Quantum Chernoff bound as a measure of distinguishability between density matrices: Application to qubit and Gaussian states

J. Calsamiglia, R. Muñoz-Tapia, Ll. Masanes, A. Acin, and E. Bagan
Phys. Rev. A 77, 032311 – Published 7 March 2008

Abstract

Hypothesis testing is a fundamental issue in statistical inference and has been a crucial element in the development of information sciences. The Chernoff bound gives the minimal Bayesian error probability when discriminating two hypotheses given a large number of observations. Recently the combined work of Audenaert et al. [Phys. Rev. Lett. 98, 160501 (2007)] and Nussbaum and Szkola [e-print arXiv:quant-ph/0607216] has proved the quantum analog of this bound, which applies when the hypotheses correspond to two quantum states. Based on this quantum Chernoff bound, we define a physically meaningful distinguishability measure and its corresponding metric in the space of states; the latter is shown to coincide with the Wigner-Yanase metric. Along the same lines, we define a second, more easily implementable, distinguishability measure based on the error probability of discrimination when the same local measurement is performed on every copy. We study some general properties of these measures, including the probability distribution of density matrices, defined via the volume element induced by the metric. It is shown that the Bures and the local-measurement based metrics are always proportional. Finally, we illustrate their use in the paradigmatic cases of qubits and Gaussian infinite-dimensional states.

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  • Received 20 September 2007

DOI:https://doi.org/10.1103/PhysRevA.77.032311

©2008 American Physical Society

Authors & Affiliations

J. Calsamiglia and R. Muñoz-Tapia

  • Grup de Física Teòrica, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

Ll. Masanes

  • Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom

A. Acin

  • ICREA and ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain

E. Bagan

  • Department of Physics and Astronomy, University of New Mexico, Albuquerque, New Mexico 87131, USA and Grup de Física Teòrica, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

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Issue

Vol. 77, Iss. 3 — March 2008

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