© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:quant-ph/0402109 · Physical Review A

Gaussian relative entropy of entanglement

Xiaoyu Chen

Published Jun 17, 2005• 32 citations

Abstract

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.

DOI: 10.1103/physreva.71.062320

Related papers

Calculating the Gaussian relative entropy of entanglement

Li-zhen Jiang, Liang-neng Wu, Yun Zhou

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.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jan 11, 2005•0 citations

Relative entropy of entanglement of two-qubit Ux-invariant states

Zhen Wang, Zhi‐Xi Wang

It is strictly proved that a two-qubit [Formula: see text]-invariant state reaches its relative entropy of entanglement (REE) by the separable state having the same matrix structure. We also formulate three quadratic equations for the corresponding closest separable state (CSS) of [Formula: see text]-invariant states by their symmetric property. Thus, the CSS of [Formula: see text]-invariant state can be provided. Furthermore, to illustrate our result we consider two concrete examples.

International Journal of Quantum Information•Dec 1, 2015•0 citations

Relative entropy as a measure of entanglement for Gaussian states

Huai-Xin Lu, Bo Zhao

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.

Chinese Physics•Aug 22, 2006•7 citations

ENTANGLEMENT BOUNDS FOR A FAMILY OF TWO-MODE GAUSSIAN STATES

Li-zhen Jiang

I study a family of bipartite quantum Gaussian states with three parameters, calculate Gaussian entanglement of formation analytically and the upper bound of relative entropy of entanglement, compare them with the coherent information of the states. Based on the numerical observation, I determine the relative entropy of entanglement and distillable entanglement of the states with infinitive squeezing.

International Journal of Quantum Information•Jun 1, 2004•7 citations

Closest separable state when measured by a quasi-relative entropy

Anna Vershynina

Abstract It is well known that for pure states the relative entropy of entanglement is equal to the reduced entropy, and the closest separable state is explicitly known as well. The same holds for Renyi relative entropy per recent results. We ask the same question for a quasi-relative entropy of entanglement, which is an entanglement measure defined as the minimum distance to the set of separable state, when the distance is measured by the quasi-relative entropy. First, we consider a maximally entangled state, and show that the closest separable state is the same for any quasi-relative entropy as for the relative entropy of entanglement. Then, we show that this also holds for a certain class of functions and any pure state. And at last, we consider any pure state on two qubit systems and a large class of operator convex function. For these, we find the closest separable state, which may not be the same one as for the relative entropy of entanglement.

arXiv:2009.04982•Journal of Physics A Mathematical and Theoretical•Jan 28, 2021•1 citation

Gaussian entanglement and quantum communication: geodesic optimization and multimode bounds

Hemant K. Mishra, Mark M. Wilde

Loss and noise limit the use of light to distribute entanglement, transmit quantum information, and establish secret keys. Gaussian states have a compact description in terms of means and covariance matrices, but their entanglement still requires global optimization. We develop a framework for computing Gaussian-restricted versions of the relative entropy of entanglement and Rains entanglement. An integral representation reveals a convex geometry under matrix geometric means and reduces Gaussian Rains entanglement to relative entropy minimized over Gaussian states with positive partial transpose. We obtain unique optimal reference states and additivity for independent Gaussian input states. Our convex-majorization algorithm has an explicit geometric convergence rate, uniform in the faithful regularization parameter and valid for pure input states. Controlled numerical integration and approximate convex subproblem solutions give a bit-complexity bound that is polynomial in accuracy bits for each fixed rational input covariance matrix, with explicit dependence on covariance size. These calculations bound extractable entanglement and, through the relative entropy of entanglement, extractable secret key. Teleportation simulation, energy allocation, and a convex optimization of passive-basis loss cores give bounds on quantum and private communication under an input-energy constraint, allowing arbitrary two-way classically assisted protocols. Two- and three-mode examples compare the bounds with achievable rates, including matching pure-loss benchmarks. Finally, we separate the Gaussian-restricted quantities from their unrestricted counterparts and give a photon-number truncation method with controlled error for computing the latter.

quant-pharXiv:2610.18827•Oct 14, 2026•0 citations

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.

Abstract It is well known that for pure states the relative entropy of entanglement is equal to the reduced entropy, and the closest separable state is explicitly known as well. The same holds for Renyi relative entropy per recent results. We ask the same question for a quasi-relative entropy of entanglement, which is an entanglement measure defined as the minimum distance to the set of separable state, when the distance is measured by the quasi-relative entropy. First, we consider a maximally entangled state, and show that the closest separable state is the same for any quasi-relative entropy as for the relative entropy of entanglement. Then, we show that this also holds for a certain class of functions and any pure state. And at last, we consider any pure state on two qubit systems and a large class of operator convex function. For these, we find the closest separable state, which may not be the same one as for the relative entropy of entanglement.

Loss and noise limit the use of light to distribute entanglement, transmit quantum information, and establish secret keys. Gaussian states have a compact description in terms of means and covariance matrices, but their entanglement still requires global optimization. We develop a framework for computing Gaussian-restricted versions of the relative entropy of entanglement and Rains entanglement. An integral representation reveals a convex geometry under matrix geometric means and reduces Gaussian Rains entanglement to relative entropy minimized over Gaussian states with positive partial transpose. We obtain unique optimal reference states and additivity for independent Gaussian input states. Our convex-majorization algorithm has an explicit geometric convergence rate, uniform in the faithful regularization parameter and valid for pure input states. Controlled numerical integration and approximate convex subproblem solutions give a bit-complexity bound that is polynomial in accuracy bits for each fixed rational input covariance matrix, with explicit dependence on covariance size. These calculations bound extractable entanglement and, through the relative entropy of entanglement, extractable secret key. Teleportation simulation, energy allocation, and a convex optimization of passive-basis loss cores give bounds on quantum and private communication under an input-energy constraint, allowing arbitrary two-way classically assisted protocols. Two- and three-mode examples compare the bounds with achievable rates, including matching pure-loss benchmarks. Finally, we separate the Gaussian-restricted quantities from their unrestricted counterparts and give a photon-number truncation method with controlled error for computing the latter.