Analysis of Quantum Error-Correcting Codes: Symplectic Lattice Codes and Toric Codes
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Pilot Quantum Error Correction for Global
Pilot Quantum Error Correction for Global- Scale Quantum Communications Laszlo Gyongyosi*1,2, Member, IEEE, Sandor Imre1, Member, IEEE 1Quantum Technologies Laboratory, Department of Telecommunications Budapest University of Technology and Economics 2 Magyar tudosok krt, H-1111, Budapest, Hungary 2Information Systems Research Group, Mathematics and Natural Sciences Hungarian Academy of Sciences H-1518, Budapest, Hungary *[email protected] Real global-scale quantum communications and quantum key distribution systems cannot be implemented by the current fiber and free-space links. These links have high attenuation, low polarization-preserving capability or extreme sensitivity to the environment. A potential solution to the problem is the space-earth quantum channels. These channels have no absorption since the signal states are propagated in empty space, however a small fraction of these channels is in the atmosphere, which causes slight depolarizing effect. Furthermore, the relative motion of the ground station and the satellite causes a rotation in the polarization of the quantum states. In the current approaches to compensate for these types of polarization errors, high computational costs and extra physical apparatuses are required. Here we introduce a novel approach which breaks with the traditional views of currently developed quantum-error correction schemes. The proposed solution can be applied to fix the polarization errors which are critical in space-earth quantum communication systems. The channel coding scheme provides capacity-achieving communication over slightly depolarizing space-earth channels. I. Introduction Quantum error-correction schemes use different techniques to correct the various possible errors which occur in a quantum channel. In the first decade of the 21st century, many revolutionary properties of quantum channels were discovered [12-16], [19-22] however the efficient error- correction in quantum systems is still a challenge. -
QIP 2010 Tutorial and Scientific Programmes
QIP 2010 15th – 22nd January, Zürich, Switzerland Tutorial and Scientific Programmes asymptotically large number of channel uses. Such “regularized” formulas tell us Friday, 15th January very little. The purpose of this talk is to give an overview of what we know about 10:00 – 17:10 Jiannis Pachos (Univ. Leeds) this need for regularization, when and why it happens, and what it means. I will Why should anyone care about computing with anyons? focus on the quantum capacity of a quantum channel, which is the case we understand best. This is a short course in topological quantum computation. The topics to be covered include: 1. Introduction to anyons and topological models. 15:00 – 16:55 Daniel Nagaj (Slovak Academy of Sciences) 2. Quantum Double Models. These are stabilizer codes, that can be described Local Hamiltonians in quantum computation very much like quantum error correcting codes. They include the toric code This talk is about two Hamiltonian Complexity questions. First, how hard is it to and various extensions. compute the ground state properties of quantum systems with local Hamiltonians? 3. The Jones polynomials, their relation to anyons and their approximation by Second, which spin systems with time-independent (and perhaps, translationally- quantum algorithms. invariant) local interactions could be used for universal computation? 4. Overview of current state of topological quantum computation and open Aiming at a participant without previous understanding of complexity theory, we will discuss two locally-constrained quantum problems: k-local Hamiltonian and questions. quantum k-SAT. Learning the techniques of Kitaev and others along the way, our first goal is the understanding of QMA-completeness of these problems. -
Reliably Distinguishing States in Qutrit Channels Using One-Way LOCC
Reliably distinguishing states in qutrit channels using one-way LOCC Christopher King Department of Mathematics, Northeastern University, Boston MA 02115 Daniel Matysiak College of Computer and Information Science, Northeastern University, Boston MA 02115 July 15, 2018 Abstract We present numerical evidence showing that any three-dimensional subspace of C3 ⊗ Cn has an orthonormal basis which can be reliably dis- tinguished using one-way LOCC, where a measurement is made first on the 3-dimensional part and the result used to select an optimal measure- ment on the n-dimensional part. This conjecture has implications for the LOCC-assisted capacity of certain quantum channels, where coordinated measurements are made on the system and environment. By measuring first in the environment, the conjecture would imply that the environment- arXiv:quant-ph/0510004v1 1 Oct 2005 assisted classical capacity of any rank three channel is at least log 3. Sim- ilarly by measuring first on the system side, the conjecture would imply that the environment-assisting classical capacity of any qutrit channel is log 3. We also show that one-way LOCC is not symmetric, by providing an example of a qutrit channel whose environment-assisted classical capacity is less than log 3. 1 1 Introduction and statement of results The noise in a quantum channel arises from its interaction with the environment. This viewpoint is expressed concisely in the Lindblad-Stinespring representation [6, 8]: Φ(|ψihψ|)= Tr U(|ψihψ|⊗|ǫihǫ|)U ∗ (1) E Here E is the state space of the environment, which is assumed to be initially prepared in a pure state |ǫi. -
Potential Capacities of Quantum Channels 2
WINTER AND YANG: POTENTIAL CAPACITIES OF QUANTUM CHANNELS 1 Potential Capacities of Quantum Channels Andreas Winter and Dong Yang Abstract—We introduce potential capacities of quantum chan- Entangled inputs between different quantum channels open the nels in an operational way and provide upper bounds for these door to all kinds of effects that are impossible in classical quantities, which quantify the ultimate limit of usefulness of a information theory. An extreme phenomenon is superactivation channel for a given task in the best possible context. Unfortunately, except for a few isolated cases, potential capac- [13]; there exist two quantum channels that cannot transmit ities seem to be as hard to compute as their “plain” analogues. quantum information when they are used individually, but can We thus study upper bounds on some potential capacities: For transmit at positive rate when they are used together. the classical capacity, we give an upper bound in terms of the The phenomenon of superactivation, and more broadly of entanglement of formation. To establish a bound for the quantum super-additivity, implies that the capacity of a quantum channel and private capacity, we first “lift” the channel to a Hadamard channel and then prove that the quantum and private capacity does not adequately characterize the channel, since the utility of a Hadamard channel is strongly additive, implying that for of the channel depends on what other contextual channels are these channels, potential and plain capacity are equal. Employing available. So it is natural to ask the following question: What these upper bounds we show that if a channel is noisy, however is the maximum possible capability of a channel to transmit close it is to the noiseless channel, then it cannot be activated information when it is used in combination with any other into the noiseless channel by any other contextual channel; this conclusion holds for all the three capacities. -
Simulation of BB84 Quantum Key Distribution in Depolarizing Channel
Proceedings of 14th Youth Conference on Communication Simulation of BB84 Quantum Key Distribution in depolarizing channel Hui Qiao, Xiao-yu Chen* College of Information and Electronic Engineering, Zhejiang Gongshang University, Hangzhou, 310018, China [email protected] Abstract: In this paper, we employ the standard BB84 protocol as the basic model of quantum key distribution (QKD) in depolarizing channel. We also give the methods to express the preparation and measurement of quantum states on classical computer and realize the simulation of quantum key distribution. The simulation results are consistent with the theoretical results. It was shown that the simulation of QKD with Matlab is feasible. It provides a new method to demonstrate the QKD protocol. QKD; simulation; depolarizing channel; data reconciliation; privacy amplification Keywords: [4] 1. Introduction 1989 . In 1993, Muller, Breguet, and Gisin demonstrated the feasibility of polarization-coding The purpose of quantum key distribution (QKD) [5] fiber-based QKD over 1.1km telecom fiber and and classical key distribution is consistent, their Townsend, Rarity, and Tapster demonstrated the difference are the realization methods. The classical key feasibility of phase-coding fiber-based QKD over 10km distribution is based on the mathematical theory of telecom fiber[6]. Up to now, the security transmission computational complexity, whereas the QKD based on [7] distance of QKD has been achieved over 120km . the fundamental principle of quantum mechanics. The At present, the study of QKD is limited to first QKD protocol is BB84 protocol, which was theoretical and experimental. The unconditionally secure proposed by Bennett and Brassard in 1984 [1].The protocol in theory needs further experimental verification. -
Experimental Observation of Coherent-Information Superadditivity in a Dephrasure Channel
Experimental observation of coherent-information superadditivity in a dephrasure channel 1, 2 1, 2 3, 1, 2 1, 2 1, 2 1, 2 Shang Yu, Yu Meng, Raj B. Patel, ∗ Yi-Tao Wang, Zhi-Jin Ke, Wei Liu, Zhi-Peng Li, 1, 2 1, 2 1, 2, 1, 2, 1, 2 Yuan-Ze Yang, Wen-Hao Zhang, Jian-Shun Tang, † Chuan-Feng Li, ‡ and Guang-Can Guo 1CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei, 230026, China 2CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, 230026, P.R. China 3Clarendon Laboratory, Department of Physics, Oxford University, Parks Road OX1 3PU Oxford, United Kingdom We present an experimental approach to construct a dephrasure channel, which contains both dephasing and erasure noises, and can be used as an efficient tool to study the superadditivity of coherent information. By using a three-fold dephrasure channel, the superadditivity of coherent information is observed, and a substantial gap is found between the zero single-letter coherent information and zero quantum capacity. Particularly, we find that when the coherent information of n channel uses is zero, in the case of larger number of channel uses, it will become positive. These phenomena exhibit a more obvious superadditivity of coherent information than previous works, and demonstrate a higher threshold for non-zero quantum capacity. Such novel channels built in our experiment also can provide a useful platform to study the non-additive properties of coherent information and quantum channel capacity. Introduction.—Channel capacity is at the heart of in- capacity [16, 17]. -
A Mathematical Elements
A Mathematical elements The language of quantum mechanics is rooted in linear algebra and functional analysis. It often expands when the foundations of the theory are reassessed, which they periodically have been. As a result, additional mathematical struc- tures not ordinarily covered in physics textbooks, such as the logic of linear subspaces and Liouville space, have come to play a role in quantum mechan- ics. Information theory makes use of statistical quantities, some of which are covered in the main text, as well as discrete mathematics, which are also not included in standard treatments of quantum theory. Here, before describ- ing the fundamental mathematical structure of quantum mechanics, Hilbert space, some basic mathematics of binary arithmetic, finite fields and random variables is given. After a presentation of the basic elements of Hilbert space theory, the Dirac notation typically used in the literature of quantum informa- tion science, and operators and transformations related to it, some elements of quantum probability and quantum logic are also described. A.1 Boolean algebra and Galois fields n A Boolean algebra Bn is an algebraic structure given by the collection of 2 subsets of the set I = {1, 2,...,n} and three operations under which it is closed: the two binary operations of union (∨) and intersection (∧), and a unary operation, complementation (¬). In addition to there being comple- ments (and hence the null set ∅ being an element), the following axioms hold. (i) Commutativity: S ∨ T = T ∨ S and S ∧ T = T ∧ S; (ii) Associativity: S ∨(T ∨U)=(S ∨T )∨U and S ∧(T ∧U)=(S ∧T )∧U; (iii) Distributivity: S ∧ (T ∨ U)=(S ∧ T ) ∨ (S ∧ U)andS ∨ (T ∧ U)= (S ∨ T ) ∧ (S ∨ U); (iv) ¬∅ = I, ¬I = ∅,S∧¬S = ∅,S∨¬S = I, ¬(¬S)=S , for all its elements S, T, U. -
Quantum Interactive Proofs and the Complexity of Separability Testing
THEORY OF COMPUTING www.theoryofcomputing.org Quantum interactive proofs and the complexity of separability testing Gus Gutoski∗ Patrick Hayden† Kevin Milner‡ Mark M. Wilde§ October 1, 2014 Abstract: We identify a formal connection between physical problems related to the detection of separable (unentangled) quantum states and complexity classes in theoretical computer science. In particular, we show that to nearly every quantum interactive proof complexity class (including BQP, QMA, QMA(2), and QSZK), there corresponds a natural separability testing problem that is complete for that class. Of particular interest is the fact that the problem of determining whether an isometry can be made to produce a separable state is either QMA-complete or QMA(2)-complete, depending upon whether the distance between quantum states is measured by the one-way LOCC norm or the trace norm. We obtain strong hardness results by proving that for each n-qubit maximally entangled state there exists a fixed one-way LOCC measurement that distinguishes it from any separable state with error probability that decays exponentially in n. ∗Supported by Government of Canada through Industry Canada, the Province of Ontario through the Ministry of Research and Innovation, NSERC, DTO-ARO, CIFAR, and QuantumWorks. †Supported by Canada Research Chairs program, the Perimeter Institute, CIFAR, NSERC and ONR through grant N000140811249. The Perimeter Institute is supported by the Government of Canada through Industry Canada and by the arXiv:1308.5788v2 [quant-ph] 30 Sep 2014 Province of Ontario through the Ministry of Research and Innovation. ‡Supported by NSERC. §Supported by Centre de Recherches Mathematiques.´ ACM Classification: F.1.3 AMS Classification: 68Q10, 68Q15, 68Q17, 81P68 Key words and phrases: quantum entanglement, quantum complexity theory, quantum interactive proofs, quantum statistical zero knowledge, BQP, QMA, QSZK, QIP, separability testing Gus Gutoski, Patrick Hayden, Kevin Milner, and Mark M. -
AQIS16 Preface Content
The Theory of Statistical Comparison with Applications in Quantum Information Science……………………………………………………………………….….……...………..… Introduction to measurement-based quantum computation…………………………. Engineering the quantum: probing atoms with light and light with atoms in a transmon circuit QED system…………………………………………………………….……….3 The device-independent outlook on quantum physics……………………..…...……. Superconducting qubit systems: recent experimental progress towards fault-tolerant quantum computing at IBM……………………………………...… Observation of frequency-domain Hong-Ou-Mandel interference……. Realization of the contextuality-nonlocality tradeoff with a qubit-qutrit photon pair…………………………………………………………………………………….….... One-way and reference-frame independent EPR-steering………...… Are Incoherent Operations Physically Consistent? — A Physical Appraisal of Incoherent Operations and an Overview of Coherence Measures………............................. Relating the Resource Theories of Entanglement and Quantum Coherence An infinite dimensional Birkhoff’s Theorem and LOCC- convertibility………..… How local is the information in MPS/PEPS tensor networks……………......….. Information-theoretical analysis of topological entanglement entropy and multipartite correlations ……................................................................................................ Phase-like transitions in low-number quantum dots Bayesian magnetometry... Separation between quantum Lovász number and entanglement-assisted zero-error classical capacity ………..……..............................................................………... -
The Private Classical Capacity of a Partially Degradable Quantum Channel
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of the Academy's Library The Private Classical Capacity of a Partially Degradable Quantum Channel Laszlo Gyongyosi 1 Quantum Technologies Laboratory, Department of Telecommunications Budapest University of Technology and Economics 2 Magyar tudosok krt, Budapest, H-1117, Hungary 2 MTA-BME Information Systems Research Group Hungarian Academy of Sciences 7 Nador st., Budapest, H-1051, Hungary [email protected] Abstract For a partially degradable (PD) channel, the channel output state can be used to simulate the degraded environment state. The quantum capacity of a PD channel has been proven to be additive. Here, we show that the private classical capacity of arbitrary dimensional PD channels is equal to the quantum capacity of the channel and also single-letterizes. We prove that higher rates of private classical communication can be achieved over a PD channel in comparison to standard degradable channels. Keywords: private classical capacity, partial degradation, quantum Shannon theory. 1 Introduction The existence of partially degradable (PD) channels has been recently investigated, and it was shown that PD channels exist in the world of those qudit channels that have entanglement- binding complementary channels [1]. The PD channel set involves conjugate-PD channels, which represent a subset in the conjugate degradable channels; this set was introduced by Bradler et al. [11]. The sets of more capable and less noisy quantum channels were defined by Watanabe [2], and it was further demonstrated that other, well-characterized channel sets could be defined be- yond the degradable and conjugate-degradable channels. -
An Experimental Implementation of Oblivious Transfer in the Noisy Storage Model
ARTICLE Received 12 Sep 2013 | Accepted 10 Feb 2014 | Published 12 Mar 2014 DOI: 10.1038/ncomms4418 An experimental implementation of oblivious transfer in the noisy storage model C. Erven1,2,N.Ng3, N. Gigov1, R. Laflamme1,4, S. Wehner3,5 & G. Weihs1,6 Cryptography’s importance in our everyday lives continues to grow in our increasingly digital world. Oblivious transfer has long been a fundamental and important cryptographic primitive, as it is known that general two-party cryptographic tasks can be built from this basic building block. Here we show the experimental implementation of a 1-2 random oblivious transfer protocol by performing measurements on polarization-entangled photon pairs in a modified entangled quantum key distribution system, followed by all of the necessary classical postprocessing including one-way error correction. We successfully exchange a 1,366 bit random oblivious transfer string in B3 min and include a full security analysis under the noisy storage model, accounting for all experimental error rates and finite size effects. This demonstrates the feasibility of using today’s quantum technologies to implement secure two-party protocols. 1 Institute for Quantum Computing and Department of Physics & Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1. 2 Centre for Quantum Photonics, H. H. Wills Physics Laboratory & Department of Electrical and Electronic Engineering, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol BS8 1UB, UK. 3 Center for Quantum Technologies, National University of Singapore, 2 Science Drive 3, Singapore 117543. 4 Perimeter Institute, 31 Caroline Street North, Waterloo, Ontario, Canada N2L 2Y5. 5 School of Computing, National University of Singapore, 13 Computing Drive, Singapore 117417. -
The Classical Capacity of a Quantum Erasure Queue-Channel
2019 IEEE 20th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC) The Classical Capacity of a Quantum Erasure Queue-Channel Prabha Mandayam Krishna Jagannathan, Avhishek Chatterjee Department of Physics Department of Electrical Engineering IIT Madras, Chennai, India IIT Madras, Chennai, India Abstract—We consider a setting where a stream of qubits is wait to be processed, and the probability of error/erasure of processed sequentially. We derive fundamental limits on the rate each qubit is modeled as a function of the time spent in the at which classical information can be transmitted using qubits queue by that qubit. We call this system a ‘queue-channel’ that decohere as they wait to be processed. Specifically, we model the sequential processing of qubits using a single server (a term borrowed from [2]), and characterise the information queue, and derive expressions for the classical capacity of such capacity of the queue-channel, for the case when decoherence a quantum ‘queue-channel.’ Focusing on quantum erasures, we leads to erasures. obtain an explicit single-letter capacity formula in terms of the stationary waiting time of qubits in the queue. Our capacity proof A. Related Work also implies that a ‘classical’ coding/decoding strategy is optimal, i.e., an encoder which uses only orthogonal product states, and An information theoretic notion of reliability of a queu- a decoder which measures in a fixed product basis, are sufficient ing system with state-dependent errors was introduced and to achieve the classical capacity of the quantum erasure queue- studied in [2], where the authors considered queue-length channel.