Experimental Quantum-Inspired Advantage in Permutation Parity Identification using Spatial Modes of Light

A permutation of n particles is either even or odd according to the parity of the number of pairwise swaps required to realize it. Classically, deterministic identification of this parity requires n distinct local labels, one for each particle. A recently proposed quantum protocol [A. Diebra et al., Phys. Rev. Lett. 135, 260603 (2025)] reduces this requirement to a local dimension dmin = ⌈√n⌉ by exploiting multipartite entanglement, while retaining deterministic parity identification after a single application of the permutation. Here, we experimentally realize the minimal nontrivial case, n = 3 and d = 2, by mapping the parity-sensitive state structure onto a coherent superposition of Hermite–Gaussian spatial modes. A programmable multi-plane light converter implements all six elements of the permutation group S3, and a single projection measurement separates the even and odd permutation classes with a visibility of V ≃ 0.78. Our experiment reproduces, with structured classical light, the operational mechanism underlying this multipartite quantum-advantage protocol and establishes programmable spatial-mode processing as a versatile platform for exploring quantum protocols based on symmetry, interference, and reduced dimensional resources.