We define the geometric measure of mixing of quantum state as a minimal Hilbert-Schmidt distance between the mixed state and a set of pure states.An explicit expression for the geometric measure is obtained.It is interesting that this expression corresponds to the squared Euclidian distance between the mixed state and the pure one in space of eigenvalues of the density matrix.As an example, geometric measure of mixing for spin-1/2 states is calculated.
We define the geometric measure of mixing of quantum state as a minimal Hilbert-Schmidt distance between the mixed state and a set of pure states.An explicit expression for the geometric measure is obtained.It is interesting that this expression corresponds to the squared Euclidian distance between the mixed state and the pure one in space of eigenvalues of the density matrix.As an example, geometric measure of mixing for spin-1/2 states is calculated.
The geometric measure of quantum entanglement of a pure state, defined by its distance to the set of pure separable states, is extended to multipartite mixed states. We characterize the nearest disentangled mixed state to a given mixed state with respect to the geometric measure by means of a system of equations. The entanglement eigenvalue for a mixed state is introduced. And we show that, for a given mixed state, its nearest disentangled mixed state is associated with its entanglement eigenvalue. Two numerical examples are used to demonstrate the effectiveness of the proposed method.
We investigate quantumness of spin-1 states, defined as the Hilbert-Schmidt distance to the convex hull of spin coherent states. We derive its analytic expression in the case of pure states as a function of the smallest eigenvalue of the Bloch matrix and give explicitly the closest classical state for an arbitrary pure state. Numerical evidence is given that the exact formula for pure states provides an upper bound on the quantumness of mixed states. Due to the connection between quantumness and entanglement we obtain new insights into the geometry of symmetric entangled 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.
This paper presents an introduction to geometric representations of quantum states in which each distinct quantum state, pure and mixed, corresponds to a unique point in a Euclidean space. Beginning with a review of some underappreciated properties of the most commonly used geometric representation, the Bloch sphere visualization of qubit states, we show how concepts, algorithms, and spatial relations viewable on this geometric representation can be extended to representations of qudit states of any finite quantum dimension d and on to the infinite-dimensional limit. A primary goal of the work is helping the reader develop a visual intuition of these spaces, which can complement the understanding of the algebraic formalism of quantum mechanics for learners, teachers, and researchers at any level. Particular emphasis is given both to understanding states in a basis-independent way and to understanding how probability amplitudes and density matrix elements used to algebraically represent states in a particular basis correspond to line segments and angles in the geometric representations. In addition to providing visualizations for such concepts as superpositions, mixtures, decoherence, and measurement, we demonstrate how the representations can be used to substitute simple geometrical calculations for more cumbersome linear algebra ones, which may be of particular use in introducing mixed states and density matrices to beginning quantum students at an early stage. The work concludes with the geometrical interpretation of some commonly used metrics such as the purity of states and their relation to real, Euclidean vectors in the infinite-dimensional limit of the space, which contains all lower-dimensional qudit spaces as subspaces.
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.