Rounding near-optimal quantum strategies for nonlocal games to strategies using maximally entangled states

TL;DR

This paper proves that near-optimal strategies for certain nonlocal games can be approximated by strategies using maximally entangled states, with implications for the structure and rigidity of quantum strategies.

Contribution

It establishes approximate rigidity results for nonlocal games, showing near-optimal strategies are close to those using maximally entangled states, and connects algebraic representations to quantum strategies.

Findings

Near-perfect strategies are approximate algebra representations.

Strategies can be approximated by those with maximally entangled states.

Results apply to BCS, XOR, and synchronous games.

Abstract

We establish approximate rigidity results for several well-known families of nonlocal games. In particular, we show that near-perfect quantum strategies for boolean constraint system (BCS) games are approximate representations of the corresponding BCS algebra. Likewise, for the class of XOR nonlocal games, we show that near-optimal quantum strategies are approximate representations of the corresponding $*$-algebra associated with optimal quantum values for the game. In both cases, the approximate representations are with respect to the normalized Hilbert-Schmidt norm and independent of the Hilbert space or quantum state employed in the strategy. We also show that approximate representation of the BCS (resp.~XOR-algebra) yields measurement operators for near-perfect (resp.~near-optimal) quantum strategies where the players employ a maximally entangled state in the game. As a corollary, every near-perfect BCS (resp.~near-optimal XOR) quantum strategy is close to a near-perfect (resp.~near-optimal) quantum strategy using a maximally entangled state. Lastly, we establish that every synchronous algebra is $*$-isomorphic to a certain BCS algebra called the SynchBCS algebra. This allows us to apply our BCS rigidity results to the class of synchronous games as well.