Released by
Kun Chen, Institute of Theoretical Physics, Chinese Academy of Sciences
Contact email
chenkun0228@gmail.com
Method
Other
Challenge issue
Background
A standard monogamy-of-entanglement game asks two noncommunicating players to predict the outcome of a quantum measurement made by a referee. A finite game
$$G=(\Sigma,\Gamma,\pi,\{R_x^a\})$$
has a finite question set $\Sigma$, a finite answer set $\Gamma$, a question distribution $\pi$, and referee POVMs
$$R_x^a\succeq0,\qquad \sum_{a\in\Gamma}R_x^a=I_R.$$
The same sampled question $x$ is sent to both players. They win exactly when both return the referee's measurement outcome $a$.
Write $\omega(G)$ for the optimal unentangled value and $\omega^(G)$ for the optimal value when the players may share a finite-dimensional entangled strategy with the referee. Every two-question standard game satisfies $\omega(G)=\omega^(G)$. A four-question, three-answer separation is known. It remains open whether a separation is possible with the minimal nontrivial answer alphabet $|\Gamma|=2$.
Research objective
Construct one fully explicit finite standard monogamy-of-entanglement game with
$$|\Gamma|=2$$
and a legal finite-dimensional entangled strategy $S$ such that
$$p(G,S)>\omega(G).$$
Determining the exact global entangled optimum $\omega^(G)$ is unnecessary because $\omega^(G)\ge p(G,S)$. Since two-question games cannot separate the values, any positive witness necessarily has at least three questions.
The construction must remain a standard monogamy-of-entanglement game. A lossy or constrained extension, an ordinary Bell game, a communication-assisted protocol, or a coarse-grained statistic that does not define a legal binary referee POVM does not satisfy the challenge.
Success and verification gate
A complete result specifies exactly:
-
$\Sigma$, $\Gamma={0,1}$, and the probability distribution $\pi$;
- the finite referee Hilbert space and every binary POVM ${R_x^0,R_x^1}$;
- the shared state and both players' POVMs for the proposed strategy; and
- exact values or rigorous certified intervals proving a strictly positive advantage.
An independent verifier should:
- check positivity, normalization, dimensions, and completeness of every state and POVM;
- compute the strategy's winning probability by exact or outward-rounded trace evaluation;
- recompute an exact or certified upper bound on $\omega(G)$—for finite data this can use deterministic-response enumeration and exact eigenvalue bounds, or an equivalent rigorous formulation; and
- confirm that the lower bound for $p(G,S)$ is strictly greater than the upper bound for $\omega(G)$.
The finite game and strategy are the answer. The solver's search process is not part of review.
Why this may lead to research output
A positive witness would locate quantum advantage at the smallest nontrivial answer alphabet and show that the growing family of binary no-advantage results is not universal. A universal negative theorem would instead prove that at least three answers are necessary. Monogamy-of-entanglement and extended nonlocal games also appear in quantum key distribution, relativistic bit commitment, and quantum position verification.
Current status and references
The binary-answer branch was audited on 28 July 2026 and assessed as likely open with medium confidence. The natural three-Pauli-basis binary game has no entangled advantage, and recent XOR/BB84 families provide additional no-advantage regimes, but no general binary impossibility theorem or positive witness was found.
- V. Russo, Extended Nonlocal Games.
- N. Johnston, R. Mittal, V. Russo, and J. Watrous, Extended non-local games and monogamy-of-entanglement games.
- L. Escolà-Farràs and F. Speelman, Lossy-and-Constrained Extended Non-Local Games with Applications to Quantum Cryptography.
- A. Broadbent and E. Culf, Rigidity for Monogamy-Of-Entanglement Games.
Released by
Kun Chen, Institute of Theoretical Physics, Chinese Academy of Sciences
Contact email
chenkun0228@gmail.com
Method
Other
Challenge issue
Background
A standard monogamy-of-entanglement game asks two noncommunicating players to predict the outcome of a quantum measurement made by a referee. A finite game
has a finite question set$\Sigma$ , a finite answer set $\Gamma$ , a question distribution $\pi$ , and referee POVMs
The same sampled question$x$ is sent to both players. They win exactly when both return the referee's measurement outcome $a$ .
Write$\omega(G)$ for the optimal unentangled value and $\omega^(G)$ for the optimal value when the players may share a finite-dimensional entangled strategy with the referee. Every two-question standard game satisfies $\omega(G)=\omega^(G)$. A four-question, three-answer separation is known. It remains open whether a separation is possible with the minimal nontrivial answer alphabet $|\Gamma|=2$ .
Research objective
Construct one fully explicit finite standard monogamy-of-entanglement game with
and a legal finite-dimensional entangled strategy$S$ such that
Determining the exact global entangled optimum $\omega^(G)$ is unnecessary because $\omega^(G)\ge p(G,S)$. Since two-question games cannot separate the values, any positive witness necessarily has at least three questions.
The construction must remain a standard monogamy-of-entanglement game. A lossy or constrained extension, an ordinary Bell game, a communication-assisted protocol, or a coarse-grained statistic that does not define a legal binary referee POVM does not satisfy the challenge.
Success and verification gate
A complete result specifies exactly:
An independent verifier should:
The finite game and strategy are the answer. The solver's search process is not part of review.
Why this may lead to research output
A positive witness would locate quantum advantage at the smallest nontrivial answer alphabet and show that the growing family of binary no-advantage results is not universal. A universal negative theorem would instead prove that at least three answers are necessary. Monogamy-of-entanglement and extended nonlocal games also appear in quantum key distribution, relativistic bit commitment, and quantum position verification.
Current status and references
The binary-answer branch was audited on 28 July 2026 and assessed as likely open with medium confidence. The natural three-Pauli-basis binary game has no entangled advantage, and recent XOR/BB84 families provide additional no-advantage regimes, but no general binary impossibility theorem or positive witness was found.