Bures measures over the spaces of two-and three-dimensional density matrices (original) (raw)

SU (4) Euler Angle Parameterization and Bipartite Density Matrices

2002

In quantum mechanics, sets of density matrices are important for numerous reasons. For example, their compact notation make them useful for describing decoherence and entanglement properties of multi-particle quantum systems. In particular, two two-state density matrices, otherwise known as two qubit density matrices, are important for their role in explaining quantum teleportation, dense coding, and computation theorems. The aim of this paper is to show an explicit parameterization for the Hilbert space of all two qubit density matrices. Such a parameterization would be extremely useful for numerical calculations concerning entanglement and other quantum information parameters. We would also like to know the properties of such parameterized two qubit density matrices; in particular, the representation of their convex sets, their subsets , and their set boundaries in terms of our parameterization. Here we present a generalized Euler angle parameterization for SU(4) and all possible two qubit density matrices as well as the corrected Haar Measures for SU(3) and SU(4) from this parameterization. The role of the parameterization in the Peres-Horodecki criteria will also be introduced as well as its usefulness in calculating entangled two qubit states.

Calculation of the Unitary part of the Bures Measure for N-level Quantum Systems

APS Meeting Abstracts, 2010

We use the canonical coset parametrization and provide a formula with the unitary part of the Bures measure for non-degenerate systems in terms of the product of even Euclidean balls. This formula is shown to be consistent with the sampling of random states through the generation of random unitary matrices.

Numerical shadow and geometry of quantum states

Journal of Physics A: Mathematical and Theoretical, 2011

The totality of normalised density matrices of order N forms a convex set Q N in R N 2 −1 . Working with the flat geometry induced by the Hilbert-Schmidt distance we consider images of orthogonal projections of Q N onto a two-plane and show that they are similar to the numerical ranges of matrices of order N . For a matrix A of a order N one defines its numerical shadow as a probability distribution supported on its numerical range W (A), induced by the unitarily invariant Fubini-Study measure on the complex projective manifold CP N −1 . We define generalized, mixed-states shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein.

Parametrizations of density matrices

Journal of Modern Optics, 2012

This article gives a brief overview of some recent progress in the characterization and parametrization of density matrices of finite dimensional systems. We discuss in some detail the Bloch-vector and Jarlskog parametrizations and mention briefly the coset parametrization. As applications of the Bloch parametrization we discuss the trace invariants for the case of time dependent Hamiltonians and in some detail the dynamics of three-level systems. Furthermore, the Bloch vector of two-qubit systems as well as the use of the polarization operator basis is indicated. As the main application of the Jarlskog parametrization we construct density matrices for composite systems. In addition, some recent related articles are mentioned without further discussion.

Bures volume of the set of mixed quantum states

Journal of Physics A: Mathematical and General, 2003

We compute the volume of the N 2 − 1 dimensional set MN of density matrices of size N with respect to the Bures measure and show that it is equal to that of a N 2 − 1 dimensional hyperhemisphere of radius 1/2. For N = 2 we obtain the volume of the Uhlmann hemisphere, 1 2 S 3 ⊂ R 4. We find also the area of the boundary of the set MN and obtain analogous results for the smaller set of all real density matrices. An explicit formula for the Bures-Hall normalization constants is derived for an arbitrary N .

Geometrical Statistics—Classical and Quantum

AIP Conference Proceedings, 2006

This is a review of the ideas behind the Fisher-Rao metric on classical probability distributions, and how they generalize to metrics on density matrices. As is well known, the unique Fisher-Rao metric then becomes a large family of monotone metrics. Finally I focus on the Bures-Uhlmann metric, and discuss a recent result that connects the geometric operator mean to a geodesic billiard on the set of density matrices.

The Hyperbolic Geometric Structure of the Density Matrix for Mixed State Qubits

2002

Density matrices for mixed state qubits, parametrized by the Bloch vector in the open unit ball of the Euclidean 3-space, are well known in quantum computation theory. We bring the seemingly structureless set of all these density matrices under the umbrella of gyrovector spaces, where the Bloch vector is treated as a hyperbolic vector, called a gyrovector. As such, this article catalizes and supports interdisciplinary research spreading from mathematical physics to algebra and geometry. Gyrovector spaces are mathematical objects that form the setting for the hyperbolic geometry of Bolyai and Lobachevski just as vector spaces form the setting for Euclidean geometry. It is thus interesting, in geometric quantum computation, to realize that the set of all qubit density matrices has rich structure with strong link to hyperbolic geometry. Concrete examples for the use of the gyrostructure to derive old and new, interesting identities for qubit density matrices are presented.

Invariant measures on multimode quantum Gaussian states

Journal of Mathematical Physics, 2012

We derive the invariant measure on the manifold of multimode quantum Gaussian states, induced by the Haar measure on the group of Gaussian unitary transformations. To this end, by introducing a bipartition of the system in two disjoint subsystems, we use a parameterization highlighting the role of nonlocal degrees of freedom -the symplectic eigenvalues -which characterize quantum entanglement across the given bipartition. A finite measure is then obtained by imposing a physically motivated energy constraint. By averaging over the local degrees of freedom we finally derive the invariant distribution of the symplectic eigenvalues in some cases of particular interest for applications in quantum optics and quantum information.

A study of two-qubit density matrices with fermionic purifications

Journal of Physics A-mathematical and Theoretical, 2008

We study 12 parameter families of two qubit density matrices, arising from a special class of two-fermion systems with four single particle states or alternatively from a four-qubit state with amplitudes arranged in an antisymmetric matrix. We calculate the Wooters concurrences and the negativities in a closed form and study their behavior. We use these results to show that the relevant entanglement measures satisfy the generalized Coffman-Kundu-Wootters formula of distributed entanglement. An explicit formula for the residual tangle is also given. The geometry of such density matrices is elaborated in some detail. In particular an explicit form for the Bures metric is given.