224 research outputs found
On determination of statistical properties of spectra from parametric level dynamics
We analyze an approach aiming at determining statistical properties of
spectra of time-periodic quantum chaotic system based on the parameter dynamics
of their quasienergies. In particular we show that application of the methods
of statistical physics, proposed previously in the literature, taking into
account appropriate integrals of motion of the parametric dynamics is fully
justified, even if the used integrals of motion do not determine the invariant
manifold in a unique way. The indetermination of the manifold is removed by
applying Dirac's theory of constrained Hamiltonian systems and imposing
appropriate primary, first-class constraints and a gauge transformation
generated by them in the standard way. The obtained results close the gap in
the whole reasoning aiming at understanding statistical properties of spectra
in terms of parametric dynamics.Comment: 9 pages without figure
Multipartite quantum correlations: symplectic and algebraic geometry approach
We review a geometric approach to classification and examination of quantum
correlations in composite systems. Since quantum information tasks are usually
achieved by manipulating spin and alike systems or, in general, systems with a
finite number of energy levels, classification problems are usually treated in
frames of linear algebra. We proposed to shift the attention to a geometric
description. Treating consistently quantum states as points of a projective
space rather than as vectors in a Hilbert space we were able to apply powerful
methods of differential, symplectic and algebraic geometry to attack the
problem of equivalence of states with respect to the strength of correlations,
or, in other words, to classify them from this point of view. Such
classifications are interpreted as identification of states with `the same
correlations properties' i.e. ones that can be used for the same information
purposes, or, from yet another point of view, states that can be mutually
transformed one to another by specific, experimentally accessible operations.
It is clear that the latter characterization answers the fundamental question
`what can be transformed into what \textit{via} available means?'. Exactly such
an interpretations, i.e, in terms of mutual transformability can be clearly
formulated in terms of actions of specific groups on the space of states and is
the starting point for the proposed methods.Comment: 29 pages, 9 figures, 2 tables, final form submitted to the journa
Concurrence of mixed bipartite quantum states in arbitrary dimensions
We derive a lower bound for the concurrence of mixed bipartite quantum
states, valid in arbitrary dimensions. As a corollary, a weaker, purely
algebraic estimate is found, which detects mixed entangled states with positive
partial transpose.Comment: accepted py PR
Separable approximation for mixed states of composite quantum systems
We describe a purely algebraic method for finding the best separable
approximation to a mixed state of a composite 2x2 quantum system, consisting of
a decomposition of the state into a linear combination of a mixed separable
part and a pure entangled one. We prove that, in a generic case, the weight of
the pure part in the decomposition equals the concurrence of the state.Comment: 13 pages, no figures; minor changes; accepted for publication in PR
Four-qubit entangled symmetric states with positive partial transpositions
We solve the open question of the existence of four-qubit entangled symmetric
states with positive partial transpositions (PPT states). We reach this goal
with two different approaches. First, we propose a
half-analytical-half-numerical method that allows to construct multipartite PPT
entangled symmetric states (PPTESS) from the qubit-qudit PPT entangled states.
Second, we adapt the algorithm allowing to search for extremal elements in the
convex set of bipartite PPT states [J. M. Leinaas, J. Myrheim, and E. Ovrum,
Phys. Rev. A 76, 034304 (2007)] to the multipartite scenario. With its aid we
search for extremal four-qubit PPTESS and show that generically they have ranks
(5,7,8). Finally, we provide an exhaustive characterization of these states
with respect to their separability properties.Comment: 5+4 pages, improved version, title slightly modifie
Symplectic geometry of entanglement
We present a description of entanglement in composite quantum systems in
terms of symplectic geometry. We provide a symplectic characterization of sets
of equally entangled states as orbits of group actions in the space of states.
In particular, using Kostant-Sternberg theorem, we show that separable states
form a unique Kaehler orbit, whereas orbits of entanglement states are
characterized by different degrees of degeneracy of the canonical symplectic
form on the complex projective space. The degree of degeneracy may be thus used
as a new geometric measure of entanglement and we show how to calculate it for
various multiparticle systems providing also simple criteria of separability.
The presented method is general and can be applied also under different
additional symmetry conditions stemming, eg. from the indistinguishability of
particles.Comment: LaTex, 31 pages, typos correcte
Global entangling properties of the coupled kicked tops
We study global entangling properties of the system of coupled kicked tops
testing various hypotheses and predictions concerning entanglement in quantum
chaotic systems. In order to analyze the averaged initial entanglement
production rate and the averaged asymptotic entanglement different ensembles of
initial product states are evolved. Two different ensembles with natural
probability distribution are considered: product states of independent
spin-coherent states and product states of arbitrary states. It appears that
the choice of either of these ensembles results in significantly different
averaged entanglement behavior. We investigate also a relation between the
averaged asymptotic entanglement and the mean entanglement of the eigenvectors
of an evolution operator. Lower bound on the averaged asymptotic entanglement
is derived, expressed in terms of the eigenvector entanglement.Comment: 11 pages, 7 figures, RevTe
Barycentric measure of quantum entanglement
Majorana representation of quantum states by a constellation of n 'stars'
(points on the sphere) can be used to describe any pure state of a simple
system of dimension n+1 or a permutation symmetric pure state of a composite
system consisting of n qubits. We analyze the variance of the distribution of
the stars, which can serve as a measure of the degree of non-coherence for
simple systems, or an entanglement measure for composed systems. Dynamics of
the Majorana points induced by a unitary dynamics of the pure state is
investigated.Comment: 11 pages, 13 figure
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