The conditional probabilities achievable by local measurements on a shared quantum state in a Bell scenario form the set of quantum behaviors, whose structure was studied by Tsirelson. In 1993, Tsirelson asked whether this set is semialgebraic, that is, describable by finite Boolean combinations of polynomial equations and inequalities. This question has remained open. We answer it in the negative: with four binary measurements per party, the set of finite-dimensional quantum behaviors, its closure, and the commuting-operator set are all nonsemialgebraic. More strongly, none admits a finite real-analytic description even locally near a particular classical behavior. These results rule out exact finite semidefinite representations and show that no finite level of the Navascués-Pironio-Acín hierarchy characterizes these sets exactly.
The conditional probabilities achievable by local measurements on a shared quantum state in a Bell scenario form the set of quantum behaviors, whose structure was studied by Tsirelson. In 1993, Tsirelson asked whether this set is semialgebraic, that is, describable by finite Boolean combinations of polynomial equations and inequalities. This question has remained open. We answer it in the negative: with four binary measurements per party, the set of finite-dimensional quantum behaviors, its closure, and the commuting-operator set are all nonsemialgebraic. More strongly, none admits a finite real-analytic description even locally near a particular classical behavior. These results rule out exact finite semidefinite representations and show that no finite level of the Navascués-Pironio-Acín hierarchy characterizes these sets exactly.
The Navascués--Pironio--Acín (NPA) hierarchy gives the standard semidefinite outer approximations to quantum behaviors. Whether \emph{any} finite level can already equal the quantum set has remained open even in the bipartite scenario with two binary measurements per party. We demonstrate that \emph{no finite level} is exact. For the symmetric doubly tilted CHSH functional hα=A0B0+A0B1+A1B0−A1B1+α(A0+B0), set T=1−α. Its quantum maximum satisfies [ωQ(1−T)−(4−2T)]/T3→4/3, whereas every fixed NPA level satisfies [ωL(1−T)−ωQ(1−T)]/T3→+∞. Under the corresponding boundary rescaling, an explicit expectation of the positive operator ωQ(1−t2)I−Ht converges to the Motzkin polynomial. A bounded fixed-level error would therefore make the Motzkin polynomial plus a nonnegative constant a sum of squares, which is impossible. Consequently, every standard NPA relaxation based on a fixed finite list of words in the measurement projectors strictly contains the complete quantum set, and its nonquantum behaviors accumulate at a local deterministic behavior. Thus, the finite-level exactness of CHSH and all one-sided tilted CHSH maxima does not extend to an exact finite-level description of the complete quantum set in the minimal scenario.
A key goal in the foundations of quantum mechanics is to identify operational constraints characterizing physical theories. Bell inequalities do so for classical probability theory, while Tsirelson's bound provides a first step for quantum mechanics. Here, we address the dual question: Can one certify that all correlations predicted by quantum theory are actually realizable? In this work, we construct a finite number of two-body correlations such that the only probabilistic theory that (1) realizes them, and (2) does so in a way that is stable under iterated teleportation, is quantum theory. For n quantum systems of dimension d each, Condition (1) can be verified using poly(d,n) measurement settings. Condition (2) may be understood as a hierarchy of tests, one for each number N of teleportation steps. There is thus a sense in which finite-dimensional quantum theory can be self-tested. In particular, one can certify the existence of Bell inequality violations larger than any that have been directly observed.
Characterizing the set of quantum nonlocal correlations through information-theoretic constraints is a central problem in quantum foundations. We show that the convex hull of no-signalling behaviors in a bipartite Bell experiment satisfying a common-overlap no-disturbance-without-uncertainty condition is exactly the finite-dimensional quantum set whenever one party has two binary measurements and the other has an arbitrary finite number of settings, each with an arbitrary finite number of outcomes. In the simplest binary input-output Bell scenario, this characterizes the full eight-dimensional set of quantum behaviors, including the local marginals. The proof constructs a quantum realization directly from the conditional probabilities and no-signalling. The characterization also reduces optimization of any linear Bell expression in these scenarios to maximization over a one-parameter family of second-order cone programs. The results establish a connection between local uncertainty-disturbance constraints and quantum nonlocality, mediated by no-signalling and shared classical randomness.
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.
Characterising the correlations that arise from locally measuring a single part of a joint quantum system is one of the main problems of quantum information theory. The seminal work [M. Navascués et al., New J. Phys. 10, 073013 (2008)], known as the Navascués–Pironio–Acín (NPA) hierarchy, reformulated this question as a polynomial optimisation problem over noncommutative variables and proposed a convergent hierarchy of necessary conditions, each testable using semidefinite programming. More recently, the problem of characterising the quantum network correlations, which arise when locally measuring several independent quantum systems distributed in a network, has received considerable interest. Several generalisations of the NPA hierarchy, such as the scalar extension [A. Pozas-Kerstjens et al., Phys. Rev. Lett. 123, 140503 (2019)], were introduced while their converging sets remain unknown. In this work, we introduce a new bilocal factorisation NPA hierarchy, prove its equivalence to a modified bilocal scalar extension NPA hierarchy, and characterise its convergence in the case of the simplest network, the bilocal scenario. We further explore its relations with the other known generalisations.
We revisit the Bell--CHSH scenario for two spin-21 particles and isolate the precise algebraic origin of the Bell contradiction. On the quantum side, spin-21 is described by a noncommutative spinor (Clifford) algebra acting on the Hilbert space of two spin-21 particles, with the singlet state yielding the usual correlation E(a,b)=−a⋅b and Tsirelson's bound 22. On the classical side, the standard Bell assumptions amount to describing all measurement outcomes as {±1}-valued random variables on a single Kolmogorov probability space, i.e.\ elements of a commutative algebra C(Λ).
We show that there is no representation of the spinor algebra of spin-21 (with its singlet state and locality structure) into any such commutative Kolmogorov algebra that preserves the {±1} spectra of local spin components and the singlet correlations entering the CHSH expression, under the standard Bell assumptions of locality (factorization) and measurement independence. In this sense, the Bell--CHSH contradiction is exhibited as an algebraic mismatch between a noncommutative spinor/Clifford description of spin and the classical assumption of a single global Kolmogorov space supporting all outcomes. In the language of quantum probability, this is a C∗-algebraic reformulation of the known fact that the singlet correlations admit no local hidden-variable model with jointly distributed outcomes on one probability space.
We also give an explicit realization of the same spinor structure within the author's Quantum Index Algebra (QIA) framework, where locality appears as disjoint index slots and the singlet state as a simple index cocycle.