Schmidt number of bipartite and multipartite states under local projections

TL;DR

This paper explores how local projections affect the Schmidt number of bipartite and multipartite quantum states, revealing the existence of PPT entangled states with any Schmidt number and introducing joint Schmidt number for multipartite states.

Contribution

It introduces the concept of joint Schmidt number for multipartite states and establishes its relation with bipartite Schmidt numbers, advancing understanding of quantum state properties under local projections.

Findings

Existence of bipartite PPT entangled states with arbitrary Schmidt number

Introduction of joint Schmidt number for multipartite states

Relation between joint Schmidt number and bipartite Schmidt numbers

Abstract

The Schmidt number is a fundamental parameter characterizing the properties of quantum states, and the local projections are a fundamental operation in quantum physics. We investigate the relation between the Schmidt numbers of bipartite states and their projected states. We show that there exist bipartite positive-partial-transpose (PPT) entangled states of any given Schmidt number. We further construct the notion of joint Schmidt number for multipartite states, and its relation with the Schmidt number of bipartite reduced density operators.