Entangled qutrits reach the quantum limit in a two-player hat game

A guessing game with a quantum twist

Alice and Bob each receive a hat whose color is chosen independently and uniformly from two possibilities. Each sees the other person’s hat, but not their own. They may agree on a strategy beforehand, then must act without further communication. Each can guess either color or pass.

The team wins if at least one person guesses and every guess is correct. Both passing counts as a loss, as does any incorrect guess.

With ordinary, classical strategies, their best winning probability is one-half. A simple optimal plan is for Alice always to guess a fixed color while Bob always passes. Seeing each other’s hats and coordinating beforehand cannot improve that classical ceiling.

In an arXiv preprint, Georgios Baziotis, Sang-Jun Park and Sang-Gyun Youn report the exact quantum ceiling: (15 + √33)/36, approximately 0.576238. Their work is a mathematical construction and proof, rather than a report of an experimental implementation. The supplied source identifies it as a preprint and gives no journal publication reference.

What the quantum strategy uses

The researchers describe the puzzle as a non-local game: separated players respond to information they receive, using shared resources but no communication during play. Each player has two possible inputs—the observed hat colors—and three possible outputs: either guess or pass.

Their winning strategy uses a pair of entangled qutrits. A qutrit is a quantum system with three basis states, rather than the two of a qubit. Entanglement supplies joint measurement correlations that the players use to coordinate their answers.

The shared state is non-maximally entangled: its three paired basis components do not all have equal weights. Depending on the hat color they see, Alice and Bob each choose a local measurement with three possible outcomes. Those outcomes determine whether they guess either color or pass.

The authors specify the state and measurements explicitly and calculate their success probability. This establishes an achievable result, not merely a numerical estimate of what might be possible. It also improves on an earlier strategy discussed in the paper that wins with probability 109/192, approximately 0.567708.

Proving that nothing does better

Finding a good strategy does not establish that it is optimal. The second part of the work therefore sets an upper bound covering every quantum strategy.

The proof handles an even broader mathematical framework called the quantum commuting model. Here, both players’ measurement operators act on a common space of arbitrary dimension, but operators belonging to different players must commute: their order of application does not matter.

The researchers construct an exact algebraic certificate showing that the winning probability cannot exceed their proposed value. Its logic resembles expressing a quantity as a sum of squares: the expression cannot be negative, so the desired bound follows.

Symmetry makes this calculation manageable. Certain exchanges of players and relabelings leave the game unchanged. These transformations form an eight-element dihedral group, denoted D8. Using representation theory—the mathematics of how symmetries act on vector spaces—the authors split a large matrix problem into smaller blocks. This reduces the independent real parameters from 325 to 52.

The resulting upper bound exactly matches the qutrit strategy. Thus ordinary quantum strategies and quantum commuting strategies have the same optimal success probability for this game.

A stronger hypothetical resource—and an open question

The paper also considers no-signalling correlations: joint response probabilities for which one player’s input cannot change the other player’s individual output distribution. This class contains quantum correlations but allows more general mathematical possibilities.

Such correlations can win the two-player game with probability two-thirds, strictly above the quantum limit. No-signalling alone therefore does not characterize what quantum mechanics permits.

For the version with n players, the authors prove a no-signalling upper bound of n/(n + 1). Known classical strategies already attain that bound when n = 2^m − 1, including three, seven and fifteen players, leaving no quantum advantage in those cases. The two-player solution does not settle every larger game: whether four players can gain a quantum advantage remains open in the paper.