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Physical Review A

Distinguishability and nonclassicality of one-mode Gaussian states

Paulina Marian, Tudor A. Marian, Horia Scutaru

Published Feb 12, 2004• 43 citations
Publisher

Abstract

We consider the class of the one-mode Gaussian states of the quantum radiation field. The relative entropy of such a state with respect to a similar one is derived. We analyze the entropic amount of nonclassicality of a Gaussian state defined as the minimal relative entropy of any classical Gaussian state with respect to it. A similar quantity built with the Hilbert-Schmidt distance is also calculated. Both nonclassicality measures are then compared with the Bures-metric degree of nonclassicality evaluated previously. The properties of the closest classical Gaussian state, as well as the decrease of the nonclassicality under thermal noise mappings are carefully examined in each case. For mixed states we find that only the Bures-distance amount of nonclassicality is equivalent to the nonclassical depth.

DOI
10.1103/physreva.69.022104

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We consider the class of the one-mode Gaussian states of the quantum radiation field. The relative entropy of such a state with respect to a similar one is derived. We analyze the entropic amount of nonclassicality of a Gaussian state defined as the minimal relative entropy of any classical Gaussian state with respect to it. A similar quantity built with the Hilbert-Schmidt distance is also calculated. Both nonclassicality measures are then compared with the Bures-metric degree of nonclassicality evaluated previously. The properties of the closest classical Gaussian state, as well as the decrease of the nonclassicality under thermal noise mappings are carefully examined in each case. For mixed states we find that only the Bures-distance amount of nonclassicality is equivalent to the nonclassical depth.
We define the degree of nonclassicality of a one-mode Gaussian state of the quantum electromagnetic field in terms of the Bures distance between the state and the set of all classical one-mode Gaussian states. We find the closest classical Gaussian state and the degree of nonclassicality using a recently established expression for the Uhlmann fidelity of two single-mode Gaussian states. The decrease of nonclassicality under thermal mapping is carefully analyzed. Along the same lines, we finally present the evolution of nonclassicality during linear amplification.
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.
The degree of nonclassicality of states of a field mode is analysed considering both phase-space and distance-type measures of nonclassicality. By working out some general examples, it is shown explicitly that the phase-space measure is rather sensitive to superposition of states, with finite superpositions possessing maximum nonclassical depth (the highest degree of nonclassicality) irrespective of the nature of the component states. Mixed states are also discussed and examples with nonclassical depth varying between the minimum and the maximum allowed values are exhibited. For pure Gaussian states, it is demonstrated that distance-type measures based on the Hilbert–Schmidt metric are equivalent to the phase-space measure. Analyzing some examples, it is shown that distance-type measures are efficient to quantify the degree of nonclassicality of non-Gaussian pure states.
We prove that the closest Gaussian state to an arbitrary NNN-mode field state through the relative entropy is built with the covariance matrix and the average displacement of the given state. Consequently, the relative entropy of an NNN-mode state to its associate Gaussian one is an exact distance-type measure of non-Gaussianity. In order to illustrate this finding, we discuss the general properties of the NNN-mode Fock-diagonal states and evaluate their exact entropic amount of non-Gaussianity.
We analyze the polarization of a quantum radiation field under a de-Gaussification process. Specifically, we consider the addition of photons to a two-mode thermal state to get mixed non-Gaussian and nonclassical states which are still diagonal in the Fock basis. Stokes-operator-based degrees of polarization and two distance-type measures defined with Hilbert-Schmidt and Bures metrics are investigated. For a better insight we here introduce a polarization degree based on the relative entropy. Polarization of the thermal states is fully investigated and simple closed expressions are found for all the defined degrees. The evaluated degrees for photon-added states are then compared to the corresponding ones for the two-mode thermal states they originate from. We present interesting findings which tell us that some popular degrees of polarization are not fully consistent. However, the most solidly defined degrees, which are based on the Bures metric and relative entropy, clearly indicate an enhancement of polarization through de-Gaussification. This conclusion is supported by the behavior of the degrees of polarization of the Fock states, which are finally discussed as a limit case.