The quantum Chernoff bound as a measure of distinguishability between density matrices: application to qubit and Gaussian states

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] and Nussbaum and Szkola [quant-ph/0607216] has proved the quantum analog of this bound, which applies when the hypotheses correspond to two quantum states. Based on the 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, and illustrate their use in the paradigmatic cases of qubits and Gaussian infinite-dimensional states.

16 pages

References in corpus (4)

• The Quantum Chernoff Bound
• Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms
• Uncertainty Principle and Quantum Fisher Information - II
• Volume of the quantum mechanical state space

Cited by in corpus (14)

• Quantum Illumination with Gaussian States
• Quantum Process Tomography: Resource Analysis of Different Strategies
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Optimal quantum estimation of the Unruh-Hawking effect
• Computable bounds for the discrimination of Gaussian states
• Correlation Matrices of Two-Mode Bosonic Systems
• Phase estimation for thermal Gaussian states
• Local discrimination of mixed states
• Quantum target detection using entangled photons
• Distinguishability measures between ensembles of quantum states
• Changepoint Problem in Quantumn Setting
• Hypothesis testing for Gaussian states on bosonic lattices
• Quantum Chernoff bound as a measure of nonclassicality for one-mode Gaussian states
• On a correspondence between regular and non-regular operator monotone functions