The NPA hierarchy does not always attain the commuting operator value

We show that it is undecidable to determine whether the commuting operator value of a nonlocal game is strictly greater than 1/2. Specifically, there is a computable mapping from Turing machines to /boolean constraint system (BCS) nonlocal games in which the halting property of the machine is encoded as a decision problem for the commuting operator value of the game. As a corollary, there is a BCS game for which the value of the Navascués-Pironio-Acín (NPA) hierarchy does not attain the commuting operator value at any finite level.

💡 Research Summary

The paper investigates the relationship between the commuting‑operator value of a nonlocal game and the Navascués‑Pironio‑Acín (NPA) hierarchy, establishing that the hierarchy does not always converge to the commuting‑operator value at any finite level. The authors focus on the decision problem (QC‑Strict): given a two‑player nonlocal game G, decide whether the commuting‑operator value ω_qc(G) is strictly greater than 1/2. They prove that this problem is RE‑hard, i.e., as hard as the halting problem, and consequently it cannot belong to coRE. This result directly answers a long‑standing open question: whether for every game there exists a finite NPA level k such that the k‑th NPA bound ω^{(k)}_NPA(G) equals ω_qc(G). Since (QC‑Strict) is RE‑hard but not in coRE, the answer must be negative.

The technical core rests on a reduction from Turing machines to Boolean constraint system (BCS) nonlocal games. Prior work (Miller‑Slofstra‑Zheng 2023) showed a computable map M ↦ α_M where α_M is a *‑polynomial that is positive (i.e., λ − Φ_G is positive semidefinite for all *‑representations) iff the machine M does not halt. The positivity of α_M corresponds to the statement ω_qc(G) ≤ 1/2 for a game whose functional Φ_G satisfies α_M = 1/2 − Φ_G. However, the construction in that work yields α_M of high degree, whereas a game functional must be a degree‑2 *‑polynomial (quadratic in the measurement operators). To bridge this gap, the authors develop a general embedding theorem for BCS algebras.

BCS algebras are *‑algebras presented by Boolean constraints (e.g., XOR, OR) that model a wide variety of algebraic relations, extending the earlier linear constraint system (LCS) algebras. The authors introduce “nested conjugacy BCS relations,” a broader class that captures the non‑group relations appearing in the α_M construction. They then devise a systematic rewriting procedure that transforms any nested conjugacy relation into a finite collection of ordinary BCS relations, while controlling the blow‑up in the number of relations. This quantitative embedding ensures that any finitely presented *‑algebra (including the one generated by the α_M construction) can be embedded into a BCS algebra with only polynomial overhead.

Applying this embedding, they obtain a computable mapping M ↦ G_M from Turing machines to two‑player BCS nonlocal games such that ω_qc(G_M) > 1/2 if and only if M halts. Consequently, deciding whether ω_qc(G) > 1/2 is RE‑hard. Since the NPA hierarchy provides a non‑increasing sequence of SDP upper bounds ω^{(k)}_NPA(G) converging to ω_qc(G), if for every game there existed a finite k with equality, the (QC‑Strict) problem would be decidable in coRE (by checking whether ω^{(k)}_NPA(G) ≤ 1/2). The RE‑hardness contradicts this, proving that there must exist games for which ω^{(k)}_NPA(G) > ω_qc(G) for all k ∈ ℕ. This is formalized as Theorem 1.3.

The paper’s contributions can be summarized as follows:

1. Complexity Result: Proves that determining whether the commuting‑operator value of a nonlocal game exceeds 1/2 is RE‑hard, establishing undecidability beyond the previously known coRE‑hardness for the “≥ 1/2” version.
2. Embedding Theorem: Provides a quantitative embedding of arbitrary finitely presented *‑algebras (with nested conjugacy relations) into BCS algebras, preserving polynomial bounds on the number of relations.
3. NPA Non‑Convergence: Shows the existence of explicit BCS games for which the NPA hierarchy never attains the commuting‑operator value at any finite level, answering Question 1.1 in the negative.
4. Methodological Bridge: Connects algebraic positivity of *‑polynomials, GNS construction, and semidefinite programming hierarchies, illustrating how undecidability in operator algebras translates into limitations of SDP relaxations used in quantum information.

The implications are twofold. From a theoretical perspective, the result deepens the understanding of the landscape of quantum nonlocality, confirming that the commuting‑operator model is strictly more powerful than any finite SDP relaxation. Practically, it cautions researchers employing the NPA hierarchy for device‑independent protocols, cryptographic soundness proofs, or quantum game analysis: one cannot rely on finite‑level convergence to certify optimal commuting‑operator performance in general. Future work may explore alternative hierarchies, identify subclasses of games where finite convergence holds, or investigate the quantitative gap between NPA bounds and commuting‑operator values for the constructed hard instances.