We construct a linear system nonlocal game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed nonlocal game provides another counterexample to the ‘middle’ Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies.
Tsirelson’s problem asks whether the commuting operator model for two-party quantum correlations is equivalent to the tensor-product model. We give a negative answer to this question by showing that there are non-local games which have perfect commuting-operator strategies, but do not have perfect tensor-product strategies. The weak Tsirelson problem, which is known to be equivalent to the Connes embedding problem, remains open. The examples we construct are instances of (binary) linear system games. For such games, previous results state that the existence of perfect strategies is controlled by the solution group of the linear system. Our main result is that every finitely-presented group embeds in some solution group. As an additional consequence, we show that the problem of determining whether a linear system game has a perfect commuting-operator strategy is undecidable.
Linear system games are a generalization of Mermin’s magic square game introduced by Cleve and Mittal. They show that perfect strategies for linear system games in the tensor-product model of entanglement correspond to finite-dimensional operator solutions of a certain set of non-commutative equations. We investigate linear system games in the commuting-operator model of entanglement, where Alice and Bob’s measurement operators act on a joint Hilbert space, and Alice’s operators must commute with Bob’s operators. We show that perfect strategies in this model correspond to possibly infinite-dimensional operator solutions of the non-commutative equations. The proof is based around a finitely presented group associated with the linear system which arises from the non-commutative equations.
Almost-quantum correlations form the
1+AB level of the Navascués--Pironio--Acín hierarchy. They strictly contain the quantum set, are characterized by a single semidefinite program, arise in Sorkin's quantum measure theory, and satisfy several principles proposed to single out quantum theory. We show that they are nevertheless unboundedly stronger than quantum correlations in a zero-error communication task. Alice and Bob receive labels
s,t∈[N],
N≥3; after one bit from Alice, Bob must decide whether
s=t, never erring but allowed to abstain. For every model of quantum correlations, from finite-dimensional tensor products to commuting operators, the optimal worst-case probability of a conclusive answer is exactly
1/N, the value attained with shared randomness, so the bound does not depend on the resolution of Tsirelson's problem. An explicit family of almost-quantum correlations attains
1/19 for every
N. For certificates of equality from binary-outcome boxes, we solve the underlying semidefinite programs in closed form. The almost-quantum optimum equals
N2/(17N2−80N+64) for
N≥8 and tends to
1/17. The first level of the hierarchy, which coincides with the set singled out by macroscopic locality, attains
N/[8(N−2)], which tends to
1/8; its optimal boxes lie outside the almost-quantum set for every
N≥5. For longer messages, entanglement saves at most one bit over shared randomness, whereas almost-quantum correlations reduce the one-way zero-error cost of deciding equality of
n-bit strings from
n bits to at most
13. The quantum bound also yields a linear witness of post-quantumness on a face of the nonsignalling polytope.
We exhibit non-closedness at perfect play for normal injective hosts: an explicit finite two-player game with 0/1 scoring has a perfect commuting-operator correlation approached by finite-dimensional quantum correlations, while no perfect strategy has a normal marginal on an injective von Neumann algebra of either player. The game has 561 Alice questions, 157 Bob questions, 3,696 sampled pairs and at most 20 answers per question. Perfect play is realized on a matrix amplification of the Bellaterra critical-KMS factor of type III with parameter 1/2. After restricting to the support of the chosen normal marginal, every perfect strategy yields a modular Cuntz pair and six self-similar relations that force the canonical trace on the free product of three groups of order two. The resulting noninjective group von Neumann algebra is an expected subalgebra of that support corner, with a faithful normal state-preserving expectation. A two-sided POVM lift through a matrix ultraproduct proves approximation by finite systems. Consequently correlations admitting normal injective hosts are not closed. Under the explicit hypotheses of Halvorson's Proposition 2.31, locally normal quantum-field-theory strategies cannot win exactly. We also formulate a KMS-face criterion for complete modular presentations. Exact arithmetic certificates check every supported losing event; the forcing proof applies to arbitrary normal strategies.