127 research outputs found
Geometry of entanglement witnesses parameterized by SO(3) group
We characterize a set of positive maps in matrix algebra of 4x4 complex
matrices. Equivalently, we provide a subset of entanglement witnesses
parameterized by the rotation group SO(3). Interestingly, these maps/witnesses
define two intersecting convex cones in the 3-dimensional parameter space. The
existence of two cones is related to the topological structure of the
underlying orthogonal group. We perform detailed analysis of the corresponding
geometric structure.Comment: 10 page
Second quantized formulation of geometric phases
The level crossing problem and associated geometric terms are neatly
formulated by the second quantized formulation. This formulation exhibits a
hidden local gauge symmetry related to the arbitrariness of the phase choice of
the complete orthonormal basis set. By using this second quantized formulation,
which does not assume adiabatic approximation, a convenient exact formula for
the geometric terms including off-diagonal geometric terms is derived. The
analysis of geometric phases is then reduced to a simple diagonalization of the
Hamiltonian, and it is analyzed both in the operator and path integral
formulations. If one diagonalizes the geometric terms in the infinitesimal
neighborhood of level crossing, the geometric phases become trivial (and thus
no monopole singularity) for arbitrarily large but finite time interval .
The integrability of Schr\"{o}dinger equation and the appearance of the
seemingly non-integrable phases are thus consistent. The topological proof of
the Longuet-Higgins' phase-change rule, for example, fails in the practical
Born-Oppenheimer approximation where a large but finite ratio of two time
scales is involved and is identified with the period of the slower system.
The difference and similarity between the geometric phases associated with
level crossing and the exact topological object such as the Aharonov-Bohm phase
become clear in the present formulation. A crucial difference between the
quantum anomaly and the geometric phases is also noted.Comment: 22 pages, 3 figures. The analysis in the manuscript has been made
more precise by including a brief account of the hidden local gauge symmetry
and by adding several new equations. This revised version is to be published
in Phys. Rev.
Stochastic evolution of classical and quantum systems
We present a basic introduction to stochastic evolution of classical and quantum finite level systems. We discuss the properties of classical and quantum states and classical and quantum channels. Moreover, we provide the description of Markovian semigroups and discuss the structure of local in time master equations. A short discussion of non-Markovian dynamics is included as well
Optimal entanglement witnesses from generalized reduction and Robertson maps
We provide a generalization of the reduction and Robertson positive maps in
matrix algebras. They give rise to a new class of optimal entanglement
witnesses. Their structural physical approximation is analyzed. As a byproduct
we provide a new examples of PPT (Positive Partial Transpose) entangled states.Comment: 14 page
Quantum Mechanics of Damped Systems II. Damping and Parabolic Potential Barrier
We investigate the resonant states for the parabolic potential barrier known
also as inverted or reversed oscillator. They correspond to the poles of
meromorphic continuation of the resolvent operator to the complex energy plane.
As a byproduct we establish an interesting relation between parabolic cylinder
functions (representing energy eigenfunctions of our system) and a class of
Gel'fand distributions used in our recent paper.Comment: 14 page
On pseudo-stochastic matrices and pseudo-positive maps
Stochastic matrices and positive maps in matrix algebras proved to be very
important tools for analysing classical and quantum systems. In particular they
represent a natural set of transformations for classical and quantum states,
respectively. Here we introduce the notion of pseudo-stochastic matrices and
consider their semigroup property. Unlike stochastic matrices,
pseudo-stochastic matrices are permitted to have matrix elements which are
negative while respecting the requirement that the sum of the elements of each
column is one. They also allow for convex combinations, and carry a Lie group
structure which permits the introduction of Lie algebra generators. The quantum
analog of a pseudo-stochastic matrix exists and is called a pseudo-positive
map. They have the property of transforming a subset of quantum states
(characterized by maximal purity or minimal von Neumann entropy requirements)
into quantum states. Examples of qubit dynamics connected with "diamond" sets
of stochastic matrices and pseudo-positive maps are dealt with.Comment: 15 pages; revised versio
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