Geometry of mixed states for a q-bit and the quantum Fisher information tensor

TL;DR

This paper explores the geometric structure of mixed quantum states for a qubit, revealing how the quantum Fisher information metric relates to the manifold of states and introducing the Fisher Tensor with its symmetric and antisymmetric components.

Contribution

It provides a geometric interpretation of the quantum Fisher information for mixed states and introduces the Fisher Tensor, connecting it to symplectic and metric structures on state manifolds.

Findings

Quantum Fisher metric coincides with the SU(2) manifold metric.

Fisher Tensor's antisymmetric part relates to the Kostant Kirillov Souriau form.

Symmetric part of Fisher Tensor is represented by the Fubini-Study metric.

Abstract

After a review of the pure state case, we discuss from a geometrical point of view the meaning of the quantum Fisher metric in the case of mixed states for a two-level system, i.e. for a q-bit, by examining the structure of the fiber bundle of states, whose base space can be identified with a co-adjoint orbit of the unitary group. We show that the Fisher Information metric coincides with the one induced by the metric of the manifold of SU(2), i.e. the 3-dimensional sphere $S^3$, when the mixing coefficients are varied. We define the notion of Fisher Tensor and show that its anti-symmetric part is intrinsically related to the Kostant Kirillov Souriau symplectic form that is naturally defined on co-adjoint orbits, while the symmetric part is nontrivially again represented by the Fubini Study metric on the space of mixed states, weighted by the mixing coefficients.