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Completely determining the relationship between quantum correlation sets is along-standing open problem, known as Tsirelson's problem. Following recentprogress by Slofstra [arXiv:1606.03140 (2016), arXiv:1703.08618 (2017)] onlytwo instances of the problem remain open. One of them is the question ofwhether the set of finite-dimensional quantum correlations is strictlycontained in the set of infinite-dimensional ones (i.e. whether $\mathcal C_{q}\neq \mathcal C_{qs}$). The usual formulation of the question assumes finitequestion and answer sets. In this work, we show that, when one allows foreither infinite answer sets (and finite question sets) or infinite questionsets (and finite answer sets), there exist correlations that are achievableusing an infinite-dimensional quantum strategy, but not a finite-dimensionalone. For the former case, our proof exploits a recent result [Nat. Comm. 8,15485 (2017)], which shows self-testing of any pure bipartite entangled stateof arbitrary local dimension $d$, using question sets of size 3 and 4 andanswer sets of size $d$. For the latter case, a key step in our proof is toshow a novel self-test, inspired by [Nat. Comm. 8, 15485 (2017)], of allbipartite entangled states of any local dimension d, using question sets ofsize $O(d)$, and answer sets of size 4 and 3 respectively.

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