Commuting Operator Value in Nonlocal Games

• Commuting operator value is defined as the supremum win probability achievable via non-tensor-product strategies in two-player nonlocal games.
• The NPA hierarchy offers semidefinite programming relaxations that converge asymptotically to the commuting operator value, yet may never reach exactness at a finite level.
• Algebraic reductions from Turing machines to Boolean constraint system games demonstrate RE-hardness and highlight fundamental limits in certifying quantum nonlocality.

The commuting operator value of a two-player nonlocal game is the supremum success probability achievable by strategies in which players implement measurements (possibly on an infinite-dimensional Hilbert space) that satisfy definite commutation relations but may not factor as tensor products. This value, denoted $\omega_{qc}(G)$ for a game $G$, is operationally the maximal win probability over all “commuting operator strategies”—specifically, joint projective measurement families $\{E^x_a\}$ (Alice) and $\{F^y_b\}$ (Bob) with $[E^x_a, F^y_b] = 0$ for all questions $(x, y)$ and all answers $(a, b)$. The NPA (Navascués–Pironio–Acín) hierarchy provides a hierarchy of semidefinite programs whose level-$k$ relaxation, $\omega^{(k)}_{npa}(G)$, yields increasingly tight upper bounds on $\omega_{qc}(G)$ as $k \to \infty$. This framework is central to characterizing quantum correlations and certifying nonlocality in device-independent settings.

1. Formal Definition and Context

The commuting operator value is given by: $$\omega_{qc}(G) = \sup \left\{ \sum_{x,y,a,b} \pi(x,y) V(a, b|x, y)\, \langle \psi, E^x_a F^y_b \psi \rangle \right\}$$ where $(\mathcal{H}, \psi)$ comprises a shared (possibly infinite-dimensional) Hilbert space with a unit vector $\psi$, $E^x_a$ and $F^y_b$ are projector families with $\sum_a E^x_a = I$, $\sum_b F^y_b = I$, and $[E^x_a, F^y_b]=0$. The focus is on strategies that only require commutativity of Alice's and Bob's operators, in contrast to the tensor product requirement in MIP$^*$ protocols.

The NPA hierarchy, at each finite level $k$, is a semidefinite programming relaxation encoding operator moment constraints up to degree $k$. These provide upper bounds: $$\omega^{(k)}_{npa}(G) \geq \omega^{(k+1)}_{npa}(G) \geq \omega_{qc}(G)$$ and converge in the limit: $$\lim_{k\to\infty} \omega^{(k)}_{npa}(G) = \omega_{qc}(G)$$. However, the present work investigates whether this convergence ever stabilizes at a finite $k$—i.e., is the true value always realized by a finite-level solution corresponding to a feasible finite-dimensional moment matrix?

2. Undecidability and Failure of Finite NPA Attainment

The main result establishes that it is recursively enumerable (RE-)hard—and thus undecidable in the Turing sense—to determine for a specific game $G$ whether $\omega_{qc}(G) > \theta$ for some fixed rational $\theta$ (notably $\theta = 1/2$). This is realized via a computable mapping from the halting problem for Turing machines to the commuting-operator value decision problem for carefully constructed BCS (Boolean constraint system) games.

The undecidability result implies that, for some games $G$, there is no $k$ such that $\omega^{(k)}_{npa}(G) = \omega_{qc}(G)$. That is, for these pathological instances, every finite level produces a strict overestimate: $$\forall k \in \mathbb{N}: \quad \omega^{(k)}_{npa}(G) > \omega_{qc}(G).$$ This demonstrates a limitation of the NPA approach—there exist games for which the hierarchy does not stabilize at any finite level.

3. Boolean Constraint System (BCS) Games as Encoding Device

A Boolean constraint system consists of variables $X$ and constraints $\{C_i\}$ (each involving a subset $U \subset X$ and a relation $R \subset \{-1, +1\}^{|U|}$). The corresponding BCS game is defined so that, on each round, Alice receives a constraint/context $U$ and Bob receives an individual variable $x \in U$. They win if their answers are consistent, and Alice’s assignment satisfies the constraint.

The algebraic structure behind a BCS game is the BCS algebra: $$A(B) = \mathbb{C}[\mathbb{Z}_2^{*X}] / \langle R(B) \rangle,$$ the quotient of the group *-algebra on the free group by the ideal generated by translations of the relations into operator form. The existence of perfect commuting-operator strategies for the game corresponds to the existence of a tracial state on $A(B)$, linking the combinatorial game theory to operator algebraic questions.

4. Computable Reductions from Turing Machines to BCS Games

The central technical construction establishes a mapping $M \mapsto G_M$ from Turing machine descriptions to BCS games such that

• If $M$ halts (i.e., $M \in L$ for some recursively enumerable $L$), then $G_M$ admits a perfect commuting operator strategy reflecting $\omega_{qc}(G_M) > \theta$ for a specified threshold.
• If $M \notin L$, then all approximate strategies for $G_M$ have an expected value for a distinguished observable $D = (1-x_D)/2$ bounded by a constant multiple of the allowed error.

The reduction is constructive and tracks an explicit variable $x_D$, whose expected value in any tracial state correlates directly with the halting of $M$. Thus, determining whether $\omega_{qc}(G_M) > \theta$ would allow the halting problem to be solved—hence, this problem is RE-hard.

5. Algebraic Techniques and Nested Conjugacy Relations

The techniques used are fundamentally algebraic, relying on embedding complex finitely presented *-algebras (including group algebras of recursively presented groups encoding Turing computations) into BCS algebras using nested conjugacy relations. In these, auxiliary variables encode words of the form

$$w = x_{i_\ell} \cdots x_{i_1} x_{i_0} x_{i_1} \cdots x_{i_\ell}$$

which can be recursively related via new constraints to enforce desired relations within the BCS framework. A theorem (Theorem 5.7) shows that algebras with these types of relations can be embedded into conventional BCS algebras, ensuring that the group-theoretic undecidability is fully reflected in the algebraic setting of nonlocal games.

These methods differ qualitatively from the techniques used in the proof of MIP$^* = \mathrm{RE}$, which relies on interactive proof systems and complexity amplification, rather than explicit algebraic embeddings and analysis of operator system representations.

6. Implications and Limitations of the NPA Hierarchy

The main implications are:

• There exist nonlocal games $G$ for which for all $k$, $\omega^{(k)}_{npa}(G) > \omega_{qc}(G)$; no finite-level hierarchy bounds exactly attain the commuting operator value.
• Deciding whether $\omega_{qc}(G) > \theta$ is RE-hard, so the problem is as hard as the halting problem for Turing machines.
• The algebraic structure of nonlocal games is sufficiently rich to reflect arbitrary computational complexity and undecidability, intimately connecting operator algebra, group theory, and quantum nonlocality.
• Device-independent certification methods that rely on finite-level NPA bounds are fundamentally limited: certification of certain quantum correlations requires infinite resources or is undecidable in general.

This work exposes a sharp boundary in quantum complexity theory and operator algebra, highlighting that even the most general convex relaxations (SDPs in the NPA hierarchy) may not suffice to exactly capture the commuting operator value in all physically meaningful settings. As a consequence, algorithmic methods for certifying quantum advantage or bounding nonlocal correlations cannot universally succeed by truncations of the NPA hierarchy.

Key Notation Summary

Symbol	Description
$\omega_{qc}(G)$	Commuting-operator value of game $G$
$\omega^{(k)}_{npa}(G)$	Level-$k$ NPA relaxation bound
$A(B)$	BCS algebra for a Boolean system $B$
$D$	Distinguished observable in reduction
$w$	Nested conjugacy word (see Def. 5.1)
$\theta$	Fixed decision threshold (e.g., $1/2$)

These results frame a fundamental obstruction to decidability in nonlocal games and operator algebra, and rigorously establish that the NPA hierarchy's convergence to $\omega_{qc}(G)$ can be only asymptotic, never exact at any finite level for certain games (Fanizza et al., 6 Oct 2025).