Orbits of quantum states and geometry of Bloch vectors for NN-level systems

Physical constraints such as positivity endow the set of quantum states with a rich geometry if the system dimension is greater than two. To shed some light on the complicated structure of the set of quantum states, we consider a stratification with strata given by unitary orbit manifolds, which can be identified with flag manifolds. The results are applied to study the geometry of the coherence vector for n-level quantum systems. It is shown that the unitary orbits can be naturally identified with spheres in R^{n^2-1} only for n=2. In higher dimensions the coherence vector only defines a non-surjective embedding into a closed ball. A detailed analysis of the three-level case is presented. Finally, a refined stratification in terms of symplectic orbits is considered.Comment: 15 pages LaTeX, 3 figures, reformatted, slightly modified version, corrected eq.(3), to appear in J. Physics

Finite-level systems, Hermitian operators, isometries, and a novel parameterization of Stiefel and Grassmann manifolds

In this paper we obtain a description of the Hermitian operators acting on the Hilbert space \C^n, description which gives a complete solution to the over parameterization problem. More precisely we provide an explicit parameterization of arbitrary $n$-dimensional operators, operators that may be considered either as Hamiltonians, or density matrices for finite-level quantum systems. It is shown that the spectral multiplicities are encoded in a flag unitary matrix obtained as an ordered product of special unitary matrices, each one generated by a complex $n-k$-dimensional unit vector, $k=0,1,...,n-2$. As a byproduct, an alternative and simple parameterization of Stiefel and Grassmann manifolds is obtained.Comment: 21 page

A Parametrization of Bipartite Systems Based on SU(4) Euler Angles

In this paper we give an explicit parametrization for all two qubit density matrices. This is important for calculations involving entanglement and many other types of quantum information processing. To accomplish this we present a generalized Euler angle parametrization for SU(4) and all possible two qubit density matrices. The important group-theoretical properties of such a description are then manifest. We thus obtain the correct Haar (Hurwitz) measure and volume element for SU(4) which follows from this parametrization. In addition, we study the role of this parametrization in the Peres-Horodecki criteria for separability and its corresponding usefulness in calculating entangled two qubit states as represented through the parametrization.Comment: 23 pages, no figures; changed title and abstract and rewrote certain areas in line with referee comments. To be published in J. Phys. A: Math. and Ge

Entropy as a function of Geometric Phase

We give a closed-form solution of von Neumann entropy as a function of geometric phase modulated by visibility and average distinguishability in Hilbert spaces of two and three dimensions. We show that the same type of dependence also exists in higher dimensions. We also outline a method for measuring both the entropy and the phase experimentally using a simple Mach-Zehnder type interferometer which explains physically why the two concepts are related.Comment: 19 pages, 7 figure

Universal Quantum Logic from Zeeman and Anisotropic Exchange Interactions

Some of the most promising proposals for scalable solid-state quantum computing, e.g., those using electron spins in quantum dots or donor electron or nuclear spins in Si, rely on a two-qubit quantum gate that is ideally generated by an isotropic exchange interaction. However, an anisotropic perturbation arising from spin-orbit coupling is inevitably present. Previous studies focused on removing the anisotropy. Here we introduce a new universal set of quantum logic gates that takes advantage of the anisotropic perturbation. The price is a constant but modest factor in additional pulses. The gain is a scheme that is compatible with the naturally available interactions in spin-based solid-state quantum computers.Comment: 5 pages, including 2 figures. This version to be published in Phys. Rev.