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, selftesting has been central to high-profile results in complexity theory as seen in the work on entangled games PCP of Natarajan and Vidick (FOCS 2018), iterated compression by Fitzsimons et al. (STOC 2019), and NEEXP in MIP* due to Natarajan and Wright (FOCS 2019). 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 non-local games that self-test non-Pauli operators resolving an open questions posed by Coladangelo and Stark (QIP 2017). Our games also provide a selftest for states other than the maximally entangled state, and hence resolves the open question posed by Cleve and Mittal (ICALP 2012). 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 (Journal of Mathematical Physics 2017) 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. * It is an interesting question to see whether this observation is always true.

Sum-of-squares decompositions for a family of noncontextuality inequalities and self-testing of quantum devices

“…In particular, we do not assume orthogonality relations between measurement effects and precise projectivity of the measurements. Moreover, we take a different approach, that is, we use the sumof-squares 'technique' that has successfully been used in the Bell scenario to derive maximal quantum violation of certain Bell inequalities as well as in making self-testing statements [4,[18][19][20][21][22][23][24], but has never been explored in the contextuality scenario.…”

“…with some non-negative parameters c k , d to be determined. By plugging the expression of M i,k (24) into it and after some rearrangement of indices, we obtain…”

“…In the context of nonlocality, sum-of-squares (SOS) decomposition of quantum operators associated with local-realist inequalities has been the key mathematical tool in recent years to obtain optimal quantum values and self-testing properties of quantum devices [4,[18][19][20][21][22][23][24]. Whether this line of study, albeit, restricted to nonlocal correlations, can further be extended to contextuality scenario is of great interest from the perspective of unified approach to non-classical correlations [9,25].…”