Heisenberg’s uncertainty relations employ commutators of observables to set
fundamental limits on quantum measurement. The information concerning
incompatibility (non-commutativity) of observables is well included but that
concerning correlation is missing. Schrödinger’s uncertainty relations remedy
this defect by supplementing the correlation in terms of anti-commutators.
However, both Heisenberg’s uncertainty relations and Schrödinger’s uncertainty
relations are expressed in terms of variances, which are not good measures
of uncertainty in general situations (e.g., when mixed states are involved ). By
virtue of the Wigner–Yanase skew information, we will establish an uncertainty
relation along the spirit of Schrödinger from a statistical inference perspective
and propose a conjecture. The result may be interpreted as a quantification of
certain aspect of the celebrated Wigner–Araki–Yanase theorem for quantum
measurement, which states that observables not commuting with a conserved
quantity cannot be measured exactly