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Journal of Lie Theory

On the Riemannian Geometry of Finite Dimensional Mixed States

J. Dittmann

Published Jan 1, 1993• 0 citations
Publisher

Abstract

We consider the Riemannian geometry of the space of nonsingular density matrices D 1 equipped with the Bures metric g B .This space is of certain physical relevance on the background of a generalization of the Berry phase to mixed states.We determine the covariant derivative and the curvature tensor field related to the Levi-Cevita connection of (D 1 , g B ), which allow us to calculate other curvature quantities .It turns out that D 1 is not a space of constant curvature and not even a locally symmetric space, in contrast to what the case of two-dimensional density matrices might suggest.Moreover, we give a local description of D 1 and explicit formulae for the Bures metric in terms of natural matrix operations containing and d only.

DOI
10.5802/jolt.58

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We consider the Riemannian geometry of the space of nonsingular density matrices D 1 equipped with the Bures metric g B .This space is of certain physical relevance on the background of a generalization of the Berry phase to mixed states.We determine the covariant derivative and the curvature tensor field related to the Levi-Cevita connection of (D 1 , g B ), which allow us to calculate other curvature quantities .It turns out that D 1 is not a space of constant curvature and not even a locally symmetric space, in contrast to what the case of two-dimensional density matrices might suggest.Moreover, we give a local description of D 1 and explicit formulae for the Bures metric in terms of natural matrix operations containing and d only.
The aim of this paper is to derive explicit formulae for the computation of the Riemannian Bures metric g on the manifold of (finite-dimensional) nonsingular density matrices . This Riemannian metric introduced by Uhlmann generalizes the Fubini-Study metric to mixed states and is the infinitesimal version of the Bures distance. Several formulae are known for computing the Bures metric in low dimensions. The formulae presented in this paper allow for computing in finite dimensions without any diagonalization procedures. The first equations we give are, essentially, of the form , where is a matrix of invariants of . A further formula, , is adapted to the local orthogonal decomposition at generic points.
The Jordan product on the self-adjoint part of a finite-dimensional C * -algebra A is shown to give rise to Riemannian metric tensors on suitable manifolds of states on A , and the covariant derivative, the geodesics, the Riemann tensor, and the sectional curvature of all these metric tensors are explicitly computed. In particular, it is proved that the Fisher-Rao metric tensor is recovered in the Abelian case, that the Fubini-Study metric tensor is recovered when we consider pure states on the algebra B ( H ) of linear operators on a finite-dimensional Hilbert space H , and that the Bures-Helstrom metric tensors is recovered when we consider faithful states on B ( H ) . Moreover, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction associated to a state is presented. In the case of pure and faithful states on B ( H ) , this alternative geometrical description clarifies the analogy between the Fubini-Study and the Bures-Helstrom metric tensor.
The quantum analog of the Fisher information metric of a probability simplex is searched and several Riemannian metrics on the set of positive definite density matrices are studied. Some of them appeared in the literature in connection with Cramér–Rao-type inequalities or the generalization of the Berry phase to mixed states. They are shown to be stochastically monotone here. All stochastically monotone Riemannian metrics are characterized by means of operator monotone functions and it is proven that there exist a maximal and a minimal among them. A class of metrics can be extended to pure states and a constant multiple of the Fubini–Study metric appears in the extension.
We examine the geometric interpretation of the quantum geometric tensor proposed in [Phys. Rev. B {\bf 110}, 035404 (2024)] for mixed quantum states, focusing on its imaginary part, which is proportional to a weighted sum of the Berry curvatures of the eigenstates of the density operator. While the real part naturally decomposes into Fisher--Rao and weighted Fubini--Study contributions, we show that the imaginary part does not, in general, coincide with the curvature of a connection on a globally defined U(1) line bundle whose holonomy yields a mixed-state geometric phase. Using a two-level system as an explicit example, we demonstrate that its surface integral depends on the choice of surface bounded by the same closed path, with ambiguities that are not integer multiples of 2π2π2π.
The theory of monotone Riemannian metrics on the state space of a quantum system was established by Dénes Petz in 1996. In a recent paper he argued that the scalar curvature of a statistically relevant – monotone – metric can be interpreted as an average statistical uncertainty. The present paper contributes to this subject. It is reasonable to expect that states which are more mixed are less distinguishable than those which are less mixed. The manifestation of this behavior could be that for such a metric the scalar curvature has a maximum at the maximally mixed state. We show that not every monotone metric fulfils this expectation, some of them behave in a very different way. A mathematical condition is given for monotone Riemannian metrics to have a local minimum at the maximally mixed state and examples are given for such metrics. 1
The theory of monotone Riemannian metrics on the state space of a quantum system was established by Dénes Petz in 1996. In a recent paper he argued that the scalar curvature of a statistically relevant—monotone—metric can be interpreted as an average statistical uncertainty. The present paper contributes to this subject. It is reasonable to expect that states which are more mixed are less distinguishable than those which are less mixed. The manifestation of this behavior could be that for such a metric the scalar curvature has a maximum at the maximally mixed state. We show that not every monotone metric fulfils this expectation, some of them behave in a very different way. A mathematical condition is given for monotone Riemannian metrics to have a local minimum at the maximally mixed state and examples are given for such metrics.