arXiv:2009.04982 · Journal of Physics A Mathematical and Theoretical

Closest separable state when measured by a quasi-relative entropy

Anna Vershynina

Published Jan 28, 2021• 1 citation

Abstract

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.

DOI: 10.1088/1751-8121/abe0d7

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arXiv:2009.04982 · Journal of Physics A Mathematical and Theoretical

Closest separable state when measured by a quasi-relative entropy

Anna Vershynina

Published Jan 28, 2021• 1 citation

arXiv HTML PDF

Abstract

DOI: 10.1088/1751-8121/abe0d7

Related papers

Attainability and Lower Semi-continuity of the Relative Entropy of Entanglement and Variations on the Theme

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Quantifying entanglement for multipartite quantum state is a crucial task in many aspects of quantum information theory. Among all the entanglement measures, relative entropy of entanglement $E_{R}$ is an important quantity due to its clear geometric meaning, easy compatibility with different system sizes, and various applications in many other related quantity calculations. Lower bounds of $E_R$ were previously found based on distance to the set of positive partial transpose states. We propose a method to calculate upper bounds of $E_R$ based on the convex hull approximation of the set of separable states. We apply our method to calculate $E_R$ for composite systems of various sizes, and compare with the previous known lower bounds, obtaining promising results. Our method adds a new tool for entanglement measure calculation and deepens our understanding for the structure of separable states.Quantifying entanglement for multipartite quantum state is a crucial task in many aspects of quantum information theory. Among all the entanglement measures, relative entropy of entanglement $E_{R}$ is an important quantity due to its clear geometric meaning, easy compatibility with different system sizes, and various applications in many other related quantity calculations. Lower bounds of $E_R$ were previously found based on distance to the set of positive partial transpose states. We propose a method to calculate upper bounds of $E_R$ based on the convex hull approximation of the set of separable states. We apply our method to calculate $E_R$ for composite systems of various sizes, and compare with the previous known lower bounds, obtaining promising results. Our method adds a new tool for entanglement measure calculation and deepens our understanding for the structure of separable states.

Sep 5, 20190 citations

Abstract The relative entropy of entanglement

E_R

E R is defined as the distance of a multipartite quantum state from the set of separable states as measured by the quantum relative entropy. We show that this optimisation is always achieved, i.e. any state admits a closest separable state, even in infinite dimensions; also,

E_R

E R is everywhere lower semi-continuous. We use this to derive a dual variational expression for

E_R

E R in terms of an external supremum instead of infimum. These results, which seem to have gone unnoticed so far, hold not only for the relative entropy of entanglement and its multipartite generalisations, but also for many other similar resource quantifiers, such as the relative entropy of non-Gaussianity, of non-classicality, of Wigner negativity—more generally, all relative entropy distances from the sets of states with non-negative

\lambda

λ -quasi-probability distribution. The crucial hypothesis underpinning all these applications is the weak*-closedness of the cone generated by free states, and for this reason, the techniques we develop involve a bouquet of classical results from functional analysis. We complement our analysis by giving explicit and asymptotically tight continuity estimates for

E_R

E R and closely related quantities in the presence of an energy constraint.

We determine the relative entropy of entanglement of a bipartite mixed state

ρ_{AB}

obtained by tracing out one subsystem of a tripartite Haar-random pure state

|ψ\rangle_{ABC}

, finding

E_R(ρ_{AB})=\log\frac{d_Ad_B}{\max(d_A,d_B,d_C)}+O(1)

. Equivalently, the relative entropy of entanglement nearly saturates the smaller of the entanglement of formation

E_F(ρ_{AB})

and the mutual information

I(A:B)

. The upper bound is achieved by an explicit separable state obtained through one-sided Schmidt dephasing, which is therefore approximately closest.

Quantifying entanglement for multipartite quantum state is a crucial task in many aspects of quantum information theory. Among all the entanglement measures, relative entropy of entanglement

E_{R}

is an important quantity due to its clear geometric meaning, easy compatibility with different system sizes, and various applications in many other related quantity calculations. Lower bounds of

E_R

were previously found based on distance to the set of positive partial transpose states. We propose a method to calculate upper bounds of

E_R

based on the convex hull approximation of the set of separable states. We apply our method to calculate

E_R

for composite systems of various sizes, and compare with the previous known lower bounds, obtaining promising results. Our method adds a new tool for entanglement measure calculation and deepens our understanding for the structure of separable states.