Purifications of multipartite states: limitations and constructive methods

Gemma De Las Cuevas,J Ignacio Cirac,Norbert Schuch,David Pérez-García

doi:10.1088/1367-2630/15/12/123021

Abstract

We analyze the description of quantum many-body mixed states using matrix product states and operators. We consider two such 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. Then we provide two constructive methods to obtain (ii) out of (i). The sum of squares (sos) polynomial method scales exponentially in the number of different eigenvalues, and its approximate version is formulated as a semidefinite program, which gives efficient approximate purifications whose D′ only depends on D. The eigenbasis method scales quadratically in the number of eigenvalues, and its approximate version is very efficient for rapidly decaying distributions of eigenvalues. 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.

Highlights

We present the question that concerns us, which is whether the MPDO form and the local purification form of a mixed state can be related
We show that the MPDO form and the local purification form are inequivalent
In this paper we have analyzed the efficiency of representing a mixed state as an MPDO and as a local purification, and we have shown that the latter can be arbitrarily more costly than the former

Summary

Introduction

We provide classical multipartite states whose MPDO form has a fixed cost, but whose local purification form has an unbounded cost. We provide two constructive purification methods applicable to all multipartite density matrices, which relate the two forms and involve the number of (different) eigenvalues. The sum of squares (sos) polynomial method has an exact version which scales exponentially in the number of different eigenvalues. The eigenbasis method has an exact version which scales multiplicatively with the number of eigenvalues, and its approximate version gives very efficient purifications for rapidly decaying distributions of eigenvalues. We present the question that concerns us, which is whether the MPDO form and the local purification form of a mixed state can be related.

Objectives

Methods

Conclusion