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Fortschritte der Physik/Progress of Physics

Gaussian Density Matrices: Quantum Analogs of Classical States

A. Mann, M. Revzen

Published Jan 1, 1993• 10 citations
Publisher

Abstract

We study quantum analogs of classical situations, i.e. quantum states possessing some specific classical attribute(s). These states seem quite generally, to have the form of gaussian density matrices. Such states can always be parametrized as thermal squeezed states (TSS). We consider the following specific cases: (a) Two beams that are built from initial beams which passed through a beam splitter cannot, classically, be distinguished from (appropriately prepared) two independent beams that did not go through a splitter. The only quantum states possessing this classical attribute are TSS. (b) The classical Cramer's theorem was shown to have a quantum version (Hegerfeldt). Again, the states here are Gaussian density matrices. (c) The special case in the study of the quantum version of Cramer's theorem, viz. when the state obtained after partial tracing is a pure state, leads to the conclusion that all states involved are zero temperature limit TSS. The classical analog here are gaussians of zero width, i.e. all distributions are δ functions in phase space.

DOI
10.1002/prop.19930410503

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We study quantum analogs of classical situations, i.e. quantum states possessing some specific classical attribute(s). These states seem quite generally, to have the form of gaussian density matrices. Such states can always be parametrized as thermal squeezed states (TSS). We consider the following specific cases: (a) Two beams that are built from initial beams which passed through a beam splitter cannot, classically, be distinguished from (appropriately prepared) two independent beams that did not go through a splitter. The only quantum states possessing this classical attribute are TSS. (b) The classical Cramer's theorem was shown to have a quantum version (Hegerfeldt). Again, the states here are Gaussian density matrices. (c) The special case in the study of the quantum version of Cramer's theorem, viz. when the state obtained after partial tracing is a pure state, leads to the conclusion that all states involved are zero temperature limit TSS. The classical analog here are gaussians of zero width, i.e. all distributions are δ functions in phase space.
The authors use the formalism of thermo field dynamics and the classical Cramer's theorem to show that if two quantum systems are prepared independently and their centre of mass is found to be a thermal squeezed state, then both systems were prepared in thermal squeezed states. This constitutes an alternative derivation of the quantum version of Cramer's theorem. Further, they consider the case of two systems that are prepared independently and their centre of mass is in a pure state in the expanded space of thermo field dynamics. In this case it is shown that the system is also separable in the centre of mass and relative coordinates, and again all the states involved are thermal squeezed states.
Coordinate wave functions for the one-mode squeezed states produced by the quantum analog of the general linear transformation in phase space are calculated. The probability density [\ensuremath{\Vert}\ensuremath{\psi}(q)${\ensuremath{\Vert}}^{2}$] for these states is Gaussian with center predicted by the classical transformation. The quantum image (which includes the traditional two-mode squeeze operator) of a three-parameter symplectic map in two-mode phase space equally generates squeezed states having Gaussian \ensuremath{\Vert}\ensuremath{\psi}(q1{q}_{1}q1​,q1{q}_{1}q1​)${\ensuremath{\Vert}}^{2}$. The center of the two-mode Gaussian is again predicted by the classical mapping.
In this paper, we generalize to 2-mode states the 1-mode state results obtained in a previous paper. We study 2-mode Gaussian density matrices (i.e., density matrices of the form: exponential of a quadratic polynomial in the creation and annihilation operators for the two modes). We find a linear transformation which maps the two annihilation operators, one for each mode, into two new annihilation operators that are uncorrelated and unsqueezed. This allows us to express the density matrix as a product of two 1mode density matrices. We find general conditions under which 2-mode Gaussian density matrices become pure states. Possible pure states include the 2-mode squeezed pure states commonly mentioned in the literature, plus other pure states never mentioned before. We discuss the entropy and thermodynamic laws (Second Law, Fundamental Equation, and Gibbs-Duhem Equation) for the 2-mode states being considered. We study the change in entropy that is produced when a 2-mode Gaussian state is subjected to a measurement of the complex amplitude of one of its two modes. We derive upper and lower bounds for the final (i.e., after the measurement) entropy of the unmeasured mode, and we give necessary and sufficient conditions for the achievement of these bounds. The existence of the bounds is shown to be a consequence of the concavity property of the entropy function.
Inspired by the definition of the non-Gaussian two-parametric continuous-variable analog of an isotropic state introduced by Mi\ifmmode \check{s}\else \v{s}\fi{}ta et al. [Phys. Rev. A 65, 062315 (2002)], we propose to take the Gaussian part of this state as an independent state by itself, which yields a simple, but with respect to the correlation structure interesting, example of a two-mode Gaussian analog of an isotropic state. Unlike conventional isotropic states which are defined as a convex combination of a thermal and an entangled density operator, the Gaussian version studied here is defined by a convex combination of the corresponding covariance matrices and can be understood as an entangled pure state with additional Gaussian noise controlled by a mixing probability. Using various entanglement criteria and measures, we study the nonclassical correlations contained in this state. Unlike the previously studied non-Gaussian two-parametric isotropic state, the Gaussian state considered here features a finite threshold in the parameter space where entanglement sets in. In particular, it turns out that it exhibits an analogous phenomenology as the finite-dimensional two-qubit isotropic state.
We obtain a classical analog of the quantum covariance matrix by performing its classical approximation for any continuous quantum state, and we illustrate this approach with the anharmonic oscillator. Using this classical covariance matrix, we propose classical analogs of the purity, linear quantum entropy, and von Neumann entropy for classical integrable systems, when the quantum counterpart of the system under consideration is in a Gaussian state. As is well known, this matrix completely characterizes the purity, linear quantum entropy, and von Neumann entropy for Gaussian states. These classical analogs can be interpreted as quantities that reveal how much information from the complete system remains in the considered subsystem. To illustrate our approach, we calculate these classical analogs for three coupled harmonic oscillators and two linearly coupled oscillators. We find that they exactly reproduce the results of their quantum counterparts. In this sense, it is remarkable that we can calculate these quantities from the classical viewpoint.
Non-Gaussian entangled states of light have been found to improve the success of quantum telepor- tation. Earlier works in the literature focussed mainly on two-mode non-Gaussian states generated by de-Gaussification of two-mode squeezed vacuum states. In the current work, we study quan- tum teleportation with a class of non-Gaussian entangled resource states that are generated at the output of a passive beam splitter (BS) with different input single mode non-Gaussian states. In particular, we consider input states that are generated under successive application of squeezing and photon addition/subtraction operations in various orders. We focus on identifying what attributes of the resource states are necessary or sufficient for quantum teleportation (QT). To this end we first evaluate two attributes considered in the literature, viz. squeezed vacuum affinity (SVA) and EPR correlation. While SVA is not non-zero for all two-mode resource states, EPR correlation is neither necessary nor sufficient of QT. We consider yet another attribute, viz. two-mode quadrature squeezing as defined by Simon et. al. [Phys. Rev. A 49, 1567 (1994)]. Our numerical results on the de-Gaussified two-mode squeezed vacuum state as well as the BS generated non-Gaussian states lead us to the conclusion that two-mode quadrature squeezing is a necessary condition for QT, in general. We further demonstrate the plausibility of this conclusion by giving an analytical proof that two-mode quadrature squeezing is a necessary condition for QT in the case of symmetric two-mode Gaussian resource states.