[Paper Review] Quantum no-signalling correlations and non-local games

Ivan G. Todorov, Lyudmila Turowska|arXiv (Cornell University)|Sep 15, 2020

Quantum Mechanics and Applications53 references9 citations

TL;DR

This paper introduces quantum non-local games as a generalization of classical non-local games, defining quantum versions of graph colouring and non-commutative graph homomorphisms. It establishes that perfect quantum strategies for composed games remain perfect under channel composition, and identifies quantum versions of the orthogonal rank and non-commutative graph homomorphisms via tensor product states on universal operator systems.

ABSTRACT

We introduce and examine three subclasses of the family of quantum no-signalling (QNS) correlations introduced by Duan and Winter: quantum commuting, quantum and local. We formalise the notion of a universal TRO of a block operator isometry, define an operator system, universal for stochastic operator matrices, and realise it as a quotient of a matrix algebra. We describe the classes of QNS correlations in terms of states on the tensor products of two copies of the universal operator system, and specialise the correlation classes and their representations to classical-to-quantum correlations. We study various quantum versions of synchronous no-signalling correlations and show that they possess invariance properties for suitable sets of states. We introduce quantum non-local games as a generalisation of non-local games. We define the operation of quantum game composition and show that the perfect strategies belonging to a certain class are closed under channel composition. We specialise to the case of graph colourings, where we exhibit quantum versions of the orthogonal rank of a graph as the optimal output dimension for which perfect classical-to-quantum strategies of the graph colouring game exist, as well as to non-commutative graph homomorphisms, where we identify quantum versions of non-commutative graph homomorphisms introduced by Stahlke.

Motivation & Objective

• To generalize classical non-local games to quantum-to-quantum settings, enabling quantum inputs and outputs.
• To define quantum versions of graph colouring and non-commutative graph homomorphisms using operator systems and C*-algebras.
• To establish closure properties of perfect strategies under game composition, particularly for quantum commuting and quantum strategies.
• To characterize quantum no-signalling (QNS) correlations via states on tensor products of universal operator systems.
• To resolve open questions on the transitivity of quantum homomorphisms and the structure of tracial and fair QNS correlations.

Proposed method

• Introduces the universal operator system for stochastic operator matrices, realized as a quotient of a matrix algebra.
• Defines three subclasses of QNS correlations—quantum commuting, quantum, and local—via states on tensor products of universal operator systems.
• Applies the framework to classical-to-quantum (CQNS) correlations, characterizing them through states on operator systems with traces.
• Introduces quantum game composition via channel composition, showing that perfect strategies are closed under composition.
• Uses trace-preserving maps and pure state decompositions to verify that composed strategies remain perfect and no-signalling.
• Applies the framework to graph colouring and non-commutative graph homomorphisms, identifying quantum orthogonal rank as the minimal output dimension for perfect strategies.

Experimental results

Research questions

• RQ1 How can non-local games be generalized to allow quantum inputs and outputs, and what are the structural properties of such quantum non-local games?
• RQ2 What is the quantum analogue of the orthogonal rank of a graph, and how does it relate to the existence of perfect quantum strategies in the graph colouring game?
• RQ3 Under what conditions are perfect quantum strategies closed under composition, and how does this relate to the transitivity of quantum graph homomorphisms?
• RQ4 How do tracial and fair QNS correlations behave under composition, and what invariance properties do they exhibit?
• RQ5 What is the role of universal operator systems and tensor product states in characterizing quantum no-signalling correlations?

Key findings

• Perfect quantum strategies for composed quantum non-local games remain perfect under channel composition, establishing closure for the classes of local, quantum, and quantum commuting strategies.
• The quantum orthogonal rank of a graph is identified as the minimal output dimension for which perfect classical-to-quantum strategies exist in the graph colouring game.
• Quantum non-commutative graph homomorphisms are characterized as those admitting perfect quantum strategies in the quantum homomorphism game, generalizing Stahlke’s notion.
• The class of tracial QNS correlations is closed under composition, with the product trace on the tensor product of C*-algebras preserving the strategy’s perfectness.
• The universal operator system for stochastic operator matrices is realized as a quotient of a matrix algebra, providing a universal framework for describing QNS correlations.
• The transitivity of quantum homomorphisms is established: if U →_x V and V →_x W, then U →_x W for x ∈ {loc, q, qc}, extending results from [71] and [57].