A generalization of Griffiths's inequalities

Hilbert–Schmidt volume of the set of mixed quantum states

“…In particular, analogous results presented by us in [33] for the measure [26,34] related to the Bures distance [35,36] allow us to investigate similarities and differences between the geometry of mixed states induced by different metrics.…”

Bures volume of the set of mixed quantum states

“…Hence Bures metric is Fubini-Study adjusted, since for pure states it induces the same geometry. Several relevant papers on this subject are due to Uhlmann [6,8,10], who showed that the topological metric of Bures is Riemannian. Hübner provided a detailed analysis of the N = 2 [11] and N = 3 case [13], while Dittmann analyzed differential geometric properties of the Bures metric [12].…”

“…This fact provides an additional motivation to investigate other possible metrics and distances in the space of mixed quantum states. Nowadays it is widely accepted that the Bures metric [5,6], is distinguished by its rather special properties: it is a Riemannian, monotone metric, which is Fisher-adjusted [1]: in the subspace of diagonal matrices it induces the statistical distance [7]. Moreover, it is also Fubini-Study-adjusted, since it agrees with this metric at the space of pure states [8].…”

“…Now with ǫ a real symmetric positive matrix we want to calculate for real symmetric ρ: 6) with real vectors ψ 1 , ..., ψ N . For ρ > 0 we may rescale ψ → √ ρψ, then the volume element is ∝ det ρ N/2 and the δ-function ∝ det ρ −(N +1)/2 .…”

“…Moreover, it is also Fubini-Study-adjusted, since it agrees with this metric at the space of pure states [8]. The Bures distance between any two mixed states is a function of their fidelity [6,9] and various properties of this distance are a subject of considerable interest (see [8,10,11,12,13,14,15,16]). …”