Near-optimal quantum strategies for nonlocal games, approximate representations, and BCS algebras

Abstract

Quantum correlations can be viewed as particular abstract states on the tensor product of operator systems which model quantum measurement scenarios. In the paradigm of nonlocal games, this perspective illustrates a connection between optimal strategies and certain representations of a finitely presented $*$-algebra affiliated with the nonlocal game. This algebraic interpretation of quantum correlations arising from nonlocal games has been valuable in recent years. In particular, the connection between representations and strategies has been useful for investigating and separating the various frameworks for quantum correlation as well as in developing cryptographic primitives for untrusted quantum devices. However to make use of this correspondence in a realistic setting one needs mathematical guarantees that this correspondence is robust to noise.

We address this issue by considering the situation where the correlations are not ideal. We show that near-optimal finite-dimensional quantum strategies using arbitrary quantum states are approximate representations of the affiliated nonlocal game algebra for synchronous, boolean constraint systems (BCS), and XOR nonlocal games. This result robustly extends the correspondence between optimal strategies and finite-dimensional representations of the nonlocal game algebras for these prominent classes of nonlocal games. We also show that finite-dimensional approximate representations of these nonlocal game algebras are close to near-optimal strategies employing a maximally entangled state. As a corollary, we deduce that near-optimal quantum strategies are close to a near-optimal quantum strategy using a maximally entangled state.

A boolean constraint system $B$ is $pp$-definable from another boolean constraint system $B'$ if there is a $pp$-formula defining $B$ over $B'$. There is such a $pp$-formula if all the constraints in $B$ can be defined via conjunctions of relations in $B'$ using additional boolean variables if needed. We associate a finitely presented $*$-algebra, called a BCS algebra, to each boolean constraint system $B$. We show that $pp$-definability can be interpreted algebraically as $*$-homomorphisms between BCS algebras. This allows us to classify boolean constraint languages and separations between various generalized notions of satisfiability. These types of satisfiability are motivated by nonlocal games and the various frameworks for quantum correlations and state-independent contextuality. As an example, we construct a BCS that is $C^*$-satisfiable in the sense that it has a representation on a Hilbert space $H$ but has no tracial representations, and thus no interpretation in terms of commuting operator correlations.