Jarosław Adam Miszczak (IITiS PAN)

• Zbigniew Puchała (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] Z. Puchała, J.A. Miszczak, "Probability measure generated by the superfidelity", J. Phys. A: Math. Theor., Vol. 44 (2011): 405301. arXiv:1107.2792.

  We study the probability measure on the space of density matrices induced by the metric defined by using superfidelity. We give the formula for the probability density of eigenvalues. We also study some statistical properties of the set of density matrices equipped with the introduced measure and provide a method for generating density matrices according to the introduced measure.

• [3] J.A. Miszczak, "Singular value decomposition and matrix reorderings in quantum information theory", J. Mod. Phys. C, Vol. 22 (2011): 897-918.

  We review Schmidt and Kraus decompositions in the form of singular value decomposition using operations of reshaping, vectorization and reshuffling. We use the introduced notation to analyze the correspondence between quantum states and operations with the help of Jamiołkowski isomorphism. The presented matrix reorderings allow us to obtain simple formulae for the composition of quantum channels and partial operations used in quantum information theory. To provide examples of the discussed operations, we utilize a package for the Mathematica computing system implementing basic functions used in the calculations related to quantum information theory.

• [4] 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.

• Project funded by Ministry of Science and Higher Education
• Number: IP2010 052270
• Dates: 24.05.2011 - 23.05.2012