Two-sided estimates of minimum-error distinguishability of mixed quantum states via generalized Holevo-Curlander bounds

TL;DR

This paper presents a concise factor-of-2 estimate for the error rate in distinguishing mixed quantum states, generalizing previous bounds and introducing a quadratically weighted measurement close to optimal.

Contribution

It introduces a new measurement method with error within a factor of 2 of optimal, generalizes Holevo and Curlander bounds, and relates these to the trace-Jensen inequality.

Findings

Provides a factor-of-2 bound for quantum state distinguishability

Introduces a quadratically weighted measurement close to optimal

Proves a bound related to Barnum and Knill's measurement bound

Abstract

We prove a concise factor-of-2 estimate for the failure rate of optimally distinguishing an arbitrary ensemble of mixed quantum states, generalizing work of Holevo [Theor. Probab. Appl. 23, 411 (1978)] and Curlander [Ph.D. Thesis, MIT, 1979]. A modification to the minimal principle of Cocha and Poor [Proceedings of the 6th International Conference on Quantum Communication, Measurement, and Computing (Rinton, Princeton, NJ, 2003)] is used to derive a suboptimal measurement which has an error rate within a factor of 2 of the optimal by construction. This measurement is quadratically weighted and has appeared as the first iterate of a sequence of measurements proposed by Jezek et al. [Phys. Rev. A 65, 060301 (2002)]. Unlike the so-called pretty good measurement, it coincides with Holevo's asymptotically optimal measurement in the case of nonequiprobable pure states. A quadratically weighted version of the measurement bound by Barnum and Knill [J. Math. Phys. 43, 2097 (2002)] is proven. Bounds on the distinguishability of syndromes in the sense of Schumacher and Westmoreland [Phys. Rev. A 56, 131 (1997)] appear as a corollary. An appendix relates our bounds to the trace-Jensen inequality.