Closing the Tsirelson Loophole in Relativistic Quantum Field Theory: Split Inclusions, Hyperfinite Local Algebras, and the Effective Dimension of the Vacuum
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
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
Whether physical reality contains an actual infinity is usually discussed philosophically. We ask a narrower, quantitative question: what would it cost a finite experiment to certify it? We combine sharp results from quantum information with physical resource bounds. (1) Certification budget. Resolving a game value to ε requires N ≳ ln(1/4δ) w(1−w)/ε² rounds (Bretagnolle–Huber), so the cost of certifying a Hilbert-space dimension D is governed by the gap ε_G(D) between the optimal value and the best dimension-D value. Take the unspecified constants in the known bounds to be one. Then the best known two-player dimension witness needs at least 10⁸⁶ rounds to certify a dimension beyond the Bekenstein bound of a 1 kg, 1 m laboratory (2.6×10⁴³ qubits). Such a laboratory can perform only 2.4×10⁶⁸ operations in the age of the universe. Beyond the de Sitter entropy (4.8×10¹²² qubits), the witness needs at least 10²⁴⁵ rounds, against the 10¹²⁰ operations of the universe. Spatial parallelisation does not help, because the Margolus–Levitin bound limits the total rate summed over all subsystems that share the energy. For the I₃₃₂₂ inequality the requirement is doubly exponential. (2) Embezzlement asymptotics. The optimal fidelity of van Dam–Hayden embezzlement of one EPR pair is F_n = (H_{n/2} + σ_{n/2})/H_n. Hence 1 − F_n = (ln 2 − σ)/H_n + O(1/(n ln n)) with σ = 0.2992041. This holds for all n. It is confirmed to 10⁻⁶ for dimensions up to 2²⁶, and to 6×10⁻⁸ for odd n = 10⁷ + 1. The infinite-dimensional resource behind these witnesses is approached only logarithmically. (3) The Tsirelson loophole. MIP*=RE implies that commuting-operator correlations can exceed every finite-dimensional value by a finite gap, which finite statistics could in principle detect. However, the set of true bounds ω*(G) ≤ t is Π⁰₁ but not computably enumerable. No consistent theory therefore proves all of them, and this route is limited by provability rather than by statistics. In practice, no separating game with small question and answer sets and a computed gap is known. (4) Thermodynamic infinities. Kaufman's exact formula for the two-dimensional Ising model on a finite torus shows that a 1 cm sample rounds the critical singularity at |K − K_c|/K_c ~ 4×10⁻⁹. The logarithmic amplitude of the specific-heat peak equals that of the bulk singularity. The orbital helium lambda-point experiment, which reached |t| ≈ 10⁻⁹, stays about four orders of magnitude above the finite-size scale of its sample. These results do not show that nature is finite. They show the following: apart from one route whose cost is logical rather than statistical, finiteness bounded by holography cannot be refuted from inside a universe that respects that bound. Code, data and figures: https://github.com/Ruqing1963/certify-infinity
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.
Tsirelson’s 2√2 ceiling is a finite-dimensional operator theorem; it does not require acontinuum quantum field theory. We formulate the exact two-site result needed by the Treoshared-closure programme and separate it from the framework-specific physical identification.In the one-excitation subspace span{|01⟩, |10⟩}, imposing a definite exchange parity uniquelyselects (|01⟩ ± |10⟩)/√2, up to phase. These are maximally entangled Bell states. For any twodichotomic observables at each site, the CHSH operator obeys ∥B∥ ≤ 2√2; suitable local axessaturate the bound exactly. No-signaling follows from the tensor-product trace rule.Two shortcuts must be avoided. Equality of the two reduced states alone does not implymaximal mixing, and the printed + Bell state does not automatically have the singlet correlationE(a, b) = −a · b in the same local basis. A local unitary maps the Bell states into one another,so their maximal CHSH value is the same, but the observable convention must be transformedwith the state. The remaining Treo burden is therefore not the algebraic bound. It is to derivethe local two-level register, the one-excitation shared-closure constraint, the allowed detectorobservables, and their spacelike operational readout from the native finite dynamics. The C9successor is not a premise of this two-site theorem.
This paper is the second and final part of a two-part study. We construct positive-energy relativistic spatial localization observables in Minkowski spacetime within standard quantum field theory, using the stress--energy--momentum tensor smeared with suitable test functions. For each fixed timelike direction, the construction gives positive operator-valued measures (POVMs) on spacelike hypersurfaces, well defined on every n-particle sector and satisfying a relativistic causality condition excluding superluminal propagation of detection probabilities.
The observables are built from local or quasi-local field-theoretic quantities, thus providing a rigorous version of earlier heuristic proposals. In the one-particle sector, the construction reduces to the observable previously introduced by the author, and its first moment gives the Newton--Wigner position operator under appropriate normalization and centering assumptions.
Because the Reeh--Schlieder theorem prevents the normally ordered stress--energy--momentum tensor from being positive on the full Fock space, we use quantum energy inequalities to obtain lower bounds controlling deviations from positivity. This leads to regularized operator families, bounded from below, which approximate the localization effects.
Finally, we define conditional localization observables for finite laboratories through modified local energy operators. By Haag duality, the corresponding conditional POVMs belong to local von Neumann algebras and commute for causally separated regions, in accordance with the Araki--Haag--Kastler framework. The results show how commutativity of localization observables is recovered for conditional measurements in finite spacetime regions.
When two entangled particles are measured light-years apart and their results correlate without fail, the question almost everyone asks —and the one almost no one answers well —is “how did the information travel between them?” This work argues that the question is badly posed from its very verb: there is no travel. What occurs is coordinated manifestation of the same Hilbert space in two causally disconnected regions —the correlations do not move, they clothe themselves in geometry where the Hilbert space that contains them permits, in both places at once, because they never ceased to be a single thing. That reinterpretation is not merely a prettier way of saying the same thing: it brings with it a precise ontological piece, with a double foundation. First, between spacelike-separated regions there is no relativistic causality to violate, because there is nothing propagating. Second, no-signaling is preserved even when the Type 3 correlations are treated as geometric information in the strict sense. From this, six concrete restrictions are derived —independent conservations, membership in the same Hilbert space, Planck separation in a spacelike relation, sufficient purity, environmental conditions, preservation of Type 1 and Type 2 properties —that the program’s apparatus imposes without the need to add a single new assumption. The result is a complete harmony with quantum entanglement as it is observed experimentally, with the locality-loophole-closed Bell experiments as the paradigmatic case. But the harmony goes beyond entanglement at distance it confirms that any change or conservation is merely a product ofthe posibilities within a Hilbert space. And the work is careful, deliberately, not to over-promise: it announces no operative teleportation nor instantaneous communication —it traces the framework that any future technological operationalization will have to respect, and it declares honestly what it does not yet know. It is the fourth article of the program, and the first that looks, with its feet firmly on the ground, toward an application that will take decades. Six specific restrictions are derived (on the independent conservations of c and quantum change, membership in the same Hilbert space, Planck separation in a spacelike relation, purity of the Hilbert space, environmental conditions, and preservation of Type 1 and Type 2 properties) that the unified apparatus of the CNMC/FCF program imposes. The work establishes complete conceptual harmony with the experimentally observed quantum entanglement, articulating the locality-loophole-closed Bell experiments as the paradigmatic instance of the operatively distinctive phenomenon. Pending explorations are declared on the specific restrictions by type of correlation (G) and by macroscopicity of the system (H). The work inaugurates the decades-long project toward the future technological operationalization of the phenomenon, establishing the formal foundational framework that this operationalization must respect.
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.