Semialgebraicity of closed quantum correlations

Problem

Is the closure of the finite-dimensional tensor-product correlation set semialgebraic in every fixed bipartite Bell scenario? Fix positive integers \(n_A,n_B,m_A,m_B\). Inputs satisfy \(1\leq x\leq n_A\) and \(1\leq y\leq n_B\), while outputs satisfy \(1\leq a\leq m_A\) and \(1\leq b\leq m_B\). The set \(\mathcal C_q(n_A,n_B,m_A,m_B)\) consists of behaviors

\begin{equation} p(a,b\mid x,y) =\operatorname{Tr}\!\left[ \rho\bigl(E_a^x\otimes F_b^y\bigr) \right] \tag{1} \end{equation}

obtained from arbitrary finite-dimensional local Hilbert spaces, a density operator \(\rho\), and local POVMs satisfying

\begin{equation} E_a^x\succeq0,\qquad F_b^y\succeq0,\qquad \sum_{a=1}^{m_A}E_a^x=I,\qquad \sum_{b=1}^{m_B}F_b^y=I. \tag{2} \end{equation}

Equation (2) fixes the measurement class in Eq. (1). Define the closed correlation set by

\begin{equation} \mathcal C_{qa}(n_A,n_B,m_A,m_B) :=\overline{\mathcal C_q(n_A,n_B,m_A,m_B)}. \tag{3} \end{equation}

For every fixed tuple, is the set in Eq. (3) a finite Boolean combination of polynomial equalities and inequalities with real coefficients? The coefficients may be arbitrary real numbers, and the description need not be one conjunction of weak inequalities.

Source

Tsirelson explicitly asks whether the quantum-correlation set in each fixed finite Bell scenario is semialgebraic [Tsi93].

Progress

In the scenario with two binary measurements per party, Jordan-type dimension reduction and finite convexification give a semialgebraic parametrization [Tsi93], [Mas05]. This does not extend to every fixed input–output tuple.
The zero-marginal correlator slice in the smallest bipartite scenario has an explicit semialgebraic description but is not basic semialgebraic. Consequently, asking for one finite conjunction of polynomial inequalities would already be too strong [LMSWZ23].
The finite-dimensional set \(\mathcal C_q\) defined by Eqs. (1)–(2) is not closed in sufficiently large fixed scenarios. This necessitates the closure in Eq. (3) [Slo19].
For sufficiently large fixed input and output sizes, membership in \(\mathcal C_{qa}\) is undecidable even for behaviors with algebraic coordinates. This excludes effective polynomial descriptions with computable algebraic coefficients, but it does not exclude a non-effective semialgebraic presentation using arbitrary real coefficients [FMS25].

Comment

Undecidable membership rules out effective descriptions of the usual kind, but semialgebraicity with unrestricted real coefficients is a set-theoretic existence question and remains unresolved for general fixed scenarios.

References

[Tsi93]: B. S. Tsirelson, “Some Results and Problems on Quantum Bell-Type Inequalities,” Hadronic Journal Supplement 8(4), 329–345 (1993). publication record; full text.link link
[Mas05]: Ll. Masanes, “Extremal Quantum Correlations for \(N\) Parties with Two Dichotomic Observables per Site,” arXiv:quant-ph/0512100 (2005).DOI arXiv
[LMSWZ23]: T. P. Le, C. Meroni, B. Sturmfels, R. F. Werner, and T. Ziegler, “Quantum Correlations in the Minimal Scenario,” Quantum 7, 947 (2023).DOI arXiv
[Slo19]: W. Slofstra, “The Set of Quantum Correlations Is Not Closed,” Forum of Mathematics, Pi 7, e1 (2019).DOI arXiv
[FMS25]: H. Fu, C. A. Miller, and W. Slofstra, “The Membership Problem for Constant-Sized Quantum Correlations Is Undecidable,” Communications in Mathematical Physics 406, 96 (2025).DOI arXiv

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