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arXiv:1901.04389 · Physical Review A

Additivity of entanglement of formation via an entanglement-breaking space

Li-Jun Zhao, Lin Chen

Published Mar 8, 2019• 8 citations
arXivHTMLPDF

Abstract

We study the entanglement-breaking (EB) space, such that the entanglement of formation (EOF) of a bipartite quantum state is additive when its range is an EB subspace. We systematically construct the EB spaces in the Hilbert space ${\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{3}$, and the two-dimensional EB space in ${\mathbb{C}}^{2}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$. We characterize the expression of two-qubit states of rank two with nonadditive EOF, if they exist. We further apply our results to construct EB spaces of arbitrarily given dimensions. We show that the example in Phys. Rev. Lett. 89, 027901 (2002) is a special case of our results. We further work out the entanglement cost of a qubit-qutrit state in terms of the two-atom system of the Tavis-Cummings model.

DOI
10.1103/physreva.99.032310

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We study the entanglement-breaking (EB) space, such that the entanglement of formation (EOF) of a bipartite quantum state is additive when its range is an EB subspace. We systematically construct the EB spaces in the Hilbert space Cm\ensuremath⨂C3{\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{3}Cm\ensuremath⨂C3, and the two-dimensional EB space in C2\ensuremath⨂Cn{\mathbb{C}}^{2}\ensuremath{\bigotimes}{\mathbb{C}}^{n}C2\ensuremath⨂Cn. We characterize the expression of two-qubit states of rank two with nonadditive EOF, if they exist. We further apply our results to construct EB spaces of arbitrarily given dimensions. We show that the example in Phys. Rev. Lett. 89, 027901 (2002) is a special case of our results. We further work out the entanglement cost of a qubit-qutrit state in terms of the two-atom system of the Tavis-Cummings model.
Abstract Entanglement-breaking (EB) subspaces determine the additivity of entanglement of formation (EOF), which is a long-standing issue in quantum information. We explicitly construct the two-dimensional EB subspaces of any bipartite system, when system dimensions are equal, and we apply the subspaces to construct EB spaces of arbitrary dimensions. We also present partial construction when system dimensions are different. Then, we present the notion and properties of EB subspaces for some systems, and in particular the absolute EB subspaces. We construct some examples of absolute EB subspaces, as well as EB subspaces for some systems by using multiqubit Dicke states.
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In several articles, the authors assume that the convex roof structure of the EoF and finite-dimensionality of subsystems AAA and BBB guarantee the existence of the (global) supporting affine functional for the EoF at any state of the system ABABAB. This means that for any state ρρρ of ABABAB there is a Hermitian operator ΛρΛ_ρΛρ​ on HAB=HA⊗HB\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_BHAB​=HA​⊗HB​ such that EF(ρ)=TrΛρρE_F(ρ)=\mathrm{Tr}Λ_ρρEF​(ρ)=TrΛρ​ρ and EF(σ)≥TrΛρσE_F(σ)\geq\mathrm{Tr}Λ_ρσEF​(σ)≥TrΛρ​σ for any state σσσ of ABABAB. We present an explicit example showing that, when ρρρ is degenerate, this is not true even in the simplest case when AAA and BBB are qubit systems. The construction is based on the fact that the existence of a supporting affine functional for the EoF at a state ρρρ is equivalent to the Lipschitz lower semicontinuity of the EoF at this state ρρρ. We use Wootters' formula and the help of Claude Fable 5 to find a state ρρρ of the system ABABAB 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 ρρρ (i.e. inequalities of the form  EF(ρ)−EF(σ)≤Cρ∥ρ−σ∥1\,E_F(ρ)-E_F(σ)\leq C_ρ\|ρ-σ\|_1EF​(ρ)−EF​(σ)≤Cρ​∥ρ−σ∥1​) with and without restrictions on the support of the state σσσ.
Entanglement of formation for arbitrary mixed states in a ($2\ensuremath{\bigotimes}2$)-dimensional Hilbert space is well known. For higher-dimensional mixed states analytical expressions for quantifying entanglement exist for a particular family of states. In this work, we calculate analytically the entanglement of formation for a family of bipartite ($2\ensuremath{\bigotimes}d$)-dimensional mixed states which can be obtained from tripartite $2\ensuremath{\bigotimes}2\ensuremath{\bigotimes}d$ pure states following the entanglement relations obtained by Koashi and Winter [Phys. Rev. A 69, 022309 (2004)].
We present observable lower and upper bounds for the entanglement of formation (EOF) and quantum discord (QD), which facilitates estimates of EOF and QD for arbitrary experimental unknown states in finite-dimensional bipartite systems. These bounds can be easily obtained by a few experimental measurements on a twofold copy $\ensuremath{\varrho}\ensuremath{\bigotimes}\ensuremath{\varrho}$ of the mixed states. Based on our results, we use the experimental measurement data of the real experiment given by Schmid et al. [Phys. Rev. Lett. 101, 260505 (2008)] to obtain the lower and upper bounds of EOF and QD for the experimental unknown state.