arXiv:quant-ph/9607009

Distillability of Inseparable Quantum Systems

Michał Horodecki, Horodecki, Pawel, Horodecki, Ryszard

Published Jul 11, 1996• 9 citations

Abstract

We apply the inseparability criterion for $2 \times 2$ systems, local filtering and Bennett et al. purification protocol [Phys. Rev. Lett. {\bf 76}, 722 (1996)] to show how to distill {\it any} inseparable $2\times 2$ system. The extended protocol is illustrated geometrically by means of the state parameters in the Hilbert-Schmidt space.

DOI: 10.48550/arxiv.quant-ph/9607009

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arXiv:quant-ph/9607009

Distillability of Inseparable Quantum Systems

Michał Horodecki, Horodecki, Pawel, Horodecki, Ryszard

Published Jul 11, 1996• 9 citations

arXiv HTML PDF

Abstract

We apply the inseparability criterion for

2 \times 2

systems, local filtering and Bennett et al. purification protocol [Phys. Rev. Lett. {\bf 76}, 722 (1996)] to show how to distill {\it any} inseparable

2\times 2

system. The extended protocol is illustrated geometrically by means of the state parameters in the Hilbert-Schmidt space.

DOI: 10.48550/arxiv.quant-ph/9607009

Related papers

Inseparable Two Spin- 12 Density Matrices Can Be Distilled to a Singlet Form

Michał Horodecki, Paweł Horodecki, Ryszard Horodecki

Physical Review LettersJan 27, 1997415 citations

Reduction criterion of separability and limits for a class of distillation protocols

Michał Horodecki, Paweł Horodecki

We analyze the problem of distillation of entanglement of mixed states in higher-dimensional compound systems. Employing the positive maps method [M. Horodecki et al., Phys. Lett. A 223, 1 (1996)] we introduce and analyze a criterion of separability that relates the structures of the total density matrix and its reductions. We show that any state violating the criterion can be distilled by suitable generalization of the two-qubit protocol that distills any inseparable two-qubit state. In particular, this means that any state $\ensuremath{\varrho}$ of two N-level systems with $〈{\ensuremath{\psi}}_{+}|\ensuremath{\varrho}|{\ensuremath{\psi}}_{+}〉>1/N$ can be distilled $({\ensuremath{\psi}}_{+}$ is the singlet state generalized to higher dimension). The criterion also singles out all the states that can be distilled by a class of protocols. The proof involves construction of the family of states that are invariant under transformation $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\varrho}}U\ensuremath{\bigotimes}{U}^{*}\ensuremath{\varrho}{U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\bigotimes}{U}^{*\ifmmode\dagger\else\textdagger\fi{}},$ where U is a unitary transformation and the asterisk denotes complex conjugation. The states are related to the depolarizing channel generalized to the nonbinary case.We analyze the problem of distillation of entanglement of mixed states in higher-dimensional compound systems. Employing the positive maps method [M. Horodecki et al., Phys. Lett. A 223, 1 (1996)] we introduce and analyze a criterion of separability that relates the structures of the total density matrix and its reductions. We show that any state violating the criterion can be distilled by suitable generalization of the two-qubit protocol that distills any inseparable two-qubit state. In particular, this means that any state $\ensuremath{\varrho}$ of two N-level systems with $〈{\ensuremath{\psi}}_{+}|\ensuremath{\varrho}|{\ensuremath{\psi}}_{+}〉>1/N$ can be distilled $({\ensuremath{\psi}}_{+}$ is the singlet state generalized to higher dimension). The criterion also singles out all the states that can be distilled by a class of protocols. The proof involves construction of the family of states that are invariant under transformation $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\varrho}}U\ensuremath{\bigotimes}{U}^{*}\ensuremath{\varrho}{U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\bigotimes}{U}^{*\ifmmode\dagger\else\textdagger\fi{}},$ where U is a unitary transformation and the asterisk denotes complex conjugation. The states are related to the depolarizing channel generalized to the nonbinary case.

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Reduction criterion of separability and limits for a class of protocols of entanglement distillation

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Ping-Xing Chen, Cheng Zu Li

This paper present a geometric diagram of a separable state: If a mixed state $σ$ is separable, there are $2^{nS(σ)}$ linearly independant product vectors which span the same Hilbert space as the $2^{nS(σ)}$ ``likely'' strings of $σ^{\otimes n}$ do. This diagram results in a criterion for separability which is strictly stronger than the inorder criterion in [M.A. Nielsen and J. Kempe, Phys. Rev. Lett. 86, 5184 (2001)]. This means that the number of product bases of states of a system has close link to the nonlocality of the system.This paper present a geometric diagram of a separable state: If a mixed state $σ$ is separable, there are $2^{nS(σ)}$ linearly independant product vectors which span the same Hilbert space as the $2^{nS(σ)}$ ``likely'' strings of $σ^{\otimes n}$ do. This diagram results in a criterion for separability which is strictly stronger than the inorder criterion in [M.A. Nielsen and J. Kempe, Phys. Rev. Lett. 86, 5184 (2001)]. This means that the number of product bases of states of a system has close link to the nonlocality of the system.

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Comment on “Inseparability Criteria for Continuous Bipartite Quantum States”

Adam Miranowicz, Marco Piani

A Comment on the Letter by E. Shchukin and W. Vogel, Phys. Rev. Lett. 95, 230502 (2005).

arXiv:quant-ph/0603239Physical Review LettersAug 2, 200655 citations

Criteria For $K$ -Separability Of Multipartite Quantum State

Zhihao Ma, Zhihua Chen, Jing‐Ling Chen

We derive criteria for $k$ -separability of multipartite Quantum stateWe derive criteria for $k$-separability of multipartite Quantum state

arXiv:1103.4889Mar 25, 20110 citations

We apply the inseparability criterion for

2 \times 2

systems, local filtering and Bennett et al. purification protocol [Phys. Rev. Lett. {\bf 76}, 722 (1996)] to show how to distill {\it any} inseparable

2\times 2

system. The extended protocol is illustrated geometrically by means of the state parameters in the Hilbert-Schmidt space.

\varrho\to U\otimes U^*\varrho U^\dagger\otimes U^{*\dagger}

where

U

is a unitary transformation and star denotes complex conjugation. The states are related to the depolarizing channel generalized to non-binary case.

This paper present a geometric diagram of a separable state: If a mixed state

σ

is separable, there are

2^{nS(σ)}

linearly independant product vectors which span the same Hilbert space as the

2^{nS(σ)}

``likely'' strings of

σ^{\otimes n}

do. This diagram results in a criterion for separability which is strictly stronger than the inorder criterion in [M.A. Nielsen and J. Kempe, Phys. Rev. Lett. 86, 5184 (2001)]. This means that the number of product bases of states of a system has close link to the nonlocality of the system.

We derive criteria for

k

-separability of multipartite Quantum state