High-dimensional Mean King’s Problem

The Mean King's Problem, introduced by Vaidman, Aharonov, and Albert in 1987, asks whether hidden measurement information can be retrodicted from a quantum system. The present problem asks for a scalable photonic realization beyond qubits that still beats the best classical strategy.

Known results

• Vaidman, Aharonov, and Albert, 1987: introduced the Mean King's Problem.
• Prime-powered theoretical solutions are known.
• A two-dimensional experiment previously reported $95.6\%$ success, but higher-dimensional implementations had not been realized.

July 2023 photonic proposal

Jaouni, Gao, Arlt, Krenn, and Karimi proposed entangled-photon schemes for prime dimensions, with extensions to prime powers. For $D=3$, $D=5$, and $D=7$, the predicted maximum success probabilities are $72.8\%$, $45.8\%$, and $34.8\%$, respectively, exceeding the classical baseline $1/D$. The construction is claimed to generalize arbitrarily, but only proposed designs—not scalable experimental demonstrations—are given.

Current status (as of July 2023): Higher-dimensional theoretical constructions and a photonic proposal exist, but scalable experimental realization beyond qubits remains open.