The Schmidt rank for the commuting operator framework

TL;DR

This paper extends the concept of Schmidt rank to the commuting operator framework in quantum information, providing algebraic and operational definitions, and analyzing various bipartite states beyond the standard tensor product setting.

Contribution

It introduces a generalized Schmidt rank for bipartite systems described by general C*-algebras, with equivalent algebraic and operational definitions, expanding the framework beyond traditional tensor products.

Findings

Computed Schmidt rank for quantum field theory vacuum states

Analyzed Araki-Woods-Powers states and ground states on spin chains

Provided open problems in the commuting operator framework

Abstract

In quantum information theory, the Schmidt rank is a fundamental measure for the entanglement dimension of a pure bipartite state. Its natural definition uses the Schmidt decomposition of vectors on bipartite Hilbert spaces, which does not exist (or at least is not canonically given) if the observable algebras of the local systems are allowed to be general C*-algebras. In this work, we generalize the Schmidt rank to the commuting operator framework where the joint system is not necessarily described by the minimal tensor product but by a general bipartite algebra. We give algebraic and operational definitions for the Schmidt rank and show their equivalence. We analyze bipartite states and compute the Schmidt rank in several examples: The vacuum in quantum field theory, Araki-Woods-Powers states, as well as ground states and translation invariant states on spin chains which are viewed as bipartite systems for the left and right half chains. We conclude with a list of open problems for the commuting operator framework.