Purifications of multipartite states: limitations and constructive methods

Purifications of multipartite states: limitations and constructive methods

Seminar author:Gemma de las Cuevas

Event date and time:10/30/2013 03:00:pm

Event location:IFAE seminar room

Event contact:

We analyze the description of quantum many-body mixed states using matrix product states and operators. We consider two descriptions: (i) as a matrix product density operator of bond dimension D, and (ii) as a purification that is written as a matrix product state of bond dimension D’. We show that these descriptions are inequivalent in the sense that D’ cannot be upper bounded by D only. We then provide two methods to obtain (ii) out of (i). The sum of squares (sos) polynomial method upper bounds D’ by D to the power of the number of different eigenvalues. Its approximate version is formulated as a Semidefinite Program, and it gives approximate purifications whose D’ only depends on D. The eigenbasis method upper bounds D’ by D times the number of eigenvalues, and its approximate version is very efficient for exponentially decaying distributions. Our results imply that a description of mixed states which is both efficient and locally positive semidefinite does not exist, but that good approximations do.

Joint work with N. Schuch, D. Pérez-García, and J. I. Cirac.