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Chinese Physics

Relative entropy as a measure of entanglement for Gaussian states

Huai-Xin Lu, Bo Zhao

Published Aug 22, 2006• 7 citations
Publisher

Abstract

In this paper, we derive an explicit analytic expression of the relative entropy between two general Gaussian states. In the restriction of the set for Gaussian states and with the help of relative entropy formula and Peres–Simon separability criterion, one can conveniently obtain the relative entropy entanglement for Gaussian states. As an example, the relative entanglement for a two-mode squeezed thermal state has been obtained.

DOI
10.1088/1009-1963/15/9/002

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In this paper, we derive an explicit analytic expression of the relative entropy between two general Gaussian states. In the restriction of the set for Gaussian states and with the help of relative entropy formula and Peres–Simon separability criterion, one can conveniently obtain the relative entropy entanglement for Gaussian states. As an example, the relative entanglement for a two-mode squeezed thermal state has been obtained.
We calculate the relative entropy between two quantum Gaussian states with given correlation matrices by demonstrating a practical method of transforming one of the correlation matrices into an exponential quadratic operator matrix. We show that the closest Gaussian separable state achieving the Gaussian relative entropy of entanglement is at the border between separable and inseparable Gaussian state sets. For a two-mode Gaussian state, the calculation of the Gaussian relative entropy of entanglement is greatly simplified by deducing a matrix with ten undetermined parameters to one with only three. The two-mode Gaussian states are classified into four types. Numerical calculations strongly suggest that the Gaussian relative entropy of entanglement for each type is realized by a Gaussian separable state within the same type. For a symmetric Gaussian state it is strictly proven that the Gaussian relative entropy of entanglement is achieved by a symmetric Gaussian separable state.
We in this paper calculate the relative entropy between the bipartite Gaussian states. By evaluating the minimization of the relative entropy of an entangled Gaussian state with respect to the separable Gaussian states, we obtain the Gaussian relative entropy of entanglement of the state. The bipartite operations of squeezing and rotations on the separable states are also addressed. We prove that for bipartite symmetric Gaussian states the separable states we sought will be symmetric Gaussian states, and we get quite simple expressions of Gaussian relative entropy of entanglement for such states.
The relative entropy of entanglement is defined in terms of the relative entropy between an entangled state and its closest separable state (CSS). Given a multipartite-state on the boundary of the set of separable states, we find an explicit expression for all the entangled states for which this state is a CSS. Our formula holds for multipartite states in all dimensions. For the bipartite case of two qubits, our formula reduce to the one given by Miranowicz and Ishizaka [Phys. Rev. A 78, 032310 (2008)].
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.
The quantum Rényi relative entropies play a prominent role in quantum information theory, finding applications in characterizing error exponents and strong converse exponents for quantum hypothesis testing and quantum communication theory. On a different thread, quantum Gaussian states have been intensely investigated theoretically, motivated by the fact that they are more readily accessible in the laboratory than are other, more exotic quantum states. In this paper, we derive formulas for the quantum Rényi relative entropies of quantum Gaussian states. We consider both the traditional (Petz) Rényi relative entropy as well as the more recent sandwiched Rényi relative entropy, finding formulas that are expressed solely in terms of the mean vectors and covariance matrices of the underlying quantum Gaussian states. Our development handles the hitherto elusive case for the Petz–Rényi relative entropy when the Rényi parameter is larger than one. Finally, we also derive a formula for the max-relative entropy of two quantum Gaussian states, and we discuss some applications of the formulas derived here.