For integers n,k≥ 2, let C_qa(n,k) and C_qc(n,k) denote the corresponding approximately quantum and commuting quantum correlation sets, and let C^fin_qc(n,k) denote the finite-dimensional commuting correlation set. Tsirelson's conjecture. One has C_qa(n,k)=C_qc(n,k), for all n,k≥ 2; equivalently, C^fin_qc(n,k) is dense in C_qc(n,k) for all n,k≥ 2. This is the strong Tsirelson conjecture, concerning whether the tensor-product and commuting-operator models agree in infinite dimensions; the source presents it as an open formulation, although its connection with the Connes Embedding Problem is discussed.

Tsirelson's conjecture on quantum correlation sets

The conjecture asserts that the approximately quantum and commuting quantum correlation sets coincide for every finite number of questions and answers. It is equivalent to the density of finite-dimensional commuting correlations in the full commuting set and is closely tied to the Connes Embedding Problem.

Known results

Tsirelson proved agreement of the tensor-product and commuting models in finite dimensions.
Slofstra proved that the tensor-product correlation set is not closed for sufficiently large parameters.
The Connes Embedding Problem and Tsirelson's problem were shown to be essentially equivalent.

December 2025 negative resolution

The $\mathrm{MIP}^*=\mathrm{RE}$ theorem of Ji, Natarajan, Vidick, Wright, and Yuen implies that the Connes Embedding Problem is false and that $C_{qa}(n,k)\ne C_{qc}(n,k)$ for some finite parameters. Thus the stated strong Tsirelson conjecture is disproved, with the conclusion also reported in later expositions.

Current status (as of August 2026): The strong conjecture is settled negatively; equality fails for some $n,k$ , although the sources do not specify the smallest such parameters.

Tsirelson's conjecture on quantum correlation sets

Primary source

Progress summary

Solutions 0