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.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.

arXiv:2510.04943Oct 6, 20250 citations

Communication and Computation by Quantum Games

Takeshi Kawakami

Lecture notes in computer scienceJan 1, 20021 citation

Quantum Entanglement Redefines Undecidability: MIP*=RE and Turing's Legacy — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: The halting problem's undecidability is foundational to computation limits; recent MIP*=RE result links it to quantum entanglement and nonlocal games, while a 2024 paper nuances Turing's original attribution. | MATH: Undecidability proof via diagonalization: assume HALT exists, construct D(x) = loop if HALT(x,x) halts, else halt; contradiction. MIP*=RE: the class of languages decidable by multiprover interactive proofs with quantum entanglement equals recursively enumerable sets — equivalently, the commuting-operator and tensor-product models diverge (Connes embedding conjecture false). No classical constants; structure is ordinal/recursion-theoretic (ω₁^CK, Turing degrees). | CONNECTION: No direct geometric ratio (0.382, 0.618, etc.) appears. However, the MIP*=RE result relies on *lattice-like* constructions of nonlocal games (Tsirelson bounds, operator algebras) and the *root system* of quantum correlations — the set of quantum strategies forms a convex body with extreme poi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: The halting problem's undecidability is foundational to computation limits; recent MIP*=RE result links it to quantum entanglement and nonlocal games, while a 2024 paper nuances Turing's original attribution. | MATH: Undecidability proof via diagonalization: assume HALT exists, construct D(x) = loop if HALT(x,x) halts, else halt; contradiction. MIP*=RE: the class of languages decidable by multiprover interactive proofs with quantum entanglement equals recursively enumerable sets — equivalently, the commuting-operator and tensor-product models diverge (Connes embedding conjecture false). No classical constants; structure is ordinal/recursion-theoretic (ω₁^CK, Turing degrees). | CONNECTION: No direct geometric ratio (0.382, 0.618, etc.) appears. However, the MIP*=RE result relies on *lattice-like* constructions of nonlocal games (Tsirelson bounds, operator algebras) and the *root system* of quantum correlations — the set of quantum strategies forms a convex body with extreme poi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Oct 15, 20260 citations

A nonlocal game whose dimension exponent is uncomputable

Seth Douglas, Nidhal Mghirbi

Can uncomputability occur in the resource law of one fixed finite nonlocal game, rather than in a problem whose input is a game? We construct a finite two-player Boolean constraint-system game with rational question probabilities, quantum value one, and no finite-dimensional perfect strategy. If $D(\varepsilon)$ is the least local dimension needed to win with probability at least $1-\varepsilon$ , then $\lim_{n\to\infty}\log_2 D(2^{-n})/n=(3+\theta_K)/4$ , where $\theta_K=\sum_{e\in K}4^{-(e+1)}$ and $K$ is the halting set. This exponent has the Turing degree of $K$ . The construction combines finite operator relations for three Cantor clocks, a counter program with forced initialization, and a conditional clock that charges dimension for halting inputs. A Boolean compiler preserves weighted precision linearly and converts flat models to strategies with quadratic loss. The lower bound holds in every density matrix. Explicit finite models attain the same rate. The flag-density lower bound uses compactness without an effective modulus; convergence to the dimension exponent has no computable certified rate.Can uncomputability occur in the resource law of one fixed finite nonlocal game, rather than in a problem whose input is a game? We construct a finite two-player Boolean constraint-system game with rational question probabilities, quantum value one, and no finite-dimensional perfect strategy. If $D(\varepsilon)$ is the least local dimension needed to win with probability at least $1-\varepsilon$, then $\lim_{n\to\infty}\log_2 D(2^{-n})/n=(3+\theta_K)/4$, where $\theta_K=\sum_{e\in K}4^{-(e+1)}$ and $K$ is the halting set. This exponent has the Turing degree of $K$. The construction combines finite operator relations for three Cantor clocks, a counter program with forced initialization, and a conditional clock that charges dimension for halting inputs. A Boolean compiler preserves weighted precision linearly and converts flat models to strategies with quadratic loss. The lower bound holds in every density matrix. Explicit finite models attain the same rate. The flag-density lower bound uses compactness without an effective modulus; convergence to the dimension exponent has no computable certified rate.

Oct 27, 20268 citations

D(\varepsilon)

is the least local dimension needed to win with probability at least

1-\varepsilon

, then

\lim_{n\to\infty}\log_2 D(2^{-n})/n=(3+\theta_K)/4

, where

\theta_K=\sum_{e\in K}4^{-(e+1)}

and

K

is the halting set. This exponent has the Turing degree of

K

. The construction combines finite operator relations for three Cantor clocks, a counter program with forced initialization, and a conditional clock that charges dimension for halting inputs. A Boolean compiler preserves weighted precision linearly and converts flat models to strategies with quadratic loss. The lower bound holds in every density matrix. Explicit finite models attain the same rate. The flag-density lower bound uses compactness without an effective modulus; convergence to the dimension exponent has no computable certified rate.