On the commutativity of states in von Neumann algebras

TL;DR

This paper generalizes the concept of commutativity of states in von Neumann algebras to multiple states, explores their joint commutativity, and examines the relationship with broadcastability.

Contribution

It extends the notion of state commutativity from pairs to arbitrary collections of states and investigates their joint properties and broadcastability relations.

Findings

Generalized commutativity to multiple states

Established conditions for joint commutativity

Explored links between commutativity and broadcastability

Abstract

The notion of commutativity of two normal states on a von Neumann algebra was defined some time ago by means of the Pedersen-Takesaki theorem. In this note we aim at generalizing this notion to an arbitrary number of states, and obtaining some results on so defined joint commutativity. Also relations between commutativity and broadcastability of states are investigated.