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Journal of Physics A Mathematical and General

Quantum Jeffreys prior for displaced squeezed thermal states

L. C. Kwek, C. H. Oh, Xiang‐Bin Wang

Published Sep 6, 1999• 12 citations
Publisher

Abstract

It is known that, by extending the equivalence of the Fisher information matrix to its quantum version, the Bures metric, the quantum Jeffreys prior can be determined from the volume element of the Bures metric. We compute the Bures metric for the displaced squeezed thermal state and analyse the quantum Jeffreys prior and its marginal probability distributions. To normalize the marginal probability density function, it is necessary to provide a range of values of the squeezing parameter or the inverse temperature. We find that if the range of the squeezing parameter is kept narrow, there are significant differences in the marginal probability density functions in terms of the squeezing parameters for the displaced and undisplaced situations. However, these differences disappear as the range increases. Furthermore, marginal probability density functions against temperature are very different in the two cases.

DOI
10.1088/0305-4470/32/37/310

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It is known that, by extending the equivalence of the Fisher information matrix to its quantum version, the Bures metric, the quantum Jeffreys prior can be determined from the volume element of the Bures metric. We compute the Bures metric for the displaced squeezed thermal state and analyse the quantum Jeffreys prior and its marginal probability distributions. To normalize the marginal probability density function, it is necessary to provide a range of values of the squeezing parameter or the inverse temperature. We find that if the range of the squeezing parameter is kept narrow, there are significant differences in the marginal probability density functions in terms of the squeezing parameters for the displaced and undisplaced situations. However, these differences disappear as the range increases. Furthermore, marginal probability density functions against temperature are very different in the two cases.
Clarke and Barron have recently established that the (classical) Jeffreys' prior yields universal codes - ones which do relatively well (in the sense of relative entropy) no matter what the true state. Twamley has recently computed the Bures metric - proportional to the quantum Fisher information (statistical distinguishability) metric, as shown by Braunstein and Caves - for squeezed thermal states. The volume elements of these metrics - that is, the quantum Jeffreys' prior - are found to be simply the product of a function of the squeeze factor and a function of the inverse temperature, the phase being irrelevant in this regard. A computational strategy to find a universal coding using the quantum Jeffreys' prior - previously implemented for the two-level quantum systems - is then discussed.
For certain infinite and finite-dimensional quantum thermal systems, we examine the prior distributions over the inverse temperature parameter beta derived from the volume elements ("quantum Jeffreys's priors") of the Bures metrics for those systems. We find that in the high-temperature limit beta -> 0, the derived priors either conform to Jeffreys' rule for variables over [0,infinity], by being proportional to 1/beta, or to the Bayes-Laplace principle of insufficient reason, by being constant. The numbers of degrees of freedom of the systems appear to determine which of these two outcomes, in fact, occurs. We study the relation of the results here to thermostatistical analyses of Lavenda in which the volume element (Jeffreys' prior) of the (classical) Fisher information metric had been employed to generate various priors over beta.
Braunstein and Caves have recently demonstrated that the Bures metric on the mixed quantum states is equivalent - up to a proportionality factor of four - to the statistical distinguishability or quantum Fisher information metric. The volume element of these metrics can then - adapting a fundamental Bayesian principle of Jeffreys to the quantum context - serve as a reparametrization-invariant prior measure over the quantum states. The implications of this line of reasoning for the two-level systems, in general, and an embedding of them into a certain set of three-level systems are investigated.
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.
In the coordinate representation of thermofield dynamics, we investigate the thermalized displaced squeezed thermal state which involves two temperatures successively. We give the wavefunction and the matrix element of the density operator at any time, and accordingly calculate some quantities related to the position and particle number operator, special cases of which are consistent with the results in the literature. The two temperatures have different correlations with the squeezing and coherence components. Moreover, different from the properties of the position, the average value and variance of the particle number operator as well as the second-order correlation function are time independent.