© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

arXiv:quant-ph/0006128 · Physical Review A

Variational characterizations of separability and entanglement of formation

Koenraad M. R. Audenaert, Frank Verstraete, Bart De Moor

Published Oct 10, 2001• 176 citations

Abstract

In this paper we develop a mathematical framework for the characterization of separability and entanglement of formation (EOF) of general bipartite states. These characterizations are variational in nature, meaning that separability and EOF are given in terms of a function that is to be minimized over the manifold of unitary matrices. A major benefit of such a characterization is that it directly leads to a numerical procedure for calculating EOF. We present an efficient minimization algorithm and apply it to the bound entangled $3\ifmmode\times\else\texttimes\fi{}3$ Horodecki states; we show that their EOF is very low and that their distance to the set of separable states is also very small. Within the same variational framework we rephrase the results by Wootters [W. Wootters, Phys. Rev. Lett. 80, 2245 (1998)] on EOF for $2\ifmmode\times\else\texttimes\fi{}2$ states and also present some progress in generalizing these results to higher-dimensional systems.

DOI: 10.1103/physreva.64.052304

Related papers

On supporting affine functionals for Entanglement of Formation

A. S. Holevo, M. E. Shirokov

In several articles, the authors assume that the convex roof structure of the EoF and finite-dimensionality of subsystems $A$ and $B$ guarantee the existence of the (global) supporting affine functional for the EoF at any state of the system $AB$. This means that for any state $ρ$ of $AB$ there is a Hermitian operator $Λ_ρ$ on $\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_B$ such that $E_F(ρ)=\mathrm{Tr}Λ_ρρ$ and $E_F(σ)\geq\mathrm{Tr}Λ_ρσ$ for any state $σ$ of $AB$. We present an explicit example showing that, when $ρ$ is degenerate, this is not true even in the simplest case when $A$ and $B$ are qubit systems. The construction is based on the fact that the existence of a supporting affine functional for the EoF at a state $ρ$ is equivalent to the Lipschitz lower semicontinuity of the EoF at this state $ρ$. We use Wootters' formula and the help of Claude Fable 5 to find a state $ρ$ of the system $AB$ for which the latter property does not hold. We also describe conditions for the existence the local and global supporting affine functionals for the EoF at a given state of both finite and infinite-dimensional bipartite quantum systems. These conditions allow us to find Lipschitz lower semicontinuity bounds for the EoF at a given finite rank state $ρ$ (i.e. inequalities of the form $\,E_F(ρ)-E_F(σ)\leq C_ρ\|ρ-σ\|_1$) with and without restrictions on the support of the state $σ$.

quant-pharXiv:2608.27363•Aug 28, 2026•0 citations

On entanglement of formation

Adam W Majewski

In this paper we present the novel qualities of entanglement of formation (EoF) for general (so also infinite-dimensional) quantum systems. A major benefit of our presentation is a rigorous description of EoF. In particular, we indicate how this description may be used to examine optimal decompositions. Illustrative examples showing the method of estimation of EoF are given.

Journal of Physics A Mathematical and General•Dec 21, 2001•17 citations

Lower bound of entanglement of formation based on correlation matrix

Jing Wang, Nan Yang, Longyu Xiao +3 more

The entanglement of formation (EOF), which quantifies the required minimally physical resources to prepare a quantum state, plays important roles in many quantum information processes. Due to the convex-roof definition, it is hard to find the explicit analytic formulae of EOF, except for some special quantum systems. The correlation matrix of a quantum state is closely related to entanglement property. In this paper, we use the Lagrange-multiplier approach to obtain an analytic lower bound of EOF for arbitrary dimensional bipartite states, which improves the previous conclusions. Numerical example shows that the bound is also valid for some PPT entangled states.

Results in Physics•Dec 19, 2023•0 citations

Determining Separability with Entanglement of Formation and Entanglement Sudden Death

Xiaoyu Chen

An algorithm is developed to calculate the entanglement of formation for bipartite quantum states to determine numerically the sufficient condition of separability. The algorithm is applied to two 3 × 3 positive partial transpose states mixed with white noise. For these two noisy states, our numerical sufficient conditions of separability are not far from the best necessary conditions of separability.

Chinese Physics Letters•Jan 1, 2015•4 citations

Universal Measure of Entanglement

M. Hossein Partovi

A general framework is developed for separating classical and quantum correlations in a multipartite system. Entanglement is defined as the difference in the correlation information encoded by the state of a system and a suitably defined separable state with the same marginals. A generalization of the Schmidt decomposition is developed to implement the separation of correlations for any pure, multipartite state. The measure based on this decomposition is a generalization of the entanglement of formation to multipartite systems, provides an upper bound for the relative entropy of entanglement, and is directly computable on pure states. The example of pure three-qubit states is analyzed in detail, and a classification based on minimal, four-term decompositions is developed.

arXiv:quant-ph/0308051•Physical Review Letters•Feb 20, 2004•23 citations

Entanglement of Formation for Gaussian States

J. Solomon Ivan, R. Simon

The entanglement of formation (EOF) is computed for arbitrary two-mode Gaussian states. Apart from a conjecture, our analysis rests on two main ingredients. The first is a four-parameter canonical form we develop for the covariance matrix, one of these parameters acting as a measure of EOF, and the second is a generalisation of the EPR correlation, used in the work of Giedke {\em et al} [Phys. Rev. Lett. {\bf 91}, 107901 (2003)], to noncommuting variables. The conjecture itself is in respect of an extremal property of this generalized EPR correlation.

arXiv:0808.1658•Aug 12, 2008•3 citations

In this paper we develop a mathematical framework for the characterization of separability and entanglement of formation (EOF) of general bipartite states. These characterizations are variational in nature, meaning that separability and EOF are given in terms of a function that is to be minimized over the manifold of unitary matrices. A major benefit of such a characterization is that it directly leads to a numerical procedure for calculating EOF. We present an efficient minimization algorithm and apply it to the bound entangled

3\ifmmode\times\else\texttimes\fi{}3

Horodecki states; we show that their EOF is very low and that their distance to the set of separable states is also very small. Within the same variational framework we rephrase the results by Wootters [W. Wootters, Phys. Rev. Lett. 80, 2245 (1998)] on EOF for

2\ifmmode\times\else\texttimes\fi{}2

states and also present some progress in generalizing these results to higher-dimensional systems.

In several articles, the authors assume that the convex roof structure of the EoF and finite-dimensionality of subsystems

A

and

B

guarantee the existence of the (global) supporting affine functional for the EoF at any state of the system

AB

. This means that for any state

ρ

of

AB

there is a Hermitian operator

Λ_ρ

on

\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_B

such that

E_F(ρ)=\mathrm{Tr}Λ_ρρ

and

E_F(σ)\geq\mathrm{Tr}Λ_ρσ

for any state

σ

of

AB

. We present an explicit example showing that, when

ρ

is degenerate, this is not true even in the simplest case when

A

and

B

are qubit systems. The construction is based on the fact that the existence of a supporting affine functional for the EoF at a state

ρ

is equivalent to the Lipschitz lower semicontinuity of the EoF at this state

ρ

. We use Wootters' formula and the help of Claude Fable 5 to find a state

ρ

of the system

AB

for which the latter property does not hold. We also describe conditions for the existence the local and global supporting affine functionals for the EoF at a given state of both finite and infinite-dimensional bipartite quantum systems. These conditions allow us to find Lipschitz lower semicontinuity bounds for the EoF at a given finite rank state

ρ

(i.e. inequalities of the form

\,E_F(ρ)-E_F(σ)\leq C_ρ\|ρ-σ\|_1

) with and without restrictions on the support of the state

σ

.

The entanglement of formation (EOF), which quantifies the required minimally physical resources to prepare a quantum state, plays important roles in many quantum information processes. Due to the convex-roof definition, it is hard to find the explicit analytic formulae of EOF, except for some special quantum systems. The correlation matrix of a quantum state is closely related to entanglement property. In this paper, we use the Lagrange-multiplier approach to obtain an analytic lower bound of EOF for arbitrary dimensional bipartite states, which improves the previous conclusions. Numerical example shows that the bound is also valid for some PPT entangled states.

A general framework is developed for separating classical and quantum correlations in a multipartite system. Entanglement is defined as the difference in the correlation information encoded by the state of a system and a suitably defined separable state with the same marginals. A generalization of the Schmidt decomposition is developed to implement the separation of correlations for any pure, multipartite state. The measure based on this decomposition is a generalization of the entanglement of formation to multipartite systems, provides an upper bound for the relative entropy of entanglement, and is directly computable on pure states. The example of pure three-qubit states is analyzed in detail, and a classification based on minimal, four-term decompositions is developed.

The entanglement of formation (EOF) is computed for arbitrary two-mode Gaussian states. Apart from a conjecture, our analysis rests on two main ingredients. The first is a four-parameter canonical form we develop for the covariance matrix, one of these parameters acting as a measure of EOF, and the second is a generalisation of the EPR correlation, used in the work of Giedke {\em et al} [Phys. Rev. Lett. {\bf 91}, 107901 (2003)], to noncommuting variables. The conjecture itself is in respect of an extremal property of this generalized EPR correlation.