Direct non-Hermitian measurement and uncertainty relation for the high-dimensional quantum domain
Phys. Rev. Applied 24, 024006 – Published 4 August, 2025
DOI: https://doi.org/10.1103/d7bd-qtpw
Abstract
Non-Hermitian dynamics in quantum systems have unveiled novel phenomena, yet the implementation of valid non-Hermitian quantum measurement remains a challenge, because a feasible quantum projective mechanism on the complete but skewed non-Hermitian eigenstates is not explicit in experiments. This limitation hinders the direct acquisition of non-Hermitian observable statistics (e.g., non-Hermitian population dynamics), and also constrains investigations of non-Hermitian quantum measurement properties such as the uncertainty relation. Here we address these challenges by presenting a non-Hermitian projective protocol and investigating the non-Hermitian uncertainty relation. We derive the uncertainty relation for pseudo-Hermitian (PH) observables under the specific non-Hermitian metric. We then investigate the projective measurement properties for general quantum states which are projected onto the complete non-Hermitian eigenvectors, and present a quantum simulation method to apply the valid non-Hermitian projective measurement on a direct-sum dilated space. Subsequently, we experimentally construct a quantum simulator in the quantum optical circuit and realize a proof-of-principle experiment involving three-dimensional non-Hermitian quantum measurement on single-photon qutrits. Using this platform, we explore the uncertainty relation experimentally with different PH metrics. Our non-Hermitian quantum measurement method is state independent and outputs directly the non-Hermitian quantum projective statistics, paving the way for studies of extensive non-Hermitian observables in the quantum domain.
