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arXiv:1707.01114 · Physical Review A

Better bounds on optimal measurement and entanglement recovery, with applications to uncertainty and monogamy relations

Joseph M. Renes

Published Oct 19, 2017• 6 citations
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Abstract

We extend the recent bounds of Sason and Verd\'u relating R\'enyi entropy and Bayesian hypothesis testing (arXiv:1701.01974.) to the quantum domain and show that they have a number of different applications. First, we obtain a sharper bound relating the optimal probability of correctly distinguishing elements of an ensemble of states to that of the pretty good measurement, and an analogous bound for optimal and pretty good entanglement recovery. Second, we obtain bounds relating optimal guessing and entanglement recovery to the fidelity of the state with a product state, which then leads to tight tripartite uncertainty and monogamy relations.

DOI
10.1103/physreva.96.042328

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We extend the recent bounds of Sason and Verd'u relating R'enyi entropy and Bayesian hypothesis testing (arXiv:1701.01974.) to the quantum domain and show that they have a number of different applications. First, we obtain a sharper bound relating the optimal probability of correctly distinguishing elements of an ensemble of states to that of the pretty good measurement, and an analogous bound for optimal and pretty good entanglement recovery. Second, we obtain bounds relating optimal guessing and entanglement recovery to the fidelity of the state with a product state, which then leads to tight tripartite uncertainty and monogamy relations.
This thesis addresses the interplay between asymptotic hypothesis testing and entropy inequalities in quantum information theory. In the first part of the thesis we focus on hypothesis testing. We consider two main settings; one can either fix quantum states while optimizing over measurements or fix a measurement and evaluate its capability to discriminate by optimizing over states. In the former setting, we prove a composite quantum Stein's Lemma. We also discuss how this gives an operational interpretation to several quantities of interest. For the latter, we give the optimal asymptotic error rates in several symmetric and asymmetric settings, and discuss properties and examples of these rates. In the second part, the focus is shifted to entropy inequalities. We start with recoverability inequalities. Using tools developed to prove the composite Stein's Lemma, we give a strengthened lower bound on the conditional quantum mutual information (CQMI). Next, we give an operational interpretation to the relative entropy of recovery via hypothesis testing. Then, we discuss some recent counterexamples, which show that the relative entropy of recovery is not a lower bound on the CQMI; we provide more counterexamples where some systems are classical. We then turn to a seemingly different type of entropy inequalities called bounds on information combining. Using a particular recoverability inequality, we give a lower bound and additionally conjecture optimal lower and upper bounds. Furthermore, we discuss implications of our bounds to the finite blocklength behavior of Polar codes. Finally, we discuss Renyi-222 entropy inequalities for Gaussian states, by exploiting their formulation as log-det inequalities to find recoverability related bounds on several quantities. We apply this to Gaussian steerability and entanglement measures, proving their monogamy and several other features.
In a unified framework, we estimate the following quantities of interest in quantum information theory: (1) the minimum-error distinguishability of arbitrary ensembles of mixed quantum states; (2) the approximate reversibility of quantum dynamics in terms of entanglement fidelity (This is referred to as “channel-adapted quantum error recovery” when applied to the composition of an encoding operation and a noise channel.); (3) the maximum overlap between a bipartite pure quantum state and a bipartite mixed-state that may be achieved by applying a local quantum operation to one part of the mixed-state; and (4) the conditional min-entropy of bipartite quantum states. A refined version of the author’s techniques [J. Tyson, J. Math. Phys. 50, 032016 (2009)] for bounding the first quantity is employed to give two-sided estimates of the remaining three quantities. We obtain a closed-form approximate reversal channel. Using a state-dependent Kraus decomposition, our reversal may be interpreted as a quadratically weighted version of that of Barnum and Knill [J. Math. Phys. 43, 2097 (2002)]. The relationship between our reversal and Barnum and Knill’s is therefore similar to the relationship between Holevo’s asymptotically optimal measurement [A. S. Kholevo, Theor. Probab. Appl. 23, 411 (1978)] and the “pretty good” measurement of Belavkin [Stochastics 1, 315 (1975)] and Hausladen and Wootters [J. Mod. Opt. 41, 2385 (1994)]. In particular, we obtain relatively simple reversibility estimates without negative matrix-powers at no cost in tightness of our bounds. Our recovery operation is found to significantly outperform the so-called “transpose channel” in the simple case of depolarizing noise acting on half of a maximally entangled state. Furthermore, our overlap results allow the entangled input state and the output target state to differ, thus obtaining estimates in a somewhat more general setting. Using a result of König et al. [IEEE Trans. Inf. Theory 55, 4337 (2009)], our maximum overlap estimate is used to bound the conditional min-entropy of arbitrary bipartite states. Our primary tool is “small angle” initialization of an abstract generalization of the iterative schemes of Ježek et al. [Phys. Rev. A 65, 060301 (2002)], Ježek et al. [Phys. Rev. A 68, 012305 (2003)], and Reimpell and Werner [Phys. Rev. Lett. 94, 080501 (2005)]. The monotonicity result of Reimpell [Ph.D. thesis, Technishe Universität, 2007] follows in greater generality.
We consider a game in which two separate laboratories collaborate to prepare a quantum system and are then asked to guess the outcome of a measurement performed by a third party in a random basis on that system. Intuitively, by the uncertainty principle and the monogamy of entanglement, the probability that both players simultaneously succeed in guessing the outcome correctly is bounded. We are interested in the question of how the success probability scales when many such games are performed in parallel. We show that any strategy that maximizes the probability to win every game individually is also optimal for the parallel repetition of the game. Our result implies that the optimal guessing probability can be achieved without the use of entanglement. We explore several applications of this result. Firstly, we show that it implies security for standard BB84 quantum key distribution when the receiving party uses fully untrusted measurement devices , i.e. we show that BB84 is one-sided device independent. Secondly, we show how our result can be used to prove security of a one-round position-verification scheme. Finally, we generalize a well-known uncertainty relation for the guessing probability to quantum side information.
Several information measures have recently been defined that capture the notion of recoverability. In particular, the fidelity of recovery quantifies how well one can recover a system AAA of a tripartite quantum state, defined on systems ABCABCABC, by acting on system CCC alone. The relative entropy of recovery is an associated measure in which the fidelity is replaced by relative entropy. In this paper we provide concrete operational interpretations of the aforementioned recovery measures in terms of a computational decision problem and a hypothesis testing scenario. Specifically, we show that the fidelity of recovery is equal to the maximum probability with which a computationally unbounded quantum prover can convince a computationally bounded quantum verifier that a given quantum state is recoverable. The quantum interactive proof system giving this operational meaning requires four messages exchanged between the prover and verifier, but by forcing the prover to perform actions in superposition, we construct a different proof system that requires only two messages. The result is that the associated decision problem is in QIP(2) and another argument establishes it as hard for QSZK (both classes contain problems believed to be difficult to solve for a quantum computer). We finally prove that the regularized relative entropy of recovery is equal to the optimal type II error exponent when trying to distinguish many copies of a tripartite state from a recovered version of this state, such that the type I error is constrained to be no larger than a constant.
One of the many interesting features of quantum nonlocality is that the states of a multipartite quantum system cannot always be distinguished as well by local measurements as they can when all quantum measurements are allowed. In this work, we characterize the distinguishability of sets of multipartite quantum states when restricted to separable measurements, those which contain the class of local measurements but nevertheless are free of entanglement between the component systems. We consider two quantities: the separable fidelity, a truly quantum quantity, which measures how well we can ``clone'' the input state, and the classical probability of success, which simply gives the optimal probability of identifying the state correctly. We obtain lower and upper bounds on the separable fidelity and give several examples in the bipartite and multipartite settings where these bounds are optimal. Moreover the optimal values in these cases can be attained by local measurements. We further show that for distinguishing orthogonal states under separable measurements, a strategy that maximizes the probability of success is also optimal for separable fidelity. We point out that the equality of fidelity and success probability does not depend on an using the optimal strategy, only on the orthogonality of the states. To illustrate this, we present an example where two sets (one consisting of orthogonal states and the other nonorthogonal states) are shown to have the same separable fidelity even though the success probabilities are different.