Properties of Squeezed-State Excitations

The photon distribution function of a discrete series of excitations of squeezed coherent states is given explicitly in terms of Hermite polynomials of two variables. The Wigner and the coherent-state quasiprobabilities are also presented in closed form through the Hermite polynomials and their limiting cases. Expectation values of photon numbers and their dispersion are calculated. Some three-dimensional plots of photon distributions for different squeezing parameters demonstrating oscillatory behaviour are given.Comment: Latex,35 pages,submitted to Quant.Semiclassical Op

How can we check the uncertainty relation?

The state-extended uncertainty relations generalizing the Robertson Schrodinger inequality are presented in the form appropriate for the experimental check by homodyne photon state detection. The method of qubit portrait of qudit states identified with the tomographic probability distributions is discussed to analyze the entanglement of two-mode field.Comment: 6 pages, 0 fugures, contribution to CEWQO in Madrid 2011, to appear in topical issue of Physica Scripta 201

Qubit portrait of the photon-number tomogram and separability of two-mode light states

In view of the photon-number tomograms of two-mode light states, using the qubit-portrait method for studying the probability distributions with infinite outputs, the separability and entanglement detection of the states are studied. Examples of entangled Gaussian state and Schr\"{o}dinger cat state are discussed.Comment: 20 pages, 6 figures, TeX file, to appear in Journal of Russian Laser Researc

Berry phase in magnetic systems with point perturbations

We study a two-dimensional charged particle interacting with a magnetic field, in general non-homogeneous, perpendicular to the plane, a confining potential, and a point interaction. If the latter moves adiabatically along a loop the state corresponding to an isolated eigenvalue acquires a Berry phase. We derive an expression for it and evaluate it in several examples such as a homogeneous field, a magnetic whisker, a particle confined at a ring or in quantum dots, a parabolic and a zero-range one. We also discuss the behavior of the lowest Landau level in this setting obtaining an explicit example of the Wilczek-Zee phase for an infinitely degenerated eigenvalue.Comment: LaTeX, 26 page

Deformed versus undeformed cat states encoding qubit

We study the possibility of exploiting superpositions of coherent states to encode qubit. A comparison between the use of deformed and undeformed bosonic algebra is made in connection with the amplitude damping errors.Comment: 6 pages, 2 eps figures, to appear in J. Opt.

Probability representation and quantumness tests for qudits and two-mode light states

Using tomographic-probability representation of spin states, quantum behavior of qudits is examined. For a general j-qudit state we propose an explicit formula of quantumness witnetness whose negative average value is incompatible with classical statistical model. Probability representations of quantum and classical (2j+1)-level systems are compared within the framework of quantumness tests. Trough employing Jordan-Schwinger map the method is extended to check quantumness of two-mode light states.Comment: 5 pages, 2 figures, PDFLaTeX, Contribution to the 11th International Conference on Squeezed States and Uncertainty Relations (ICSSUR'09), June 22-26, 2009, Olomouc, Czech Republi

MuSR method and tomographic probability representation of spin states

Muon spin rotation/relaxation/resonance (MuSR) technique for studying matter structures is considered by means of a recently introduced probability representation of quantum spin states. A relation between experimental MuSR histograms and muon spin tomograms is established. Time evolution of muonium, anomalous muonium, and a muonium-like system is studied in the tomographic representation. Entanglement phenomenon of a bipartite muon-electron system is investigated via tomographic analogues of Bell number and positive partial transpose (PPT) criterion. Reconstruction of the muon-electron spin state as well as the total spin tomography of composed system is discussed.Comment: 20 pages, 4 figures, LaTeX, submitted to Journal of Russian Laser Researc

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

Symmetric informationally complete positive operator valued measure and probability representation of quantum mechanics

Symmetric informationally complete positive operator valued measures (SIC-POVMs) are studied within the framework of the probability representation of quantum mechanics. A SIC-POVM is shown to be a special case of the probability representation. The problem of SIC-POVM existence is formulated in terms of symbols of operators associated with a star-product quantization scheme. We show that SIC-POVMs (if they do exist) must obey general rules of the star product, and, starting from this fact, we derive new relations on SIC-projectors. The case of qubits is considered in detail, in particular, the relation between the SIC probability representation and other probability representations is established, the connection with mutually unbiased bases is discussed, and comments to the Lie algebraic structure of SIC-POVMs are presented.Comment: 22 pages, 1 figure, LaTeX, partially presented at the Workshop "Nonlinearity and Coherence in Classical and Quantum Systems" held at the University "Federico II" in Naples, Italy on December 4, 2009 in honor of Prof. Margarita A. Man'ko in connection with her 70th birthday, minor misprints are corrected in the second versio

Unitary and Non-Unitary Matrices as a Source of Different Bases of Operators Acting on Hilbert Spaces

Columns of d^2 x N matrices are shown to create different sets of N operators acting on $d$-dimensional Hilbert space. This construction corresponds to a formalism of the star-product of operator symbols. The known bases are shown to be partial cases of generic formulas derived by using d^2 x N matrices as a source for constructing arbitrary bases. The known examples of the SIC-POVM, MUBs, and the phase-space description of qubit states are considered from the viewpoint of the developed unified approach. Star-product schemes are classified with respect to associated d^2 x N matrices. In particular, unitary matrices correspond to self-dual schemes. Such self-dual star-product schemes are shown to be determined by dequantizers which do not form POVM.Comment: 12 pages, 1 figure, 1 table, to appear in Journal of Russian Laser Researc