A characterization of freeness by invariance under quantum spreading

TL;DR

This paper introduces quantum spreadability for noncommutative random variables and proves it characterizes free independence and identical distribution, extending classical results into the quantum setting.

Contribution

It defines quantum spreadability and establishes its equivalence to free independence and identical distribution, providing a new characterization in free probability theory.

Findings

Quantum spreadability is equivalent to free independence.

Constructed spaces of quantum increasing sequences.

Extended classical invariance results to the quantum setting.

Abstract

We construct spaces of quantum increasing sequences, which give quantum families of maps in the sense of Soltan. We then introduce a notion of quantum spreadability for a sequence of noncommutative random variables, by requiring their joint distribution to be invariant under taking quantum subsequences. Our main result is a free analogue of a theorem of Ryll-Nardzewski: for an infinite sequence of noncommutative random variables, quantum spreadability is equivalent to free independence and identical distribution with respect to a conditional expectation.