A generalization of CHSH and the algebraic structure of optimal strategies

Abstract

Self-testing has been a rich area of study in quantum information theory. It allows an experimenter to interact classically with a black box quantum system and to test that a specific entangled state was present and a specific set of measurements were performed. Recently, self-testing has been central to high-profile results in complexity theory as seen in the work on entangled games PCP of Natarajan and Vidick \cite{low-degree}, iterated compression by Fitzsimons et al. \cite{iterated-compression}, and NEEXP in MIP* due to Natarajan and Wright \cite{neexp}. The most studied self-test is the CHSH game which features a bipartite system with two isolated devices. This game certifies the presence of a single EPR entangled state and the use of anti-commuting Pauli measurements. Most of the self-testing literature has focused on extending these results to self-test for tensor products of EPR states and tensor products of Pauli measurements.In this work, we introduce an algebraic generalization of CHSH by viewing it as a linear constraint system (LCS) game, exhibiting self-testing properties that are qualitatively different. These provide the first example of LCS games that self-test non-Pauli operators resolving an open questions posed by Coladangelo and Stark \cite{RobustRigidityLCS}. Our games also provide a self-test for states other than the maximally entangled state, and hence resolves the open question posed by Cleve and Mittal \cite{BCSTensor}. Additionally, our games have 1 bit question and log⁡n bit answer lengths making them suitable candidates for complexity theoretic application. This work is the first step towards a general theory of self-testing arbitrary groups. In order to obtain our results, we exploit connections between sum of squares proofs, non-commutative ring theory, and the Gowers-Hatami theorem from approximate representation theory. A crucial part of our analysis is to introduce a sum of squares framework that generalizes the solution group of Cleve, Liu, and Slofstra \cite{BCSCommuting} to the non-pseudo-telepathic regime. Finally, we give the first example of a game that is not a self-test. Our results suggest a richer landscape of self-testing phenomena than previously considered.

Rigidity for Monogamy-of-Entanglement Games

“…Note that the above definition and theorem have a slightly different form from how they were presented in previous work [Vid18,CMMN20].…”

“…In this way, we may study the strategies of a game by studying the positivity of this operator-valued polynomial. This technique expands upon one that has been used previously to study nonlocal games [CMMN20].…”

“…Similarly to what is often done for nonlocal games [CMMN20], we transform the expression for the winning probability into an expression in terms of observables rather than measurements. Definition 3.10.…”

“…The rigidity of the CHSH game has found many applications: for example it was used to construct a protocol for quantum delegated computation, which was then used to show the equivalence of complexity classes QMIP = MIP * [RUV13]. Other examples of nonlocal games include the Mermin-Peres magic square game -which can always be won and self-tests two copies of the maximally entangled state, and was used to show MIP * = RE [JNV + 20] -and more generally linear constraint games [CMMN20]. They form a pair of bases analogous to the conjugate-coding bases, but rotated by π 4 so that each vector is located at the midpoint between a vector from the computational basis and one from the Hadamard basis.…”

Noncommutative Nullstellensätze and Perfect Games

“…The problem of establishing uniqueness of solution to the game has been studied and there is a standard technique available. cf [CMMN20]. It can be tried on a game for which the bias and optimal value satisfy ω * − Φ G = SOS exactly (as opposed to approximately as in [NPA08, DLTW08, HM04]).…”