Entanglement transformation and the structure of matrix space

QuICS Special Seminar

Yinan Li (UTS)
October 11, 2017
CSS 3100A

Entanglement is arguably the most important physical resources in quantum information processing, and a fundamental task is to transform a multipartite entangled pure state into a bipartite entangled pure state shared between two specific parties. In this talk, we study the following scenario: given a pure tripartite state shared by Alice, Bob and Charlie, what kind of pure bipartite entangled state can be recovered with a nonzero probability by Alice and Bob with the help of Charlie under local operations and classical communication (a.k.a. SLOCC transformation)? Such a problem can be characterized by the maximal rank of a matrix space, i.e., the largest rank of matrices of a matrix space (linear space of matrices). We study this quantity and find a number of interesting properties, such as super-multiplicativity. By studying the behavior of maximal rank under tensor product, we obtain explicit formulas to compute the asymptotic entanglement transformation rate for a large family of tripartite states. Notably, utilizing certain results of the classification of matrix spaces, including the study of matrix semi-invariants in geometric invariant theory, we obtain a sufficient and necessary condition to decide whether a tripartite state can be transformed to the bipartite maximally entangled state by SLOCC, in the asymptotic setting. Interestingly, based on the recent seminal progress on the non-commutative rank problem, our characterization can be verified in deterministic polynomial time.

 

Reference:

Y Li, Y Qiao, X Wang and R Duan, “Tripartite-to-bipartite Entanglement Transformation by Stochastic Local Operations and Classical Communication and the Classification of Matrix Spaces,” arXiv:1612.06491, 2016