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arXiv:1909.02748

Entanglement distillation in terms of a conjectured matrix inequality

Yize Sun, Lin Chen

Published Sep 6, 2019• 0 citations
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Abstract

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 $K$.

DOI
10.48550/arxiv.1909.02748

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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 rank(∑iRi⊗Si)≤Krank(∑iRiT⊗Si)\mathop{\rm rank}(\sum_i R_i \otimes S_i)\leq K \mathop{\rm rank}(\sum_i R_i^T \otimes S_i)rank(∑i​Ri​⊗Si​)≤Krank(∑i​RiT​⊗Si​) holds for any bipartite matrix M=∑iRi⊗SiM=\sum_i R_i \otimes S_iM=∑i​Ri​⊗Si​ and Schmidt rank KKK. We prove that the conjecture holds for MMM with K=3K=3K=3 and some special MMM with arbitrary KKK.
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.
The pure entangled state is of vital importance in the field of quantum information. The process of asymptotically extracting pure entangled states from many copies of mixed states via local operations and classical communication is called entanglement distillation. The entanglement distillability problem, which is a long-standing open problem, asks whether such process exists. The 2-copy undistillability of 4×44\times44×4 undistillable Werner states has been reduced to the validness of the a matrix inequality, that is, the sum of the squares of the largest two singular values of matrix A⊗I+I⊗BA\otimes I + I \otimes BA⊗I+I⊗B does not exceed (3d−4)/d2(3d-4)/d^2(3d−4)/d2 with A,BA,BA,B traceless d×dd\times dd×d matrices and ∣∣A∣∣F2+∣∣B∣∣F2=1/d||A||_F^2+||B||_F^2=1/d∣∣A∣∣F2​+∣∣B∣∣F2​=1/d when d=4d=4d=4. The latest progress, made by Ł.~Pankowski~ et al~[IEEE Trans. Inform. Theory, 56, 4085 (2010)], shows that this conjecture holds when both matrices AAA and BBB are normal. In this paper, we prove that the conjecture holds when one of matrices AAA and BBB is normal and the other one is arbitrary. Our work makes solid progress towards this conjecture and thus the distillability problem.
Characterizing the relations among the three bipartite reduced density operators$ \rho _{AB},,, \rho _{AC}andandand \rho _{BC}ofatripartitemixedstateof a tripartite mixed stateofatripartitemixedstate \rho _{ABC}hasbeenanopenprobleminquantuminformation.Oneofsuchrelationshasbeenreducedby[Cadneyetal,LAA.452,153,2014]toaconjecturedinequalityintermsofmatrixrank,namelyhas 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, namelyhasbeenanopenprobleminquantuminformation.Oneofsuchrelationshasbeenreducedby[Cadneyetal,LAA.452,153,2014]toaconjecturedinequalityintermsofmatrixrank,namelyr(\rho _{AB}) \cdot r( \rho _{AC})\ge r( \rho _{BC})foranyfor anyforany \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.
The bipartite quantum states ρ, with rank strictly smaller than the maximum of the ranks of the reduced states ρ A and ρ B , are distillable by local operations and classical communication (Horodecki P, Smolin J A, Terhal B M and Thapliyal A V 2003 Theor. Comput. Sci. 292 589–96; 1999 arXiv:quant-ph/9910122). Our first main result is that this is also true for NPT states with rank equal to this maximum. (A state is PPT if the partial transpose of its density matrix is positive semidefinite, and otherwise it is NPT.) This was conjectured first in 1999 in the special case when the ranks of ρ A and ρ B are equal (see (Horodecki P, Smolin J A, Terhal B M and Thapliyal A V 2003 Theor. Comput. Sci. 292 589–96; 1999 arXiv:quant-ph/9910122). Our second main result provides a complete solution of the separability problem for bipartite states of rank 4. Namely, we show that such a state is separable if and only if it is PPT and its range contains at least one product state. We also prove that the so-called checkerboard states are distillable if and only if they are NPT.
We extend the idea of entanglement concentration protocol for pure states (Phys. Rev. Lett. 88 , 187903) to the case of entanglement distillation for mixed states. The scheme works only with particle statistics and local operations, without the need of any other interactions. We show that the maximally entangled state can be distilled out when the initial state is pure, otherwise the entanglement distilled depends on the off-diagonal element of density matrix of the initial state. Because of the requirement for density matrix, the entanglement distilled is always less than one, and this result is the same for both fermionic and bosonic particles. The distillation efficiency is a product of the diagonal elements of the initial state, it takes the maximum 50%, the same as that in the case for pure states.
We study the relation between violation of Bell inequalities and distillability properties of quantum states. Recently, Dür [Phys. Rev. Lett. 87, 230402 (2001)] has shown that there are some multiparticle bound entangled states which are nonseparable and nondistillable, that violate a Bell inequality. We prove that for all the states violating this inequality there exists at least one splitting of the parties into two groups such that some pure-state entanglement can be distilled, obtaining a connection between Bell inequalities and bipartite distillable entanglement.