111 research outputs found
Prime decomposition and correlation measure of finite quantum systems
Under the name prime decomposition (pd), a unique decomposition of an
arbitrary -dimensional density matrix into a sum of seperable density
matrices with dimensions given by the coprime factors of is introduced. For
a class of density matrices a complete tensor product factorization is
achieved. The construction is based on the Chinese Remainder Theorem and the
projective unitary representation of by the discrete Heisenberg group
. The pd isomorphism is unitarily implemented and it is shown to be
coassociative and to act on as comultiplication. Density matrices with
complete pd are interpreted as grouplike elements of . To quantify the
distance of from its pd a trace-norm correlation index is
introduced and its invariance groups are determined.Comment: 9 pages LaTeX. Revised version: changes in the terminology, updates
in ref
Phase-space-region operators and the Wigner function: Geometric constructions and tomography
Quasiprobability measures on a canonical phase space give rise through the action of Weyl's quantization map to operator-valued measures and, in particular, to region operators. Spectral properties, transformations, and general construction methods of such operators are investigated. Geometric trace-increasing maps of density operators are introduced for the construction of region operators associated with one-dimensional domains, as well as with two-dimensional shapes (segments, canonical polygons, lattices, etc.). Operational methods are developed that implement such maps in terms of unitary operations by introducing extensions of the original quantum system with ancillary spaces (qubits). Tomographic methods of reconstruction of the Wigner function based on the radon transform technique are derived by the construction methods for region operators. A Hamiltonian realization of the region operator associated with the radon transform is provided, together with physical interpretations
Group Theory and Quasiprobability Integrals of Wigner Functions
The integral of the Wigner function of a quantum mechanical system over a
region or its boundary in the classical phase plane, is called a
quasiprobability integral. Unlike a true probability integral, its value may
lie outside the interval [0,1]. It is characterized by a corresponding
selfadjoint operator, to be called a region or contour operator as appropriate,
which is determined by the characteristic function of that region or contour.
The spectral problem is studied for commuting families of region and contour
operators associated with concentric disks and circles of given radius a. Their
respective eigenvalues are determined as functions of a, in terms of the
Gauss-Laguerre polynomials. These polynomials provide a basis of vectors in
Hilbert space carrying the positive discrete series representations of the
algebra su(1,1)or so(2,1). The explicit relation between the spectra of
operators associated with disks and circles with proportional radii, is given
in terms of the dicrete variable Meixner polynomials.Comment: 11 pages, latex fil
Geometric phase in open systems
We calculate the geometric phase associated to the evolution of a system
subjected to decoherence through a quantum-jump approach. The method is general
and can be applied to many different physical systems. As examples, two main
source of decoherence are considered: dephasing and spontaneous decay. We show
that the geometric phase is completely insensitive to the former, i.e. it is
independent of the number of jumps determined by the dephasing operator.Comment: 4 pages, 2 figures, RevTe
Semiclassical dynamics of a spin-1/2 in an arbitrary magnetic field
The spin coherent state path integral describing the dynamics of a
spin-1/2-system in a magnetic field of arbitrary time-dependence is considered.
Defining the path integral as the limit of a Wiener regularized expression, the
semiclassical approximation leads to a continuous minimal action path with
jumps at the endpoints. The resulting semiclassical propagator is shown to
coincide with the exact quantum mechanical propagator. A non-linear
transformation of the angle variables allows for a determination of the
semiclassical path and the jumps without solving a boundary-value problem. The
semiclassical spin dynamics is thus readily amenable to numerical methods.Comment: 16 pages, submitted to Journal of Physics
An expectation value expansion of Hermitian operators in a discrete Hilbert space
We discuss a real-valued expansion of any Hermitian operator defined in a
Hilbert space of finite dimension N, where N is a prime number, or an integer
power of a prime. The expansion has a direct interpretation in terms of the
operator expectation values for a set of complementary bases. The expansion can
be said to be the complement of the discrete Wigner function.
We expect the expansion to be of use in quantum information applications
since qubits typically are represented by a discrete, and finite-dimensional
physical system of dimension N=2^p, where p is the number of qubits involved.
As a particular example we use the expansion to prove that an intermediate
measurement basis (a Breidbart basis) cannot be found if the Hilbert space
dimension is 3 or 4.Comment: A mild update. In particular, I. D. Ivanovic's earlier derivation of
the expansion is properly acknowledged. 16 pages, one PS figure, 1 table,
written in RevTe
Factorizations and Physical Representations
A Hilbert space in M dimensions is shown explicitly to accommodate
representations that reflect the prime numbers decomposition of M.
Representations that exhibit the factorization of M into two relatively prime
numbers: the kq representation (J. Zak, Phys. Today, {\bf 23} (2), 51 (1970)),
and related representations termed representations (together with
their conjugates) are analysed, as well as a representation that exhibits the
complete factorization of M. In this latter representation each quantum number
varies in a subspace that is associated with one of the prime numbers that make
up M
Linear canonical transformations and quantum phase:a unified canonical and algebraic approach
The algebra of generalized linear quantum canonical transformations is
examined in the prespective of Schwinger's unitary-canonical basis. Formulation
of the quantum phase problem within the theory of quantum canonical
transformations and in particular with the generalized quantum action-angle
phase space formalism is established and it is shown that the conceptual
foundation of the quantum phase problem lies within the algebraic properties of
the quantum canonical transformations in the quantum phase space. The
representations of the Wigner function in the generalized action-angle unitary
operator pair for certain Hamiltonian systems with the dynamical symmetry are
examined. This generalized canonical formalism is applied to the quantum
harmonic oscillator to examine the properties of the unitary quantum phase
operator as well as the action-angle Wigner function.Comment: 19 pages, no figure
Non-adiabatic geometrical quantum gates in semiconductor quantum dots
In this paper we study the implementation of non-adiabatic geometrical
quantum gates with in semiconductor quantum dots. Different quantum information
enconding/manipulation schemes exploiting excitonic degrees of freedom are
discussed. By means of the Aharanov-Anandan geometrical phase one can avoid the
limitations of adiabatic schemes relying on adiabatic Berry phase; fast
geometrical quantum gates can be in principle implementedComment: 5 Pages LaTeX, 10 Figures include
Geometric phases of mesoscopic spin in Bose-Einstein condensates
We propose a possible scheme for generating spin-J geometric phases using a
coupled two-mode Bose-Einstein condensate (BEC). First we show how to observe
the standard Berry phase using Raman coupling between two hyperfine states of
the BEC. We find that the presence of intrinsic interatomic collisions creates
degeneracy in energy that allows implementation of the non-Abelian geometric
phases as well. The evolutions produced can be used to produce interference
between different atomic species with high numbers of atoms or to fine control
the difference in atoms between the two species. Finally, we show that errors
in the standard Berry phase due to elastic collisions may be corrected by
controlling inelastic collisions between atoms.Comment: 6 pages, 2 figure
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