paper

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

arXiv:0907.3386 · doi:10.1063/1.3463451

Abstract

In a unified framework, we obtain two-sided estimates of 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 also referred to as "channel-adapted quantum error recovery" when the reversed channel is 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. 4. The conditional min-entropy of bipartite quantum states. A refined version of the author's techniques [J. Math. Phys. 50, 032016] for bounding the first quantity is employed to give two-sided estimates of the remaining three quantities. Our primary tool is "small angle" initialization of an abstract generalization of the iterative schemes for computing optimal measurements and quantum error recoveries introduced by Jezek-Rehacek-Fiurasek [Phys. Rev. A 65, 060301], Jezek-Fiurasek-Hradil [Phys. Rev. A 68, 012305], and Reimpell-Werner [Phys. Rev. Lett 94, 080501].

Extensively revised & new content added. Improved min-entropy bounds. Notation made more accessible. Minimax theorem used to clarify relationship between "worst case" bounds and "single instance" bounds. Improved motivation of the choice of "small angle" guess. Eliminated spurious factor appearing when overlap bounds are applied to state distinction. Work connected to that of Beny and Oreshkov

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