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.
On the other hand, a previous work by the present authors REF provides sequences of correlations, on either infinite question sets (and finite answer sets) or infinite answer sets (and finite question sets), that cannot be attained in finite-dimensions but can be attained explicitly in infinite dimensions.
Separation of finite and infinite-dimensional quantum correlations, with infinite question or answer sets
arXiv: Quantum Physics
Mathematics Physics