Fisher Information of Wavefunctions: Classical and Quantum

Abstract

A parametric quantum mechanical wavefunction naturallyinduces parametric probability distributions by taking absolute square, and we canconsider its classical Fisher information. On the other hand, it also inducesparametric rank-one projections which may be viewed as density operators, and wecan talk about its quantum Fisher information. Among many versions of quantum Fisherinformation, there are two prominent ones. The first, defined via a quantum scorefunction, was introduced by Helstrom in 1967 and is well known. The second, definedvia the square root of the density operator, has its origin in the skew informationintroduced by Wigner and Yanase in 1963 and remains relatively unnoticed. This studyis devoted to investigating the relationships between the classical Fisherinformation and these two versions of quantum Fisher information for wavefunctions.It is shown that the two versions of quantum Fisher information differ by a factor 2and that they dominate the classical Fisher information. The non-coincidence ofthese two versions of quantum Fisher information may be interpreted as amanifestation of quantum discord. We further calculate the difference between theHelstrom quantum Fisher information and the classical Fisher information, and showthat it is precisely the instantaneous phase fluctuation of the wavefunctions.