The Physics of Quantum Information: Complementarity, Uncertainty, and Entanglement

Abstract

1 The 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 imperfectlyentangled 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 reconciling 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 the second party on the other. The latter is known as privacy amplification, and it turns out 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 understood 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.1 Interestingly, Vannevar Bush had already used the phrase of "bits of information" in 1936 to describe information enco...

Identifying the Information Gain of a Quantum Measurement

“…The protocol is based on the observation from [8,45] that state merging is a by-product of an entanglement distillation protocol in which Alice measures the stabilizers of a Calderbank-ShorSteane (CSS) code such that, given the resulting (classical) syndrome results, Bob could determine both the amplitude (logical X value) and phase (logical Z value) 8 of Alice's remaining encoded system by using his systems. Indeed, for state merging of classically coherent states such as ρ X A X B BR in Lemma 2, the situation is considerably simpler since Bob can already determine the amplitude of Alice's system X A by measuring X B .…”

“…Thus, from the analysis of [8,45], all that remains is for Alice to measure a sufficient number of phase stabilizers from an error-correcting code to enable Bob to determine the phase of her encoded systems by using the syndromes and his systems X B and B, with probability of error at most . Use of a linear code ensures that Alice does not damage Bob's amplitude information in the course of trying to increase his phase information.…”

Uncertainty relations: An operational approach to the error-disturbance tradeoff

“…This link has been used by Shor and Preskill to prove the security of quantum key distribution [58] and by Devetak to determine the quantum channel capacity [59]. Entropic state-preparation uncertainty relations from [6,44] can be used to understand both results, as shown in [60,61].…”

Quantum Information as a Non-Kolmogorovian Generalization of Shannon’s Theory

“…The role of the complementarity principle in quantum information theory was discussed in [107], where it is shown that it is crucial for understanding the main features of quantum information protocols. One of the most important formal expressions of the complementarity principle is that of the non-commutativity of operators representing physical observables.…”