248 research outputs found
Coherent states and the reconstruction of pure spin states
Coherent states provide an appealing method to reconstruct efficiently the pure state of a quantum mechanical spin s. A Stern-Gerlach apparatus is used to measure (4s + 1) expectations of projection operators on appropriate coherent states in the unknown state. These measurements are compatible with a finite number of states which can be distinguished, in the generic case, by measuring one more probability. In addition, the present technique shows that the zeros of a Husimi distribution do have an operational meaning: they can be identified directly by measurements with a Stem-Gerlach apparatus. This result comes down to saying that it is possible to resolve experimentally structures in quantum phase space which are smaller than (h) over bar
Reconstructing a pure state of a spin s through three Stern-Gerlach measurements: II
The density matrix of a spin s is fixed uniquely if the probabilities to obtain the value s upon measuring n.S are known for 4s(s+1) appropriately chosen directions n in space. These numbers are just the expectation values of the density operator in coherent spin states, and they can be determined in an experiment carried out with a Stern-Gerlach apparatus. Furthermore, the experimental data can be inverted providing thus a parametrization of the statistical operator by 4s(s+1) positive parameters
Discrete Moyal-type representations for a spin
In Moyal’s formulation of quantum mechanics, a quantum spin s is described in terms of continuous symbols, i.e., by smooth functions on a two-dimensional sphere. Such prescriptions to associate operators with Wigner functions, P or Q symbols, are conveniently expressed in terms of operator kernels satisfying the Stratonovich-Weyl postulates. In analogy to this approach, a discrete Moyal formalism is defined on the basis of a modified set of postulates. It is shown that appropriately modified postulates single out a well-defined set of kernels that give rise to discrete symbols. Now operators are represented by functions taking values on (2s+1)2 points of the sphere. The discrete symbols contain no redundant information, contrary to the continuous ones. The properties of the resulting discrete Moyal formalism for a quantum spin are worked out in detail and compared to the continuous formalism
Quantum State Tomography Using Successive Measurements
We describe a quantum state tomography scheme which is applicable to a system
described in a Hilbert space of arbitrary finite dimensionality and is
constructed from sequences of two measurements. The scheme consists of
measuring the various pairs of projectors onto two bases --which have no
mutually orthogonal vectors--, the two members of each pair being measured in
succession. We show that this scheme implies measuring the joint
quasi-probability of any pair of non-degenerate observables having the two
bases as their respective eigenbases. The model Hamiltonian underlying the
scheme makes use of two meters initially prepared in an arbitrary given quantum
state, following the ideas that were introduced by von Neumann in his theory of
measurement.Comment: 12 Page
Contracting the Wigner kernel of a spin to the Wigner kernel of a particle
A general relation between the Moyal formalisms for a spin and a particle is established. Once the formalism has been set up for a spin, the phase-space description of a particle is obtained from contracting the group of rotations to the oscillator group. In this process, turn into a spin Wigner kernel turns into the Wigner kernel of a particle. In fact, only one out of 22s different possible kernels for a spin shows this behavior
On the Convergence to Ergodic Behaviour of Quantum Wave Functions
We study the decrease of fluctuations of diagonal matrix elements of
observables and of Husimi densities of quantum mechanical wave functions around
their mean value upon approaching the semi-classical regime (). The model studied is a spin (SU(2)) one in a classically strongly chaotic
regime. We show that the fluctuations are Gaussian distributed, with a width
decreasing as the square root of Planck's constant. This is
consistent with Random Matrix Theory (RMT) predictions, and previous studies on
these fluctuations. We further study the width of the probability distribution
of -dependent fluctuations and compare it to the Gaussian Orthogonal
Ensemble (GOE) of RMT.Comment: 13 pages Latex, 5 figure
Inverse spin-s portrait and representation of qudit states by single probability vectors
Using the tomographic probability representation of qudit states and the
inverse spin-portrait method, we suggest a bijective map of the qudit density
operator onto a single probability distribution. Within the framework of the
approach proposed, any quantum spin-j state is associated with the
(2j+1)(4j+1)-dimensional probability vector whose components are labeled by
spin projections and points on the sphere. Such a vector has a clear physical
meaning and can be relatively easily measured. Quantum states form a convex
subset of the 2j(4j+3) simplex, with the boundary being illustrated for qubits
(j=1/2) and qutrits (j=1). A relation to the (2j+1)^2- and
(2j+1)(2j+2)-dimensional probability vectors is established in terms of spin-s
portraits. We also address an auxiliary problem of the optimum reconstruction
of qudit states, where the optimality implies a minimum relative error of the
density matrix due to the errors in measured probabilities.Comment: 23 pages, 4 figures, PDF LaTeX, submitted to the Journal of Russian
Laser Researc
Inverted spectroscopy and interferometry for quantum-state reconstruction of systems with SU(2) symmetry
We consider how the conventional spectroscopic and interferometric schemes
can be rearranged to serve for reconstructing quantum states of physical
systems possessing SU(2) symmetry. The discussed systems include a collection
of two-level atoms, a two-mode quantized radiation field with a fixed total
number of photons, and a single laser-cooled ion in a two-dimensional harmonic
trap with a fixed total number of vibrational quanta. In the proposed
rearrangement, the standard spectroscopic and interferometric experiments are
inverted. Usually one measures an unknown frequency or phase shift using a
system prepared in a known quantum state. Our aim is just the inverse one,
i.e., to use a well-calibrated apparatus with known transformation parameters
to measure unknown quantum states.Comment: 8 pages, REVTeX. More info on
http://www.ligo.caltech.edu/~cbrif/science.htm
Minimal Informationally Complete Measurements for Pure States
We consider measurements, described by a positive-operator-valued measure
(POVM), whose outcome probabilities determine an arbitrary pure state of a
D-dimensional quantum system. We call such a measurement a pure-state
informationally complete (PSI-complete) POVM. We show that a measurement with
2D-1 outcomes cannot be PSI-complete, and then we construct a POVM with 2D
outcomes that suffices, thus showing that a minimal PSI-complete POVM has 2D
outcomes. We also consider PSI-complete POVMs that have only rank-one POVM
elements and construct an example with 3D-2 outcomes, which is a generalization
of the tetrahedral measurement for a qubit. The question of the minimal number
of elements in a rank-one PSI-complete POVM is left open.Comment: 2 figures, submitted for the Asher Peres festschrif
Evidence for the Validity of the Berry-Robnik Surmise in a Periodically Pulsed Spin System
We study the statistical properties of the spectrum of a quantum dynamical
system whose classical counterpart has a mixed phase space structure consisting
of two regular regions separated by a chaotical one. We make use of a simple
symmetry of the system to separate the eigenstates of the time-evolution
operator into two classes in agreement with the Percival classification scheme
\cite{Per}. We then use a method firstly developed by Bohigas et. al.
\cite{BoUlTo} to evaluate the fractional measure of states belonging to the
regular class, and finally present the level spacings statistics for each class
which confirm the validity of the Berry-Robnik surmise in our model.Comment: 15 pages, 9 figures available upon request, Latex fil
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