Experimental Solutions to the High-Dimensional Mean King's Problem

In 1987, Vaidman, Aharanov, and Albert put forward a puzzle called the Mean King's Problem (MKP) that can be solved only by harnessing quantum entanglement. Prime-powered solutions to the problem have been shown to exist, but they have not yet been experimentally realized for any dimension beyond two. We propose a general first-of-its-kind experimental scheme for solving the MKP in prime dimensions ($D$). Our search is guided by the digital discovery framework PyTheus, which finds highly interpretable graph-based representations of quantum optical experimental setups; using it, we find specific solutions and generalize to higher dimensions through human insight. As proof of principle, we present a detailed investigation of our solution for the three-, five-, and seven-dimensional cases. We obtain maximum success probabilities of $72.8 \%$, $45.8\%$, and $34.8 \%$, respectively. We, therefore, posit that our computer-inspired scheme yields solutions that exceed the classical probability ($1/D$) twofold, demonstrating its promise for experimental implementation.