7 research outputs found
Orbits of quantum states and geometry of Bloch vectors for -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
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
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 -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 -dimensional unit vector, . As a
byproduct, an alternative and simple parameterization of Stiefel and Grassmann
manifolds is obtained.Comment: 21 page
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.