents will correspond to the amplitude quadrature (x) of the input signal again with additional shot noise from the auxiliary beam. Thus using this approach we will be able to measure two quadratures of the ﬁeld simultaneously. 2.6 Summary 35

This approach is thus equivalent to ﬁrst splitting the input signal into two beams and performing homodyning of the conjugate quadratures on the two beams. This simpliﬁed approach was taken when heterodyning bright beams.

2.6 Summary

In this chapter the quantized electromagnetic field was introduced and then the quantum states of the light field were explained. Experimental components used and methods in order to measure the quantum states were investigated. The chapter finished with the discussion of various detection methods for the light field in quantum optics. 36 Quantum Optics Formalism Chapter 3

The Quantum Ampliﬁer

In this chapter we will explain quantum amplification which is a fundamental information protocol. First we will give the basics of quantum limited amplification, and then present our scheme on the continuous variable quantum limited amplifier. We will also give the experimental results on the quantum noise limited amplification of coherent states.

3.1 The Basics

A phase insensitive quantum amplifier is a device which isotropically amplifies the input quantum state in phase space, independent of any relative phase apparent in the state. Basic laws of quantum mechanics prevent the possibility of phase insensitive amplification of a quantum state without adding noise to it. Caves [30] developed fundamental theorems which specify the ultimate limits imposed by quantum mechanics on amplifiers. This intrinsic quantum noise, closely linked with measurement theory and the no-cloning theorem, gives rise to many inextricable restrictions on the manipulations of quantum states. For example, microscopic quantum objects cannot be amplified perfectly into macroscopic objects for detailed inspection [66]: for phase-insensitive amplification nonclassical features of quantum states, such as squeezing or oscillations in phase space, will be gradually washed out, and the signal- to-noise ratio of an information carrying quantum state will be reduced during the course of amplification. Despite these limitations, the universal phase-insensitive amplifier is, however, rich of applications, in particular, in optical communication. Amplifiers operating at the quantum noise limit are of particular importance for 38 The Quantum Amplifier

The process adds noise to the state (grey shaded region) P

Amplification X

Figure 3.1: Phase insensitive or isotropic amplification of a coherent state. The quantum amplification process is noisy and this is depicted by the gray shaded region in the amplified state. This noise will have the effect of reducing the signal to noise ratio of the signal.

quantum communication where information is encoded in fragile quantum states, thus extremely vulnerable to noise. Numerous apparatuses accomplish, in principle, ideal phase-insensitive amplification as, for instance, solid state laser amplifiers [124], parametric down-converters [139], and schemes based on four wave mixing processes [142]. However, to date phase-insensitive amplification at the quantum limit has been only partially demonstrated [105, 88]: a number of difficulties are indeed involved in practice, especially for low gain applications. These difficulties mainly lie in the fact that the amplified field has to be efficiently coupled, mediated by a nonlinearity, to a pump field. Following a recent trend in quantum information science, where nonlinear media are efficiently replaced by linear optics [47], we show in this chapter that universal phase-insensitive amplification can also be achieved using only linear optics and homodyne detection. The simplicity and the robustness of this original scheme enable us to achieve near quantum noise limited amplification of coherent states, even in the low gain regime.

Let us first briefly summarize the basic formalism describing a phase-insensitive amplifier [30]. Because of the symmetry of such an amplifier, it can be described by the following input-output transformation: √ aôut = Gaîn + Nˆ (3.1) 3.1 The Basics 39

wherea în,out represent the input (output) annihilation bosonic operators, G is the power gain, and Nˆ the operator associated with noise addition. Even for an ideal amplifier, this noise term must be added to ensure the preservation of the commutation relations † [âi, aî ] = 1 (3.2) and must satisfy [N,ˆ Nˆ †]=1− G. (3.3)

Thus it can be divided in two parts: a fundamental quantum part given by √ ˆ − † Nq = G 1ˆaint, (3.4)

† wherea ˆint is associated with the unavoidable ﬂuctuations of the internal bosonic mode, and a scalar classical part denoted Ncl. Therefore, a phase-insensitive ampli-

fier working at the quantum noise limit (i.e., Ncl =0)obeystherelation: √ √ − † aôut = Gaîn + G 1âint (3.5)

† The intrinsic quantum noise, described by aint, can be traced back to different physical processes. For instance, spontaneous emission is unavoidably introduced in laser amplification, whereas, in parametric amplifiers and four wave mixers, vacuum fluctuations of the idler mode are added to the output signal. In Raman amplifiers and Brillouin amplifiers zero-point fluctuations of, respectively, lattice vibrational modes (optical phonon) and acoustic phonon modes cause the noise [92]. The efficiency of a phase-insensitive amplifier is typically quantified by the noise figure (NF), which is defined by:

NF = SNRout/SNRin. (3.6)

Here SNRin,out is the signal-to-noise ratio of the input (output) field. For coherent state amplification, the noise figure(NF) then reads

G NF = 2 , (3.7) 2G − 1+Δ Ncl

2 which is maximized for quantum noise limit operation corresponding to Δ Ncl =0.

However, provided that the spurious technical noise Ncl is constant or has only a 40 The Quantum Ampliﬁer weak dependence on G, the noise ﬁgure still approaches the quantum limit of 3 dB in the high gain regime: G 1 lim = . (3.8) G→∞ 2G − 1 2

Figure 3.2: The scheme for the quantum limited ampliﬁer.

The situation is different at low gains, as technical noise or internal losses become devastating for quantum noise limited performance [105]. To date, these effects have hitherto prevented the full demonstration of quantum noise limited phase-insensitive amplification in the low gain regime, which is the domain of interest in the context of quantum information science.

3.2 The Scheme

We now show that the amplifier transformation can be realized using only linear optics, homodyne detection, and feedforward, rendering the complex coupling between a strong pump and the signal inside a nonlinear crystal superfluous. Our scheme is illustrated schematically inside the dashed box in Fig. 3.2, and runs as follows. The input signal, represented bya în, is impinging on a beam splitter with transmission T and reflectivity R, and hence transformed into √ √ † − aîn = T aîn Rvˆ1, (3.9) where the annihilation operatorv ˆ1 represents the vacuum mode entering the dark port of the beam splitter. Conjugate quadrature amplitudes, e.g., the amplitude 3.2 The Scheme 41 xˆ and the phase quadraturep ˆ, are simultaneously measured on the reflected part by dividing it on a 50/50 beam splitter and subsequently performing homodyne measurements on the two output beams. The measured quadratures are

1 √ √ xˆm = √ ( Rxˆin + T xˆv1 +ˆxv2), (3.10) 2 and 1 √ √ pˆm = √ ( Rpˆin + T pˆv1 − pˆv2). (3.11) 2

Herex ˆv(1,2) andp ˆv(1,2) denote the quadratures of the uncorrelated vacuum modes entering at the two beam splitters. These projective measurements (with outcomes represented by their eigenstates and corresponding eigenvalues xm and pm) are then used to control a unitary displacement operation on the remaining system [14]. The feedforward loop can be described without any measurement by considering the unitary operator,by choosing an electronic gain of: g = 2R/T, (3.12) we arrive at 1 1 † aˆ = aˆ + − 1 vˆ (3.13) out T in T 2

1 Setting G = T , we exactly recover the transformation for an ideal phase-insensitive ampliﬁer, where the ampliﬁcation factor is controlled by the beam splitting ratio.

Note that the noise that enters from the vacuum fluctuations on v1 is automatically canceled out in the output via the feedforward. We also note that a related scheme, where noise entering a beam splitter was canceled via feedforward, was used in Ref. [82] to build a noiseless amplifier (with NF = 1 for the amplitude quadrature). However, in contrast to our proposal, this scheme, apart from being phase sensitive, was not fully operating at the fundamental quantum limit. A truly quantum noise limited phase sensitive amplifier based on the same principles was recently proposed [53], but it requires a nonclassical resource, namely, a squeezed vacuum state. Interestingly, the fundamental amplifier noise, represented byv ˆ2, arises from the vacuum fluctuations that enter through the dark port of the 50/50 beam splitter used forx ˆ andp ˆ quadrature measurements. The amplifier noise is therefore directly related to the noise penalty associated with simultaneous measurement of conjugate 42 The Quantum Amplifier quadratures. The close link between amplification and measurement theory is thus particularly emphasized by our scheme. The amplifier proposed in this chapter is phase insensitive, and, in principle, amplifies any input state at the quantum limit. Our analysis assumed a perfect lossless feedforward loop. Of course there are some imperfections in the system such as: Nonunity quantum efficiency of photodiodes, nonideal displacer, electronic noise in the detectors and noise in the electronic feedforward loop. The electronic noise in the system was ultra low, much lower than the standard quantum noise limit, and the components used in the feedforward loop such as the amplifiers were low noise amplifiers having excellent performances which allowed a clean feedforward loop. The displacer was implemented with a 99/1 beam splitter and a bright (classical) auxiliary beam matched more than 99% on to the signal beam which meant that the displacer could be treated ideally. We can however include the fact that we have nonunity quantum efficiency in the photodiodes. In order to do this we assume there is beam splitter with transmission η infront of the feedforward detectors. The tranmission η of this beam splitter is equal to the efficiency of the feedforward detectors. Now we will have the following input-output relations for our setup [4]: 1 1 † aˆ = aˆ + − 1 vˆ , (3.14) out T in T 2

R √ T √ aôut = g η + T aîn + g η − R aˆv1 2 2 (3.15) +g η/2âv2 + g (1 − η)âv3. aˆv3 arises from the nonunity quantum efficiency of the photodiodes.√ Now notice that in order to have the gain infront ofa în to be equal to G as for the standard amplifier we obtain the electronical gain to be: 2R g = . (3.16) ηT

Now we will get the input-output relation for the system as: √ √ 2(1 − η)(G − 1) aˆ = Gaˆ + G − 1ˆa 2 + aˆ 3. (3.17) out in v η v 3.3 The Experimental Demonstration and Results 43

For gains close to unity and low losses in the feedforward loop the ineﬃciencies become negligible in the ampliﬁer.

3.3 The Experimental Demonstration and Results

In the following, we demonstrate experimentally the amplification of a particular quantum state, namely, the coherent state. The experimental setup is shown in Fig. 3.2. The laser source was a monolithic continuous wave Nd:YAG laser at 1064 nm. A small part of the laser beam was tapped off to serve as an input signal to the amplifier and the rest was used as local oscillator beams. Since the output from a laser is not a perfect coherent state due to low frequency technical noise, we define our coherent state to reside at a certain sideband frequency which we chose to be 14.3 MHz, within a bandwidth of 100 kHz. At this frequency the laser was found to be shot noise limited, and by applying modulations at 14.3 MHz (by independently controlling an amplitude and a phase modulator), the sidebands are excited and thus serve as a perfect coherent state. The coherent state is then directed to the amplifier where it is divided by a beam splitter; the reflected part is measured and the transmitted part is displaced according to the measurement outcomes. Simultaneous measurements of the amplitude and phase quadrature are performed by combining the reflected signal beam with an auxiliary beam, v2,withaπ/2 phase shift, and balanced intensities. The sum and difference of the photocurrents generated by two high quantum efficiency photodiodes then provide the simultaneous measurement of amplitude and phase quadrature (this strategy is not shown explicitly in the figure). The outcomes are sent to electronic amplifiers with appropriate gains and then finally fed into independent modulators. The modulators are placed in an auxiliary beam which is coupled to the signal beam via an asymmetric beam splitter which transmits 99% of the signal and reflects 1% of the auxiliary beam, thus leading to a negligible small noise addition. After displacement, the amplified signal is directed into a homodyne measurement system for verification. The performance of the system is characterized by measuring the spectral noise properties of the signal before and after amplification. Since the quadrature statistics of the involved fields 2 are Gaussian, measurements of the first [xôut] and second moments [Δ xôut]oftwo conjugate observables, such as the amplitude and phase quadrature, suffice to fully characterize the states. Both quadratures are measured at the sideband frequency 44 The Quantum Amplifier using standard homodyne detection techniques. To ensure consistent comparison between the input and output signal, these measurements are realized by the same homodyne detector.

Figure 3.3: Power spectra showing the operation of the amplifier for conjugate quadratures. The beam splitter was set to 1:2 enabling an optical gain of G=1.5. The mean value of the field is amplified by 1.8 dB and the noise is consequently increased by 3.2 dB, which is very close to the ideal noise level of 3 dB above the shot noise. The noise figure is NF=0.7. The resolution bandwidth is 10 kHz and the video bandwidth is 30 Hz.

An example of a specific amplifier run is shown in Fig 3.3. Here we set the beam splitting ratio to 1:2 in order to reach an optical gain of 1.5. The spectral densities of the amplitude and phase are shown over a 100 kHz frequency span for the input signal and the amplified output signal. Considering the whole span as a part of the quantum state, the heights of the peaks correspond to the coherent mean values whereas the noise floor can be regarded as the actual noise in the state. Therefore, the amplification factor, which is roughly the difference between the input and output peaks, as well as the added noise, which is the difference between the shot noise limit and the noise floor, can be easily estimated. It is evident from the plots that additional noise has been added to the signal as a result of the amplification process. To evaluate the noise figure, we estimated accurately the gain and the added noise at 14.3 MHz. This was realized in a zero span measurement 3.3 The Experimental Demonstration and Results 45

Figure 3.4: The noise figure, NF, as a function of the gain, G. The black ( gray) dots represent the experimental data for the amplitude (phase) quadrature. The solid line represents the quantum noise limit, whereas the predicted noise figure for our device with imperfect detectors is shown by the dashed line. For comparison, the dotted line corresponds to an amplifier with two vacuum units of extra technical noise. Errors mainly stem from the inaccuracy in determining the quantum efficiency of the photodiodes. over 2 seconds by subsequently switching on and off the modulation. Moreover, to avoid erroneously underestimation of the noise power, all the measurement have been corrected for losses occurring in the homodyne detection. The total efficiency, including mode matching and photodiode quantum efficiency, has been carefully estimated to ηhd =0.83. In Fig. 3.4, we report the noise figure of our amplifier for a whole range of gains (corresponding to different transmission coefficients and optimized electronic gains). By comparing the experimental results with the ideal ones [indicated by the solid line], we clearly see that the amplifier operates close to the fundamental limit even for low amplification factors. The small deviation to the ideal amplifier performance is due to imperfections in the in-line homodyne detector and feedforward electronics. These limiting factors were partly overcome by paying special attention to the construction and alignment of the system. The efficiency of the homodyne detector amounted to 93% (95% photodiode efficiency and 99% mode 46 The Quantum Amplifier overlap efficiency) and the electronic noise of the detectors was overcome by using newly designed ultrasensitive detectors. Taking these imperfections into account, the theoretically expected noise figure is given by:

ηG NF = (3.18) 2G − 2+η where η is the overall efficiency of the detector system. This expression, which tends to the limit NFl =0.46 corresponding to -3.3 dB for high gains, is shown in Fig. 4.4 by the dashed line: it is in good agreement with the experimental results, demonstrating that basically no additional technical noise is invading the amplifying process. The challenge of realizing such a quantum noise limited amplifier in the low gain regime is highlighted by considering the behavior of an amplifier that exhibits 2 only two vacuum units of extra technical noise[Δ Ncl = 2] ]. As mentioned earlier and clearly illustrated by in Fig. 3.4, even such a small amount of background noise, which is quite common for amplifiers, leads to a strong deviation from the quantum noise limit at low gains. To complete the investigation of the system, we finally

Figure 3.5: Proposed scheme for a phase conjugating amplifier with the nonlinearity put off-line. The displacements, indicated by D, can be performed as shown in Fig. 4.2. focus here on the existence of the phase conjugate amplified output state. This state, mirrored about the amplitude quadrature axis in phase space with respect 3.3 The Experimental Demonstration and Results 47 to the input state, must be present in all amplifiers to ensure unitarity [30]. In down-converters, this mode is the idler output and thus easily accessible for further processing. However, it is not always directly accessible: e.g., in a laser amplifier this mode is scattered into vibrational modes of the atoms. But where is the phase conjugate output in our scheme? It turns out that it can be extracted by the introduction of an entangled ancilla, as shown in Fig. 3.5. The amplifier settings are not changed [The fundamental equation still holds], but now, in addition, one half of the entangled ancilla is injected into the empty port of the variable beam splitter and the other half is displaced according to the classical measurement outcomes. The amplification noise is not affected by this since the noise due to the entangled ancilla is canceled out, as mentioned earlier. We are interested in extracting the amplified phase conjugate output of the coherent state. Phase conjugation corresponds to keeping the sign of the amplitude quadraturex ˆ the same and flipping the sign in front of the phase quadraturep ˆ. The continuous variable perfectly entangled state has the following quadrature relations:

xˆent1 +ˆxent2 =0, (3.19) and

pˆent1 − pˆent2 =0. (3.20)

The electronic gains of the classical currents for the amplitude and phase quadra- 2 − 2 ture before displacement are chosen as λx = T and λp = T . When we write the idler output fields for the two quadratures for the case of perfect entanglement we obtain: R 1 xˆ = xˆ + xˆ 2, (3.21) ph T in T v and R 1 pˆ = − pˆ + xˆ 2. (3.22) ph T in T v Clearly the output field is phase conjugated since the sign in front of the amplitude quadrature is the same and the sign in front of the phase quadrature is flipped. We can also right the following input-output relation for the additional output mode: R † 1 aˆ = aˆ + aˆ 2 (3.23) ph T in T v 48 The Quantum Amplifier

The input output relations of our quantum ampliﬁer mimic the ones of a down- converter and allow us to interpret ain and aout as being the input and output signal modes and v2 and aph as being the input and output idler modes.

3.4 Conclusion

In conclusion, we have proposed and experimentally demonstrated that a phase- insensitive amplifier can be constructed from simple linear optical components, homodyne detectors, and feedforward. Quantum noise limited performance was exhib- ited, in particular, at low gains, only limited by inefficiencies of the in-line detection process. The fact that our amplifier exhibits nearly quantum noise limited performance at low gains suggests that it can be used to amplify nonclassical states (such as squeezed states and Schroedinger cat states) and still maintain some of their nonclassical features such as squeezing and interference in phase space. Furthermore, we believe that such an amplifier can find usage in the field of quantum communication, where optimal amplification of information carrying quantum states is needed partly to compensate for downstream losses of a quantum channel and partly to enable an arbitrary quantum cloning function [21]. One particular cloning transformation of a coherent state was recently demonstrated with a fixed gain amplifier [3]. Chapter 4

The Quantum Cloner

This chapter is about the concept of quantum cloning which actually is forbidden by the basic law of quantum mechanics known as the no-cloning theorem. Thanks to this no go theorem superluminal communication via quantum entanglement is prevented. Although it is forbidden it is still of practical importance to devise machines which will make the best quality clones. We will start the chapter with the concept of cloning and introduce the no cloning theorem. We will then present a universal continuous variable cloning machine as the fundamental Gaussian cloner. We will explain how a superior cloning machine can be realized by combining phase conjugation with a clever measurement strategy. At the end of the chapter we will consider cloning coherent states which are picked from certain subsets of states in phase space.

4.1 Introduction

Quantum cloning is not allowed as a result of the linearity of quantum mechanics. This situation is widely referred to as the no-cloning theorem of quantum mechanics as just mentioned. The no-cloning theorem is easy to understand intuitively. In order to make a clone (copy) of a quantum state we ﬁrst need to make a measurement of the state. The Heisenberg Uncertainty principle restricts us from extracting simultaneously precise information of the conjugate (noncommuting) variables, and since we don’t have exact information about the state we want to duplicate we will fail to copy it. One is then tempted to think that the no cloning theorem being a no go theorem will provide us nothing useful, indeed due to this theorem 50 The Quantum Cloner classical error correction techniques cannot be applied on quantum states. However secure quantum communication relies heavily on the no cloning theorem. The no- cloning theorem rules out the possibility of creating copies of a transmitted quantum cryptographic key, making an eavesdroppers task impossible. Lets now prove the no-cloning theorem [107]. We have an unknown quantum state v we send into the perfect cloner which will at its output give two identical copies of v; the machine has an input Φ which ensures unitarity as depicted in Figure 4.1. F | > Cloner |v> |v> U |v> Figure 4.1: A perfect cloning machine is not allowed by quantum mechanics.

Φ ⊗ v → U(Φ ⊗ v)=Φ ⊗ v ⊗ v, (4.1)

Φ ⊗ w → U(Φ ⊗ w)=Φ ⊗ w ⊗ w, (4.2) taking the inner product of these two equations will give

Φ, Φv, w = Φ, Φv, wv, w (4.3) since 0 < |v, w| < 1 (4.4) and Φ, Φ =1. (4.5)

From what we would need Φ, Φv, w = 1, however this is not possible, which means that it is not possible to have a machine which can copy unknown quantum states perfectly. This intriguing result of quantum mechanics stating that an unknown quantum state cannot be exactly cloned was shown in [138, 41]. This fact stems from the inherent linearity of quantum mechanics, and is one of the most discussed features in recent years since it enables diﬀerent quantum information 4.2 A universal CV cloner 51 protocols such as quantum key distribution and secret sharing [116, 32]. Because quantum cloning is, in general, imperfect one is led to the construction of optimal, but imperfect, quantum cloning machines. Such machines based on qubits have been experimentally realized in several diﬀerent settings . On the other hand, cloning machines based on continuous variables have only very recently been implemented [3, 79].

4.2 A universal CV cloner

We have proven that quantum mechanics forbids an unknown nonorthogonal quantum state to be copied perfectly. But a quantum cloning machine which produces clones which are nonperfect copies of the original is possible. Buzek and Hillery in their seminal paper considered such a quantum cloning machine for the first time, where they went beyond the no-cloning theorem by considering the possibility of producing approximate clones for qubits [28]. An extension was performed first into the finite-dimensional regime of quantum states [29] and later on to the continuous variable (CV) regime [36]. The CV regime holds many advantages especially in the ease of preparing and manipulating quantumstatesintheCVregime[24].Wemust also note that in the CV regime every prepared state is used. Quantum cloning is believed to be the optimal eavesdropping attack for a certain class of quantum key distribution protocols employing coherent states [69]. Quantum cloning also may be utilized to boost the performance of some quantum computation algorithms [61]. One can also use quantum cloning to create a copy of quantum information either for backup purposes or in order to distribute the quantum information content to different nodes (members) of a network [20]. Experimental realizations of quantum cloning have been first performed successfully in the two-dimensional qubit regime where the polarization state of single photons has been conditionally cloned [83] and later in the CV regime [3]. Although the theoretical proposals of CV quantum cloning were based on at least one parametric amplifier [49, 40, 22] the experimental demonstration relied only on linear optics combined with a feedforward strategy [3]. A1→ 2 Gaussian cloner was built using a simple scheme consisting of simple unitary beam splitter transformations and homodyne detection which cloned coherent states of light at the quantum limit. For coherent states two canonical conjugate quadratures characterize the state - e.g. the amplitude,x ˆ, and phase,p ˆ -andthey 52 The Quantum Cloner have Gaussian statistics. The unknown coherent state to be cloned is then uniquely described by 1 |α = | (x + p ) (4.6) in 2 in in where xin and pin are the expectation values ofx în andp în. The outputs of the cloning machine are Gaussian mixed states with the expectation values xclone and pclone and characterized by the density operator ρclone. The efficiency of the cloning |r> |a Amplifier G=2 |r > 50:50 > Cloner

Figure 4.2: The optimal Gaussian cloner of unknown coherent states is a quantum limited amplifier having a gain factor of 2 (G=2) followed by a symmetric beam splitter ensuring unity gain between the clones and the input state. machine is typically quantified by the fidelity, which is defined by the phase space overlap of the output and input states. It is mathematically shown as:

F = αin|ρclone|αin , (4.7)

and for the particular case of unity cloning gains (corresponding to xclone = xin and pclone = pin)itreads

2 F = 2 2 . (4.8) (1 + Δ xclone)(1 + Δ pclone)

2 2 Δ xclone and Δ pclone stand for variances. When only Gaussian transformations are allowed the best strategy to clone a completely unknown coherent state is to send it through an amplifier working at the quantum limit and having a gain factor of two and subsequently directing it through a beam splitter which will generate the 4.2 A universal CV cloner 53 two clones. The quantum fidelity(phase space overlap) of the clones will then be 2/3. The classical intuitive way to produce clones would be the so called measure- and-prepare strategy. In this approach the coherent states both quadraturesx în andp în are measured simultaneously and the classical information extracted from this measurement is used to generate the clones of the input state. In this cloning transformation two additional units of quantum noise are added to the clones arising from the measurement of two non-commuting variables simultaneously and at the the construction of the clones. The fidelity of this classical cloner is limited to 1/2. However since we have stored the information classically we are able to produce an infinite number of clones (1 →∞cloner) using this method. However the best strategy when cloning unknown coherent states using Gaussian operations is to use a quantum amplifier with gain equal to 2 followed by a beam splitter which will reflect one clone and transmit the other. Experimentally a quantum cloning machine can be then constructed following the idea of the quantum amplifier. The quantum approach to cloning uses intrinsic correlations, and runs as follows (see Fig 4.3). At the input side of the cloning machine the unknown quantum state is divided by a 50/50 beam splitter. At one output we perform an optimal estimation of the coherent state: the state is split at another 50/50 beam splitter and the amplitude and the phase quadratures are measured simultaneously using ideal homodyne detection. These measurement outcomes are then used to displace the other other half of the input state. We can write the displaced field as:

1 g 1 g g † aˆdisp =(√ + )âin +(√ − )ˆv1 − √ vˆ2, (4.9) 2 2 2 2 2 where g just represents√ the tunable gain in the feedforward loop. We can choose the gain to be g = 2, which will allow the cancellation of the noise contribution from the first tap beam splitter and ensure unity gain in the clone generation. Herev ˆ1 and vˆ2 refer to the annihilation operators associated with the uncorrelated vacuum modes entering the two beam splitters (see Fig 4.3), anda în anda ˆdisp are the annihilation operators for the input and displaced states. In a final step the displaced state is sent onto a symmetric beamsplitter which reflects one clone and transmits the other:

1 † aˆclone1 =ˆain + √ (ˆv3 − vˆ2) (4.10) 2 54 The Quantum Cloner

Figure 4.3: The experimental realization of an optimal Gaussian universal cloner, ≈ 2 which successfully copies unknown coherent states with a ﬁdelity 3 .

1 † aˆclone2 =ˆain − √ (ˆv3 +ˆv2) (4.11) 2 The above relations correspond to the input output relations for an optimal Gaussian cloning machine. Notice that the equations do not depend on displacement or rotations making the cloning transformation a universal unconditional one. We can write the variances of the clones for the amplitude and phase quadratures as:

2 2 Δ xclone =Δxin + 1 (4.12) and 2 2 Δ pclone =Δpin + 1 (4.13)

, respectively. One can immediately see that this approach adds one unit of quantum noise to the state, which means that we are beating the classical cloning strategy. Using the fidelity formula we can conclude that this method generates clones with a 2/3, the same fidelity as the optimal Gaussian cloner. This way one can successfully clone coherent states with a fidelity of 2/3, which is the theoretical boundary for universal coherent state cloning. However there is a way to surpass this limit and clone coherent states with a higher fidelity using a clever measurement strategy, utilizing this superior cloning method we will generate clones with higher fidelities. 4.3 Towards a superior cloning machine 55

4.3 Towards a superior cloning machine

An interesting feature of quantum mechanics was discovered in 1999 by Gisin and Popescu. They realized that more quantum information can be encoded into pairs of anti-parallel spins than in parallel ones [64]. The continuous variable analogue of this eﬀect was further addressed by Cerf and Iblisdir who showed that more CV quantum information can be encoded into pairs of phase conjugate coherent states |α, |α∗, than in pairs of identical coherent states, |α, |α [34]. Phase conjugation coherent state refers to a phase ﬂip with respect to the x axis, whereas the x quadrature remains the same as the original coherent state as depicted in Fig. 4.4. Because of the |a>>|a* P P

X X

| 1 | | ∗ 1 | − Figure 4.4: α = 2 x + ip and α = 2 x ip The phase conjugate coherent states phase is flipped with respect to the x axis, whereas the x quadrature remains the same as the original coherent state. existence of a strong link between cloning and measurement theory, the superiority of using anti-parallel spins or phase conjugate coherent states led Cerf and Iblisdir to suggest that cloning machines with such inputs perform better than conventional cloning machines. This was indeed the case as shown theoretically in ref. [35] for phase conjugate coherent states and in ref. [56] for anti-parallel spin states. Recently, these results were generalized to d-dimensional systems [145]. Cerf and Iblisdir also found that the physical implementation of the N→M 56 The Quantum Cloner cloning of phase conjugate input states is composed of a sequence of beam splitters, a nonlinear process and another sequence of beam splitters [35]. Here we propose and experimentally realize a much more elegant approach for phase conjugate cloning which is not relying on a non-linear parametric process. A simple combination of beam splitters, detectors and feedforwards suffice to enable optimal N→M Gaussian cloning with phase conjugate input states, where N/2copiesof|α and N/2copies of |α∗ serve as inputs while M clones are produced. Theoretically, we treat the general case of N→M cloning while the 2→3 cloning of phase conjugate inputs is implemented experimentally. We note that the cloning protocol with phase conjugate inputs realized here has never been implemented before in any quantum system. Therefore to the best of our knowledge, this is the first example of a continuous variable quantum information processing experiment for which there is no experiment with discrete variables. Before introducing the cloning experiment with phase conjugate inputs we will present a new encode/measure strategy which allows us to extract more information out of coherent states. We now demonstrate the experiment which highlights the superiority of phase conjugation combined with joint measurements. This phase conjugate encoding combined with joint measurement strategy results in lesser added noise which in return will play a crucial role in the cloning transformation which will enable generation of clones with higher fidelities. In this experiment the laser used was a monolithic Nd:YAG laser producing a field at 1064 nm, which is split into two parts and subsequently directed into the coherent state preparation stage (see Fig. 4.5). To ensure that the information is encoded as pure coherent states, the states are as before assumed to be residing at a radio frequency sideband defined within a certain bandwidth of the laser beam. Note therefore that the two beams are bright although the particular sidebands in question are vacuum states before the encoding. The production of the two phase conjugate coherent states, |α and |α∗, is then performed by displacing the vacuum sidebands using an amplitude modulator (AM) and a phase modulator (PM) in each arm as shown in Fig. 4.5. The two states are prepared by using the same signal generator, that is by communicating classically correlated information between the two preparation stations. The relative phase shift of π between the phase quadratures was established by adjusting the cable lengths appropriately. This then accomplished the sign flip in front of the phase quadrature hence generation of the phase conjugated coherent state. First, the prepared states are characterized by measuring the two copies individually. This is done by successive use of a heterodyne detector yielding information about the 4.3 Towards a superior cloning machine 57

Local preparation a) Local measurements (times two) p Aux osc

x +/- {|0>} {|a>} AM PM {|a>}or {|a*>} Classical communication p } b) Global measurements {|a>} {|0>} {|a*>} AM PM p +/- x

{|a*>}

Figure 4.5: Schematic of the experimental setup. The diagrams show the phase space contours of |α (upper diagram) and |α∗ (lower diagram). The states are measured using a) a local strategy and b) a non-local strategy. AM: Amplitude modulator, PM: Phase modulator. amplitude and phase quadratures simultaneously. The coherent state is combined with a phase stabilized auxiliary beam at a 50:50 beam splitter with a π/2relative phase shift and balanced intensities. They interfere with a contrast of 99% and the two output beams are detected with high quantum efficiency (95%) photodiodes. Subsequently the photocurrents are subtracted and added which provides information about the phase and amplitude quadratures, respectively. Finally the spectral densities of the quadratures are recorded on a spectrum analyzer. Using the fact that the heterodyne detector projects the signal under investigation onto a vacuum state, the spectral densities of the prepared copies is easily inferred. Furthermore the measurements have also been corrected to account for the detection losses and electronic dark noise in order to avoid an erroneous underestimation. The inferred results for the spectral densities are shown by the solid horizontal lines in Fig. 4.5. These measurements for characterization of the prepared copies are in fact identical to the measurements associated with an optimal local estimation strategy. How- ever, in contrast to the characterization, for the estimation of unknown coherent states the results are not corrected for detector losses and electronic dark noise. The individual spectral densities for local measurements of |α and |α∗ are shown 58 The Quantum Cloner in column a) and c) of Fig. 4.5. From these measurements the added noise is found to be Δx =Δp =1.12 ± 0.04 for the amplitude and phase quadratures. Assuming a flat distribution of coherent states, the fidelity is given by

2 F = (4.14) (2 + Δx)(2 + Δp) and calculated to

FLocal =64.0 ± 1. (4.15)

This is close to the theoretical maximum of 2/3. We now discuss the experimental

Figure 4.6: The superior measurement strategy gives a better signal to noise ratio, resulting in extracting more information content. Spectral power densities (normalized to the quantum noise level) of the local and joint strategies. The resolution bandwidth is 100kHz and the video bandwidth is 30Hz. realization of the optimal nonlocal measurement of the phase conjugate copies. The optimal nonlocal estimation strategy is to combine the two copies at a 50:50 beam splitter and subsequently measure the amplitude quadrature in one output and the phase quadrature in the other output port of the beam splitter. This strategy measures the combinationsx ˆ1 +ˆx2 andp ˆ1 − pˆ2 where the indices refer to the two input modes. This combination can, however, be accessed using an experimentally simpler approach since the information is encoded onto sidebands of two equally 4.3 Towards a superior cloning machine 59 intense bright beams (with the power 60μW). The two classically correlated copies are carefully mode-matched ( 99%) at a 50:50 beam splitter and actively locked to have balanced intensities at the outputs of the beam splitter. Directly measuring the two outputs yield the quadrature combinations:

ˆi1 =(ˆx1 +ˆx2 +ˆp1 +ˆp2) (4.16)

î2 =(ˆx1 +ˆx2 − pˆ1 +ˆp2) , (4.17) by taking the summation and subtraction of these two contributions we obtain the required combinationsx ˆ1 +ˆx2 andp ˆ1 − pˆ2 . The spectral densities of these measurements are shown in column b) and d) of Fig. 4.6. The upper traces in Fig. 4.6 correspond to the coherent amplitudes of the input states and of the joint estimates, whereas the lower traces are the powers associated with the noise levels, all of which are at the shot noise level. The signal-to-noise ratio of the estimate is clearly larger than that of the prepared states; the coherent amplitudes of the amplitude and phase quadratures are increased by 3.0 dB and 2.9 dB, respectively, which effectively correspond to noise equivalent power of Δx =0.51 ± 0.02 and Δp =0.52 ± 0.02 shot noise units. The fidelity is then calculated to be F ∗ =79.5 ± 0.7 thus clearly surpassing the classical local fidelity of 2/3 and close to the theoretical value 4/5. It is possible to give an intuitive explanation to the fact that the strategy employing phase conjugate states outperforms the optimal measurement on identical copies. Imagine to apply this measurement strategy to the |α|α case. In this scenario the two modes are concentrated on one output port, so that it becomes a necessity to perform heterodyning in order to measure x and p simultaneously. This means that the state will be inflicted by one extra shot noise unit of vacuum at another beam splitter (BS). However if we apply this same strategy to the |α|α∗ state, we can directly access the entire information by homodyning each of the two output modes of the BS. This measurement will not introduce extra vacuum noise in this setup while detecting the same mean signal, we have less noise and a better signal to noise ratio which will result in a higher fidelity. We have experimentally demonstrated that a set of phase-conjugated pairs of coherent states can be better discriminated with an entangled measurement than with any sequence of local operations and classical communications. Experimentally more quantum information was extracted from the phase conjugation encoding strategy combined with the joint measurement technique. 60 The Quantum Cloner

4.4 A superior cloning machine

In the following we will use the technique shown in the previous section in the demonstration of superior cloning of coherent states. The figure of merit normally used to quantify the quality of a cloning transformation is the average fidelity, which is a measure of the similarity between the input states and the clones as previously discussed. When considering a flat distribution of coherent states as input states and assuming that the cloning transformation conserves the Gaussian nature of the input states, the optimal cloning machine yields the average fidelity [33]

2MN F = (4.18) C 2MN + M − 2N

On the other, it has been found that the optimal cloning machine with phase conjugate input states produces clones with ﬁdelity [35]

4M 2N F = (4.19) PC 4M 2N +(M − N)2

Here N refers to the total number of inputs 2N consisting of N copies of |α and N copies of |α for 4.18 and N copies of |α and N copies of |α∗ for 4.19. M is the number of output clones generated. The difference in the two fidelities for various combinations of the number of outputs for the 2 input case is depicted in Fig. 4.7, and it is clearly seen that the phase conjugate cloning machine outperforms the conventional one for M>2. As an example, considering the case where a single pair of phase-conjugate inputs is transformed into three clones the fidelities are

FPC = 90% and FC =85.7%. A schematic of the cloning machine is presented in Fig 4.8. The entanglement source, variable beam splitter at the input and the phase conjugate input at the heterodyne measurement are the diﬀerences to the universal Gaussian cloner introduced earlier. The EPR source is only necessary when extracting the phase conjugate output of the clones as was the case in the ampliﬁer setup. Lets consider a scenario where we have a coherent state and its phase conjugate at the input and generate two clones. The scheme will be similar to

Fig 4.3 except that where v2 enters we will now have the conjugate coherent state coming in and the ﬁrst beam splitter is a variable beam splitter. If we perform an 4.4 A superior cloning machine 61

0,12 F-FPC C 0,10 0,08 0,06 0,04 0,02 0,00 2345678 -0,02 -0,04 M( number of copies)

Difference in Fidelity -0,06

Figure 4.7: The diﬀerence of the two ﬁdelities for N = 1 and various numbers of the outputs is depicted, and it is clearly seen that the phase-conjugate cloning machine outperforms the conventional one for M>2 analysis for the amplitude quadrature of the clone we will obtain:

1 √ √ g √ √ xˆclone = √ ( Txˆin1 − Rv1 + √ ( Rxˆin1 + Tv1 +ˆxin2)+v3) (4.20) 2 2

In order to have unity gain cloning we choose: √ 2 − 2T g = √ . (4.21) R +1

The output clones variance is then:

2 1 − T/2 Δx2 =1+2 √ (4.22) R +1

The output variance is minimized for R=1/9. And for R=1/9 the output variance is Δx2 =9/8 which corresponds to a ﬁdelity of F=16/17 for the generated clones. Using formula 4.19 for N=1 and M=2 one obtains also F=16/17, which means that 62 The Quantum Cloner our cloner is the optimum Gaussian phase conjugate cloner.

* |a1 > |a * > 2 HD |a * > P 3 . . * . |aN/2 > g1

HDX

|a >

1 r .

. M

|a > . .

2 . . g1 . |a3 > r . v 3 . 3 r . v 2 |a > . 2 N/2 D(x) D(p) r1

BS(R,T) v1

g2p r . . M

. g2x . . .

r3 v3 r2 v2 D(x) D(p) r EPR 1

Figure 4.8: Scheme for phase conjugate cloning, N + N → Mcloner. BS(R,T): Variable beam splitter with transmission T and reﬂection R. HDx,p:Homodyne detector measuringx ˆ,ˆp. g1, g2x,andg2p: Electronic gains. D(x,p): Displacers ofx ˆ andp ˆ. EPR: Einstein-Podolsky-Rosen entanglement source. vi denotes the vacuum inputs while ρi denotes the density operators of the outputs.

In the general case the input signal is contained in an ensemble consisting of N identical coherent states and N identical phase conjugate coherent states. The two sets of states are uniquely described by the relations |α⊗N/2 = |x + ip⊗N/2 and |α∗⊗N/2 = |x − ip⊗N/2,wherex is the amplitude quadrature, p is the phase quadrature and [x, p]=2i. Each of the two sets of states are then collected into two single states using two arrays of (N/2-1) beam splitters. In the Heisenberg picture, the amplitude quadratures after collection can be written as 1 N 1 N xc1 = √ xl = √ Nx + δxl (4.23) N N l=0 l=0 1 N 1 N xc2 = √ xk = √ Nx + δxk (4.24) N k=0 N l=0 4.4 A superior cloning machine 63

where xl,k have been decomposed as xk,l = x+δxk,l and x = xk = xl.Thecol- lective coherent state, |α⊗N/2, impinges on a beam splitter (with transmission T and reflection 1 − T ) and the reflected part is measured jointly with the collective phase conjugated coherent state, |α∗⊗N/2, using a symmetric beam splitter and two homodyne detectors measuring the quadratures x and p. The measurement outcomes are electronically amplified with gain g and used to displace the remaining part of the collective coherent state. Such a combination of linear optics, measurements and feedforward enables a shot noise limited amplification if g = 2(1 − T )/T,and the input-output relation is simply [78] 1 1 − T x = x 1 + x 2 (4.25) out T c T c

Finally, the resulting state is divided into M clones using an M-splitter. The input- output relation between the inputs and any output clone is thus 1 √ xcl = √ (N + N 1 − T)x + (4.26) TNM N N √ 1 δx + 1 − T δx + 1 − x l k M v l=0 k=0

Universal cloning, that is, cloning with a preservation of the mean value of x,is 4MN obtained for T = (M+N)2 . In that case the variance of the amplitude quadrature noise of any clone is (M − N)2 Δx2 =1+ (4.27) cl 2M 2N The same analysis applies for the phase quadrature and thus it is readily veriﬁed 2 2 that the cloning transformation is symmetric in x and p:Δpcl =Δxcl.Now,by using the relation for universal cloning ﬁdelity of coherent states;

2 F = 2 2 (4.28) (1 + Δxcl)(1 + Δpcl) we immediately retain the optimal ﬁdelity in (4.19). Hence, optimal Gaussian cloning of phase conjugate coherent states can be obtained using simple linear optics, homodyne detection and feedforward.

We now proceed with the experimental demonstration of the production of 2 and 3 clones from a single pair of phase conjugate coherent states. The experimental 64 The Quantum Cloner setup is shown in Fig. 4.9. The laser beam was split into three parts; two parts

Figure 4.9: AM: amplitude modulator; PM: phase modulator; g: electronic gains. ρ1, ρ2,andρ3 are the density operators of the clones. A half wave plate(λ/2) followed by a polarizing beam splitter(PBS) is used as a variable beam splitter BS(R,T) served as input modes whereas the last part was used as auxiliary and local oscillator beams. Coherent states are generated at sidebands of bright laser modes. These sidebands, originally in the vacuum state, are excited by the use of amplitude and phase modulators driven by a signal generator. One of the phase modulators is driven π out of phase with respect to the other phase modulator to ensure the production of phase conjugate beams. In contrast, the amplitude modulators are driven in phase. The phase relation between the two input states is verified by interfering the two states on a 50/50 beam splitter and subsequently measuring the amplitude and phase quadratures in the two outputs of the beam splitter. Extinction of the amplitude (phase) quadrature in the difference (sum) output port of the beam splitter is a clear signature of the preparation of states with proper phase relation and identical amplitudes. After preparation of the pair of phase conjugate coherent states (|α and |α∗), they are injected into the cloning machine. A tunable beam splitter, consisting of a half wave plate and a polarizing beam splitter, separates the coherent state into two parts. For the production of two or three clones the transmission was set to 8/9 or 3/4, respectively in order to optimize the cloning fidelity. The reflected part of the input state interferes with the phase conjugate state on a balanced beam splitter. The carrier power of the input modes has been tailored such that the powers of the two states in the joint measurement are balanced. This enables the joint measurements of x and p using the simplified setup shown 4.4 A superior cloning machine 65 in Fig. 4.9 (rather than the standard heterodyne setup consisting of two homodyne detectors): After interference at the beam splitter with a π/2 relative phase shift, the outputs are directly detected. The difference and sum currents from the two

Clone 1 Clone 2 4 4

/ Coherent State 3 3 Cloning Limit 2 2 1 1 Phase Conjugate Cloning Limit 0

0 ative noise power

ative noise power

-1 Re Re -1 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Time/Seconds Time/Seconds

Figure 4.10: Experimental result for the generation of 2 clones. Red and blue traces correspond to the added noise of the clones with respect to the input state for the phase and amplitude quadrature respectively. The cloning limit for coherent states as well as the ideal phase conjugate cloner limit are depicted by straight lines. The data was corrected taking into account the detection efficiencies which were measured to be 83% and 85% for clone 1 and 2. The optical gains for this particular measurement run for the phase and amplitude quadratures of clone 1 and clone 2 were: gp1 =1, 01±0, 01, gx1 =1, 02±0, 01 and gp2 =1, 00±0, 01, gx2 =0, 99±0, 01. 2 2 The noises were measured to be Δ x1 =1.15 ± 0.2, Δ p1 =1.18 ± 0.2 for clone 1 2 2 and Δ x2 =1.19 ± 0.2andΔp2 =1.19 ± 0.2 for clone 2. detectors then represent the sum of amplitude quadratures and the difference of the phase quadratures of the two inputs, respectively. These results are used to modulate an auxiliary beam which subsequently is coupled with the remaining part of |α using a very asymmetric beam splitter (with splitting ratio 99/1). This accomplishes the displacement operation, and finally the two or three clones are produced by using a single symmetric beam splitter or a tritter, consisting of two beam splitters with ratios 2:3 and 1:1 respectively. Knowing that the Wigner functions of the input states and the output clones are Gaussian, they are fully characterized by measuring the first and second order moments of conjugate quadratures. This is done by using homodyne detection by which the mean and the variance of the two quadratures of the clones were measured accurately by using electronic locking loops. Switching between measuring the spectral properties of the input state or the clone was achieved by turning the 66 The Quantum Cloner

Figure 4.11: Experimental result for the generation of 3 clones. The first and second order moments of the generated clones for two conjugate quadratures is measured. The straight line at the 1,25 dB and 0,87 level represents the 2 → 3 cloning limit for identical coherent states and 1+1 → 3 cloning respectively. The data was corrected taking into account the detection efficiencies which were measured to be 83%, 85% and 80% for clone 1, clone 2 and clone 3. In this measurement run the noises were 2 2 2 2 measured to be: Δ x1 =1.26 ± 0.2, Δ p1 =1.27 ± 0.2,Δ x2 =1.22 ± 0.2,Δ p2 = 2 2 1.28±0.2, Δ x3 =1.23±0.2,Δ p3 =1.28±0.2. The optical gains for this particular measurement run for the phase and amplitude quadratures of the three clones were: Gp1 =1, 00 ± 0, 01, Gx1 =0, 95 ± 0, 01; Gp2 =0, 95 ± 0, 01, Gx2 =0, 96 ± 0, 01; Gp3 =1, 01 ± 0, 01, Gx3 =1, 00 ± 0, 01 modulations off or on in the auxiliary beam. A measurement run for the production of 2 or 3 clones is given in Fig. 4.10 and Fig. 4.11 respectively. In these measurements the Spectrum Analyser settings were: 14,3 MHz Center Frequency, Zero Span, 100 kHz Resolution Bandwidth, 30 Hz Video Bandwidth and 2 seconds Sweep Time. In order to determine accurately the optical gains for the various clones, we measured the signal power of the input and output. After the gain measurements, the input modulators were switched off in order to precisely measure the cloning noise at 14.3 MHz and thus quantify the cloning performance. Typical examples of measurement runs for the production of two and three clones are shown in Figs. 4.10 and 4.11, respectively. Here the added cloning noises relative to the shot noise of the input state for the amplitude and phase quadratures are displayed. In the 1 + 1 → 2 clone generation experiment the fidelities were calculated to be 92.4% ± 1 and 91.3% ± 1. For the 1 + 1 → 3 clone generation experiment the clones were created with fidelities 88.3% ± 1,88.9% ± 1and88.7% ± 1. This performance proves that the 1 + 1 → 3 machine with phase-conjugate inputs operates very close to the optimum of 90% and it outperforms the conventional cloning machine which ideally yields a fidelity of 85.7% . The close to optimal performance is a result of the high quality of the feedforward loop. The quantum efficiencies of the detectors 4.4 A superior cloning machine 67

(including the interference visibility) were measured to 93% and the electronic noise was negligible. The 1 + 1 → 2 cloning fidelities are of course not surpassing the fidelity for the conventional 2 → 2 cloning machine which trivially yields a fidelity of 100%. However, using phase-conjugate inputs, there exists also the possibility of producing optimal copies of |α∗ in addition to copies of |α, which is not possible with the conventional cloning approach.

We have now experimentally proved the surprising fact that a cloning machine with phase-conjugate input states performs better than a cloning machine with identical inputs. The fact that a pair of phase-conjugate coherent states is more in- formative than identical ones led to a suppression of the noise induced by the cloning action. This close relation between the cloning noise and the information content of the input states is easily understood from the part of the setup executing a joint measurement of phase conjugated coherent states. Such a measurement strategy has recently been proven to be superior for information retrieval of phase-conjugate states. In contrast, for identical coherent states such nonlocal measurement strategy has no advantage over the standard local strategy. Thus the phase conjugation combined with the joint measurement strategy yields more information which in turn leads to less noise in the displacement operation and subsequently less noise added to the clones. It is also clear from the setup that the production of inﬁnitely many clones (M →∞) coincides with optimal estimation as proved by Bae and Acin [7]. For M →∞, the transmission T → 0 which results in a complete joint measurement of the phase conjugate coherent states. This measurement strategy coincides with the optimal one for estimating the information in phase conjugate coherent states, thus illustrating the strong link between cloning and measurement theory. Finally, we mention that the combined use of phase conjugation and joint measurement for improving the performance of a quantum protocol has many applications beyond the cloning action. Recently it was realized that such a strategy can be used to execute a minimum disturbance measurement [5], enable an optimal individual eavesdropping attack [99], and increase the sensitivity of optical coherence tomography beyond the coherent state limit and ultimately reaching the entanglement based limit [48]. Another proposal, but without an experimental veriﬁcation, was recently published [37]. 68 The Quantum Cloner

4.5 Cloning of partial quantum information

The ignorance about a given quantum state is what makes quantum protocols difficult to execute in practice or even impossible in principle. For example, high efficiency and deterministic teleportation of a quantum state with no prior information is only possible in the unrealistic limit of perfect entanglement. Furthermore, it is well known that perfect cloning of an arbitrary quantum state is impossible as formulated in the no-cloning theorem[138, 41]. Luckily, in all practical quantum communication or computation schemes we are not completely ignorant about the set of possible input states which in turn greatly facilitates the execution of these protocols: The more prior information one has about the input alphabet the less resources are needed for the process. An interesting example demonstrating the influence of partial quantum information is cloning. For example, the optimised continuous variable (cv) cloner of an arbitrary state (also known as the cv universal cloner) yields a cloning fidelity of 1/2 corresponding to a standard classical distributor [22]. However, if the input states are a-priori known to be coherent states (but with unknown amplitude and phase), the fidelity increases to 2/3 [36, 89]. By further limiting the number of possible input states the fidelity increases even further as theoretically analysed in ref. [39] for a symmetric Gaussian distribution of coherent states, in refs. [39, 99, 44] for coherent states with known phase and in ref. [115] for phase covariant cloning where the mean amplitude is fixed but the phase random. Cloning of displaced thermal states and squeezed states has also been theoretically analyzed [104]. Despite this high interest in cloning of partial cv quantum information, there has been no experimental demonstrations. Experimental studies have been entirely devoted to cloning of qubits with partial information, e.g. phase covariant cloning [45, 119, 11]. In this work we investigate, theoretically and experimentally, the optimization of a continuous variable quantum cloning machine with respect to two different coherent state alphabets using a simple setup based entirely on linear optics, homodyne detection and feedforward. In particular we propose and experimentally realise an optimal Gaussian cloning machine tailored to clone a symmetric Gaussian alphabet of coherent states as well as coherent states with known phases. In addition, we prove the optimality of the latter scheme and find that a fidelity as large as 96.1% can in principle be achieved. A common measure of the quality of a cloning operation is the fidelity which is defined as follows. Consider the protocol where the 4.5 Cloning of partial quantum information 69 coherent state, |α, to be cloned is chosen from an ensemble defined by {p(α), |α} where p(α) denotes the probability that the state |α was chosen. This state under- goes a cloning transformation, thus results in a density matrix Γ(α) associated with the input state, α.Theoverlapα|Γ(α)|α then quantifies the quality of cloning a specific member, |α, of the alphabet. The average fidelity of the cloning action thus reads F¯ = p(α)α|Γ(α)|αd2α (4.29)

Using the ﬁdelity as a measure, the cloning transformation is optimal when this expression is maximized. Such a maximization normally yields a non-Gaussian solution, that is, the optimal map Γ is non-Gaussian. However, since the Gaussian cloning transformation is known to be near optimal, we will mainly focus on such maps.

In ref. [3] a 1 → 2 cloning map based on linear optics, homodyne detection and feedforward was proposed. A generalized version of this map is illustrated in fig. 4.12 and described in the figure caption. Setting the transmittivity to T2 =1/2 and the electronic gains to gx = gp = 2(1 − T1)/T1 the input-output relation for one of the clones in the Heisenberg picture reads 1 1 1 † aˆclone1 = √ ( aîn + − 1â2 +â3). (4.30) 2 T1 T1 wherea ˆclone1, âin, aˆ2 anda ˆ3 are the field operators associated with the output, the input and the ancilla fields respectively. The field operators are given bya ˆ = xˆ + ipˆ wherex ˆ andp ˆ are the amplitude and phase quadratures, respectively. The map in eq.4.30 coincides with the one in ref. [3] for T1 =1/2 which was found to be the optimal Gaussian cloning map for completely unknown coherent states corresponding to a flat input alphabet. If on the other hand the number of possible input coherent states is finite the transformation in ref. [3] is no longer optimal. For a symmetric Gaussian distribution of coherent states with variance V :p(α)= 1/2πV exp(−|α|2/2V ), the fidelity becomes:

¯ √ 2T1 F = 2 (4.31) 2V (1 − 2T1) + T1 +1

It is clear from this expression that the ﬁdelity is a function of the knowledge of the input states through the variance, V , of the Gaussian alphabet. For a given 70 The Quantum Cloner variance, V , the maximized ﬁdelity is ⎧ ⎪ 4V +2 ≥ 1 1 ⎨⎪ 6V +1 ,V 2 + 2 ¯ F = ⎪ (4.32) ⎩⎪ √1 ,V ≤ 1 + 1 (3−2 2)V +1 2 2

1 2 which is obtained using the scheme in Fig. 4.12 with T1 = 2 (1/(2V )+1) and T1 = 1, corresponding to the upper and lower inequalities, respectively. These fidelities are identical to the ones found in ref. [39] for optimized Gaussian cloning using an OPA. Let us consider these expressions in two extreme cases: If no a priori information is available about the distribution of coherent states, V →∞and the fidelity averages to F¯ =2/3. In the other extreme, where complete information about the input state is at hand, V = 0 and the fidelity is unity. In the following, we investigate experimentally the realistic intermediate regime where the width of the input distribution is finite and non-zero.

Figure 4.12: Schematic of the proposed 1→2 cloning protocol. The signal, ain,is reﬂected of a beam splitter with transmittance T1 and detected using a beamsplitter with transmittance T2 and two homodyne detectors measuring the amplitude, x,and phase, p, quadratures. The measurement outcomes are scaled with the gains gx and gp and used to displace the transmitted signal. The displaced state is subsequently split on a symmetric beam splitter, thus producing two clones denoted by aclone1 and aclone2. a1, a2 and a3 are ancilla states.

We prepare as usual the input coherent states by modulating a continuous wave laser beam (1064nm) at the frequency of 14.3 MHz. Two electro-optical modulators inserted in the beam path were used to control the mean phase pin and amplitude xin quadratures independently by separate low voltage function generators. 4.5 Cloning of partial quantum information 71

Through the modulation, photons were transferred from the carrier into the sidebands, thus producing a pure coherent state at the modulation frequency. We set the modulation frequency to 14.3 MHz and deﬁned the bandwidth of the coherent state to be 100 kHz.

Figure 4.13: Experimental cloning setup. AM: Amplitude Modulator, PM: Phase Modulator, EOM: electro-optic modulator; LO: Local Oscillator; AUX: Auxiliary state; T1: Variable beam splitter.

The pure coherent states are subsequently injected into the cloning machine. First the states are split into two parts using a variable beam splitter which is consisting of a half wave plate and a polarising beam splitter; thus any T1 in eq. (4.30) is easily accessed by a simple phase plate rotation. The reflected part of the state is measured using heterodyne detection where x and p are simultaneously measured. This is done by interfering the signal with an auxiliary beam (AUX1) with a π/2 phase shift and balanced intensities; subsequently the two output are measured with high efficiency and low noise detectors, and the sum and the difference currents are constructed to provide a measure of x and p. The outcomes are scaled with low noise electronic amplifiers and used to modulate the amplitude and phase of an auxiliary beam (AUX2), and subsequently combined by the remaining part of the signal employing a 99/1 beam splitter. This accomplishes a high efficiency displacement operation. Finally, the displaced state is divided into two clones using a symmetric beam splitter and the two outputs are characterized using two homodyne detectors with intense local oscillator beams (LO1 and LO2). The signal power and variances of the input state as well as the output states are then measured using a spectrum analyzer with resolution bandwidth set at 100kHz and video bandwidth at 300 Hz. Such measurements suffice to fully characterise the states due to the 72 The Quantum Cloner

Gaussian statistics of x and p. Active electronic feedback loops were implemented at all interferences to ensure stable relative phases. From the power and variance measurements, we estimate the gain as well as the added noises associated with the cloning transformation. Using these values we calculate the ﬁdelity for a given input alphabet using the expression

¯ 2 F = 2 2 2 2 (4.33) (1 + Δx +4V (1 − λx) )(1 + Δp +4V (1 − λp) ) which is obtained by inserting an arbitrary Gaussian state (with variance V )inre- placement of Γ(α) in the ﬁdelity expression. λx = xclone/xin and λp = pclone/pin are the cloning amplitude gains. As an example, we consider a Gaussian input dis-

√ Figure 4.14: The average fidelity is plotted against the width ( V ) of the distribution of input states. The solid line corresponds to the theory and the red dot and black square correspond to average fidelities for clones 1 and 2. The dashed line takes into account that the amplifier used in the scheme was nonideal and some technical noise entered the cloning process. The dotted line corresponds to a measure and prepare strategy. The grey shaded area corresponds the region where the solution T1 = 1 is optimal. The red line extends this solution into the region where it is not optimal. 4.5 Cloning of partial quantum information 73 tribution with V =1.72 shot noise units (SNU). For this alphabet, the cloning machine is optimized by setting T1 =0.83 and gx = gp =0.64 corresponding to a cloning gain of λx = λp =0.775. We adjusted the beam splitter transmittance to this value and tuned the electronic gains to the optimized value while monitor- ing the optical gain of a test signal (through comparison between the input power and output power of the signal). For this specific experimental run, we measured an optical cloning gain of 0.775 ± 0.005 valid for all input states. After adjusting the gain to this value, the associated added noises in x and p were measured to 1.21 ± 0.02 and 1.26 ± 0.02. Inserting these values in eqn. (4.33) we find an average cloning fidelity of F =0.775 ± 0.01 which is very close to the optimal value of F =0.785 (see eqn. (4.32)). This experiment was repeated with different gains corresponding to different widths of the input alphabet and the results are summarized in Fig. 4.14. The solid curve in Fig. 4.14 represents the ideal average fidelity given by eqn. 4.32. Small deviations from ideal performance is caused by small inefficiencies of the heterodyne detector in the feedforward loop: The mode overlap between the auxiliary beam AUX1 and the signal beam was 99% and the quantum efficiency of the associated detectors were 95%±2%. Taking these parameters into account, the expected average fidelity follows the dashed curve which agrees well with the measured data. Note that all the measured data were corrected for the detection inefficiencies of the verifying detectors (amounting to 83% and 85%) to avoid an erroneous underestimation of the added√ noise and thus an overestimation of the fidelity. Note also that for V<1/2+1/ 2 (corresponding to the gray shaded region in Fig. 4.14), the best cloning strategy is a simple beam splitter operation, which is obtained√ in the present setup by setting T =1andgx = gp =0,thus λx = λp =1/ 2. In this case ideal performance is naturally achieved and the ideal solid curve and real dashed curve in Fig. 4.14 are identical. Since the detection efficiency is inferred out of the results, the actual measured performance will be only limited by the errors in estimating these efficiencies. We now proceed by considering another input alphabet. The coherent states are assumed to have a known and constant average phase but completely random amplitude. This input distribution was also considered theoretically in ref. [39] and [99] where two different strategies were suggested for the experimental realizations. In the latter reference, however, the proposed strategy was not optimal and in the former reference the method relied on squeezing transformations to surpass the classical cloning strategy. Furthermore, the optimality of the suggested schemes were not proven in these references. In the 74 The Quantum Cloner following we show that the transformation depicted in Fig. 4.12 is optimal for special choices of the ancilla states a1 and a3, and the transmittances T1 and T2. We start by setting T1 =1/2andT2 = 1 and thus get the following transformation for one of the output clones

1 xclone1 = xin + √ x3 (4.34) 2 1 1 1 pclone1 = pin − p1 + √ p3 (4.35) 2 2 2

First assuming that the input ancillas(a1, a2,anda3) are vacuum states, the fidelity for this transformation is easily found using the expression for the fidelity√ and inserting a distribution with the above mentioned properties. We find F =2/ 5 ≈ 0.894. This should be compared with the optimised measure and prepare strategy, which we conjecture to be associated with single quadrature detection followed by displacement of an optimally squeezed ancilla state in the quadrature direction corresponding the constant phase. The optimised squeezing factor is 1/2ofthe undisplaced phase quadrature,√ and this measure and prepare strategy yields a fidelity of F =2/ 3+ 2 ≈ 0.828. Remarkably, our proposed scheme surpasses this value without the use of squeezed states. Although this cloning protocol surpasses the measure and prepare protocol, it is not the optimal Gaussian cloning machine for this input alphabet. If the input state a1 is infinitely squeezed in the amplitude quadrature and a3 is squeezed√ by a factor of 8/5, the cloning machine is optimal yielding a fidelity, F =4( 10 − 1)/9 ≈ 0.961. Hence, knowing the phase of the input coherent states, the cloning fidelity can be exceptionally high using a very simple scheme.

Let us now prove the optimality of this scheme. A generic Gaussian cloning transformation is casted as

xclone1 = λx1(xin + nx1) pclone1 = λp1(pin + np1)

xclone2 = λx2(xin + nx2) pclone2 = λp2(pin + np2)

where ni are noise operators. Since the two clones are assumed to be identical we set λx = λx1 = λx2 and λp = λp1 = λp2 and because the amplitude of the input is completely random we must have λx = 1 to maximize the cloning ﬁdelity. 4.5 Cloning of partial quantum information 75

Furthermore by using the fact that the amplitude (phase) quadrature of clone 1 commutes with the phase (amplitude) quadrature of clone 2 we ﬁnd

[xc1,pc2]=g([xin,pin]+[nx1,np2]) = 0

⇒ [nx1,np2]=−2i and

[nx2,np1]=−2i (4.36) which yield the uncertainty relations

2 2 Δ (nx1)Δ (np2) ≥ 1 2 2 Δ (nx2)Δ (np1) ≥ 1

Due to symmetry we get 2 2 Δ npΔ nx ≥ 1 (4.37)

2 2 2 2 2 2 with Δ nx ≡ Δ nx1 =Δnx2 and Δ np ≡ Δ np1 =Δnp2. Now considering the commutation relation between conjugate quadratures of a single clone we ﬁnd

[xc1,pc1]=g([xin,pin]+[nx1,np1]) = 2i 1 − g ⇒ [n 1,n 1]=2i x p g

2 2 ≥|1−λp |2 which results in the uncertainty product Δ nxΔ np λp for both clones. This product is minimized for λp =1/2 while satisfying the relation (4.37). The minimum variances of the two output clones are therefore

2 2 Δ x =1+Δnx 1 Δ2p = (1 + Δ2n ) 4 p

By evaluating the Gaussian ﬁdelity for these clones we ﬁnd

2 F = (2 + 1/Δ2p)(5/4+Δ2p/4) 2 which is maximized if the ancilla state is squeezed such that Δ np = 5/2and 76 The Quantum Cloner √ 2 Δ nx = 2/5. The maximum fidelity is thus found to be F =4( 10 − 1)/9. We now demonstrate cloning of coherent states with constant phases using the scheme in Fig. 4.12 with a1 and a3 in vacuum states. The experimental setup is slightly modified with respect to the one in Fig. 4.13. To enable direct detection of the amplitude quadrature in the feedforward loop, the auxiliary beam AUX1 is blocked and the sum of the currents produced in the two detectors is taken. This yields the amplitude quadrature and is correspondingly used to generate the amplitude displacement. The feedforward gain driving the phase displacement is set to zero, thus the phase quadrature is unaffected by the feedforward action. Since the amplitude quadrature of the input states is completely unknown, the electronic gain, gx is set such that the overall optical amplitude quadrature gain is unity. This maximises the average fidelity for this set of states. Finally, the clones are generated

Figure 4.15: Spectral amplitude quadrature noise densities of the two clones (upper black traces) relative to the quantum noise level (lower red trace). The measurement was taken over a period of 2 seconds. The settings of the spectrum analyser were 14.3 MHz central frequency, 100 kHz bandwidth and 300 Hz video bandwidth. The added noise contributions are 1.8 ± 0.1dB and 1.85 ± 0.1dB for clone 1 and clone 2 respectively. The optimal cloning limit (1.75 dB above the shot noise level) is pointed out by the solid green line. at the output of the third beam splitter. The verification procedure is the same as before, and a measurement run is depicted in Fig. 4.15. Making use of eqn. (4.33) we calculated the fidelity of the generated clones to be 89.1 ± 0.2% and 88.7 ± 0.2%. In this particular measurement run the gains for the amplitude quadratures were measured to be λx1 =0.98 ± 0.01 and λx2 =0.99 ± 0.01 for clone 1 and clone 2. The experimental cloning fidelity greatly exceed the classical fidelity of 82.8% and 4.6 Conclusion 77 is close to the optimal value of 89.4% for non-squeezed ancillas.

4.6 Conclusion

In this chapter we discussed various experimental quantum cloning protocols. We first introduced a universal cloner which produces clones of unknown coherent states. Then we explained that the combined use of phase conjugation and joint measurement can be used to build a superior cloner with improved performance over the universal cloning machine, and in fact we successfully implemented such a cloner. Later we illustrated the intriguing relationship between cloning fidelity and prior partial information by proposing and experimentally demonstrating the state dependent cloning transformation of coherent states with superior fidelities. We found that the more prior information about the input states the greater is the cloning fidelities. This relationship is not only valid for cloning protocols, but also for other protocols such as teleportation and purification of quantum information. Since prior partial information is common in quantum information networks, we believe that the state-dependent cloning strategies presented in this chapter as well as similar strategies for other protocols will have a vital role in future quantum informational systems. 78 The Quantum Cloner Chapter 5

Minimal Disturbance Measurements

In this chapter we will investigate a crucial concept of quantum measurement theory, namely minimal disturbance measurements. We propose and experimentally demonstrate an optimal nonunity gain Gaussian scheme for partial measurement of an unknown coherent state that causes minimal disturbance of the state. The information gain and the state disturbance are quantified by the noise added to the measurement outcomes and to the output state, respectively. We derive the optimal trade-off relation between the two noises and we show that the tradeoff is saturated by nonunity gain teleportation. Optimal partial measurement is demonstrated experimentally using a linear optics scheme with feedforward. In addition, we show that such a scheme can be used to enhance the transmission fidelity of a class of noisy channels.

5.1 Introduction

One of the most counter-intuitive concepts of quantum mechanics is the fact that any attempt to gain information on an unknown quantum state of a physical system will inevitably result in a noisy feedback to the measured system. No matter how cleverly the measurement is performed, the state will always be disturbed to some extent: The more information obtained about a quantum state from a measurement, the more it will be altered, and vice versa. Although this measurement-disturbance concept is very old and originally only of fundamental interest, it has recently received 80 Minimal Disturbance Measurements renewed interest due to its direct application in the flourishing field of quantum information science, and in particular, quantum key distribution. The study of the interplay between the quality of the estimation of a quantum state and the disturbance of the post-measurement state has been extensively carried out in finite-dimensional systems, where optimal trade-off relations have been established for various cases [10, 9, 96, 81, 114, 59, 93] and realized recently in an experiment [118]. In contrast, much less effort has been devoted to the study of this trade-off in infinitely-dimensional systems [5, 81, 62, 103] where quantum information is carried by observables with a continuous spectrum, important examples being the canonically conjugate quadrature amplitudes. Gaussian states which belong to continuous variable states have played a key role in various experimental realizations of quantum information protocols, thanks to the ease in generating and handling them in a quantum optics lab [98, 24]. Completely unknown pure quantum states described in an infinitely dimensional Hilbert space can be estimated only very poorly based on a single measurement. However, in quantum communication systems, a priori knowledge is often given. It is e.g. normally known that the state belongs to a certain set each occurring with a certain a priori probability. In this work we assume that the states are taken from a flat distribution of coherent states. With this a priori information at hand, it was recently proven that the state can be optimally estimated using a setup where conjugate quadratures are measured simultaneously using a symmetric beam splitter and two homodyne detectors [71]. However, employing this strategy the coherent state is maximally disturbed. On the contrary, if the unknown coherent state is left untouched, our guess will be completely random but the state will be intact. In the following we will investigate the intermediate cases and hence address the question: For a given information gain what is the minimum disturbance to the coherent state? Let us define the problem that will be addressed in this chapter. The task is to perform a minimal disturbance measurement on a coherent state which is taken from an unknown distribution (see Fig. 5.1). That is, a completely random coherent state will be received by our measurement device. The question that will be raised and answered in this chapter is: What is the optimal information disturbance trade-off for this scenario? The answer to that question depends on the figure of merit used to quantify the information gain and the measurement disturbance. For Gaussian states, a useful and practical measure of the quality of the measurement is the phase insensitive added noise, since it directly determines the Shannon information opti- 5.1 Introduction 81 mally extracted by the measurement. Thus the optimal trade-off between the added noises determines the maximal information that can be gained from the Gaussian measurement represented by a channel with a given additive noise. A second parameter of high relevance for describing the measurement is the gain (attenuation or amplification) of the channel, since the minimization of the added noise is done with respect to that gain. For example in the experiment on minimal disturbance measurement [5], the added noise was minimized under the constraint that the channel gain was unity (corresponding to a conservation of the mean values). For Gaussian measurements and Gaussian channels this optimization procedure corresponds to a maximization of the fidelity over all possible input states drawn from the unknown coherent state alphabet. It should however be noted that by using the fidelity as a measure the optimal solution is non-Gaussian [81] due to specific properties of fidelity.

Classical Information (xcl,p cl )

r |a>=x+ip in MDM UnknownCoherent State Disturbed state

Figure 5.1: The principles of a minimal disturbance measurement of coherent states. The input state is drawn from an unknown distribution of coherent states, say |αin = |x + ipin, and the task is to acquire information about the state (through a measurement) in such a way that the state is minimally disturbed according to quantum mechanics. This is the essence of a minimal disturbance measurement. There are two outputs of the protocol; a classical one yielding information about the input state (in form of two numbers, sayx ¯cl andp ¯cl) and a quantum one, namely the post measurement state ρ. In this chapter we consider only cases where the classical data as well as the disturbed quantum state are inﬂicted by additive phase insensitive Gaussian noise. The gain, g, of the protocol is deﬁned by the ratio between the input and output mean values: g = tr(xρ)/αin|x|αin = tr(pρ)/αin|p|αin.

In this chapter we investigate theoretically and experimentally the optimal trade- off relations in terms of added noises using two different strategies. In the first approach the channel gain is a free parameter that is optimised to minimize the trade-off between the added noises associated with the measurement and disturbance. This trade-off relation was derived by Ralph [109] who also found that the relation could be experimentally demonstrated employing an ideal teleportation 82 Minimal Disturbance Measurements scheme with tunable entanglement. Here we propose and experimentally realise a different approach which is not relying on entanglement but solely on linear optics, Gaussian measurements and feed-forward similar to the one employed in Ref. [5]. In the second approach that will be carefully addressed in this chapter, the channel gain of the minimal disturbance measurement is fixed to a certain value associated with a particular realisation (the unity gain operation demonstrated in Ref. [5] being a special case). For this case we derive a trade-off relation for arbitrary gains and prove its optimality using two different complementary proofs. As in the previous case, we also find here that a scheme similar to the one in Ref. [5] can be used to implement the optimal trade-off for fixed but non-unity gain operation. This is demonstrated and near optimal performance is achieved. The experimental scheme is not only of fundamental interest but it can be also applied to perform optimal individual Gaussian attacks in a continuous-variable quantum key distribution scheme based on heterodyne detection [136, 91]. The chapter is organized as follows. Section 5.2 deals in general with trade-off between added noises and in section 5.3 the trade-off is exemplified by the non-unity gain teleportation scheme. In Section 5.4 we give two different proofs of optimality of the trade-off. Section 5.5 is dedicated to linear optical scheme saturating the trade-off. The experimental demonstration of the scheme is given in Section 5.6. At the end we discuss the possibility of using the minimal disturbance measurement as an eavesdropping attack, we give the unity gain minimum disturbance experiment and finally we conclude the section.

5.2 Gaussian minimal disturbance measurements

We consider Gaussian quantum operation that acts on a single mode of an optical ﬁeld “in” described by the canonically conjugate amplitude and phase quadratures xin and pin ([xin,pin]=2i). We assume that the output mode of the operation is characterized by a pair of quadratures xout,pout ([xout,pout]=2i) related to the input quadratures by the formulas

xout = g(xin + nout,x),pout = g(pin + nout,p), (5.1)

where the quantity g>0 is the gain of the operation. The operators nout,x and nout,p are standard operators of noises added to the input state. The operation also 5.2 Gaussian minimal disturbance measurements 83

outputs a pair of mutually commuting variables xcl and pcl that depend linearly on the input quadratures xin and pin and that therefore can be used for simultaneous measurement of these quadratures. These variables can be expressed, after a suitable scaling transformation, as

xcl = xin + ncl,x,pcl = pin + ncl,p, (5.2) and satisfy the commutation rules

[xcl,pcl]=[xout,xcl]=[xout,pcl]=

=[pout,xcl]=[pout,pcl]=0. (5.3)

The operators ncl,x and ncl,p describe noises added to the outcomes of simultaneous measurement of input quadratures xin and pin by homodyne detection of the variables xcl and pcl. Naturally, the operators nout,x and ncl,x (nout,p and ncl,p) are independent of the input quadrature xin (pin) and hence

[nout,x,pin]=[ncl,x,pin]=[nout,p,xin]=

=[ncl,p,xin]=0. (5.4)

In addition, the gains of the operation are assumed to be ﬁxed for all input states, i.e. xout/xin = pout/pin = g, xcl/xin = pcl/pin = 1, which implies that

[nout,x,xin]=[nout,p,pin]=[ncl,x,xin]=

=[ncl,p,pin]=0. (5.5)

Substituting Eqs. (5.1) and (5.2) into the commutation rules [xout,pout]=2i and (5.3) one ﬁnds using the latter commutation rules (5.4) and (5.5) that the noise operators nout,x,nout,p,ncl,x and ncl,p must satisfy

(1 − g2) [nout x,nout p]=2i , [ncl x,ncl p]=−2i, , , g2 , ,

[nout,p,ncl,x]=[ncl,p,nout,x]=2i,

[ncl,x,nout,x]=[ncl,p,nout,p]=0. (5.6) 84 Minimal Disturbance Measurements

The noise operators represent the noise by which the outcomes of the homodyne detections of the variables xcl and pcl as well as the output state are contaminated. The commutation rules (5.6) and the Heisenberg uncertainty relations then impose fundamental bounds on the noises that have to be satisﬁed by any Gaussian operation. Since we are interested in partial measurements on coherent states, it is convenient to quantify the two noises by the following sums:

2 2 2 2 nout,x + nout,p ncl,x + ncl,p νout ≡ ,νcl ≡ , (5.7) 2 2 for which the respective bounds read as

|1 − g2| νcl ≥ 1,νout ≥ ,νclνout ≥ 1. (5.8) g2

The use of noises (5.7) is advantageous since they are a simple function of the added 2 2 2 2 noises nout,x , nout,p , ncl,x and ncl,p that can be directly measured experimentally. We shall see that the operations that for a given νcl and g minimize νout add 2 2 noise symmetrically to the x and p quadratures, which means that nout,x = nout,p 2 2 and ncl,x = ncl,p holds. In this case the quantities νout and νcl are exactly the noises added to the input state quadratures and to the measurement outcomes, respectively. This symmetry and isotropy is a natural feature of optimal partial measurement on coherent states that exhibit the same variances for all quadrature components. The interpretation of noises (5.7) is particularly simple for symmetric operations with unity gain (g = 1). In this case the quantity νout/2coincideswith the mean number of thermal photons added by the operation to the input state. The interpretation of the quantity νcl is a little bit more involved. The classical measurement outcomesx ¯cl andp ¯cl obtained when measuring the variables xcl and pcl can be used to prepare a classical guess |αcl = |(¯xcl + ip¯cl)/2 of the input coherent state

|α = |(x + ip)/2in. By repeating this procedure many times with the same input state we thus prepare on average a mixed quantum state called the estimated state of the input state. Similarly as in the previous case for the symmetric unity gain operation the quantity (νcl +1)/2 equals to the mean number of thermal photons in the estimated state. 5.3 Quantum teleportation as a minimal disturbance measurement 85

5.3 Quantum teleportation as a minimal disturbance measurement

In the following we show that one of the most celebrated quantum information protocols - quantum teleportation - enables a minimal disturbance measurement in the sense of saturating the inequalities in (5.8). Using teleportation as an example we arrive at a very useful equality deﬁning the optimum trade-oﬀ. The optimality will then be rigorously proven in the following two sections. The protocol in question is the standard continuous variable teleportation scheme [127, 23, 60] operating in the non-unity gain regime [16]. An unknown state of an optical mode “in” described by the quadratures xin and pin is teleported by a sender Alice (A) to a receiver Bob (B). At the beginning, Alice and Bob share an entangled state of two other modes A and B produced by the two-mode squeezing transformation of two vacuum states

(0) − (0) xA =cosh(r)xA sinh(r)xB , (0) (0) pA =cosh(r)pA +sinh(r)pB , (0) − (0) xB =cosh(r)xB sinh(r)xA , (0) (0) pB =cosh(r)pB +sinh(r)pA , (5.9)

(0) (0) (0) (0) where xA , pA , xB and pB denote the vacuum quadratures of modes A and B and r is the squeezing parameter. Alice then mixes the input mode with mode A on a balanced√ beam splitter and performs√ homodyne detection of the variables x1 =(xin + xA)/ 2andp2 =(pin −pA)/ 2 at the outputs of the beam splitter. She then communicates the measurement outcomesx ¯1 andp ¯2 via a classical channel√ to → Bob who displaces√ his part of the shared state as xB xout = xB + g 2¯x1 and pB → pout = pB +g 2¯p2,whereg>0 stands for the gain of the transformation from photocurrents to the output optical ﬁeld. At Bob’s site we thus have the output quadratures (5.1), where

xB pB nout x = x + ,nout p = −p + . (5.10) , A g , A g

At Alice’s location we have two commuting√ variables (5.2) obtained by rescaling of the variables x1 and p2 by the factor of 2 and the operators of added noises ncl,x 86 Minimal Disturbance Measurements

and ncl,p read as

ncl,x = xA,ncl,p = −pA. (5.11)

Substituting now from Eqs. (5.10) and (5.11) the noise operators nout,x,nout,p,ncl,x and ncl,p in the commutation rules (5.6) one ﬁnds the operators in the non-unity gain teleportation indeed satisfy the commutation algebra (5.6). Making use of Eqs. (5.9), (5.10) and (5.11) one obtains the noises (5.7) for the non-unity gain teleportation in the form,

(1 + g2) 2 νcl =cosh(2r),νout = cosh(2r) − sinh(2r). g2 g (5.12)

2 2 2 2 It holds that nout,x = nout,p = νout and ncl,x = ncl,p = νcl hence the added noise is isotropic. Eliminating now the parameter r from the second equation (5.12) using the first one one finds the trade-off between the noises (5.7) in the non-unity gain teleportation to be 2 2 2 g νout =(1+g )νcl − 2g νcl − 1. (5.13)

In the plane of the noises νcl and νout the trade-off relation determines a certain quadratic curve that turns out to be a fraction of a hyperbola whose exact shape depends on the gain g. By changing the squeezing r one can continuously move along the whole trade-off curve from one extreme point to the other one. In the first extreme point one has νcl = 1, i.e. the first of inequalities (5.8) is saturated, while 2 2 νout =(1+g )/g and the point is reached for r = 0. In the second extreme point 2 2 the noise νout attains the minimal possible value νout = |1 − g |/g , i.e. the second 2 1+g of inequalities (5.8) is saturated, whereas νcl = 1−g2 and the point is reached for g>1(g<1) by choosing r such that coth r = g (tanh r = g).

Based on the previous results we arrive at an important property of Gaussian quantum operations described by the transformation rules (5.1) and (5.2). Namely, in the plane (νcl,νout) the optimal operations lie in the rectangle deﬁned by the 5.3 Quantum teleportation as a minimal disturbance measurement 87 inequalities 2 1+g 1 ≤ νcl ≤ , (5.14) 1 − g2 |1 − g2| 1+g2 ≤ νout ≤ . (5.15) g2 g2

The left-hand sides of the inequalities follow from the commutation rules (5.6) and cannot be overcome by any operation. On the other hand, the operations that violate either of the right-hand sides of the inequalities add too much noise and therefore they are suboptimal. This can be shown as follows. Consider a quantum operation 2 2 2 1+g for which νout > (1 + g )/g (νcl > 1−g2 ). The inequalities (5.8) then reveal that 2 2 at most νcl =1(νout = |1 − g |/g ). Then, however, we have a better quantum operation given by the teleportation operating in the first (second) extreme point 2 2 2 2 for which νcl =1(νout = |1 − g |/g ) but simultaneously νout =(1+g )/g <νout 1+g2 cl (ν = 1−g2 <νcl). The formulas (5.13), (5.14) and (5.15) are one of the main theoretical results of the present chapter. This is because as we will show in the following section the trade-off (5.13) is optimal on the set of all Gaussian operations described by Eqs. (5.1), (5.2) and (5.6). The trade-off is depicted for several values of the gain g in Fig. 5.2. Before going to the proof of optimality we can answer another important question based on the trade-off (5.13). Up to now we considered Gaussian quantum operations with a fixed gain g. Provided that the trade-off (5.13) is optimal its right-hand side 2 then gives us (after division by g ) the least possible noise νout that can be attained for a given value of noise νcl by any such operation. The fundamental question that can be risen in this context is that if the gain g of operation can be adjusted freely what is its optimal value gopt that gives for a given value of the noise νcl the least possible value of the noise νout. The task was already solved by Ralph [109] who showed that in the non-unity gain teleportation one can adjust for a given value of the noise νcl the gain such that the third of inequalities (5.8) is saturated and therefore such teleportation protocol realizes the sought optimal operation. The trade-off relation (5.13) contains Ralph’s result as a particular instance and can be used to rederive it: Expressing νout as a function of g and νcl using Eq. (5.13) and minimizing it with respect to g one finds the optimal gain for νcl =1tobe 2 gopt = νcl/ νcl − 1thatgivesνout =1/νcl and thus the fundamental quantum 88 Minimal Disturbance Measurements

Figure 5.2: Optimal trade-oﬀ between the output noise νout and the noise in measurement outcomes νcl for a single-mode Gaussian operation with optimal gain 2 g = νcl/ νcl − 1 (solid curve), amplifying operation with g = 2 (dashed curve), unity gain operation with g = 1 (dotted curve) and attenuating operation with g =0.8 (dash-dotted curve). See text for details. mechanical limit given by the third of inequalities (5.8) is indeed saturated. For

νcl = 1 the optimal gain is inﬁnitely large (gopt = ∞) for which one has νout =1.In this case, all the inequalities (5.8) are saturated simultaneously but the operation achieving this regime is unphysical.

5.4 Proofs of optimality

In this section we prove the optimality of the inequalities derived above using two different methods. The optimization task we want to solve can be generally formulated as follows: Find a Gaussian operation described by Eqs. (5.1), (5.2) and (5.6) that 5.4 Proofs of optimality 89 for a given gain g and a given amount of added noise in the measurement outcomes adds the least possible amount of noise into the input state. The optimal operation will in general depend on the quantities used to quantify the two noises. Here we are interested in optimal operations that add noise symmetrically into the amplitude 2 2 2 2 and phase quadrature, i.e. for which nout,x = nout,p and ncl,x = ncl,p .As we will show below, this requirement is satisﬁed if we take the sums (5.7) of the variances of the noise operators ncl,x,ncl,p,nout,x and nout,p to quantify the noise in the measurement outcomes and the noise added into the input state, respectively.

This is a consequence of the fact that for any Gaussian operation that is asym- 2 2 2 2 metric in x and p variables, i.e. for which nout,x = nout,p and ncl,x = ncl,p , there is always a symmetric Gaussian operation giving the same values of νcl and

νout. This statement can be proved in the following way. Suppose we have the asymmetric operation described by the formulas

xout = g(xin + nout,x),xcl = xin + ncl,x,

pout = g(pin + nout,p),pcl = pin + ncl,p. (5.16)

Assume that, in addition, we have at our disposal another asymmetric operation that is obtained from the previous one by placing it in between one phase shifter at the input and two phase-shifters at the outputs. The ﬁrst phase shifter interchanges the input quadratures as follows, xin →−pin and pin → xin, and the two phase shifters on the output modes perform the inverse transformation xi → pi and pi →−xi, i =out, cl. Taking all the above transformation rules together the entire operation is described by the following rules:

xout = g(xin + nout,p),xcl = xin + ncl,p, − − pout = g(pin nout,x),pcl = pin ncl,x. (5.17)

The prime was used merely to express that the noise operators in Eq. (5.17) are completely independent on and therefore completely uncorrelated with the unprimed noise operators in Eq. (5.16). The variances of the primed and unprimed noise opera- 2 2 2 2 tors are, however, identical, nout,i = (nout,i) and ncl,i = (ncl,i) , i = x, p.The desired symmetric operation can then be constructed from the operations (5.16) and (5.17) by placing them into two arms of a balanced Mach-Zehnder interferometer. At the ﬁrst balanced beam splitter of the interferometer the input quadratures are 90 Minimal Disturbance Measurements √ − mixed with the quadratures√ x0 and p0 of an√ auxiliary mode 0 as√xin =(xin x0)/ 2, pin =(pin − p0)/ 2andx0 =(xin + x0)/ 2, p0 =(pin + p0)/ 2. The quadratures xin, pin and x0, p0 are then used as inputs into the operation (5.16) and (5.17), respectively. The quadratures at outputs of the operations xin, pin, x0 and p0 are ﬁnally superimposed on the second balanced beam splitter of the interferometer at one outcome of which one has

xin + x0 xout = √ = g(xin +ñout,x), 2 pin + p0 pout = √ = g(pin +ñout,p), (5.18) 2 √ √ − wheren õut,x =(nout,x + nout,p)/ 2andñout,p =(nout,p nout,x)/ 2. Further, two pairs of the commuting variables xcl,in pcl,in and xcl,0 pcl,0 representing the output of the operation (5.16) on mode “in” of the interferometer and the operation (5.17) on mode 0, respectively, give after averaging a new pair of commuting variables

xcl,in + xcl,0 xcl = √ = xin +ñcl,x, 2 pcl,in + pcl,0 pcl = √ = pin +ñcl,p, (5.19) 2 √ √ − wheren ˜cl,x =(ncl,x + ncl,p)/ 2andñcl,p =(ncl,p ncl,x)/ 2. As the primed and 2 the unprimed noise operators are uncorrelated one immediately finds that nõut,x = 2 2 2 nõut,p as well as n˜cl,x = n˜cl,p and therefore the new operation described by Eqs. (5.18) and (5.19) is symmetric with respect to x and p. Moreover, calculating 2 2 the noises (5.7) for the new operation yieldsν õut = nõut,x = νout andν ˜cl = n˜cl,x = νcl which completes the proof.

5.4.1 Proof I

For the sake of simplicity of mathematical formulas occurring in the proofs of optimality of the trade-oﬀ (5.13) we will work with rescaled operators of added noises

mout,x ≡ gnout,x,mout,p ≡ gnout,p. (5.20) 5.4 Proofs of optimality 91

2 Using these new operators one can write νout = σout/g ,where

2 2 mout,x + mout,p σout ≡ , (5.21) 2 and the trade-oﬀ (5.13) whose optimality is to be proved then reads 2 2 σout =(1+g )νcl − 2g νcl − 1. (5.22)

T It is convenient to introduce the column vector τ =(ncl,x,mout,x,ncl,p,mout,p) . In this notation all the commutators (5.6) can be rewritten in the compact form

[τi,τj]=2iΓij,where 0 −G 1 g Γ= ,G= . (5.23) G 0 g −(1 − g2)

Since the gains of considered Gaussian operations are ﬁxed the ﬁrst moments of the noise operators vanish, i.e. τ = 0, where the symbol denotes averaging over the input state ρaux of the auxiliary modes. Consequently, the studied operations are completely characterized by the 4 × 4 real symmetric noise matrix N with elements

Nij = {τi,τj},where{A, B}≡(AB + BA)/2. The commutation rules (5.6) then impose a speciﬁc uncertainty principle on the noise matrix N that reads

N + iΓ ≥ 0. (5.24)

Now we want to ﬁnd such of the considered quantum operations that gives for a given noise νcl minimum possible noise σout. This task can be equivalently reformu- lated as follows:

minimize f(N)=aνcl + bσout, (5.25) N under the constraint (5.24). The coefficients a, b ≥ 0 (except for the case a = b = 0) control the ratio between the noise in the measurement outcomes and in the output state. The optimization task (5.25) is a typical example of the so-called semidefinite programme (SDP) [130]. Recently, also other important problems in quantum information theory were formulated and solved as semidefinite programmes ranging from separability criteria [42, 75] and optimization of completely positive 92 Minimal Disturbance Measurements maps [6] to optimization of teleportation with a mixed entangled state [131] or finding optimal POVMs for quantum state discrimination [76].

The SDPs are generally difficult to solve analytically and we are often forced to use numerical methods. However, in the case of the problem (5.25) we are able to find the solution analytically. This can be done in two steps following the standard strategy employed, for instance, in [6, 57]. In the first step we guess the analyt- ical form of the solution of the problem (5.25) while in the second step we prove its optimality. The first step has been already done in the previous section where we surmised the solution of the problem (5.25) to be given by the non-unity gain teleportation described by Eqs. (5.9)-(5.11). Calculating the operators (5.20) using Eqs. (5.10) and substituting them together with the operators (5.11) into the defi- nition of the noise matrix N we arrive at the noise matrix for the teleportation in the form:

Ntel = A ⊕ A, (5.26) where A is the symmetric 2 × 2 matrix with elements 2 A11 = νcl,A12 = A21 = gνcl − νcl − 1, 2 2 A22 =(1+g )νcl − 2g νcl − 1, (5.27) where νcl =cosh(2r). Since the matrix (5.26) is manifestly invariant under the exchange of subscripts x and p the noise is added symmetrically into the amplitude and phase quadrature as required and the non-unity gain teleportation is indeed a good candidate for the optimal operation.

In order to prove optimality of the matrix (5.26) we can proceed along the lines of the proof of optimality of multicopy asymmetric cloning of coherent states [57]. The proof relies on finding a certain Hermitean positive semidefinite 4 × 4 matrix Z that satisfies for any admissible matrix N the condition Tr(ZN)=f(N). From the condition Z ≥ 0 and the constraint N + iΓ ≥ 0 then immediately follows a lower bound on the functional f(N) that is to be minimized, f(N)=Tr(ZN) ≥ −iTr(ZΓ). If, in addition, Z satisfies the condition

Z(Ntel + iΓ) = 0, (5.28) the lower bound is saturated by the matrix (5.26) and therefore the corresponding 5.4 Proofs of optimality 93 quantum operation is optimal.

The matrix Z we are looking for can be taken in the block form [57] 1 PiQ Z = , (5.29) 2 −iQ P where P and Q are real symmetric 2 × 2 matrices. The condition (5.28) gives rise to the following set of equations for the matrices P and Q

QΓ=PA, QA = P Γ. (5.30)

The matrix P is determined solely by the condition Tr(ZN)=f(N)thatgives P = diag(a, b). This is because Eqs. (5.30) do not impose any further restriction on P as they provide the equation P (AΓ−1A − Γ) = 0 for the matrix P that is satisﬁed by any P due to the equality AΓ−1A = Γ. Having the matrix P in hands one can now substitute it into the equation Q = PAΓ−1 derived from the ﬁrst of Eqs. (5.30) that leads to the matrix Q in the form: ⎛ ⎞ 2 2 a νcl − g νcl − 1 a νcl − 1 Q = ⎝ ⎠ . 2 2 2 b 2gνcl − (1 + g ) νcl − 1 b g νcl − 1 − νcl (5.31)

For our guess (5.26) the coefficients a and b are not independent but instead they are tied together by a specific relation that can be calculated by minimizing the functional f(N) under the constraint (5.13). Using the standard method of Lagrange multipliers one then finds the relation to be 2 2 2 a νcl − 1=b 2gνcl − (1 + g ) νcl − 1 , (5.32) which reveals that the matrix Q is indeed symmetric. It remains to check the positive semidefiniteness of the matrix (5.29). Since we require a, b ≥ 0 (except for the case a = b = 0) the expression in the square brackets on the right hand side of Eq. (5.32) must be nonnegative. This condition is not satisfied exactly by those operations 1+g2 cl ≤ which violate the inequality ν 1−g2 . Since, however, these operations have been already ruled out from our considerations as being suboptimal it is sufficient to restrict ourselves to operations satisfying this inequality. For these operations 94 Minimal Disturbance Measurements three different cases must be distinguished in dependence on the value of the noise

νcl.

1) If νcl = 1 then Eq. (5.32) implies b = 0 whence P = Q = diag(a, 0). The eigenvalues of the matrix Z then read as α1,2,3 =0andα4 = a>0 and therefore Z ≥ 0. 2 1+g 2) For νcl = 1−g2 one ﬁnds a = 0 using Eq. (5.32) that gives P = diag(0,b)and Q = diag(0, ±b) where the upper (lower) sign holds for g>1(g<1). One can again directly calculate the eigenvalues of the matrix Z in the form β1,2,3 =0and β4 = b>0 and therefore Z ≥ 0. 1+g2 3) In the intermediate case when 1 <νcl < 2 one has a>0 and simultaneously 1−g √ − 1 − 1 b>0. This allows one to introduce the matrix V = 2diag(P 2 ,P 2 )andto transform the matrix Z as − 1 − 1 † IiP2 QP 2 Z1 = V ZV = − 1 − 1 , (5.33) −iP 2 QP 2 I where I is the 2 × 2 identity matrix. The speciﬁc feature of the transformation

(5.33) is that if Z1 is positive semidefinite then also Z is positive semidefinite and it is thus sufficient to prove the positive semidefiniteness of the matrix Z1.Performing † the similarity transformation Z2 = UZ1U where 1 IiI U = √ , (5.34) 2 iI I

− 1 − 1 the matrix Z1 is brought into the block diagonal matrix Z2 = diag(I+P 2 QP 2 ,I− − 1 − 1 P 2 QP 2 ) whose eigenvalues are easy to ﬁnd in the form γ1,2 =0andγ3,4 =2. Consequently, Z2 ≥ 0 and therefore also Z ≥ 0 which completes the proof.

5.4.2 Proof II

There is an alternative way of proving the optimality of the noise matrix (5.26).

The proof relies on the mapping of the noise operators mout,x, mout,p, ncl,x and ncl,p onto the quadratures in the non-unity gain teleportation. The search for the optimal noise matrix then boils down to searching a suitable two-mode state shared in the non-unity gain teleportation. Suppose we have a noise matrix N, i.e. a real symmetric 4 × 4 matrix satisfying 5.4 Proofs of optimality 95 the uncertainty principle N + iΓ ≥ 0. Assume in addition, there is a real regular 4 × 4 matrix M satisfying the condition

Γ=MΩM T, (5.35) which means that M realizes mapping between the commutation rules for noise operators [τi,τj]=2iΓij and the standard canonical commutation rules [ξi,ξj]= T 2iΩij,whereξ =(xA,pA,xB,pB) is the vector of quadratures and 01 Ω=J ⊕ J, J = (5.36) −10 is the standard symplectic matrix. Provided that such a matrix M exists we can associate with any admissible noise matrix N a certain real symmetric 4 × 4 matrix −1 T −1 VAB = M N M , (5.37)

that can be shown to satisfy the standard Heisenberg uncertainty principle VAB + iΩ ≥ 0 and therefore to be a covariance matrix of a two-mode state. This can be shown as follows. Expressing the left hand side of the Heisenberg uncertainty principle using the formulas (5.35) and (5.37) one finds V + iΩ=M −1(N + AB T −1 | | iΓ)(M ) . Taking now the spectral decomposition N + iΓ= i μi μi μi ,where μ ≥ 0 are eigenvalues and |μ are corresponding eigenvectors of the matrix N + i i | | | | −1| |2 ≥ | iΓ, one finds that ψ VAB + iΩ ψ = i μi ψ M μi 0 for any vector ψ and therefore VAB + iΩ is indeed positive semidefinite whence VAB is a two-mode covariance matrix. Since VAB is a covariance matrix its elements can be written as

(VAB)ij =Tr[ρAB{ξi,ξj}], i, j =1,...,4, where ξi, i =1,...,4 are components of the vector ξ of standard quadratures and ρAB is a state of two modes A and B.A natural realization of the vector of quadratures ξ is by the linear relation

ξ = M −1τ. (5.38)

In this case the state ρAB coincides with the state ρaux over which the averaging in the deﬁnition of the noise matrix N is performed.

It remains to show that a regular matrix M satisfying the condition (5.35) exists. Non-unity gain teleportation provides such a matrix that, in addition, proves to be 96 Minimal Disturbance Measurements suitable for minimization of the functional (5.25). By writing Eqs. (5.10) and (5.11) in the matrix form (5.38) one ﬁnds the matrix M in the non-unity gain teleportation to be a regular matrix of the form: ⎛ ⎞ 1000 ⎜ ⎟ ⎜ ⎟ ⎜ g 010⎟ M = ⎜ ⎟ . (5.39) ⎝ 0 −100⎠ 0 −g 01

The problem of minimization of the functional (5.25) over the noise matrices N is then transformed into the problem of ﬁnding a two-mode state ρAB (with covariance matrix VAB) shared in the non-unity gain teleportation that minimizes the functional (5.25). Substituting from Eqs. (5.10) and (5.11) into Eq. (5.25) using the deﬁnitions

(5.7) one can express the functional (5.25) as the trace f(N)=Tr(ρABO), where 1 2 2 { } 2 2 2 O = a g xA +2g xA,xB + xB + g pA 2 − { } 2 2 2 2g pA,pB + pB + b xA + pA . (5.40)

The operator O is lower bounded by O ≥ min[eig(O)] which implies that the functional f(N) is lower bounded by f(N) ≥ min[eig(O)]. This lower bound is saturated if the state ρAB is an eigenstate of the operator (5.40) corresponding to its lowest eigenvalue. The operator (5.40) can be diagonalized by the two-mode squeezing transformation S described in the Heisenberg picture by Eq. (5.9). Choosing the squeezing parameter as 2ga tanh (2r)= (5.41) a (g2 +1)+b the operator (5.40) is diagonalized to the form

† O = SOS =2(xnA + ynB)+x + y, (5.42)

2 2 − where ni =(xi + pi )/4 1/2, i = A, B are standard photon number operators and x, y ≥ 0. Inserting the formula (5.42) into the expression for f(N)=Tr(ρABO)one † † obtains that f(N)=Tr(SρABS O ) which is obviously minimized if SρAB,optS =

|0000|,where|00 is the vacuum state. Hence, the optimal state ρAB,opt is the two-mode squeezed vacuum state with the squeezing parameter given by the formula (5.41). Thus, we arrived in a diﬀerent way at the conclusion that the non-unity gain 5.5 Linear optics scheme 97 teleportation with shared two-mode squeezed vacuum state with a properly chosen squeezing represents optimal Gaussian quantum operation that for a given gain g and noise νcl introduces the least possible noise νout.

5.5 Linear optics scheme

Non-unity gain teleportation is not the only scheme that saturates the optimal trade-oﬀ (5.13). As shown in Ref. [5] there are at least two other schemes that can accomplish a minimal disturbance measurement. One other strategy is to use optimal 1→2 Gaussian cloning followed by a joint measurement between one of the clones and the anti-clone. However, a much simpler strategy which will be investigated in the following achieves the optimal bound using only linear optics, homodyne detection and feed-forward. The setup is depicted in Fig. 5.3. In this

xcl pcl

PM G PM ESA - + B G AM AM

50:50 BS - |a> PBS 99/1BS r in IN BS HWP D(x,p) BS(R,T) A LO Minimum Disturbance Measurement

Figure 5.3: The Experimental Scheme. AM: Amplitude Modulator; PM: Phase Modulator; |α: The incoming coherent state to be measured; HWP: Half Wave Plate; PBS: Polarizing Beam Splitter Cube; BS: Beam Splitter; D(x,p): Displace- ment operation; B: Auxiliary beam; BS(R,T): Variable beam splitter with reflectivity R and transmissivity T for intensity; A: Vacuum mode; G: electronic gains; xcl,pcl: classical measurement outcomes; ρ: output state; LO: Local Oscillator Beam; ESA: Electronic Spectrum Analyzer. 98 Minimal Disturbance Measurements scheme the input mode “in” is mixed with an auxiliary√ vacuum mode A on√ an unbalanced beam splitter with amplitude reflectivity R and transmissivity T (R + T = 1). The reflected mode “in” and the transmitted mode A are described by the following quadratures √ √ √ √ (0) (0) x = Rxin + Tx ,p= Rpin + Tp , in √ √ A in √ √ A − (0) − (0) xA = Txin RxA ,pA = Tpin RpA . (5.43)

The mode “in” is then superimposed with another vacuum mode√ B on a balanced (0) beam splitter and the quadratures x1 ≡ x =(x + x )/ 2andp2 ≡ p = √ in in B A − (0) (pin pB )/ 2 are measured at its outputs. After rescaling the measured quadratures x1 and p2 by the factor 2/R we arrive at the variables xcl and pcl in the form (5.2), where √ √ (0) (0) (0) − (0) TxA + xB TpA pB ncl,x = √ ,ncl,p = √ . (5.44) R R

The outcomes of the measurement√ x ¯1 andp ¯2 are then used√ to displace the mode → → A as xA xout = xA + 2Gx¯1 and pA pout = pA + 2Gp¯2,whereG is the normalized electronic gain. The output quadratures xout and pout are then of the form (5.1), where the gain reads as √ √ g = T + G R, (5.45) and √ √ − (0) (0) G T R xA + GxB nout x = , , g √ √ − (0) − (0) G T R pA GpB nout p = . (5.46) , g

Inserting now Eqs. (5.44) and (5.46) into the deﬁnitions (5.7) one ﬁnds

√ √ 2 − 2 1+T G T R + G νcl = ,νout = . (5.47) 1 − T g2 5.6 Experiment 99

2 Expressing g νout using the second of equations (5.47), substituting in the obtained formula for G employing Eq. (5.45) and making use of the formula R + T =1and the first of equations (5.47) we finally confirm that the noises (5.47) indeed satisfy the optimal trade-off (5.13).

Equations (5.47) immediately allow us to derive the output noise νout for the case with optimized gain. Making use of the ﬁrst of equations (5.47) in the formula 2 gopt = νcl/ νcl − 1 one gets the optimal gain

1+T gopt = √ . (5.48) 2 T

Substituting further the latter expression√ for√ gopt into Eq. (5.45) one ﬁnds the elec- − tronic gain Ggopt to be Ggopt = 1 T/2 T . Inserting now Eq. (5.48) and the obtained expression for Ggopt into the second of equations (5.47) we ﬁnally arrive at the output noise in the linear optics scheme with the feed-forward in the form

1 − T νout = . (5.49) 1+T

Comparison of the formula for νout just obtained with the formula for the noise

νcl given in Eq. (5.47) reveals that νclνout = 1 holds and therefore the third of inequalities (5.8) is saturated.

5.6 Experiment

After the theoretical part where we proved the optimality of the scheme depicted in Fig. 5.3 we now proceed by describing the actual experiments demonstrating minimal disturbance measurements (MDMs) on coherent states. As mentioned above, a MDM was recently performed on coherent states [5] in which the output signal had the same mean value as the input. This is a particular case and will be reported on after he present work which extends this previous work to a more complete experimental study of MDMs of coherent states namely to the cases where the mean value of the input state is not preserved. We systematically investigated two cases. First, we considered the non-unity gain phase-insensitive MDM where the gain was optimized according to the formula (5.48). In the second case we studied the non-unity gain MDM where the optical gain was fixed. The experimental setup is depicted in Fig. 5.3. In both experiments we used a 100 Minimal Disturbance Measurements stable continuous wave Nd:YAG laser from Innolight oscillating at 1064 nm wave- length which could deliver up to 500 mW of power in one transversal mode. The signal beam, local oscillator (LO) beam and the auxiliary beam (B) were all obtained from this source enabling high quality mode matching between the beams and hence allowing very efficient quantum measurements. We generated the coherent states by placing concatenated electro-optical phase and amplitude modulators in the beam path. We applied a signal to the modulators at 14.3 MHz which created sidebands with respect to the laser carrier, meaning that some of the photons from the carrier were transferred to these sidebands. Hence we defined our coherent state to reside at the sideband frequency of 14.3 MHz and having a 100 kHz bandwidth. In this operating window the dark noise of the detectors was negligible and the locking loops for the amplitude and phase quadrature measurement at the homodyne detector were optimized to operate stably. In addition, the feed-forward loop was chosen to function most efficiently inside this window. After the preparation, the coherent state impinges on a beam splitter BS(R,T) with variable beam splitting ratio. The variable beam splitter is realized by placing a polarizing beam splitter behind a half wave plate in the beam path. The reflected part of the input state is mixed with an auxiliary beam of equal intensity on a 50:50 beam splitter. By directly measuring the output of the beam splitter and subsequently constructing the sum and difference photocurrents, the amplitude and phase quadratures of the reflected light are simultaneously measured [87], thus information about the input is acquired. The classical measurement outcome is amplified electronically and fed to another pair of amplitude and phase modulators which are traversed by another auxiliary beam. This beam is then coupled into the remaining part of the signal beam thus accomplishing a lossless displacement operation. The output state ρ is finally analyzed by making use of a homodyne detector. A bright local oscillator beam interferes with the signal beam on a beam splitter, and the conjugate quadratures, amplitude and phase, are stably measured by locking the relative phase between the two beams employing standard electronic feedback techniques. The first and second moments of the amplitude and phase quadratures of the output state as well as the input state are thus measured using an electronic spectrum analyzer. The input state was characterized by switching off the displacement operation, measuring the resulting output state and inferring the input state by carefully characterizing all the losses including the detector and beam splitter losses. In all the measurements the central frequency was 14.3 MHz, the resolution bandwidth was 100 kHz and the 5.6 Experiment 101 video bandwidth was 30 Hz. In the first experiment the optical gain of our measurement device depended on the transmission of the beam splitter BS(R,T) according to Eq. (5.48) thereby reducing the noise νout to a minimum possible value. This was achieved by tuning the variable beam splitter to various transmission/reflection ratios and correspondingly adjusting the feed-forward electronic gain to obtain the desired optimal optical gain. In order to quantify our measurement device and to verify that it indeed measures the coherent state optimally with minimal disturbance we needed to determine the 2 2 2 2 added noises nout,x , nout,p , ncl,x and ncl,p . For this purpose we measured the

Signal to Noise Ratio of the input RS/N,in and the Signal to Noise Ratio of the output RS/N,out for the conjugate amplitude and phase quadratures, respectively. 2 2 The variances of added noises in the output quadratures nout,x and nout,p then read as

RS N in x RS N in p 2 / , , − 2 / , , − nout,x = 1, nout,p = 1. RS/N,out,x RS/N,out,p (5.50)

2 2 The variances ncl,x and ncl,p of the noises added into the classical measurement outcomes were calculated from the measured transmittance T of the variable beam splitter. By construction, the device should exhibit identical transmittance for the amplitude and phase quadratures, and this was explicitly confirmed by measurement. We thus have 2 2 1+T νcl = n = n = . (5.51) cl,x cl,p 1 − T Simultaneously changing the transmittance T and the electronic gain G enabled us to adjust at will the degree of disturbance of the measured quantum state. The experimental results are summarized in Fig. 5.4. We get excellent agreement between theory and experiment and we conclude that the measurement apparatus operates at the fundamental limits imposed by quantum theory. In our second experiment we demonstrated the MDM for a fixed optical gain. After analyzing our input state as in the previous experiment we connected the feedforward loop and adjusted the appropriate electronic gain which guaranteed the desired fixed optical gain. This means that for a particular beam splitter transmission the feed-forward electronic gain will increase as the desired optical gain increases. If, in addition, the desired optical gain is less than the beam splitter transmission 102 Minimal Disturbance Measurements

2 Figure 5.4: Experimental results for MDM with optimized gain. Variances nout,x 2 (left figure) and nout,p (middle figure) of the added noises in the amplitude and 2 2 phase quadratures are plotted against the quantity ncl,x and ncl,p characterizing the noise added into the outcomes of simultaneous measurement of amplitude and phase quadratures. We also make use of Eq.(7) to plot νout against νcl (right figure). The solid line represents the theoretical relation νout =1/νcl.Theexperimental data were obtained by taking into account the detection efficiency of 83% at the homodyne detector. The error bars in the x-axis steam from the uncertainty in the measurement of the beam splitter transmission (2% deviation). The error bars in the y-axis are caused by 0.1 dB relative measurement accuracy of the Electronic Spectrum Analyzer and 0.1 dB deviation of the homodyning efficiency.

then a deamplification of the optical signal is required which was achieved by adding a π phase shift in the electronic feed-forward loop and by making use of destructive interference which resulted in optical deamplification. This particular operation is actually suboptimal meaning that the measurement outcomes lie outside the opti-√ mality window which follows directly from the fact that in this case where g< T , 2 2 calculation of νcl reveals that νcl > (1 + g )/|1 − g |. Using a similar procedure as before we measured the first and second moments for the amplitude and phase quadrature of the output signal. Having measured the input and output states we then calculate all the necessary added noises by means of Eqs. (5.50) and (5.51).

We performed this kind of measurement for three fixed optical gains of g2 = 0.5, 0.8, 1.3 and the results are summarized in Fig. 5.5. Here we have to stress again that optimal MDMs with a fixed optical gain lie only within a specific region of the

(νcl,νout)-plane (gray shaded region in the ﬁgures). The boundaries of these regions depend solely on the optical gain and can be easily determined from Eqs. (5.14) and (5.15). For the optical gains considered here these optimality windows explicitly 5.6 Experiment 103

(a) 4 4 4 Optimality Window 3 3

out,x

out,p

2 2

out

n 2 2 n 2

1 1 1

0 0 12345 12345 0 12345 2 2 n ncl,x ncl,p cl

(b) 3 3 3 Optimality Window

2 2 2

out,x 2

out

out,p

n 2

n 1 1 1

0 0 0 1 5 101520 1 5 1015 20 1 5 101520 2 2 n ncl,x cl ncl,p (c) 2 2 2

Optimality Window

out,x

out,p

2 2

1 1 1out

n n

0 0 0 1 246810 1 246810 1 246810 2 n 2 n cl,x ncl,p cl

Figure 5.5: Experimental results for MDM with fixed gains g2 =0.5(a),g2 =0.8 2 2 2 (b) and g =1.3 (c). Variances nout,x (left figure) and nout,p (middle figure) of the added noises in the amplitude and phase quadratures are plotted against the 2 2 ncl,x and ncl,p characterizing the noise added into the outcomes of simultaneous measurement of amplitude and phase quadratures. We also make use of Eq.(7) to plot νout against νcl (right figure). The optimality windows (gray shaded regions) are determined by Eqs. (5.14) and (5.15). The solid line represents the theoretical trade-off (5.13). The experimental data were obtained by taking into account the detection efficiency of 83% at the homodyne detector. The error bars in the x-axis stem from the uncertainty in the measurement of the beam splitter transmission (2% deviation). The error bars in the y-axis are caused by 0.1 dB relative measurement accuracy of the Electronic Spectrum Analyzer and 0.1 dB deviation of the homodyning efficiency. 104 Minimal Disturbance Measurements read as follows:

2 1 ≤ νcl ≤ 3, 1 ≤ νout ≤ 3forg =0.5, 1 9 2 1 ≤ νcl ≤ 9, ≤ νout ≤ for g =0.8, 4 4 23 3 23 2 1 ≤ νcl ≤ , ≤ νout ≤ for g =1.3. 3 13 13

The left hand sides of these inequalities are dictated by the commutation relations (5.6) and cannot be overcome by any operation, i.e. the theoretical trade-off in the figures never lies below or to the left of the optimality window. However, the trade-off (5.13) lies to the right of the optimality window for sufficiently large noise

νcl. In this case the operation saturating the trade-oﬀ is suboptimal and a better performance is obviously obtained by the operation corresponding to the minimum of the trade-oﬀ which also corresponds to the bottom right corner of the optimality window. Note that the only exception occurs in the unity gain regime (g =1) demonstrated in [5] where the optimality window is not bounded from the right, i.e.

1 ≤ νcl ≤∞, 0 ≤ νout ≤ 2. (5.52) 2 Thus, since νcl − νcl − 1 ≤ 1 for any νcl ≥ 1 the optimal output noise νout for g = 1 always satisfies the second inequality in (5.52) and therefore in the unity gain regime the trade-off (5.13) never leaves the optimality window. As can be seen in Fig. 5.5, the obtained experimental trade-offs are in very good agreement with the theory which shows that our measuring apparatus indeed realizes optimal non-unity gain Gaussian partial estimation of coherent states. In particular, the noises added to the phase and amplitude quadratures are practically the same, which confirms that the measurement procedure introduces isotropic phase- independent noise into the estimated state as well as the post-measurement state.

5.7 Discussion and conclusions

Our experimental minimal disturbance measurement with fixed non-unity gain finds a direct application in the context of optimal Gaussian individual attacks on coherent state quantum key distribution (QKD) with heterodyne detection and direct reconciliation [136]. The optimal trade-off demonstrated by us determines the min- 5.8 Unity Gain MDM with an application 105 imum added noise in the outcomes of simultaneous measurement of complementary quadratures a potential eavesdropper can reach for a Gaussian quantum channel with a fixed gain and a fixed phase-insensitive added noise. In the QKD terminol- ogy it means that the minimal disturbance measurement provides an eavesdropper with maximum possible information that can be gained from an individual Gaussian attack in the heterodyne-based coherent state QKD protocol with direct reconciliation. Recently, the similar problem has been studied theoretically directly in the context of QKD [90] and another form of the above mentioned trade-off was found (Eq. (11) of Ref. [90]). The proofs of optimality presented here, however, follow completely different strategies in comparison with those presented in [90] and more importantly we saturate the optimal trade-off between added noises experimentally. In this section we have extended the concept of the phase-insensitive MDM for coherent states to the non-unity gain regime. We have given a complete theoretical as well as experimental study of this MDM for two different scenarios. First, we have found the non-unity gain MDM assuming fixed optical gain. In the second scenario we considered MDM with optimized gain. We have shown that both MDMs can be realized by a scheme consisting of only linear optical elements and a feedforward and we implemented the scheme experimentally. We have experimentally reached theoretical limits in both scenarios. Our results give answer to a fundamental question of how much noise will be in the measurement outcomes from the non-destructive measurement of a coherent state provided that it is represented by a single-mode Gaussian channel with a given optical gain and isotropic added noise.

5.8 Unity Gain MDM with an application

In this section we investigate the tradeoff between information gain and state disturbance for completely unknown coherent states in the special case of the unity gain regime. Under the assumption that the Gaussian statistic must be preserved we derive the optimal tradeoff, stated in terms of an inequality using appropriate measures for the information gain and state disturbance. We also show that this scheme is capable of increasing the transmission fidelity of some noisy channels, an improvement that will be also demonstrated in the end of this section. Completely unknown pure quantum states described in an infinitely dimensional Hilbert space can be estimated only very poorly based on a single measurement. 106 Minimal Disturbance Measurements

However, in quantum communication systems, a priori knowledge is often given. It is e.g. normally known that the state belongs to a certain set each occurring with a certain a priori probability. In this work we assume that the states are taken from a flat distribution of coherent states. With this a priori information at hand, it was recently proven that the state can be optimally estimated using a setup where conjugate quadratures are measured simultaneously using a symmetric beam splitter and two homodyne detectors [71]. However, employing this strategy the coherent state is maximally disturbed. On the contrary, if the unknown coherent state is left untouched, our guess will be completely random but the state will be intact. In the following we will investigate the intermediate cases and hence address the question: For a given information gain what is the minimum disturbance to the coherent state? Consider a coherent state characterised by the amplitude and phase quadrature xîn andp în with [ˆxin, pîn]=i. The state is injected into a machine with a classical output and a quantum output. Assuming phase insensitive operation (meaning that conjugate quadratures are equally well measured and equally disturbed), we derive the tradeoff relation for the added noises from the commutation relations to be [109]

Δ2nˆΔ2mˆ ≥ 1 (5.53) with the constraints Δ2mˆ ≥ 1andΔ2nˆ ≥ (|1−g2|)/g2.Δ2nˆ denotes the uncertainty with which the coherent state can be estimated and Δ2mˆ is the variance of the noise added to the quantum state caused by the measurement. Therefore the inequality in (5.53) dictates that the optimum tradeoff for the added noises is achieved for Δ2nˆΔ2mˆ = 1. However, this relation can only be satisfied when the gain g can be chosen freely to minimize the added noises. Universality of the protocol, however, requires unity gain operation (g = 1) which imposes another restriction to the achievable tradeoff: By noting that the commutation relations for the added noise to the measurement resembles that of quadrature operators, we setm ˆ x =ˆx1 and mˆ p =ˆp1 wherex ˆ1 andp ˆ1 are quadratures of the ancilla. The commutation relations forces the remaining noise operators into the superpositionsn ˆx =ˆx1 +ˆx2 andn ˆp = pˆ1 − pˆ2,whereˆx2 andp ˆ2 likewise are quadratures of the other ancilla. The task is now to simultaneously minimise the variance ofx ˆ1 +ˆx2 andp ˆ1 − pˆ2 for given variances ofx ˆ1 andp ˆ1. For quantification of the information gain, we use the estimation fidelity G, 5.8 Unity Gain MDM with an application 107 which is the phase space overlap between the input state and the state that can be prepared based on the classical information and the a priori information [60, 71]. The disturbance is quantified by the transfer fidelity F , which quotes the overlap between the input state and the post measurement state [60, 71]. With unity gain the fidelities are simply given by G =2/(3 + Δ2mˆ )andF =2/(2 + Δ2nˆ). In terms of the fidelities the tradeoff relation therefore reads:

G F ≤ (5.54) 2 1 − G − (1 − G)(1 − 2G)

The optimal tradeoff between classical and quantum information can be implemented using various systems. One strategy is to use teleportation [60] as discussed earlier, where the measurement outcomes at the Bell state analyzer yield the classical information whereas the teleported state serves as the quantum state after disturbance (see Fig. 5.6a). The whole range of optimal tradeoffs can then be achieved by tuning the amount of entanglement. Alternatively, a continuous variable asymmetric quantum cloner [55] can be employed as illustrated in Fig. 5.6b: The anti-clone is mixed with one of the clones on a symmetric beam splitter, and subsequently amplitude and phase quadratures are measured in the two output ports to retrieve some classical information. The other clone is left unchanged and serves as the output quantum state. By tuning the asymmetry of the cloning machine, and the beam splitting ratio of the measuring beam splitter accordingly, the whole range of maximal tradeoffs can be accessed.

We already showed that a simpler approach can realise the optimal tradeoff. The scheme relying entirely on simple linear optical components and homodyne detectors which was shown earlier is once again depicted inside the dashed box of Fig. 5.6c: The quantum state is partially reflected off a beam splitter with transmission coefficient T . The reflected part of the state is optimally estimated by simultaneously measuring two conjugate quadratures, i.e.x ˆ andp ˆ, and the resulting classical information is partly used to guess the state and partly used to displace the transmitted part of the quantum state after an appropriate scaling of the classical data. Using this√ very simple scheme, the added noises to the quantum state are Δ2nˆ =2(1− T)2/(1 − T )andΔ2mˆ =(1+T )/(1 − T ) and the estimation and 108 Minimal Disturbance Measurements

Figure 5.6: Three methods by which the optimal tradeoﬀ can be established. a) Teleportation. b) Asymmetric cloning followed by a joint measurement. c) Simple feed forward approach. C(1,2): Clones, AC: Anti-clone, D:Displacer, AM: Am- plitude modulator, PM: Phase modulator, PBS: Polarizing beam splitter, BS(T): Variable beam splitter with transmission T, AUX(1,2): Auxiliary beams and LO: Local oscillator. transfer ﬁdelities are found to be

1 − T 1 − T G = F = √ (5.55) 2 − T 2 − 2 T which is saturating inequality (5.54). A distinct difference between the first two approaches in Fig. 5.6a and 5.6b (teleportation and cloning) and the simple approach in Fig. 5.6c is that the latter one does not require any nonlinear interaction. Some similarities between the telepor- 5.8 Unity Gain MDM with an application 109 tation scheme and the feed forward scheme were pointed out in ref. [73], and both schemes have been suggested as potential eavesdropping attacks [109]. However we note that the cloning approach in Fig. 5.6b might be superior for an eavesdropper, since in this protocol the classical information can be extracted at any instance if a quantum memory is available. The experimental procedure is the same as the nonunity gain minimal disturbance measurements. After the information retrieval and displacement, the resulting quantum state is characterised using a standard homodyne detector with a strong local oscillator (LO). The signal and the noise variances of the phase and the amplitude quadratures are measured using a spectrum analyzer with resolution and video bandwidth set to 100 kHz and 30 Hz, respectively. We compute the transfer fidelity by comparing the output to the input state, which was measured using the same homodyne detector in order to make a consistent comparison [60]. The estimation fidelity is calculated from the carefully measured reflectivity of the variable beam splitter. In order to avoid erroneously overestimations of the fidelities the values are corrected to account for detection inefficiencies.

0.9

0.8

F 0.7

0.6

0.5 0 0.1 0.2 0.3 0.4 0.5 G

Figure 5.7: Quantum state fidelity as a function of the estimation fidelity. The optimal tradeoff given by Eq. 5.54 is represented by the solid curve, whereas the dashed curve is associated with the tradeoff taking into account detection inefficiencies. The error bars stem from the inaccuracy in determining the detector efficiencies. 110 Minimal Disturbance Measurements

Many different tradeoffs were realised and the results are shown in Fig. 5.7. The solid curve in Fig. 5.7 represents the optimal tradeoff given by the saturation of inequality (5.54). Deviation from optimality is caused by the inefficiency of the in-loop detector which partly degrades the classical guess and partly imposes additional non-fundamental noise onto the quantum state after measurement. There- fore we paid special attention to the optimization of this measurement: The mode matching efficiency between the auxiliary beam (AUX1) and the signal state was carefully optimized to yield a visibility of 99% and the quantum efficiency of the photo diodes were 95%. Despite the high quality of the homodyne setup, the curve for the optimum tradeoff achievable with the experimental setup is slightly shifted and represented by the dashed curve in Fig. 5.7. In the last part of the chapter we discuss how optimal partial state estimation can be exploited to enhance some communication tasks. Let us consider the following protocol. Alice wants to transmit quantum information which is encoded into a coherent state, and after the transmission Bob receives the quantum state. Such a communication task is always inflicted by loss or noise, hereby corrupting the quantum state and as a result reducing the transmission fidelity. Let us first consider a lossy channel characterized by the transmission coefficient η. In such a channel the transmission loss must be compensated by an amplifier in order to ensure maximal transmission fidelity. The highest transmission fidelity is achieved by amplifying the state before it is injected into the lossy channel. However, by considering the power constraint scenario [31], amplification prior to transmission is not possible and as a result the amplifier must be placed at Bob’s receiving station. This yields a transmission fidelity of F = η. However, by optimally separating the input state into a classical and a quantum channel as demonstrated in this article, the fidelity can be increased: The optimal post measurement quantum state is sent through the lossy channel, whereas the classical information is sent through a classical channel (see Fig. 5.8). At the receiving station Bob displaces the corrupted quantum state based on the information he gains from the classical channel. Obviously there is an optimal separation ratio between the classical and the quantum information for a given attenuation in the channel. This ratio is optimized by setting T = η and we find the optimised fidelity to be F =1/(2 − η). This fidelity is for all values of η larger than the fidelity achievable when only the quantum channel is used. We demonstrate this idea by inserting an attenuator with η = 31% into the channel, which is placed between the variable beam splitter (BS(T)) and the displacement 5.8 Unity Gain MDM with an application 111

Alice p Bob Classical channel

x Noisy/lossy quantum channel D T

Figure 5.8: Schematic illustration of a protocol capable of increasing the fidelity of noisy channels. The displacement operation, D, is similar to the one shown in the long shaded box of Fig. 5.6c, and other experimental details about the preparation and verification follow those of Fig. 5.6c. operation (D) in Fig. 5.6c (or Fig. 5.8). If an amplifier is employed after the channel to compensate for these losses the fidelity is F = 31%. Now using our strategy of dividing the information into classical and quantum as shown in Fig. 5.8, we measure a quantum state fidelity of F =63± 1%, which clearly surpasses the standard amplifier approach. In this experimental run we measured the gains to be 1.00 ± 0.01 and 1.01 ± 0.01 for the amplitude and phase respectively. We now consider a fully transparent channel which adds noise to the signal. We first assume that the nature of this noise is additive and deterministic. If the added noise of the quantum channel exceeds two vacuum units, pure classical communication maximises the fidelity and is therefore the better alternative. However if the added noise is less than two vacuum units, then pure quantum communication becomes advantageous. Therefore, only the two extreme schemes will be relevant. From here on we assume that the noise in the channel is additive and probabilistic. In this case the intermediate scheme also becomes important [110]: We consider a channel which is perfectly transmitting the signal with probability p and fails to transmit it with probability (1 − p). The average fidelity of such a channel is given by F = p. If we now apply the partial estimation approach in front of the channel as illustrated in Fig. 5.8, then the fidelity is given by F = pF +(1− p)G. We find that for 0

5.9 Conclusion

In this section we have extended the discussion on the optimal information-disturbance tradeoff to the continuous variable regime and derived a tradeoff relation for coherent states. In summary we proposed and successfully demonstrated an optimal nonunity gain Gaussian scheme for partial measurement of an unknown coherent state that caused minimal disturbance of the state. The information gain and the state disturbance were quantified by the noise added to the measurement outcomes and to the output state, respectively. We derived the optimal trade-off relation between the two noises and we show that the tradeoff is saturated by nonunity gain teleportation. Optimal partial measurement was demonstrated experimentally using a linear optics scheme with feedforward. Finally, we have demonstrated that our scheme can be used to enhance the transmission fidelity of some noisy channels, rendering our approach as a useful tool in future quantum communication networks. Chapter 6

Noise Erasure

In this chapter we propose and successfully demonstrate a novel approach in which the information leakage to the environment is used to combat excess Gaussian noise coupled to a quantum communication link for continuous variables. In the limit of detecting the outgoing noise completely and knowing the losses in the system, the channel can even approach ideal noiseless transmission. We also demonstrate that a partial measurement on the environment enables preventing a channel from entanglement braking.

6.1 Introduction

Quantum communication boosted by quantum key distribution is fundamentally more secure than traditional classical communication [12]. This fact alone has recently triggered a lot of research and development in quantum communication [65]. However the fragile nature of quantum signals make the realization of many quantum communication tasks far from trivial [5]. When a quantum state interacts with the environment decoherence processes start which gradually wash away the quantum features of the state [146]. This trend pushes one toward the classical regime making decoherence the main obstacle in quantum information protocols. A realistic quantum communication link is lossy and noisy because of the imperfection of the devices being used, thus a transmitted fragile quantum signal will face decoherence. As a consequence we will lose quantum properties of the signal and the advantages of using a quantum channel instead of a classical channel. There are some techniques in which one can maintain the ”‘quantumness”’ of the signals as 114 Noise Erasure they propagate through noisy environments such as quantum error correcting codes or entanglement purification protocols, where one protects quantum states from in- teracting with the environment [13]. However these methods will lose power when the channel is not entanglement preserving. Then such a channel in principle could be substituted by the measurement and repreperation approach of the quantum state. An interesting technique known as quantum erasing enables posteriori recovery of quantum states which were disturbed through a coupling with an external system [121]. The idea of quantum erasing was first tested using single photons [117, 126] and, later extended to the continuous variable regime in Ref. [51], and demonstrated using a squeezed light [2]. By employing erasing with the help of squeezed light, an irreducible loss of continuous variable quantum information can be corrected [94]. In ref. [117], it was demonstrated that quantum erasing is possible even if the coupled system is a strongly mixed state. Quantum erasing has been also recently discussed as a method of environmental assisted quantum state correction for a qubit (qudit) channel [68]. Remarkably, any random-unitary CP map is invertible by quantum erasing which allows to correct any qubit and qutrit channel. Furthermore, it was proved that for higher dimensional spaces, (although the environment is in a pure state) perfect state reconstruction is not generally possible [27]. In practice, it is unfeasible to completely control the environment, especially the complex part which has been evolving before it ever coupled to the system. Rather, a realistic measurement could only access part of the environment which directly couples to the system and average over all the rest of the environment. In general, this will result in reduction of the quality of quantum state correction [26]. Consequently, the most common linear optical coupling process between an unknown coherent state information and a noisy environment cannot be perfectly inverted. But the partial information from the environment can suffice to restore the information content in the complementary quadratures simultaneously with minimum noise penalty [54]. This method is then environmentally assisted quantum information correction, which basically corresponds to a phase-insensitive quantum erasing (without any squeezing) based only on the classical correlations between the system and environment which enables restoration. In principle, it allows decoupling of any disturbing Gaussian noise such that the channel will remain secure for the quantum key distribution employing coherent states. Recently, it was demonstrated experimentally that detection of the environmental noise can protect single-photon quantum channels against entangle- 6.2 Theory 115 ment loss caused by any linear coupling and even reveal entanglement after it was completely lost [120]. A coherent state of modulated laser beam is a common simple information carrier for continuous variable quantum communication, which is encoded simultaneously in both of the complementary amplitude and phase quadratures. In a receiver station, information is easily and efficiently detectable by either homodyne or heterodyne measurement. A common source of decoherence process can be simulated by making use of a basic linear passive coupling mechanism between a signal mode S and noisy mode N which represents Gaussian excess noise in the communication channel. If there is more than two shot units of excess noise then the channel is certainly entanglement breaking. In this chapter we propose a method in which the information leakage into the environment is used to regain the continuous variable quantum information which was buried under loss and noise induced through a linear coupling. Under certain conditions such as knowing the losses in the channel beforehand the noise can be completely decoupled from the channel at the cost of amplification of the state. Having complete access to the environmental leakage modes we show that it is possible to regain the quantum information content encoded into the coherent state perfectly and deterministically by heterodyne detection. If the link is used for quantum key distribution the method allows to reestablish security for this kind of lossy and noisy channel. If no prior information is known about the channel still decoupling of noise can be accomplished, though now only probabilistically and not as good as before, meaning that still some of the noise will be traveling in the channel. We furthermore investigate the possibility of noise erasure in the realistic case of not having complete access to the environment. The result is that if the channel is very well approximating additive Gaussian channel then even very weak portion of the environmental modes is enough to reduce the excess noise in order to restore security. The security is restored although the erasing operation is considered to be untrusted.

6.2 Theory

Lets start with the formal discussion of the scenario as depicted in ﬁgure 6.1a, and modeled in ﬁgure 6.1b. The phase-insensitive Gaussian channel is represented by a mixing process between the signal mode S and scattered noisy mode N in a perfect 116 Noise Erasure

Figure 6.1: a) Many modes from the reservoir interact with the signal causing decoherence. A part of the leakage to the environmental is then measured. This measurement is used for noise erasure. b) The coupling of a signal mode S with noisy mode N in a perfect passive unitary coupling. The environmental measurement is either fedforward or or used for postselection at the receiver in order to accomplish noise erasure. passive unitary coupling as shown in figure 6.1b. With only a partial attempt to mode N by a measurement, we can describe the output modes of the coupling (for any quadrature) by: √ − − XS = ηXS 1 ηXN , √ √ − − XN = γ( 1 ηXS + ηXN )+ 1 γX0, (6.1) where η<1 is the coupling strength and X is the standard amplitude quadrature of the optical field, satisfying the commutation relation [X, P]=2i with the corresponding complementary quadrature P . γ stands for the efficiency of the detection of the noisy mode N in that quadrature. The auxiliary quadrature operator X0 is used to correctly describe the process of a partial measurement, an artificial mode 0 is assumed in the vacuum state for a detector with low electronic noise. For the complementary quadrature, the coupling transformation given by Eq. (6.1) is identical, thus we can discuss below only a single quadrature. If the noisy mode is completely accessible by ideal homodyne (heterodyne) detection then γ =1(γ =1/2). With- out any access to the noisy mode after the coupling, the phase insensitive gain of 1−η the channel is η<1 and phase-insensitive added noise is ΔV = η VN .Itissuffi- cient If ΔV > 2 then such a channel will not be able to preserve entanglement and the security is broken. A more strictly, if ΔV>0.8 the security against collective attacks is lost. ¯ A homodyne measurement of the mode N results in a measured value XN ,which → can be used to make a displacement operation on the signal variable: XS XS + 6.2 Theory 117

¯ gX XN ,wheregX is an arbitrary electronic gain of the feed-forward loop. Optimizing the gain gX in the electronic correction loop, the noisy mode N can be completely decoupled and the amplified signal after the correction becomes √ √ √ − XS = G XS + α G 1X0, (6.2) where G =1/η is the gain of the amplification and α =(1− γ)/γ corresponds to the normalized inefficiency of the detection of the mode N. The added noise after the correction is then ΔV =(1−η)(1−γ)/γ. To practically approach this, without prior manipulation with the noisy mode, the attenuation factor η has to be known.

Remarkably, for arbitrary γ>0, the decoupling is independent of the variance VN and the external excess noise from mode N is substituted by vacuum noise of the mode 0 in vacuum state. Thus, the external excess noise completely erased and cannot propagate further in the channel. If γ>η/(VN + η)thenΔV < ΔV and the added noise is even reduced. In order to decouple noise in both of the quadratures simultaneously, heterodyne detection (γ ≤ 1/2) has to be used. The external excess noise can be also completely decoupled for an arbitrary efficiency γ, now at a cost of an phase-insensitive amplification. In the limit of perfect heterodyne detection (γ =1/2), the added noise approaches ΔV =1− η, corresponding to an ideal phase-insensitive amplification. It was proved that such the channel is not only entanglement preserving but mainly is security preserving for any gain [52]. The added noise cannot vanish in both the quadratures but is for any η always smaller than even the original channel with VN = 1. Thus the unknown original quantum state unfortunately cannot be completely reconstructed by this method. But if the signal after the decoupling is finally measured simultaneously in both the quadratures by an ideal heterodyne detection, then the added noise equals

(1 − η)(1 − γ) ΔV = η + (6.3) het γ

with the gain Ghet =1/(2η). For

1 γ> 2 (6.4) 1+(1−η)VN −η 1+ η(1−η)

− the added noise ΔVhet is smaller than ΔVhet = ((1 η)VN +1)/η associated with 118 Noise Erasure the original direct channel with heterodyne detection. Remarkably, for an ideal erasing measurement (γ =1/2), the added noise is only single vacuum unit. This is exactly equal to the added noise of an ideal lossless and noiseless coherent state communication with heterodyne measurement. Thus not only the external excess noise is perfectly decoupled, but also, remarkably, ideal coherent-state communication in both the quadratures simultaneously can be approached. Thus, even if the quantum state cannot be reconstructed by the feedforward decoupling method, the information content carried by the quantum state which was buried under noise can be regained perfectly. The added noise can be further reduced by optimizing the electronic gain gX to get

(1 − η)(1 − γ)VN ΔVopt = (6.5) η(1 − γ)+γVN

1−η which is now, for arbitrary γ>1, always smaller than ΔV = η VN for any VN > 1 and η>0. The corresponding optical gain is

2 1 (1 − γ)η + γVN Gopt = (6.6) η 1 − γ + γVN

But, in this case, the result depends on VN , thus the noise is not decoupled perfectly and is further propagating through the channel.

There is an interesting alternative to at least probabilistically, decouple the excess noise in the channel without any knowledge about η and VN . Instead of applying deterministic correction, the output signal is post-selected only if the measurement result of the homodyne detection lies in a certain interval −B,B.Ofcourse, the success rate of this probabilistic decoupling method decreases rapidly as B approaches zero.

As the variance VS of the input signal increases, the probabilistic method works less eﬃciently; for information encoded into both the quadratures, the best result is for a coherent state signal (VS = 1). In the limit of post selection interval approach- ing zero B → 0, the added noise approaches

2 1 − η (1 − γ)γη(VN − 1) + VN ΔVpost = 2 (6.7) η (1 + γ(VN − 1)) 6.3 Experiment 119 and the channel gain is going to be close to

2 η(1 − γ + γVN ) Gpost = 2 . (6.8) (1 − γ + γ(ηVN +(1− η)))

The result of the probabilistic method were calculated by employing the covariance matrix formalism [63]. This result fully corresponds to the deterministic correction scheme, in which the total variance of the output noise after feed-forward correction is minimized over electronic gain. The added noise is now a function of the variance

VN (for any γ) in contrast to the previous deterministic method. Using this approach it is impossible to generally decouple the excess noise from the signal. For any γ>0 and VN > 1 it is always smaller than ΔV = VN (1 − η)/η for the original direct channel without decoupling method.

6.3 Experiment

The experimental setup is shown in Fig. 6.2. In the experiment we used a Nd:YAG laser emitting continuous waves at 1064 nm. The experiment consists of four main parts, the sender where the coherent state information is encoded via two EOMs, the noisy quantum channel, the measurement on the leakage on the environment and finally the receiver station. Generation of the coherent states is accomplished by employing concatenated amplitude and phase electro optical modulators in the beam path excited with function generators at 14.3 MHz creating a sideband with respect to the laser carrier. This method has been successfully used before in various experiments to generate pure coherent states. In the experiment we used heterodyne measurement both to measure the leakage to the environment as well as to measure the signal at the receiver station. The coherent state couples to noisy modes through a variable beam splitter with transmission η which is composed of a half wave plate and a polarizing beam splitter. By a simple phase plate rotation we can tune the strength of the coupling between the noise modes and the coherent state. The noise modes are generated by connecting noise sources to another set of amplitude and phase modulators. The amount of excess noise can easily be adjusted by the voltage output amplitude of the noise sources. The leakage modes are measured by detectors having diodes with high quantum efficiency (95%). The halvwaveplate and polarizing beam splitter placed infront of these detectors enable us to control the efficiency of the environmental 120 Noise Erasure

Figure 6.2: Schematic of the proposed protocol. AM: Amplitude Modulator, PM: Phase Modulator, F-gen: Function generator; η, γ: Variable beam splitter. The signal at the sender is generated at the rf sideband by low voltage controlled electro optical modulators. Noise modes from the environment are imposed on to the signal through a beam splitter with transmission η. The reflected portion of the beam is detected by a heterodyne measurement with tunable efficiency controlled by a beam splitter (Environment measurement). The receiver is also a heterodyne detector measuring the two quadratures X and P simultaneously. The data taken from the receiver are processed by the measurement outcome from the environmental measurement. We use either feedforward or postselection strategies to erase the noise in the channel. measurement. We make use of heterodyne measurements both at the environment and the receiver station. For this purpose we make use of auxiliary beams equally bright as the signal we probe. Interfering the signal with an auxiliary beam with a π/2 phase shift and balanced intensities; subsequently the two output are measured with high efficiency and low noise detectors, and the sum and the difference currents are constructed to provide a measure of x and p simultaneously. The information extracted from the environment using this method is then used at the receiver end for the noise erasure procedure. 6.4 Results 121

2 2 a) Amplitude b) Amplitude Phase Phase oise N oise

1 N 1 Added Added

0 0 10 15 20 25 30 35 40 45 0123456789 10

VN VN

Figure 6.3: The results for the deterministic (feedforward) approach. The added noise(ΔVhet) is plotted against amount of excess noise from the environment(VN ). a) The strong coupling regime with η =0.9 b) Weak coupling regime with η =0.1 The solid line depicts the ideal noise erasure, and the dashed line takes into account the measurement imperfections on the environment.The nonunity measurement ef- ﬁciency of the environment plays a bigger role for the weak coupling regime, due to the fact that more information is leaking to the environment when η is small. Without employing erasure by direct detection at the receiver the added noise would have been between 1.11 and 5 for 10

The theoretical concepts developed in the first section of the chapter will now be tested experimentally. First we start with the deterministic erasure protocol introduced as the first method. The feedforward correction scheme: This method assumes that one has prior information about the loss (η) inside the channel. We set the variable beam splitter(η) to two specific values (0.1 and 0.9) corresponding to the analysis of the weak and strong coupling regimes respectively. These transmis- sions were easily obtained by rotating a halfwaveplate located in front of a polarizing beam splitter. Depending on the beam splitter value we use the environment measurement to amplify the signal at the receiver end such that we measure an optical a gain equal to G =1/2η. The total measurement efficiency in the environment was measured to be γ = 46% . Taking this into account we calculate the added noises to be ΔVhet =1.16 for η=0.1 and ΔVhet =1.02 for η=0.9 respectively. These values are slighlty higher than the ideal decoupling of noise from the channel which 122 Noise Erasure

corresponds to a ΔVhet =1.

Excess noise(VN ) could be tuned for detailed analysis by adjusting the amplitude of the noise source generators connected to the two EOMs in the noise mode. The signal power and variances of the signal at the receiver with and without feedforward are then measured using a spectrum analyzer with zero span, 2 second sweep, 10kHz resolution bandwidth and 30 Hz video bandwidth . Such measurements suffice to fully characterise the states due to the Gaussian statistics of x and p. Active electronic feedback loops were implemented at all interferences to ensure stable relative phases. From the power and variance measurements, we estimate the excess noise as well as the added noises associated in the characterisation of our erasure protocol. We correct the data taken into account a measurement efficiency of 80%. The dark noise of the detectors was negligible. The results of the measurements are summarized in Fig. 6.3. Applying the feedforward one can totally decouple the excess noise from the channel when we have a heterodyne communication channel, independent of how high the excess noise contribution VN is. This noise erasure is completely deterministic, one only needs the coupling strength η of the system beforehand and almost ideal noise removal can be achieved by appropriate feedforward manipulation of the data at the receiver. The improvement of the protocol increases as VN increases. Optimized Gain: Now we will investigate the dependence of added noise on the measurement efficiency at the environment. We fixed the excess noise(VN )to25 SNU(Shot-Noise-Unit). By rotating the halfwave plate before the beam splitter in the environment measurement we were able to tune γ, which corresponds to the efficiency of the measurement performed on to the environment. We then optimized the feedforward gain for every γ such that for this particular γ value we minimized the added noise at the receiver end. We performed some measurements by varying γ, and the results are summarized in Fig. 6.4. The solid line corresponds to the optimized theoretical prediction whereas the blue squares and red dots correspond to the amplitude and phase measurements respectively. As seen from the figures perfect decoupling of the noise from the channel is not possible and the noise is still propagating after the optimized feedforward. The probabilistic scheme: Now we consider the case where we don’t have any apriori information about the communication channel. When the losses η in the channel are not known, it is not possible to make use of the feedforward approach for noise erasure. Even now 6.4 Results 123

Figure 6.4: The added noise(ΔVhom) is plotted against the efficiency of the environmental measurement γ. The blue squares and red dots correspond to the amplitude and phase quadrature measurements respectively. The solid line corresponds to the theoretical prediction which is calculated for VN =25andη =0.9. The added noise when measured directly without any noise decoupling procedure would lie at ΔVhom =2.77 The inset graph depicts the channel gain change as a function of γ. it is still possible to achieve decoupling of the excess noise up to some extend by utilizing postselection based techniques. For the postselection approach we took the amplitude and phase quadrature measurements from the environment and receiver in time domain. The radio frequency outputs from the detectors were downmixed at an electronic mixer with a strong electronic local oscillator centered at 14.3 MHz having a power of around 7 dBm. The downmixed signal is then amplified by 50 dB and low pass filtered with cutoff frequency at 150 khz, and the output recorded on the computer (10 million points recorded at 5Ms/s with a measurement window of 500ms.) If the measurement outcome lies in a certain window at the leakage measurement then we tell the receiver 124 Noise Erasure to keep the measurement, otherwise the measurement outcomes are discarded. The measurement window corresponded to data centered around the origin at the environmental station. By changing the bandwidth of the postselection window we change the success rate of the protocol, as well as the amount of decoupling of the excess noise from the communication link. The coherent state level was calibrated by switching off the noise generators which imposed the excess noise on to the coherent input signal. As seen from the results in Fig. 6.5 and 6.6, perfect decoupling is not possible however the protocol holds the following advantages: firstly we do not require any apriori information about the channel and secondly we are not revealing any information when communicating with the receiver we are just telling when to measure and when not to.

Figure 6.5: The added noise(ΔVhet) plotted against the logarithm of success probability for the amplitude quadrature. The solid curve corresponds to the theoretical calculations. The straight green line corresponds to measuring directly at the receiver station with heterodyning ΔVhet =4.55 for VN = 31. The blue line across the unity added noise line corresponds to a perfect deterministic decoupling of excess noise. 6.4 Results 125

Figure 6.6: The added noise(ΔVhet plotted against the logarithm of success probability for the phase quadrature. The solid curve corresponds to the theoretical calculations.The straight green line corresponds to measuring directly at the receiver station with heterodyning ΔVhet =3forVN = 17. The blue line across the unity added noise line corresponds to a perfect deterministic decoupling of excess noise.

By only having access to a portion of the leakage modes one can still achieve erasure of the noise suﬃcient to preserve the quantum properties which directly demonstrates the applicability of the method for quantum communication. Let us assume the described Gaussian channel characterized by the transmission T =0.9= η and added noise ΔV =2.78. Such a channel is certainly entanglement breaking and therefore also the security is broken [100]. We assume that somebody has an access to just 50% of the noise leaking from the coupling. By optimization, the channel has been transformed to a slightly amplifying channel with G =1.06 and ΔVX =0.34 and ΔVP =0.42. It is suﬃcient to achieve a secure key rate transmission if direct reconciliation is used [52]. It is enough if the receiver uses homodyne 126 Noise Erasure detection and the sender makes modulation of coherent state with the variances

σX > 0.002 and σP > 0.018. Maximal secure key rates approach KX =0.535 and

KP =0.4 as the modulation variances increase. An ideal key rate Kopt =0.62 can be achieved for this kind of scenario. Generally the secure key rate can be optimized directly but for this data there is no signiﬁcant improvement with respect to the optimization of the added noise, the improvement will become visible as γ reduces.

6.5 Conclusion

We presented an experiment in which coherent states inﬂicted by Gaussian noise in a realistic quantum communication channel are decoupled from the noise via deterministic as well as probabilistic strategies. We obtain near ideal noise erasure, thus demonstrating that measurements in the environment may recover fully the quantum information. These strategies will play an important role in preventing security breaking due to excess noise in a lossy quantum communication link. Chapter 7

Conclusion and Outlook

In this thesis we investigated and realized various continuous variable quantum information protocols such as the Quantum Amplifier, Cloning Machines, Minimal Dis- turbance Measurements and Noise Erasure. We for the first time to our knowledge proposed and experimentally demonstrated that a phase-insensitive amplifier could be constructed from simple linear optical components, homodyne detectors, and feedforward [78]. Quantum noise limited performance was achieved. Our amplifier can be used to amplify nonclassical states (such as squeezed states and Schroedinger cat states) and still maintain some of their nonclassical features such as squeezing and interference in phase space since it exhibits quantum noise limited performance at low gains. Especially if the Schroedinger cats size is small it may become important to amplify it [106]. This was considered by Filip where he considered a Schrodinger-cat state with a small amplitude, which was then partially amplified with only a weak decoherence effect [50]. Furthermore, we believe that such an amplifier can find usage in the field of quantum communication, where optimal amplification of information carrying quantum states is needed partly to compensate for downstream losses of a quantum channel and partly to enable an arbitrary quantum cloning function [21]. One particular cloning transformation of a coherent state was demonstrated with a fixed gain amplifier [3]. We then discussed various experimental quantum cloning protocols for continuous variables. We first introduced a universal cloner which produced clones of unknown coherent states. This cloner was based on a fixed gain amplifier as constructed in chapter 4 followed by a linear symmetric beam splitter. After having explained this general cloner we showed that the combined use of phase conjuga- 128 Conclusion and Outlook tion and joint measurement can be used to build a superior cloner with improved performance over the universal cloning machine, and in fact we successfully implemented such a cloner [111, 102]. This quantum information protocol is to our knowledge the first quantum protocol for which the demonstration in the CV domain was performed before the DV. We mention that the combined use of phase conjugation and joint measurement for improving the performance of a quantum protocol has many applications beyond the cloning action. It was realized that such a strategy can be used to execute a minimum disturbance measurement [5], enable an optimal individual eavesdropping attack [99], and increase the sensitivity of optical coherence tomography beyond the coherent state limit and ultimately reaching the entanglement based limit [48]. Another proposal, but without an experimental verification, was recently published [37]. Later we illustrated the intriguing relationship between cloning fidelity and prior partial information by proposing and experimentally demonstrating the state dependent cloning transformation of coherent states with superior fidelities [113]. We found that the more prior information about the input states the greater is the cloning fidelities. This relationship is not only valid for cloning protocols, but also for other protocols such as teleportation and purification of quantum information. Since prior partial information is com- moninquantuminformationnetworks,webelieve that the state-dependent cloning strategies presented here as well as similar strategies for other protocols will have a vital role in future quantum informational systems. Recently there has been a proposal on the quantum cloning of continuous-variable entangled states [135]. We thoroughly analysed minimal disturbance measurements (MDM) and extended the discussion on the optimal information-disturbance trade-off to the continuous variable regime and derived a tradeoff relation for coherent states. We proposed and successfully demonstrated an optimal nonunity gain Gaussian scheme for partial measurement of an unknown coherent state that caused minimal disturbance of the state [112]. The information gain and the state disturbance were quantified by the noise added to the measurement outcomes and to the output state, respectively. We derived the optimal trade-off relation between the two noises and showed that the tradeoff is saturated by nonunity gain teleportation. Optimal partial measurement was demonstrated experimentally using a linear optics scheme with feedforward. Finally, we also demonstrated that our scheme can be used to enhance the transmission fidelity of some noisy channels, rendering our approach as a useful tool in future quantum communication networks [5]. One can also in- 129 vestigate the minimum disturbance measurements for coherent states with a priori information. For example minimal disturbance measurements on coherent states having known phases can be analysed. Recent research directions include MDM without postselection [8] and MDM for maximally entangled states [144]. The MDM protocol has direct application in the field of quantum cryptography or more specifically quantum key distribution. Quantum Key Distribution (QKD) uses the fundamental laws of quantum physics to ensure secure communication and has one of the biggest application potentials in quantum information. Once key generation speed plus transmission distances are increased, and error rate reduced QKD will become very practical indeed. We presented an experiment in which coherent states inflicted by Gaussian noise in a realistic quantum communication channel are decoupled from the noise via deterministic as well as probabilistic strategies. We obtain near ideal noise erasure, thus demonstrating that measurements in the environment may recover fully the quantum information. These strategies will play an important role in preventing security breaking due to excess noise in a lossy quantum communication link [120]. There are main directions of continuous variable quantum optics spanning, quantum dense coding, teleportation, quantum computation, quantum networks, decoherence and quantum error correction and entanglement purification. Some recent research accomplishments include the distillation entanglement of mesoscopic quantum states [43], demonstration of a quantum nondemolition sum gate [141], generation of four-mode continuous-variable cluster states [143], optimal discrimination of optical coherent states [137] and entangling the spatial properties of Laser Beams [133]. Another direction of quantum information protocols with continuous variables which was not investigated in this thesis is the field of atomic continuous variables which use multi-atomic ensembles that serve as efficient storage and processing units for quantum information. At this point it is difficult to say whether the research will end with the successful construction of a quantum computer. It would of course be great to have such a ”super” machine which is capable of solving problems that a classical computer cannot. However even if such a machine is not realised we can still expect that future advances in quantum information will give new prospects in quantum state engineering, quantum computing and also quantum communication. 130 Conclusion and Outlook Bibliography

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