A Characterisation of Wigner–yanase Skew Information Among Statistically Monotone Metrics

Abstract

Let Mn = Mn(C) be the space of n × n complex matrices endowed with the HilbertSchmidt scalar product, let Sn be the unit sphere of Mn and let Dn ⊂ Mn be the space of strictly positive density matrices. We show that the scalar product over Dn introduced by Gibilisco and Isola 3 (that is the scalar product induced by the map Dn ρ → √ ρ ∈ Sn)coincides with the Wigner-Yanase monotone metric.

Wigner–Yanase information on quantum state space: The geometric approach

“…Indeed Wigner-Yanase information appears as the pull-back of the square root map. 18 Next we prove the formula for the scalar curvature. One proof, due to Dittmann, uses the general formula 13 and requires a long calculation.…”

“…Despite the existence of general results for the theory 13,17,19,26,27,28,30,40,41 a number of open problems resists investigation. For example, it does not exist yet a general formula for the geodesic path and the geodesic distance associated to an arbitrary monotone metric.…”

Inequalities for quantum Fisher information

“…Several quantum versions of Fisher information have been studied. Among the first examples one has the Wigner-Yanase information (see [24] or [6], [7], [8], [9] for a recent treatment) and the SLD-information (see [1], [23], [13]) that are defined as follows. As usual [·, ·] denotes the commutator.…”

“…For the first equality see [12] or [6], [11]. For the second equality remember that f SLD (x) := 1+x 2 .…”

Direct approach to quantum extensions of Fisher information

“…Some related investigations along this line are Refs. [22][23][24][25][26][27][28][29][30][31][32]. The present paper is devoted to this purpose by entirely intuitive and elementary methods.…”