Quantum Chernoff bound as a measure of nonclassicality for one-mode Gaussian states.pdf

arXiv: [quant-ph] 11 Jun 2008 Quantum Cherno? bound as a measure of nonclassicality for one-mode Gaussian states M?ad?alina Boca, Iulia Ghiu, Paulina Marian, and Tudor A. Marian Centre for Advanced Quantum Physics, University of Bucharest, MG-11, R-077125 Bucharest-M?gurele, Romania (Dated: June 11, 2008) Abstract We evaluate the degree of nonclassicality of a single-mode Gaussian state in terms of its distance to the set of all classical Gaussian states. The employed distance is de?ned in terms of the recently discovered quantum Cherno? bound. The general properties of the quantum Cherno? bound and its relation to the Uhlmann ?delity are interestingly displayed by our approach. We thus give a ?rst example of discriminating between a state and a set of states by using the quantum Cherno? bound. PACS numbers: .-a; 1 Nonclassicality of a quantum state is usually de?ned with respect to the behavior of its diagonalPrepresentation. A negative or a highly singularPrepresentation (more singular than Dirac’sδ) describes a nonclassical state. On the contrary, states possessing a well- behavedPrepresentation are termed classical [1]. When dealing withnon-Gaussian states, nonclassicality can be also identi?ed through the negativity of the Wigner function [2] or by analyzing the characteristic function and the higher-order moments of the state [3]. A quantitative measure of nonclassicality was ?rst proposed by Hillery [4] as a prop- erly de?ned distance between the given nonclassical state and the convex set of all classical states. The trace distance employed in Refs.[4] proved however to be di?cult to deal with analytically. Therefore the original de?nition ofa nonclassical distancewas subsequently modi?ed twofold: restricting the set of all classical states to a tractable subset identi?ed by a classicality criterion and using more convenient distances such as Hilbert-Schmidt [5] and Bures [6, 7] ones and the relative entropy measure in R