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理论物理通讯：英文版

Sub-Fidelity and Super-Fidelity Between Gaussian States

Qi Xiao

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

In this paper, we analyze the sub-fidelity and super-fidelity of an arbitrary pair of n-mode Gaussian states.Particularly, an explicit formula for the sub-fidelity and super-fidelity between any two-mode Gaussian states is obtained.

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In the theory of quantum transmission of information the concept of fidelity plays a fundamental role. An important class of channels, which can be experimentally realized in quantum optics, is that of Gaussian quantum channels. In this work we present a general formula for fidelity in the case of two arbitrary Gaussian states. From this formula one can get a previous result [H. Scutaru, J. Phys. A 31, 3659 (1998)], for the case of a single mode. The concept of fidelity used in this paper is the standard one [D. Bures, Trans. Am. Math. Soc. 135, 199 (1969); A. Uhlmann, Rep. Math. Phys. 9, 273 (1976); R. Josza, J. Mod. Opt. 41, 2314 (1994)]. It can be defined by $F({\ensuremath{\rho}}_{1},{\ensuremath{\rho}}_{2}{)}^{\underset{=}{\mathrm{def}}}{\mathrm{max}}_{|{\ensuremath{\psi}}_{1}〉,|{\ensuremath{\psi}}_{2}〉}|〈{\ensuremath{\psi}}_{1}|{\ensuremath{\psi}}_{2}〉{|}^{2},$ where $|{\ensuremath{\psi}}_{i}〉,$ i=1,2i=1,2i=1,2 are purifications of the density matrices ${\ensuremath{\rho}}_{i}.$
We analyze the Uhlmann fidelity of a pair of nnn-mode Gaussian states of the quantum radiation field. This quantity is shown to be the product of an exponential function depending on the relative average displacement and a factor fully determined by the symplectic spectrum of the covariance matrix of a specific Gaussian state. However, it is difficult to handle our general formula unless the Gaussian states commute or at least one of them is pure. On the contrary, in the simplest cases n=1n=1n=1 and n=2n=2n=2, it leads to explicit analytic formulas. Our main result is a calculable expression of the fidelity of two arbitrary two-mode Gaussian states. This can be applied to build reliable measures of quantum correlations between modes in various branches of quantum physics.
In this paper, we give the definition of k th sub-fidelity and k th super-fidelity between quantum states ρ and ϱ, and investigate the relationship among k th partial fidelity ( F k (ρ, ϱ)), k th sub-fidelity ( E k (ρ, ϱ)) and k th super-fidelity ( G k (ρ, ϱ)). In particular, it is shown that E k (ρ, ϱ) ⩽ F k (ρ, ϱ) ⩽ G k (ρ, ϱ). In addition, we show that the partial super-fidelity is a super-multiplicative map, which is G kl (ρ 1 ⊗ϱ 1 , ρ 2 ⊗ϱ 2 ) ⩾ G k (ρ 1 , ρ 2 ) G l (ϱ 1 , ϱ 2 ), for 1 ⩽ k , l ⩽ d .
We derive several bounds on fidelity between quantum states. In particular we show that fidelity is bounded from above by a simple to compute quantity we call super--fidelity. It is analogous to another quantity called sub--fidelity. For any two states of a two--dimensional quantum system (N=2) all three quantities coincide. We demonstrate that sub-- and super--fidelity are concave functions. We also show that super--fidelity is super--multiplicative while sub--fidelity is sub--multiplicative and design feasible schemes to measure these quantities in an experiment. Super--fidelity can be used to define a distance between quantum states. With respect to this metric the set of quantum states forms a part of a N2−1N^2-1N2−1 dimensional hypersphere.