Minimal separation of quantum correlation models

Finite-dimensional tensor-product correlations form $C_q$; their closure is $C_{qa}$; commuting-operator correlations form $C_{qc}$. Nonclosure of $C_q$ and separation of $C_{qa}$ from $C_{qc}$ are different established phenomena, not one undifferentiated separation.

Slofstra proved nonclosure of finite-dimensional quantum correlations. Dykema–Paulsen–Prakash give an explicit five-input, two-output nonclosure example. MIP*=RE establishes $C_{qa}\ne C_{qc}$ and refutes the corresponding Tsirelson/Connes conjectures. Its construction is effective: calling all these proofs nonconstructive is inaccurate, although the resulting examples can be complicated.

The residual benchmark must specify which two sets are compared and what “smallest” means. Setting counts and output counts form a partial order, so distinct Pareto-minimal scenarios may matter. Useful advances are smaller certified examples, lower bounds excluding separation in specified scenarios and transparent separating correlations.

Source catalogue review: 4 September 2026. Improved indicates partial progress, including qualified preprint claims.