Zbigniew Puchała (IITiS PAN)

• 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)

• [1] Ł. Pawela, Z. Puchała, "Quantum control with spectral constraints", preprint (2012). arXiv:1204.6557.

  Various constraints concerning control fields can be imposed in the realistic implementations of quantum control systems. One of the most important is the restriction on the frequency spectrum of acceptable control parameters. It is important to consider the limitations of experimental equipment when trying to find appropriate control parameters. Therefore, in this paper we present a general method of obtaining a piecewise-constant controls, which are robust with respect to spectral constraints. We consider here a Heisenberg spin chain, however the method can be applied to a system with more general interactions. To model experimental restrictions we apply an ideal low-pass filter to numerically obtained control pulses. The usage of the proposed method has negligible impact on the control quality as opposed to the standard approach, which does not take into account spectral limitations.

• [2] Z. Puchała, "Local controllability of quantum systems", Quantum Information Processing (2012). arXiv:1203.4056.

  We give a criterion that is sufficient for controllability of multipartite quantum systems. We generalize the graph infection criterion to the quantum systems that cannot be described with the use of a graph theory. We introduce the notation of hypergraphs and reformulate the infection property in this setting. The introduced criterion has a topological nature and therefore it is not connected to any particular experimental realization of quantum information processing.

• [3] Z. Puchała, J.A. Miszczak, P. Gawron, C.F. Dunkl, J.A. Holbrook, K. Życzkowski, "Restricted numerical shadow and geometry of quantum entanglement", preprint (2012). arXiv:1201.2524.

  The restricted numerical range W_R(A) of an operator A acting on a D-dimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset R of the of set of pure quantum states of dimension D. One considers for instance the set of real states, or in the case of composite spaces, the set of product states and the set of maximally entangled states. Combining the operator theory with a probabilistic approach we introduce the restricted numerical shadow of A -- a normalized probability distribution on the complex plane supported in W_R(A). Its value at point z in C is equal to the probability that the expectation value <\psi|A|\psi> is equal to z, where |\psi> represents a random quantum state in subset R distributed according to the natural measure on this set, induced by the unitarily invariant Fubini--Study measure. Studying restricted shadows of operators of a fixed size D=N_A N_B we analyse the geometry of sets of separable and maximally entangled states of the N_A x N_B composite quantum system. Investigating trajectories formed by evolving quantum states projected into the plane of the shadow we study the dynamics of quantum entanglement. A similar analysis extended for operators on D=2^3 dimensional Hilbert space allows us to investigate the structure of the orbits of GHZ and W quantum states of a three--qubit system.

• [4] B. Gardas, Z. Puchała, "Notes on the Riccati operator equation in open quantum systems", J. Math. Phys., Vol. 53 (2012): 012106. arXiv:1111.4380.

  Recent problem [B. Gardas, J. Math. Phys. {bf 52}, 042104 (2011)] concerning an antilinear solution of the Riccati equation is solved. We also exemplify that a simplification of the Riccati equation, even under reasonable assumptions, can lead to a not equivalent equation.

• [5] 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.

• [6] 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.

• [7] J.A. Miszczak, P. Gawron, Z. Puchała, "Qubit flip game on a Heisenberg spin chain", 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.

• Project funded by National Science Centre
• Number: N N514 513340
• Dates: 04.04.2011 - 03.04.2014