Ryszard Winiarczyk (IITiS PAN)

• Jan Sładkowski (University of Silesia)
• Piotr Gawron (IITiS PAN)
• Radosław Grzymkowski (IITiS PAN and Silesian University of Technology)
• Jerzy Klamka (IITiS PAN and Silesian University of Technology)
• Jarosław Adam Miszczak (IITiS PAN)
• Zbigniew Puchała (IITiS PAN)

• 14/01/2011 - Katowice, Institute of Physics
• 27/01/2011 - Gliwice, IITiS PAN

• [1] J.A. Miszczak, "Generating and using truly random quantum states in Mathematica", Computer Physics Communications, Vol. 183 (2012): 118-124.

  The problem of generating random quantum states is of a great interest from the quantum information theory point of view. In this paper we present a package for Mathematica computing system harnessing a specific piece of hardware, namely Quantis quantum random number generator (QRNG), for investigating statistical properties of quantum states. The described package implements a number of functions for generating random states, which use Quantis QRNG as a source of randomness. It also provides procedures which can be used in simulations not related directly to quantum information processing.

• [2] P. Gawron, P. Głomb, J.A. Miszczak, Z. Puchała, "Eigengestures for natural human computer interface", Man-Machine Interactions 2 / Proceedings of the International Conference on Man-Machine Interactions ICMMI 2011, Vol. 103 (2011): 49-56. arXiv:1105.1293.

  We present the application of Principal Component Analysis for data acquired during the design of a natural gesture interface. We investigate the concept of an eigengesture for motion capture hand gesture data and present the visualisation of principal components obtained in the course of conducted experiments. We also show the influence of dimensionality reduction on reconstructed gesture data quality.

• [3] C.F. Dunkl, P. Gawron, J.A. Holbrook, J.A. Miszczak, Z. Puchała, K. Życzkowski, "Numerical shadow and geometry of quantum states", J. Phys. A: Math. Theor., Vol. 44 (2011): 335301. arXiv:1104.2760.

  The totality of normalised density matrices of order N forms a convex set Q_N in R^(N^2-1). Working with the flat geometry induced by the Hilbert-Schmidt distance we consider images of orthogonal projections of Q_N onto a two-plane and show that they are similar to the numerical ranges of matrices of order N. For a matrix A of a order N one defines its numerical shadow as a probability distribution supported on its numerical range W(A), induced by the unitarily invariant Fubini-Study measure on the complex projective manifold CP^(N-1). We define generalized, mixed-states shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein.

• [4] B. Gardas, Z. Puchała, "Stationary states of two-level open quantum systems", J. Phys. A: Math. Theor., Vol. 44 (2011): 215306. arXiv:1006.3328.

  A problem of finding stationary states of open quantum systems is addressed. We focus our attention on a generic type of open system: a qubit coupled to its environment. We apply the theory of block operator matrices and find stationary states of two-level open quantum systems under certain conditions applied on both the qubit and the surrounding.

• [5] C.F. Dunkl, P. Gawron, J.A. Holbrook, Z. Puchała, K. Życzkowski, "Numerical shadows: measures and densities on the numerical range", Linear Algebra Appl., Vol. 434 (2011): 2042-2080. arXiv:1010.4189.

  For any operator M acting on an N-dimensional Hilbert space H_N we introduce its numerical shadow, which is a probability measure on the complex plane supported by the numerical range of M. The shadow of M at point z is defined as the probability that the inner product (Mu,u) is equal to z, where u stands for a random complex vector from H_N, satisfying ||u||=1. In the case of N=2 the numerical shadow of a non-normal operator can be interpreted as a shadow of a hollow sphere projected on a plane. A similar interpretation is provided also for higher dimensions. For a hermitian M its numerical shadow forms a probability distribution on the real axis which is shown to be a one dimensional B-spline. In the case of a normal M the numerical shadow corresponds to a shadow of a transparent solid simplex in R^{N-1} onto the complex plane. Numerical shadow is found explicitly for Jordan matrices J_N, direct sums of matrices and in all cases where the shadow is rotation invariant. Results concerning the moments of shadow measures play an important role. A general technique to study numerical shadow via the Cartesian decomposition is described, and a link of the numerical shadow of an operator to its higher-rank numerical range is emphasized.

• [6] Miszczak, J.A., "Models of quantum computation and quantum programming languages", Bulletin of the Polish Academy of Sciences - Technical Sciences (2011).

  The goal of this report is to provide an introduction to the basic computational models used in quantum information theory. We various review models of quantum Turing machine, quantum circuits and quantum random access machine (QRAM) along with their classical counterparts. We also provide an introduction to quantum programming languages, which are developed using the QRAM model. We review the syntax of several existing quantum programming languages and discuss their features and limitations.

B. Gardas, Exact reduced dynamics for a qubit in a precessing magnetic field and in the contact with a heat-bath, Phys. Rev. A 82, 042115 (2010)

• Project funded by Ministry of Science and Higher Education
• Number: N N519 442339
• Dates: 25.08.2010 - 24.08.2013