  Non-Hermitian dynamics in quantum systems have unveiled novel phenomena, yet
the implementation of valid non-Hermitian quantum measurement remains a
challenge, because a universal quantum projective mechanism on the complete but
skewed non-Hermitian eigenstates is not explicit in experiment. This limitation
hinders the direct acquisition of non-Hermitian observable statistics (e.g.,
non-Hermitian population dynamics), also constrains investigations of
non-Hermitian quantum measurement properties such as 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 that is
generalized beyond the Hermitian ones. We then investigate the projective
properties of general quantum states onto complete non-Hermitian eigenvectors,
and present a quantum simulating 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 the 3-dimensional non-Hermitian quantum measurement on the
single-photon qutrit. Employing 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 observable in quantum domain.
