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

TL;DR

This paper develops a unified framework providing two-sided bounds on key quantum information measures, including state distinguishability, reversibility of quantum dynamics, and maximum overlap, using iterative schemes and refined techniques.

Contribution

It introduces a unified approach with two-sided estimates for multiple quantum information quantities using directional iterates and refined bounding techniques.

Findings

Provides two-sided bounds for minimum-error quantum state discrimination.

Estimates the reversibility of quantum channels via entanglement fidelity.

Bounds the maximum overlap achievable by local quantum operations.

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].