338 research outputs found
Tomography on f-oscillators
Symplectic tomographies of classical and quantum states are shortly reviewed.
The concept of nonlinear f-oscillators and their properties are recalled. The
tomographic probability representations of oscillator coherent states and the
problem of entanglement are then discussed. The entanglement of even and odd
f-coherent states is evaluated by the linear entropy
Tomographic map within the framework of star-product quantization
Tomograms introduced for the description of quantum states in terms of
probability distributions are shown to be related to a standard star-product
quantization with appropriate kernels. Examples of symplectic tomograms and
spin tomograms are presented.Comment: LATEX plus sprocl.sty, to appear in the Proceedings of the conference
``Quantum Theory and Symmetries'' (Krakow, July 2001), World Scietifi
Stochastic evolution of finite level systems: classical vs. quantum
Quantum dynamics of the density operator in the framework of a single
probability vector is analyzed. In this framework quantum states define a
proper convex quantum subset in an appropriate simplex. It is showed that the
corresponding dynamical map preserving a quantum subset needs not be stochastic
contrary to the classical evolution which preserves the entire simplex.
Therefore, violation of stochasticity witnesses quantumness of evolution.Comment: 8 page
Metric on the space of quantum states from relative entropy. Tomographic reconstruction
In the framework of quantum information geometry, we derive, from quantum
relative Tsallis entropy, a family of quantum metrics on the space of full
rank, N level quantum states, by means of a suitably defined coordinate free
differential calculus. The cases N = 2, N = 3 are discussed in detail and
notable limits are analyzed. The radial limit procedure has been used to
recover quantum metrics for lower rank states, such as pure states. By using
the tomographic picture of quantum mechanics we have obtained the Fisher- Rao
metric for the space of quantum tomograms and derived a reconstruction formula
of the quantum metric of density states out of the tomographic one. A new
inequality obtained for probabilities of three spin-1/2 projections in three
perpendicular directions is proposed to be checked in experiments with
superconducting circuits.Comment: 31 pages. No figures. Abstract and Introduction rewritten. Minor
corrections. References adde
Alternative structures and bi-Hamiltonian systems on a Hilbert space
We discuss transformations generated by dynamical quantum systems which are
bi-unitary, i.e. unitary with respect to a pair of Hermitian structures on an
infinite-dimensional complex Hilbert space. We introduce the notion of
Hermitian structures in generic relative position. We provide few necessary and
sufficient conditions for two Hermitian structures to be in generic relative
position to better illustrate the relevance of this notion. The group of
bi-unitary transformations is considered in both the generic and non-generic
case. Finally, we generalize the analysis to real Hilbert spaces and extend to
infinite dimensions results already available in the framework of
finite-dimensional linear bi-Hamiltonian systems.Comment: 11 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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