Coupling capacity in C*-algebras

TL;DR

This paper introduces parameters to measure how well positive operators in tensor products of C*-algebras align with couplings of states, establishing duality formulas and applications to quantum entanglement detection.

Contribution

It defines new parameters for operators in C*-algebra tensor products, proves a duality formula, and connects these to quantum entanglement and classical optimal transport.

Findings

Parameters relate to Null Set Theorem and optimal transport dualities.

Duality formula shows equality of two parameters for certain operators.

Parameters can detect maximal entanglement and separability in quantum states.

Abstract

Given two unital C*-algebras equipped with states and a positive operator in the enveloping von Neumann algebra of their minimal tensor product, we define three parameters that measure the capacity of the operator to align with a coupling of the two given states. Further we establish a duality formula that shows the equality of two of the parameters for operators in the minimal tensor product of the relevant C*-algebras. In the context of abelian C*-algebras our parameters are related to quantitative versions of Arveson's Null Set Theorem and to dualities considered in the theory of optimal transport. On the other hand, restricting to matrix algebras we recover and generalise quantum versions of Strassen's Theorem. We show that in the latter case our parameters can detect maximal entanglement and separability.