Characterization of mixed quantum states represented by density operator is one of the most important task in quantum information processing. In this work we will present a geometric approach to characterize the density operator in terms of fiber bundle over a quantum phase space. The geometrical structure of the quantum phase space of an isospectral mixed quantum states can be realized as a co-adjoint orbit of a Lie group equipped with a specific Kähler structure. In particular we will briefly discuss the construction of a fiber bundle over the quantum phase space based on symplectic reduction and purification method. We will also show that the map is a Riemannian submersion which enable us to provide some applications of the geometric framework such as geometric phase and quantum speed limit.
Characterization of mixed quantum states represented by density operator is one of the most important task in quantum information processing. In this work we will present a geometric approach to characterize the density operator in terms of fiber bundle over a quantum phase space. The geometrical structure of the quantum phase space of an isospectral mixed quantum states can be realized as a co-adjoint orbit of a Lie group equipped with a specific Kähler structure. In particular we will briefly discuss the construction of a fiber bundle over the quantum phase space based on symplectic reduction and purification method. We will also show that the map is a Riemannian submersion which enable us to provide some applications of the geometric framework such as geometric phase and quantum speed limit.
Abstract. The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this work we discuss a geometric formulation of mixed quantum states represented by density operators. Our formulation is based on principal fiber bundles and purifications of quantum states. In our construction, the Riemannian metric and symplectic form on the total space are induced from the real and imaginary parts of the Hilbert-Schmidt Hermitian inner product, and we define a mechanical connection in terms of a locked inertia tensor and moment map. We also discuss some applications of our geometric framework. 1.
The geometric formulation of quantum mechanics is a very interesting field of research which has many applications in the emerging field of quantum computation and quantum information, such as schemes for optimal quantum computers. In this work we discuss a geometric formulation of mixed quantum states represented by density operators. Our formulation is based on principal fiber bundles and purifications of quantum states. In our construction, the Riemannian metric and symplectic form on the total space are induced from the real and imaginary parts of the Hilbert–Schmidt Hermitian inner product, and we define a mechanical connection in terms of a locked inertia tensor and moment map. We also discuss some applications of our geometric framework.
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.
In this paper we introduce a geometric framework for mixed quantum states based on a Kähler structure. The geometric framework includes a symplectic form, an almost complex structure, and a Riemannian metric that characterize the space of mixed quantum states. We argue that the almost complex structure is integrable. We also in detail discuss a visualizing application of this geometric framework by deriving a geometric uncertainty relation for mixed quantum states. The framework is computationally effective and it provides us with a better understanding of general quantum mechanical systems.
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, which can also be identified with a co-adjoint orbit of the unitary group. We show that the full quantum Fisher tensor has a symmetric part that coincides with the metric of the manifold of SU(2), i.e. the 3-dimensional sphere S3, and an anti-symmetric part that is intrinsically related to the KKS-symplectic form that is naturally defined on co-adjoint orbits.
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.