Unconditional separation of finite and infinite-dimensional quantum correlations

Determining the relationship between quantum correlation sets is a long-standing open problem. The most well-studied part of the hierarchy is captured by the chain of inclusions $\mathcal C_q \subseteq \mathcal C_{qs} \subsetneq \mathcal C_{qa} \subseteq \mathcal C_{qc}$. The separation $\mathcal C_{qs} \neq \mathcal C_{qa}$, showing that the set of quantum spatial correlations is not closed, was proven in breakthrough work by Slofstra [arXiv:1606.03140 (2016), arXiv:1703.08618 (2017)]. Resolving the question of $\mathcal C_{qa} = \mathcal C_{qc}$ would resolve the Connes Embedding Conjecture and would represent major progress in the mathematical field of operator algebras. In this work, we resolve the ambiguity in the first inclusion, showing that $\mathcal{C}_q \neq \mathcal{C}_{qs}$. We provide an explicit construction of a correlation that can be attained on a tensor product of infinite-dimensional Hilbert spaces but not finite-dimensional ones. This property is also conjectured to be possessed by any correlation which maximally violates the $I_{3322}$ inequality.