94 research outputs found
Symmetries, group actions, and entanglement
We address several problems concerning the geometry of the space of Hermitian
operators on a finite-dimensional Hilbert space, in particular the geometry of
the space of density states and canonical group actions on it. For quantum
composite systems we discuss and give examples of measures of entanglement.Comment: 21 page
Segre maps and entanglement for multipartite systems of indistinguishable particles
We elaborate the concept of entanglement for multipartite system with bosonic
and fermionic constituents and its generalization to systems with arbitrary
parastatistics. The entanglement is characterized in terms of generalized Segre
maps, supplementing thus an algebraic approach to the problem by a more
geometric point of view.Comment: 16 pages, the version to appear in J. Phys. A. arXiv admin note: text
overlap with arXiv:1012.075
Unitary quantum gates, perfect entanglers and unistochastic maps
Non-local properties of ensembles of quantum gates induced by the Haar
measure on the unitary group are investigated. We analyze the entropy of
entanglement of a unitary matrix U equal to the Shannon entropy of the vector
of singular values of the reshuffled matrix. Averaging the entropy over the
Haar measure on U(N^2) we find its asymptotic behaviour. For two--qubit quantum
gates we derive the induced probability distribution of the interaction content
and show that the relative volume of the set of perfect entanglers reads 8/3
\pi \approx 0.85. We establish explicit conditions under which a given
one-qubit bistochastic map is unistochastic, so it can be obtained by partial
trace over a one--qubit environment initially prepared in the maximally mixed
state.Comment: 14 pages including 6 figures in eps, version 4, title changed
according to a suggestion of the editor
On the relation between states and maps in infinite dimensions
Relations between states and maps, which are known for quantum systems in
finite-dimensional Hilbert spaces, are formulated rigorously in geometrical
terms with no use of coordinate (matrix) interpretation. In a tensor product
realization they are represented simply by a permutation of factors. This leads
to natural generalizations for infinite-dimensional Hilbert spaces and a simple
proof of a generalized Choi Theorem. The natural framework is based on spaces
of Hilbert-Schmidt operators and
the corresponding tensor products of
Hilbert spaces. It is proved that the corresponding isomorphisms cannot be
naturally extended to compact (or bounded) operators, nor reduced to the
trace-class operators. On the other hand, it is proven that there is a natural
continuous map
from trace-class operators on
(with the nuclear norm) into compact operators mapping the space of all bounded
operators on into trace class operators on
(with the operator-norm). Also in the infinite-dimensional context, the Schmidt
measure of entanglement and multipartite generalizations of state-maps
relations are considered in the paper.Comment: 19 page
Tensor Products of Random Unitary Matrices
Tensor products of M random unitary matrices of size N from the circular
unitary ensemble are investigated. We show that the spectral statistics of the
tensor product of random matrices becomes Poissonian if M=2, N become large or
M become large and N=2.Comment: 23 pages, 2 figure
Geometry of quantum dynamics in infinite dimension
We develop a geometric approach to quantum mechanics based on the concept of
the Tulczyjew triple. Our approach is genuinely infinite-dimensional and
including a Lagrangian formalism in which self-adjoint (Schroedinger) operators
are obtained as Lagrangian submanifolds associated with the Lagrangian. As a
byproduct we obtain also results concerning coadjoint orbits of the unitary
group in infinite dimension, embedding of the Hilbert projective space of pure
states in the unitary group, and an approach to self-adjoint extensions of
symmetric relations.Comment: 32 page
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