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Physical Review A

Bures distance as a measure of entanglement for two-mode squeezed thermal states

Paulina Marian, Tudor A. Marian, Horia Scutaru

Published Dec 10, 2003• 91 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.68.062309

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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.
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 Uhlmann fidelity of a pair of two-mode squeezed thermal states is evaluated as the maximal transition probability between two four-mode purifications. By applying the Peres-Simon criterion of separability we find the closest separable state. This enables us to derive an insightful expression of the amount of entanglement. Then we apply this measure of entanglement to the study of the Braunstein-Kimble protocol of teleportation. We use as input state in teleportation a mixed one-mode Gaussian state. The entangled state shared by the sender (Alice) and the receiver (Bob) is taken to be a two-mode squeezed thermal state. We find that the properties of the teleported state depend on both the input state and the entanglement of the two-mode resource state. As a measure of the quality of the teleportation process, we employ the Uhlmann fidelity between the input and output mixed one-mode Gaussian states.
We evaluate a Gaussian entanglement measure for a symmetric two-mode Gaussian state of the quantum electromagnetic field in terms of its Bures distance to the set of all separable Gaussian states. The required minimization procedure was considerably simplified by using the remarkable properties of the Uhlmann fidelity as well as the standard form II of the covariance matrix of a symmetric state. Our result for the Gaussian degree of entanglement measured by the Bures distance depends only on the smallest symplectic eigenvalue of the covariance matrix of the partially transposed density operator. It is thus consistent to the exact expression of the entanglement of formation for symmetric two-mode Gaussian states. This non-trivial agreement is specific to the Bures metric.
Entangled states play a crucial role in quantum information protocols, thus the dynamical behaviour of entanglement is of great importance. In this letter, we consider a two-mode squeezed vacuum state coupled to one thermal reservoir as a model of an entangled state embedded in an environment. As a criterion for entanglement we use a continuous-variable equivalent of the Peres–Horodecki criterion, namely the Simon criterion. To quantify entanglement we use the logarithmic negativity. We derive a condition, which assures that the state remains entangled in spite of the interaction with the reservoir. Moreover for the case of interaction with vacuum as an environment we show that a state of interest after infinitely long interaction is not only entangled, but also pure. For comparison we also consider a model in which each mode is coupled to its own reservoir.
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.
Considering the propagation of a two-mode optical field that is initially in a squeezed vacuum state in a thermal environment, we obtain an intuitive expression for inseparability condition of the two-mode mixed state which is given in the coherent state representation. This condition shows that the two modes have quantum entanglement if and only if the coefficient of the correlation term between the two modes is larger than that of the off-diagonal term of each mode in the density matrix. We find that even if the quantum channel is dynamically coupled to the thermal environment, the fidelity for teleporting coherent states larger than 1/2 is still the criterion for quantum teleportation. We also show that the entanglement, squeezing and quantum teleportation conditions are always consistent with each other.