Two-sided bounds on minimum-error quantum measurement, on the reversibility of quantum dynamics, and on maximum overlap using directional iterates

Abstract

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.