Topological and geometrical properties of the set of mixed quantum states in the N -dimensional Hilbert space are analysed. Assuming that the corresponding classical dynamics takes place on the sphere we use the vector SU (2) coherent states and the generalized Husimi distributions to define the Monge distance between two arbitrary density matrices. The Monge metric has a simple semiclassical interpretation and induces a non-trivial geometry. Among all pure states the distance from the maximally mixed state ρ * , proportional to the identity matrix, admits the largest value for the coherent states, while the delocalized `chaotic' states are close to ρ * . This contrasts the geometry induced by the standard (trace, Hilbert-Schmidt or Bures) metrics, for which the distance from ρ * is the same for all pure states. We discuss possible physical consequences including unitary time evolution and the process of decoherence. We introduce also a simplified Monge metric, defined in the space of pure quantum states and more suitable for numerical computation.
Topological and geometrical properties of the set of mixed quantum states in the N -dimensional Hilbert space are analysed. Assuming that the corresponding classical dynamics takes place on the sphere we use the vector SU (2) coherent states and the generalized Husimi distributions to define the Monge distance between two arbitrary density matrices. The Monge metric has a simple semiclassical interpretation and induces a non-trivial geometry. Among all pure states the distance from the maximally mixed state ρ * , proportional to the identity matrix, admits the largest value for the coherent states, while the delocalized `chaotic' states are close to ρ * . This contrasts the geometry induced by the standard (trace, Hilbert-Schmidt or Bures) metrics, for which the distance from ρ * is the same for all pure states. We discuss possible physical consequences including unitary time evolution and the process of decoherence. We introduce also a simplified Monge metric, defined in the space of pure quantum states and more suitable for numerical computation.
We define a metric in the space of quantum states taking the Monge distance between corresponding Husimi distributions ( Q -functions). This quantity fulfils the axioms of a metric and satisfies the following semiclassical property : the distance between two coherent states is equal to the Euclidean distance between corresponding points in the classical phase space. We compute analytically distances between certain states (coherent, squeezed, Fock and thermal) and discuss a scheme for numerical computation of Monge distance for two arbitrary quantum states.
Abstract. We define a metric in the space of quantum states taking the Monge distance between corresponding Husimi distributions (Q-functions). This quantity fulfils the axioms of a metric and satisfies the following semiclassical property: the distance between two coherent states is equal to the Euclidean distance between corresponding points in the classical phase space. We compute analytically distances between certain states (coherent, squeezed, Fock and thermal) and discuss a scheme for numerical computation of Monge distance for two arbitrary quantum states. 1.
When studying the geometry of quantum states, it is acknowledged that mixed states can be distinguished by infinitely many metrics. Unfortunately, this freedom causes metric-dependent interpretations of physically significant geometric quantities such as the complexity and volume of quantum states. In this paper, we present an insightful discussion on the differences between the Bures and the Sjöqvist metrics inside a Bloch sphere. First, we begin with a formal comparative analysis between the two metrics by critically discussing three alternative interpretations for each metric. Second, we explicitly illustrate the distinct behaviors of the geodesic paths on each one of the two metric manifolds. Third, we compare the finite distances between an initial state and the final mixed state when calculated with the two metrics. Interestingly, in analogy with what happens when studying the topological aspects of real Euclidean spaces equipped with distinct metric functions (for instance, the usual Euclidean metric and the taxicab metric), we observe that the relative ranking based on the concept of a finite distance between mixed quantum states is not preserved when comparing distances determined with the Bures and the Sjöqvist metrics. Finally, we conclude with a brief discussion on the consequences of this violation of a metric-based relative ranking on the concept of the complexity and volume of mixed quantum states.
Any set of pure states living in a given Hilbert space possesses a natural and unique metric—the Haar measure—on the group U(N) of unitary matrices. However, there is no specific measure induced on the set of eigenvalues, Δ, of any density matrix ρ. Therefore, a general approach to the global properties of mixed states depends on the specific metric defined on Δ. In the present work we shall employ a simple measure on Δ that has the advantage of possessing a clear geometric visualization whenever discussing how arbitrary states are distributed according to some measure of mixedness. The degree of mixture will be that of the participation ratio R=1/Tr(ρ2) and the concomitant maximum eigenvalue, λm. The cases studied will be the qubit–qubit system and the qubit–qutrit system, whereas some discussion will be made on higher-dimensional bipartite cases in both the R-domain and the λm-domain.
We examine the geodesic between two mixed states of arbitrary dimension by means of their geometric mean operator. We utilize the fiber bundle approach by which the distance between two mixed state density operators ρ1 and ρ2 in the base space M is given by the shortest distance in the (Hilbert Schmidt) bundle space E of their purifications. The latter is well-known to be given by the Bures distance along the horizontal lift in E of the geodesic between the ρ1 and ρ2 in M. The horizontal lift is that unique curve in E that orthogonally traverses the fibers F⊂E above the curve in M, and projects down onto it. We briefly review this formalism and show how it can be used to construct the intermediate mixed quantum states ρ(s) along the base space geodesic parameterized by affine parameter s between the initial ρ1 and final ρ2 states. We emphasize the role played by geometric mean operator M(s)=ρ1−1/2ρ11/2ρ(s)ρ11/2ρ1−1/2, where the Uhlmann root fidelity between ρ1 and ρ(s) is given by F(ρ1,ρ(s))=Tr[M(s)ρ1]=Tr[ρ11/2ρ(s)ρ11/2], and ρ(s)=M(s)ρ1M(s). We give examples for the geodesic between the maximally mixed state and a pure state in arbitrary dimensions, as well as for the geodesic between Werner states ρ(p)=(1−p)I/N+p∣Ψ⟩⟨Ψ∣ with ∣Ψ⟩={∣GHZ⟩,∣W⟩} in dimension N=23. For the latter, we compare expressions in the limit p→1 to the infinite number of possible geodesics between the orthogonal pure states ∣GHZ⟩ and ∣W⟩. Lastly, we compute the analytic form for the density matrices along the geodesic that connects two arbitrary endpoint qubit density matrices within the Bloch ball for dimension N=2.
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 , i.e. the three-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 non-trivially again represented by the Fubini–Study metric on the space of mixed states, weighted by the mixing coefficients.