Self-testing from maximal four-dimensional correlations

Determine whether the composite state with correlation matrix $C=1_4$ and vanishing local marginals self-tests four anticommuting involutions.

Background

The paper constructs a locally tomographic, no-signalling composite of two hyperbits whose states are described by local expectation vectors $x$ and $y$ and a $4\times4$ correlation matrix $C$. It identifies a unique state with $C=1_4$ and $x=y=0$, corresponding to the joint $+1$ eigenstate of four commuting products of anticommuting involutions. This state has maximal trace-norm correlation, $\|C\|_1=4$, and reaches the usual Tsirelson bound for a suitable CHSH expression.

The unresolved problem asks whether these maximal correlations uniquely characterize, up to the appropriate notion of equivalence, the underlying four anticommuting involutions. In other words, it concerns whether the correlation data provide a self-test of the operator realization rather than merely witnessing strong entanglement.