The Physics of Quantum Information

The Physics of Quantum Information Complementarity, Uncertainty, and Entanglement Joseph M. Renes 12 December 2012 arXiv:1212.2379v1 [quant-ph] 11 Dec 2012 Habilitation TU Darmstadt, Fachbereich Physik 2011 Contents Preface 1 1 Introduction: What is Quantum Information? 5 1.1 Understanding Classical Information via the InformationGame ................ 6 1.2 ComplementarityintheInformationGame . .................. 7 1.3 EntanglementintheInformationGame . .................. 8 2 Illustrations and Motivations 11 2.1 TeleportationandSuperdenseCoding . ................... 11 2.2 QuantumError-Correction......................... ................... 12 2.3 EntanglementDistillation ........................ .................... 17 2.4 QuantumKeyDistribution.......................... .................. 18 3 Characterizing Quantum Information 23 3.1 Amplitude and Phase Predictability & Entanglement . ..................... 23 3.2 Duality&Decoupling.............................. .................. 28 3.3 EntropicCharacterizations. .. .. .. .. .. .. .. .. .. .. ..................... 31 3.4 SecretKeys&PrivateStates ........................ ................... 33 4 Processing Quantum Information 37 4.1 OptimalEntanglementDistillation . ..................... 37 4.2 OptimalStateMerging............................. .................. 43 4.3 Secret Key Distillation and Private Communication . ...................... 45 5 Duality of Protocols 47 5.1 Duality of Privacy Amplification and Information Reconciliation ................ 47 5.2 Different Approaches to Entanglement Distillation . ....................... 51 5.3 ClassicalChannelCoding.......................... ................... 54 6 Security of Quantum Key Distribution 59 6.1 NotionsofSecurity ............................... .................. 59 6.2 EntanglementinPrepareandMeasureQKD . ................. 60 6.3 PrivateStatesinQuantumKeyDistribution . .................... 64 7 Summary and Outlook 67 Bibliography 69 Included Papers 79 i Preface Complementarity is one of the central mysteries of quantum mechanics. First put forth by Bohr [1, 2, 3], complementarity holds that the attributes of a physical system familiar from classical mechanics do not all simultaneously exist and are not entirely independent of how they are measured. Fa- mously, if the momentum of a particle is known then its position must be unknown, and vice versa, a fact encapsulated in Heisenberg’s uncertainty relation ∆x∆p hħ/2 [4]. Even more dramatic is ≥ the wave-particle duality encountered in Young’s double-slit experiment, which illustrates the important role of observation. Light passing through the double slit setup produces an interference pattern on a screen beyond the slits, as would be characteristic of a wave. But a closer examina- tion reveals that light arrives in particle-like “packets” at the screen, and the interference pattern only arises as a statistical average of these particle arrival events. This particle picture tempts us to observe which slit the light went through, which we find destroys the interference pattern! Feyn- man regarded this bizarre phenomena as characteristic of all the seemingly-paradoxical quantum behavior, claiming that the double-slit experiment is “impossible, absolutely impossible to describe classically, [and which] has in it the heart of quantum mechanics”, and that “in reality, it contains the only mystery” (emphasis original) [5]. The overarching goal of this thesis is to demonstrate that complementarity is also at the heart of quantum information theory, that it allows us to make (some) sense of just what information "quantum information" refers to, and that it is useful in understanding and constructing quantum information processing protocols. The detailed research results which form the basis of these claims are to be found in the included papers, and the aim here is to present an overview comprehensible to a more general audience.1 As we shall see in Chapter 1, quantum information can heuristically be thought of as a kind of combination of two types of normal “classical” information, specifically, classical information about the result of measuring one of two complementary observables. Due to the uncertainty principle, we can expect both pieces of information are not simultaneously realizable, and indeed the uncertainty principle will play a central quantitative role throughout this work. Particularly relevant will be the entropic uncertainty relation of [RB09] and its generalization in [BCC+10], which state that the more that can be known by one party about one observable, the less can be known by another party about a complementary observable. That complementary observables play an important role in quantuminformation theoryis not new tothis thesis, and Chapter 2 discusses several fundamental quantum information processing tasks based on their use, such as teleportation and quantum error-correction. This chapter also provides some relevant formal background for the remainder of this work and establishes the notation used herein. Chapter 3 begins the overview of the new results obtained in the included papers. Here we show that information about complementary observables not only plays an important role, but indeed a central one, and that possession of both complementary pieces of classical information is strictly equivalent to the existence of entanglement between the physical system the information pertains to and the system in which the information is stored. Moreover, the uncertainty principle provides a dual characterization, saying that entanglement between these two systems exists when the “environment”, i.e. any and all other degrees of freedom, has no information about either complementary observable. Both characterizations can be modified to describe secret keys useful in cryptography instead of entangled states. Because Chapter 3 gathers and mixes results from several 1The included papers are referenced in alphabetical style, while references to other works are numeric. 1 PREFACE of the included papers, it is entirely self-contained, whereas subsequent chapters do not go into as much detail. In Chapter 4 we show that this complementary approach is also useful in constructing quantum information processing protocols and understanding why they work. Especially relevant is the process of entanglement distillation, that is, extracting maximal entanglement from a imperfectly- entangled bipartite resource system. The entanglement distillation process can be built up from two instances, one for each of two complementary observables, of a simpler distillation process for classical information called information reconciliation or data compression with side information. Here partial classical correlation between two systems is refined into maximal correlation, and rec- onciling classical information about two complementary observables. Protocols for entanglement distillation can then be adapted to a large variety of quantum information processing tasks, such as quantum communication over noisy channels or distillation of secret keys. Chapter 5 extends the duality in characterizing entanglement afforded by the uncertainty principle to two fundamental information processing tasks, the information reconciliation task of establishing correlations with the first party on the one hand, and the task of removing all correlations with thesecond partyon theother. Thelatteris known as privacy amplification, and it turnsout that the ability to perform one protocol implies the ability to perform the other in certain circumstances. This duality also implies alternative methods of entanglement distillation, in particular one which proceeds by destroying all classical correlations with the environment that pertain to two complementary observables. We shall also see that information reconciliation and privacy amplification can be combined to enable classical communication over noisy quantum channels. Finally, Chapter 6 describes the usefulness of this approach to establishing the security of quantum key distribution (QKD). QKD is perhaps the most natural setting in which the uncertainty principle and corresponding issues of complementarity are immediately relevant, as the goal of this protocol is to establish a secret key between two spatially-separated parties, a shared piece of classical information which no one else should know. Since the uncertainty principle can be under- stood as a limitation on who can know how much about what sorts of information, we shall see that complementarity-based arguments form the basis for the security of QKD protocols. These allow us to increase the security threshold, the maximum amount of tolerable noise, of several protocols beyond the previously-known values. The following table summarizes which included papers form the basis for the various sections. 2 Preface Chapter Section 1 Introduction 1.1 1.2 1.3 [RB09], [BCC+10] 2 Illustrations & Motivations — 3 Characterizing Quantum Information 3.1 3.2 3.3 3.4 [RB08] [Ren11] [RB09] [RB08] 4 Processing Quantum Information 4.1 4.2 4.3 [RB08] [BR09] [RB08] 5 Duality of Protocols 5.1 5.2 5.3 [Ren11] new [RR11] 6 Security of QKD 6.1 6.2 6.3 — [RG06] [RS07] [SRS08] [KR08] 3 Introduction: What is Quantum Information? 1 At a stroke, Shannon’s landmark 1948 publication A Mathematical Theory of Communication [6] es- tablished the field of information theory, laying out the fundamental lines of inquiry and answering some of the important basic questions. The fundamental problem, according to

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