Journal of Physics A Mathematical and Theoretical

The distillability of entanglement of bipartite reduced density matrices of a tripartite state

Yize Sun, Lin Chen

Published Jun 1, 2020• 4 citations

Abstract

Abstract The relation between the distillability of entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in Hayashi and Chen [2011 Phys. Rev. A 84 012325]. We extend this result to the tripartite mixed state of rank at most three. In particular we show that if the state has two bipartite reduced density operators with rank two, then the third bipartite reduced density operator additionally having non positive partial transpose (non-PPT) is distillable. In contrast, we show that the tripartite PPT state with two reduced density operators of rank two is a three-qubit fully separable state. We obtain these facts by proving a conjectured matrix inequality for the bipartite matrix M with Schmidt rank at most three. This is one of the main results of this paper. We also prove it for some M with arbitrary Schmidt rank.

DOI: 10.1088/1751-8121/ab985b

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Journal of Physics A Mathematical and Theoretical

The distillability of entanglement of bipartite reduced density matrices of a tripartite state

Yize Sun, Lin Chen

Published Jun 1, 2020• 4 citations

Publisher

Abstract

DOI: 10.1088/1751-8121/ab985b

Related papers

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Entanglement distillation is a basic task in quantum information, and the distillable entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in [Phys. Rev. A 84, 012325 (2011)]. We extend this result to tripartite mixed states by studying a conjectured matrix inequality, namely $\mathop{\rm rank}(\sum_i R_i \otimes S_i)\leq K \mathop{\rm rank}(\sum_i R_i^T \otimes S_i)$ holds for any bipartite matrix $M=\sum_i R_i \otimes S_i$ and Schmidt rank $K$ . We prove that the conjecture holds for $M$ with $K=3$ and some special $M$ with arbitrary $K$ .Entanglement distillation is a basic task in quantum information, and the distillable entanglement of three bipartite reduced density matrices from a tripartite pure state has been studied in [Phys. Rev. A 84, 012325 (2011)]. We extend this result to tripartite mixed states by studying a conjectured matrix inequality, namely $\mathop{\rm rank}(\sum_i R_i \otimes S_i)\leq K \mathop{\rm rank}(\sum_i R_i^T \otimes S_i)$ holds for any bipartite matrix $M=\sum_i R_i \otimes S_i$ and Schmidt rank $K$. We prove that the conjecture holds for $M$ with $K=3$ and some special $M$ with arbitrary $K$.

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arXiv:1903.05373QuantumDec 2, 20199 citations

\mathop{\rm rank}(\sum_i R_i \otimes S_i)\leq K \mathop{\rm rank}(\sum_i R_i^T \otimes S_i)

holds for any bipartite matrix

M=\sum_i R_i \otimes S_i

and Schmidt rank

K

. We prove that the conjecture holds for

M

with

K=3

and some special

M

with arbitrary

K

Assume that Alice, Bob, and Charlie share a tripartite pure state$ \vert \psi _{ABC}\rangle

. We prove that if Alice cannot distill entanglement with either Bob or Charlie using

\vert \psi _{ABC}\rangle

and local operations with any one of the following configurations for classical communication:

(A\to B, A\leftrightarrow C), (A\leftrightarrow B, A\to C)

, and

(A\leftrightarrow B, A\leftrightarrow C)

, then the same is also true for the other two configurations. Moreover, this happens precisely when the state is such that both its reductions on systems

and

AC$are separable, which is further equivalent to the reductions being PPT. This, in particular, implies that any NPT bipartite state is such that either the state itself or its complement is 2-way distillable. In proving these results, we first obtain an explicit lower bound on the 2-way distillable entanglement of low rank bipartite states. Furthermore, we show that even though not all low rank states are 1-way distillable, a randomly sampled low rank state will almost surely be 1-way distillable.

Characterizing the relations among the three bipartite reduced density operators$ \rho _{AB}

,

\rho _{AC}

and

\rho _{BC}

of a tripartite mixed state

\rho _{ABC}

has been an open problem in quantum information. One of such relations has been reduced by [Cadney et al, LAA. 452, 153, 2014] to a conjectured inequality in terms of matrix rank, namely

r(\rho _{AB}) \cdot r( \rho _{AC})\ge r( \rho _{BC})

for any

\rho _{ABC}$. It is denoted as open problem 41 in the website “Open quantum problems-IQOQI Vienna”. We prove the inequality, and thus establish a complete picture of the four-party linear inequalities in terms of the 0-entropy. Our proof is based on the construction of a novel canonical form of bipartite matrices under local equivalence. We extend our result to inequalities in multipartite systems, as well as the condition when the inequality is saturated.