In 2020, Ji et al. [arXiv:2001.04383 and Comm.~ACM 64}, 131 (2021)] provided a proof that the complexity classes $\text{MIP}^\ast$ and $\text{RE}$ are equivalent. This result implies a negative resolution of Tsirelson's problem, that is, $C_{qa}$ (the closure of the set of tensor product correlations) and $C_{qc}$ (the set of commuting correlations) can be separated by a hyperplane (that is, a Bell-like inequality). In particular, there are correlations produced by commuting measurements (a finite number of them and with a finite number of outcomes) on an infinite-dimensional quantum system which cannot be approximated by sequences of finite-dimensional tensor product correlations. Here, we point out that there are four logical possibilities of this result. Each possibility is interesting because it fundamentally challenges the nature of spacially separated systems in different ways. We list open problems for making progress for deciding which of the possibilities is correct.
In 2020, Ji et al. [arXiv:2001.04383 and Comm.~ACM 64}, 131 (2021)] provided a proof that the complexity classes MIP∗ and RE are equivalent. This result implies a negative resolution of Tsirelson's problem, that is, Cqa (the closure of the set of tensor product correlations) and Cqc (the set of commuting correlations) can be separated by a hyperplane (that is, a Bell-like inequality). In particular, there are correlations produced by commuting measurements (a finite number of them and with a finite number of outcomes) on an infinite-dimensional quantum system which cannot be approximated by sequences of finite-dimensional tensor product correlations. Here, we point out that there are four logical possibilities of this result. Each possibility is interesting because it fundamentally challenges the nature of spacially separated systems in different ways. We list open problems for making progress for deciding which of the possibilities is correct.
Connes’ Embedding Problem is a deep question on approximability of certain tracial von Neumann algebras by finite-dimensional matrix algebras. We survey the connections between operator algebras, quantum information and theoretical computer science that enabled the recent resolution of this problem. The resolution goes through an equivalent formulation, known as Tsirelson’s problem, in terms of separating convex sets whose definition is motivated by the study of nonlocality in quantum mechanics. We construct an explicit separating hyperplane using the theory of two-player games from complexity theory.
MIP*=RE implies that correlations of commuting observables on an infinite-dimensional Hilbert space (C_qc) can lie at a finite distance from every limit of finite-dimensional tensor-product correlations (C_qa). Such correlations would let a finite experiment detect an actual infinity with a modest number of rounds. In earlier work this was identified as the only route that escapes the resource bounds on certifying Hilbert-space dimension. Here we ask whether relativistic quantum field theory allows Bell correlations in this gap. Under standard structural assumptions, it does not. Suppose the theory has the split property, which follows from the Buchholz–Wichmann nuclearity condition and holds for free fields. Then the correlations of local measurements in strictly spacelike separated regions are tensor-product correlations and lie in C_qa. If the local algebras are hyperfinite, the same holds for regions that are merely spacelike, with no gap between them. Buchholz, D'Antoni and Fredenhagen showed this hyperfiniteness under a scaling-limit assumption. We give a self-contained proof based on semidiscreteness. Alice's measurements are compressed through matrix algebras by unital completely positive maps, and the resulting correlations converge to the original ones. The same applies to the type II₁ observer algebra of a de Sitter static patch, constructed by Chandrasekaran, Longo, Penington and Witten. This algebra is a corner of a crossed product of such an algebra by ℝ, so it is again injective. Within these frameworks, local measurements in spacelike separated regions cannot realise the gap found by MIP*=RE. This excludes the fourth of the logical possibilities listed by Cabello, Quintino and Kleinmann. We then bound, from the state side, how much dimension a Bell test can certify. Compressing each party to the support of its ε-smoothed reduced state reproduces all correlations to within 2(√ε_A + √ε_B) + ε_A + ε_B. Since N rounds resolve correlations only to N^(-1/2), an experiment with N rounds certifies at most the smooth max-entropy of the local state at smoothing ε ~ 1/N. We compute this effective dimension exactly for the vacuum of a lattice free scalar field. The single-particle entanglement energies form two towers, one per block endpoint, with a spacing set by the mass, so only a handful of modes are active. The effective dimension follows an area law: it is at most 9 bits for a 128-site block at ε = 10⁻⁶, against 553 bits from naive counting. As ε decreases from 10⁻¹ to 10⁻¹², it grows only from 2.3 to 13.1 bits. A product bound over thermal modes shows that the growth is polylogarithmic in 1/ε for any regularised block. In the continuum, the corresponding object is the canonical splitting type I factor of Doplicher and Longo, whose vacuum entropy coincides with the reflected entropy. Evaluating its smooth max-entropy as a function of the splitting distance is left for future work. Together with the certification budget of the companion paper, the result gives a coherent picture. Physics uses infinite-dimensional Hilbert spaces and type III algebras. Yet, within these frameworks, the correlations that local measurements can produce are reproduced to any finite precision by finite-dimensional models. The size of these models is controlled by entanglement entropy. Code, data and figures: https://github.com/Ruqing1963/qft-tsirelson-loophole
We construct an explicit binary linear system game that separates Cqa, the closure of the set of finite dimensional quantum correlations, from Cqc, the set of commuting operator correlations. The game admits a perfect commuting operator strategy, while every correlation in Cqa has a success probability strictly less than 1. This provides a concrete counterexample to Tsirelson's problem in its approximation form. The defining system has 1417152 equations in 1889684 variables, with exactly three nonzero coefficients per equation and a single nonzero entry on the right hand side. We also compute the classical value exactly. The complete system is specified in Lean 4, and the separation and exact classical value are formalized using Mathlib.
In a previous work, we showed that real quantum theory (RQT) cannot be experimentally distinguished from ordinary quantum theory (QT). In this paper, we investigate whether this indistinguishability survives two natural extensions of the setting: allowing separated measurements to be represented by commuting rather than tensor-product observables, and imposing superselection rules, as in fermionic information theory. We show that it does in both cases.
Our first result is a positive answer to the Tsirelson problem in RQT: commuting and tensor-product measurements generate the same finite-dimensional correlations. This is surprisingly not a trivial consequence of the standard complex proof, as it does not carry over to the reals. The proof obstruction becomes particularly relevant in the presence of superselection rules, which is why we first establish the equivalence in RQT before extending the framework to superselected RQT.
Our second result is a local, real quantum formulation of superselected theories. We extend the RQT framework to accommodate a representation of superselection over the reals and show, in particular, that it admits a fermionic version whose predictions are experimentally indistinguishable from those of fermionic information theory. More generally, the construction extends to all superselected QTs.
We show that the negation of the Tsirelson’s conjecture can be understood as, and in fact follows from, the ZFC - genericity of infinite sequences of the QM outcomes. We extend the result of Landsman that there holds the equivalence of the statistics of the entire sequences with the statistic of the single QM measurements in such sequences, over stratified infinite tensor products of the finite dimensional spaces. The stratification is based on uncomputable Turing classes and also on the Solovay genericity and the equivalence is now corrected by the appearance of certain self-adjoint operator. From studies on algorithmic randomness follows that these two properties are in a sense orthogonal one to the other for random sequences. We show two models of ZFC which indeed separate them. This is based on the classic results for Cohen and random forcings in set theory and also on Takeuti’s early results on Boolean-valued models and quantum mechanics. This separation is the tool for showing that not all correlations between commuting operators on infinite dimensional Hilbert spaces can be reproduced on a tensor product of two spaces with the corresponding self-adjoint operators. We discuss the findings also from the perspective of eventual practical applications. The supplementary subsections contain discussion of QM with the set theory component along with Solovay randomness of the outcomes.
In 1964, John Bell showed that any local hidden‑variable theory must satisfy a specific inequality on correlations between distant measurements. In its CHSH form (Clauser, Horne, Shimony, Holt 1969), the inequality reads |S| = |E(a,b) + E(a,b′) + E(a′,b) − E(a′,b′)| ≤ 2, where E(a,b) is the correlation between measurement outcomes at two separated parties using settings a and b. Any local realist theory respects this bound. Quantum mechanics violates it. Experiments confirm the violation. The standard quantum prediction for a maximally entangled state with optimal measurement angles is |S| = 2√2 ≈ 2.828. In 1980, Boris Tsirelson proved that 2√2 is not just one quantum prediction — it is the *maximum* value any quantum correlation can produce. No quantum state, no measurement scheme, no clever protocol can push |S| beyond 2√2. The bound is sharp, the value is exact, and it has been measured in laboratories around the world to experimental precision. The Tsirelson bound is one of the deepest empirical structures in physics. It tells us that quantum correlations exceed classical correlations by a specific, finite amount — there is room to violate Bell’s inequality, but only so much room. The number 2√2 has no obvious physical interpretation. It does not appear in the formulation of quantum mechanics. It emerges from the mathematics of the bipartite Hilbert space when the CHSH operator is constructed and bounded. The question is where it comes from. In standard QM, the answer is that it falls out of the Hilbert‑space structure plus Cauchy–Schwarz plus an operator‑norm bound on the CHSH operator. Technically correct, and Tsirelson’s 1980 proof remains the canonical derivation — but it leaves the deeper structure unanswered. The Hilbert space is postulated. The inner product that makes Cauchy–Schwarz available is postulated. The bipartite tensor‑product structure on H_A ⊗ H_B is postulated. The 2√2 value is what those postulates produce when run through the CHSH operator. This walkthrough shows how the Event Density framework derives the Tsirelson bound from a smaller set of structural commitments. The Hilbert‑space structure, the inner product, and the bipartite tensor‑product structure all emerge as forced consequences of the substrate ontology rather than being postulated. With those structures forced rather than assumed, Tsirelson’s 1980 argument applies directly, and the 2√2 value is no longer the output of a chain of postulates but the output of a chain of derived theorems. Like Heisenberg, Bell–Tsirelson is a place where the framework’s contribution is upstream. The bound itself is mathematical physics already in the literature; what changes is the foundational status of the structures the argument requires. ED does not produce a new derivation step beyond standard QM here — it removes the postulational status of the structures the standard derivation already uses. The 2√2 value is now a derived consequence of the substrate ontology, all the way down.