The uncertainty principle, originally formulated by Heisenberg 1 , clearly illustrates the difference between classical and quantum mechanics. The principle bounds the uncertainties about the outcomes of two incompatible measurements, such as position and momentum, on a particle. It implies that one cannot predict the outcomes for both possible choices of measurement to arbitrary precision, even if information about the preparation of the particle is available in a classical memory. However, if the particle is prepared entangled with a quantum memory, a device that might be available in the not-too-distant future 2 , it is possible to predict the outcomes for both measurement choices precisely. Here, we extend the uncertainty principle to incorporate this case, providing a lower bound on the uncertainties, which depends on the amount of entanglement between the particle and the quantum memory. We detail the application of our result to witnessing entanglement and to quantum key distribution.Uncertainty relations constrain the potential knowledge one can have about the physical properties of a system. Although classical theory does not limit the knowledge we can simultaneously have about arbitrary properties of a particle, such a limit does exist in quantum theory. Even with a complete description of its state, it is impossible to predict the outcomes of all possible measurements on the particle. This lack of knowledge, or uncertainty, was quantified by Heisenberg 1 using the standard deviation (which we denote by R for an observable R). If the measurement on a given particle is chosen from a set of two possible observables, R and S, the resulting bound on the uncertainty can be expressed in terms of the commutator 3 :In an information-theoretic context, it is more natural to quantify uncertainty in terms of entropy rather than the standard deviation. Entropic uncertainty relations for position and momentum were derived in ref. 4 and later a relation was developed that holds for any pair of observables 5 . An improvement of this relation was subsequently conjectured 6 and then proved 7 . The improved relation iswhere H (R) denotes the Shannon entropy of the probability distribution of the outcomes when R is measured. The term 1/c quantifies the complementarity of the observables. For non-degenerate observables, c := max j,k | ψ j |φ k | 2 , where |ψ j and |φ k are the eigenvectors of R and S, respectively. One way to think about uncertainty relations is through the following game (the uncertainty game) between two players, Alice and Bob. Before the game commences, Alice and Bob agree on two measurements, R and S. The game proceeds as follows. Bob prepares a particle in a quantum state of his choosing and sends it to Alice. Alice then carries out one of the two measurements and announces her choice to Bob. Bob's task is to minimize his uncertainty about Alice's measurement outcome. This is illustrated in Fig. 1.Equation (1) bounds Bob's uncertainty in the case that he has no quantum memory-all information Bob hold...

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Heisenberg's uncertainty principle forms a fundamental element of quantum mechanics. Uncertainty relations in terms of entropies were initially proposed to deal with conceptual shortcomings in the original formulation of the uncertainty principle and, hence, play an important role in quantum foundations. More recently, entropic uncertainty relations have emerged as the central ingredient in the security analysis of almost all quantum cryptographic protocols, such as quantum key distribution and two-party quantum cryptography. This review surveys entropic uncertainty relations that capture Heisenberg's idea that the results of incompatible measurements are impossible to predict, covering both finite-and infinite-dimensional measurements. These ideas are then extended to incorporate quantum correlations between the observed object and its environment, allowing for a variety of recent, more general formulations of the uncertainty principle. Finally, various applications are discussed, ranging from entanglement witnessing to wave-particle duality to quantum cryptography.

Abstract:The Quantum Reverse Shannon Theorem states that any quantum channel can be simulated by an unlimited amount of shared entanglement and an amount of classical communication equal to the channel's entanglement assisted classical capacity. In this paper, we provide a new proof of this theorem, which has previously been proved by Bennett, Devetak, Harrow, Shor, and Winter. Our proof has a clear structure being based on two recent information-theoretic results: one-shot Quantum State Merging and the Post-Selection Technique for quantum channels.

We provide a security analysis for continuous variable quantum key distribution protocols based on the transmission of two-mode squeezed vacuum states measured via homodyne detection. We employ a version of the entropic uncertainty relation for smooth entropies to give a lower bound on the number of secret bits which can be extracted from a finite number of runs of the protocol. This bound is valid under general coherent attacks, and gives rise to keys which are composably secure. For comparison, we also give a lower bound valid under the assumption of collective attacks. For both scenarios, we find positive key rates using experimental parameters reachable today.

If a quantum system A, which is initially correlated to another system, E, undergoes an evolution separated from E, then the correlation to E generally decreases. Here, we study the conditions under which the correlation disappears (almost) completely, resulting in a decoupling of A from E. We give a criterion for decoupling in terms of two smooth entropies, one quantifying the amount of initial correlation between A and E, and the other characterizing the mapping that describes the evolution of A. The criterion applies to arbitrary such mappings in the general one-shot setting. Furthermore, the criterion is tight for mappings that satisfy certain natural conditions. One-shot decoupling has a number of applications both in physics and information theory, e.g., as a building block for quantum information processing protocols. As an example, we give a one-shot state merging protocol and show that it is essentially optimal in terms of its entanglement consumption/production.

This paper establishes several converse bounds on the private transmission capabilities of a quantum channel. The main conceptual development builds firmly on the notion of a private state, which is a powerful, uniquely quantum method for simplifying the tripartite picture of privacy involving local operations and public classical communication to a bipartite picture of quantum privacy involving local operations and classical communication. This approach has previously led to some of the strongest upper bounds on secret key rates, including the squashed entanglement and the relative entropy of entanglement. Here we use this approach along with a "privacy test" to establish a general meta-converse bound for private communication, which has a number of applications. The meta-converse allows for proving that any quantum channel's relative entropy of entanglement is a strong converse rate for private communication. For covariant channels, the meta-converse also leads to second-order expansions of relative entropy of entanglement bounds for private communication rates. For such channels, the bounds also apply to the private communication setting in which the sender and receiver are assisted by unlimited public classical communication, and as such, they are relevant for establishing various converse bounds for quantum key distribution protocols conducted over these channels. We find precise characterizations for several channels of interest and apply the methods to establish converse bounds on the private transmission capabilities of all phase-insensitive bosonic channels.

Hilbert's projective metric in quantum information theory J. Math. Phys. 52, 082201 (2011) The conditional quantum mutual information I(A; B|C) of a tripartite state ρ ABC is an information quantity which lies at the center of many problems in quantum information theory. Three of its main properties are that it is non-negative for any tripartite state, that it decreases under local operations applied to systems A and B, and that it obeys the duality relation I(A; B|C) = I(A; B|D) for a four-party pure state on systems ABCD. The conditional mutual information also underlies the squashed entanglement, an entanglement measure that satisfies all of the axioms desired for an entanglement measure. As such, it has been an open question to find Rényi generalizations of the conditional mutual information, that would allow for a deeper understanding of the original quantity and find applications beyond the traditional memoryless setting of quantum information theory. The present paper addresses this question, by defining different α-Rényi generalizations I α (A; B|C) of the conditional mutual information, some of which we can prove converge to the conditional mutual information in the limit α → 1. Furthermore, we prove that many of these generalizations satisfy non-negativity, duality, and monotonicity with respect to local operations on one of the systems A or B (with it being left as an open question to prove that monotonicity holds with respect to local operations on both systems). The quantities defined here should find applications in quantum information theory and perhaps even in other areas of physics, but we leave this for future work. We also state a conjecture regarding the monotonicity of the Rényi conditional mutual informations defined here with respect to the Rényi parameter α. We prove that this conjecture is true in some special cases and when α is in a neighborhood of one. C 2015 AIP Publishing LLC. [http://dx