Jarosław Adam Miszczak (IITiS PAN)

• Ryszard Winiarczyk (IITiS PAN)
• Jan Sładkowski (University of Silesia)
• Piotr Gawron (IITiS PAN)
• Jerzy Klamka (IITiS PAN and Silesian University of Technology)
• Zbigniew Puchała (IITiS PAN)

• 27/10/2011, University of Silesia

• [1] P. Zawadzki, Z. Puchała, J.A. Miszczak, "Increasing the security of the ping-pong protocol by using mutually unbiased bases", preprint (2012). arXiv:1201.3230.

  In this paper we propose an extended version of the ping-pong protocol and study its security. The proposed protocol incorporates the usage of mutually unbiased bases in the control mode. We show that, by increasing the number of bases, it is possible to improve the security of this protocol. We also provide the upper bounds on eavesdropping average non-detection probability and propose a control mode modification that increases the attack detection probability.

• [2] 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.

• [3] Z. Puchała, J.A. Miszczak, "Symbolic integration with respect to the Haar measure on the unitary group in Mathematica", preprint (2011). arXiv:1109.4244.

  We present IntU package for Mathematica computer algebra system. The presented package performs a symbolic integration of polynomial functions over the unitary group with respect to unique normalized Haar measure. We describe a number of special cases which can be used to optimize the calculation speed for some classes of integrals. We also provide some examples of usage of the presented package.

• [4] 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.

• [5] J.A. Miszczak, P. Gawron, Z. Puchała, "Qubit flip game on a Heisenberg spin chain", Published on-line in Quantum Information Processing (2011). arXiv:1108.0642.

  We study a quantum version of a penny flip game played using control parameters of the Hamiltonian in the Heisenberg model. Moreover, we extend this game by introducing auxiliary spins which can be used to alter the behaviour of the system. We show that a player aware of the complex structure of the system used to implement the game can use this knowledge to gain higher mean payoff.

• [6] 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.

• [7] 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.

• Project funded by National Science Centre
• Number: N N516 475440
• Dates: 20.04.2011-19.04.2013