Separation of finite and infinite-dimensional quantum correlations, with infinite question or answer sets

Abstract

Completely determining the relationship between quantum correlation sets is a long-standing open problem, known as Tsirelson's problem. Following recent progress by Slofstra [Slo16, Slo17], only two instances of the problem remain open. One of them is the question of whether the set of finite-dimensional quantum correlations is strictly contained in the set of infinite-dimensional ones (i.e. whether Cq = Cqs). The usual formulation of the question assumes finite question and answer sets. In this work, we show that, when one allows for either infinite answer sets (and finite question sets) or infinite question sets (and finite answer sets), there exist correlations that are achievable using an infinite-dimensional quantum strategy, but not a finite-dimensional one. For the former case, our proof exploits a recent result [CGS17], which shows self-testing of any pure bipartite entangled state of arbitrary local dimension d, using question sets of size 3 and 4 and answer sets of size d. For the latter case, a key step in our proof is to show a novel self-test, inspired by [CGS17], of all bipartite entangled states of any local dimension d, using question sets of size O(d), and answer sets of size 4 and 3 respectively.

Generalization of the Clauser-Horne-Shimony-Holt inequality self-testing maximally entangled states of any local dimension

“…2 , and this holds both when wm = 0 (and pm is the ideal qubit CHSH correlation) and when wm = 0. Likewise, we have that Π 2m+1 A robust version of the self-testing result via the correlations of [CGS17] was shown in [CS17b], where, informally, the authors prove that a strategy producing a correlation that is -close to the ideal one, must be O(d 3 1 4 )-close (according to some measures of distance) to the ideal strategy from Lemma 1. However, this does not trivially translate to a robust self-test via our Bell inequality, for which we require that a close-to-maximal violation certifies a close-to-ideal strategy.…”

“…There is also a notion of robust self-testing, when one can approximately characterize strategies that are close to achieving the ideal correlation [MYS12,Kan16]. For a precise definition, we refer the reader to [CS17b]. We remark that, technically, we don't need to assume that the original strategy uses a pure state |ψ , but rather our proofs can be directly translated to the case of a mixed state (see [CGS17] for a more precise account of this).…”

Verification of Quantum Computation: An Overview of Existing Approaches

“…This means that if the referee observes a near maximal win rate for the players, in the game, she can conclude that they are using the optimal strategy and can therefore characterise their shared state and their observables, up to a local isometries. More formally, we give the definition of self-testing, adapted from [70] and using notation similar to that of [23]: Definition 4 (Self-testing). Let G denote a game involving n non-communicating players denoted {P i } n i=1 .…”

Constant-sized correlations are sufficient to robustly self-test maximally entangled states with unbounded dimension

“…To generate a correlation in the family, Alice and Bob use constant-sized input alphabets but size-d output alphabets, where d is the local dimension of the entangled state. The robustness of this family of correlations is proved in [CS17b].…”