Quantum no-signalling bicorrelations

TL;DR

This paper introduces and characterizes classical and quantum no-signalling bicorrelations, linking them to operator systems, quantum automorphisms, and quantum graph isomorphism, advancing the understanding of quantum symmetries and correlations.

Contribution

It defines new classes of no-signalling bicorrelations, characterizes them via operator system tensor products, and connects quantum bicorrelations to quantum automorphisms and the quantum graph isomorphism game.

Findings

Quantum no-signalling bicorrelations characterized by operator system states.

Concurrent bicorrelations of quantum commuting type linked to tracial states on universal C*-algebras.

Application to quantum graph isomorphism game and algebraic quantum graph notions.

Abstract

We introduce classical and quantum no-signalling bicorrelations and characterise the different types thereof in terms of states on operator system tensor products, exhibiting connections with bistochastic operator matrices and with dilations of quantum magic squares. We define concurrent bicorrelations as a quantum input-output generalisation of bisynchronous correlations. We show that concurrent bicorrelations of quantum commuting type correspond to tracial states on the universal C*-algebra of the projective free unitary quantum group, showing that in the quantum input-output setup, quantum permutations of finite sets must be replaced by quantum automorphisms of matrix algebras. We apply our results to study the quantum graph isomorphism game, describing the game C*-algebra in this case, and make precise connections with the algebraic notions of quantum graph isomorphism, existing presently in the literature.