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Journal of Physics A Mathematical and General

Bures fidelity for diagonalizable quadratic Hamiltonians in multi-mode systems

Xiang‐Bin Wang, L. C. Kwek, C. H. Oh

Published Jun 27, 2000• 15 citations
Publisher

Abstract

Fidelity, as a measure of the distinguishability of states, is an important concept in quantum mechanics, quantum optics and quantum information theory. Recently, the explicit expressions of fidelity for single-mode squeezed states have been given. However, in experimental studies, especially in non-degenerate parametric down-conversion, two photons are generated and one studies two- or more-mode systems. In this paper we study the Bures fidelity for thermal states of a diagonalizable quadratic Hamiltonian in multi-mode Fock space. To the best of our knowledge, no one has yet attempted to give an explicit general formula of fidelity of mixed states in multi-mode systems.

DOI
10.1088/0305-4470/33/27/310

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Fidelity, as a measure of the distinguishability of states, is an important concept in quantum mechanics, quantum optics and quantum information theory. Recently, the explicit expressions of fidelity for single-mode squeezed states have been given. However, in experimental studies, especially in non-degenerate parametric down-conversion, two photons are generated and one studies two- or more-mode systems. In this paper we study the Bures fidelity for thermal states of a diagonalizable quadratic Hamiltonian in multi-mode Fock space. To the best of our knowledge, no one has yet attempted to give an explicit general formula of fidelity of mixed states in multi-mode systems.
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 ${\ensuremath{\rho}}_{i}.$
Fidelity has always been an important concept in quantum optics. Recently, it was found that fidelity can also play a key role in quantum information and communication theory. Fidelity can be interpreted as the probability that a decoded message possesses the same information content as the message prior to coding and transmission. In this paper, we give a formula of Bures fidelity for displaced squeezed thermal states directly by the displacement and squeezing parameters and briefly discuss how the results can apply to quantum information theory.
We introduce and investigate a distance-type measure of non-Gaussianity based on the quantum fidelity. This new measure can readily be evaluated for all pure states and the mixed ones that are diagonal in the Fock basis. In particular, for an M -photon-added thermal state, an analysis of the Bures degree of non-Gaussianity is performed in comparison with two previous measures built with the Hilbert–Schmidt metric and the relative entropy. We obtain a compact analytic formula for the Hilbert–Schmidt measure of non-Gaussianity and find good consistency between the three examined measures.
We investigate two special classes of two-mode Gaussian states of light that are important from both the experimental and theoretical points of view: the mode-mixed thermal states and the squeezed thermal ones. Aiming to a parallel study, we write the Uhlmann fidelity between pairs of states belonging to each class in terms of their defining parameters. The quantum Fisher information matrices on the corresponding four-dimensional manifolds are diagonal and allow insightful parameter estimation. The scalar curvatures of the Bures metric on both Riemannian manifolds of special two-mode Gaussian states are evaluated and discussed. They are functions of two variables, namely, the mean numbers of photons in the incident thermal modes. Our comparative analysis opens the door to further investigation of the interplay between geometry and statistics for Gaussian states produced in simple optical devices.
We propose a reliable entanglement measure for a two-mode squeezed thermal state of the quantum electromagnetic field in terms of its Bures distance to the set of all separable states of the same kind. The requisite fidelity of a pair of two-mode squeezed thermal states is exactly evaluated. By applying the Peres-Simon criterion of separability, we find the closest separable state. This enables us to establish an insightful expression of the amount of entanglement. We also derive the relative entropy of a two-mode squeezed thermal state with respect to another one. This quantity is exploited in the pure-state case by undertaking a similar approximation for the entropic entanglement of a two-mode squeezed vacuum state. Comparison with previous work lends support to our approach that replaces the reference set of all separable Gaussian states by its subset consisting only of the squeezed thermal ones.
One of the key issues in problems related to quantum information theory is concerned with the distinguishability of quantum states. In this context, Bures distance serves as one of the foremost choices among various distance measures. It also relates to fidelity, which is another quantity of immense importance in quantum information theory. In this work we derive exact results for the average fidelity and variance of the squared Bures distance between a fixed density matrix and a random density matrix and also between two independent random density matrices. These results go beyond the recently obtained results for the mean root fidelity and mean of the squared Bures distance. The availability of both mean and variance also enables us to provide a gamma-distribution-based approximation for the probability density of the squared Bures distance. The analytical results are corroborated using Monte Carlo simulations. Furthermore, we compare our analytical results with the mean and variance of the squared Bures distance between reduced density matrices generated using coupled kicked tops and a correlated spin chain system in a random magnetic field. In both cases, we find good agreement.