Duality of privacy amplification against quantum adversaries and data compression with quantum side information

We show that the tasks of privacy amplification against quantum adversaries and data compression with quantum side information are dual in the sense that the ability to perform one implies the ability to perform the other. These are two of the most important primitives in classical information theory, and are shown to be connected by complementarity and the uncertainty principle in the quantum setting. Applications include a new uncertainty principle formulated in terms of smooth min- and max-entropies, as well as new conditions for approximate quantum error correction.Comment: v2: Includes a derivation of an entropic uncertainty principle for smooth min- and max-entropies. Discussion of the Holevo-Schumacher-Westmoreland theorem remove

Convex-split and hypothesis testing approach to one-shot quantum measurement compression and randomness extraction

We consider the problem of quantum measurement compression with side information in the one-shot setting with shared randomness. In this problem, Alice shares a pure state with Reference and Bob and she performs a measurement on her registers. She wishes to communicate the outcome of this measurement to Bob using shared randomness and classical communication, in such a way that the outcome that Bob receives is correctly correlated with Reference and Bob's own registers. Our goal is to simultaneously minimize the classical communication and randomness cost. We provide a protocol based on convex-split and position based decoding with its communication upper bounded in terms of smooth max and hypothesis testing relative entropies. We also study the randomness cost of our protocol in both one-shot and asymptotic and i.i.d. setting. By generalizing the convex-split technique to incorporate pair-wise independent random variables, we show that our one shot protocol requires small number of bits of shared randomness. This allows us to construct a new protocol in the asymptotic and i.i.d. setting, which is optimal in both the number of bits of communication and the number of bits of shared randomness required. We construct a new protocol for the task of strong randomness extraction in the presence of quantum side information. Our protocol achieves error guarantee in terms of relative entropy (as opposed to trace distance) and extracts close to optimal number of uniform bits. As an application, we provide new achievability result for the task of quantum measurement compression without feedback, in which Alice does not need to know the outcome of the measurement. This leads to the optimal number of bits communicated and number of bits of shared randomness required, for this task in the asymptotic and i.i.d. setting.Comment: version 5: 29 pages, 1 figure. Added applications to randomness extraction (against quantum side information) and measurement compression without feedbac

Tight Finite-Key Analysis for Quantum Cryptography

Despite enormous progress both in theoretical and experimental quantum cryptography, the security of most current implementations of quantum key distribution is still not established rigorously. One of the main problems is that the security of the final key is highly dependent on the number, M, of signals exchanged between the legitimate parties. While, in any practical implementation, M is limited by the available resources, existing security proofs are often only valid asymptotically for unrealistically large values of M. Here, we demonstrate that this gap between theory and practice can be overcome using a recently developed proof technique based on the uncertainty relation for smooth entropies. Specifically, we consider a family of Bennett-Brassard 1984 quantum key distribution protocols and show that security against general attacks can be guaranteed already for moderate values of M.Comment: 11 pages, 2 figure

The apex of the family tree of protocols: Optimal rates and resource inequalities

We establish bounds on the maximum entanglement gain and minimum quantum communication cost of the Fully Quantum Slepian-Wolf protocol in the one-shot regime, which is considered to be at the apex of the existing family tree in Quantum Information Theory. These quantities, which are expressed in terms of smooth min- and max-entropies, reduce to the known rates of quantum communication cost and entanglement gain in the asymptotic i.i.d. scenario. We also provide an explicit proof of the optimality of these asymptotic rates. We introduce a resource inequality for the one-shot FQSW protocol, which in conjunction with our results, yields achievable one-shot rates of its children protocols. In particular, it yields bounds on the one-shot quantum capacity of a noisy channel in terms of a single entropic quantity, unlike previously bounds. We also obtain an explicit expression for the achievable rate for one-shot state redistribution.Comment: 31 pages, 2 figures. Published versio

Identifying the Information Gain of a Quantum Measurement

We show that quantum-to-classical channels, i.e., quantum measurements, can be asymptotically simulated by an amount of classical communication equal to the quantum mutual information of the measurement, if sufficient shared randomness is available. This result generalizes Winter's measurement compression theorem for fixed independent and identically distributed inputs [Winter, CMP 244 (157), 2004] to arbitrary inputs, and more importantly, it identifies the quantum mutual information of a measurement as the information gained by performing it, independent of the input state on which it is performed. Our result is a generalization of the classical reverse Shannon theorem to quantum-to-classical channels. In this sense, it can be seen as a quantum reverse Shannon theorem for quantum-to-classical channels, but with the entanglement assistance and quantum communication replaced by shared randomness and classical communication, respectively. The proof is based on a novel one-shot state merging protocol for "classically coherent states" as well as the post-selection technique for quantum channels, and it uses techniques developed for the quantum reverse Shannon theorem [Berta et al., CMP 306 (579), 2011].Comment: v2: new result about non-feedback measurement simulation, 45 pages, 4 figure

A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks

We consider two fundamental tasks in quantum information theory, data compression with quantum side information as well as randomness extraction against quantum side information. We characterize these tasks for general sources using so-called one-shot entropies. We show that these characterizations - in contrast to earlier results - enable us to derive tight second order asymptotics for these tasks in the i.i.d. limit. More generally, our derivation establishes a hierarchy of information quantities that can be used to investigate information theoretic tasks in the quantum domain: The one-shot entropies most accurately describe an operational quantity, yet they tend to be difficult to calculate for large systems. We show that they asymptotically agree up to logarithmic terms with entropies related to the quantum and classical information spectrum, which are easier to calculate in the i.i.d. limit. Our techniques also naturally yields bounds on operational quantities for finite block lengths.Comment: See also arXiv:1208.1400, which independently derives part of our result: the second order asymptotics for binary hypothesis testin