A matrix inequality related to the entanglement distillation problem

Abstract

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 $4\times 4$ 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 $A\otimes I+I\otimes B$ does not exceed $(3d-4)/d^2$ with $A,B$ traceless $d\times d$ matrices and $||A|{|}_F^2+||B|{|}_F^2=1/d$ when $d=4$. The latest progress, made by Ł.~Pankowski~ et al~[IEEE Trans. Inform. Theory, 56, 4085 (2010)], shows that this conjecture holds when both matrices $A$ and $B$ are normal. In this paper, we prove that the conjecture holds when one of matrices $A$ and $B$ is normal and the other one is arbitrary. Our work makes solid progress towards this conjecture and thus the distillability problem.

References in corpus (24)

• Quantum entanglement
• Quantum Cryptography
• Experimental quantum teleportation
• Separability Criterion for Density Matrices
• Concentrating Partial Entanglement by Local Operations
• Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels
• Persistent entanglement in arrays of interacting particles
• Quantum privacy amplification and the security of quantum cryptography over noisy channels
• Mixed-state entanglement and distillation: is there a ``bound'' entanglement in nature?
• Topological Quantum Distillation
• An improved bound on distillable entanglement
• Evidence for Bound Entangled States with Negative Partial Transpose
• Distillability and partial transposition in bipartite systems
• Superdense coding of quantum states
• Remote information concentration using a bound entangled state
• A few steps more towards NPT bound entanglement
• Qudit Entanglement
• Study of the Distillability of Werner States Using Entanglement Witnesses and Robust Semidefinite Programs
• Rank three bipartite entangled states are distillable
• Non-positive-partial-transpose quantum states of rank four are distillable
• New classes of n-copy undistillable quantum states with negative partial transposition
• Separability of Completely Symmetric States in Multipartite System
• Separability of Multipartite Quantum States with Strong Positive Partial Transpose
• On a matrix inequality related to the distillability problem

Cited by in corpus (1)

• Five open problems in quantum information