0 Zbigniew Pucha{\l}a Jaros{\l}aw Adam Miszczak Piotr Gawron Bart{\l}omiej Gardas 2010 Quantum Information Processing http://www.springerlink.com/content/hl0642v53379167k/ 0 Miszczak J. A. Pucha{\l}a Z. Horodecki P. Uhlmann A. {\.Z}yczkowski K. 2009 Quantum Information & Computation Rinton Press 9 0103-0130 01/2009 quantum fidelity quantum states Bures distance distances in state space We derive several bounds on fidelity between quantum states. In particular we show that fidelity is bounded from above by a simple to compute quantity we call super--fidelity. It is analogous to another quantity called sub--fidelity. For any two states of a two--dimensional quantum system ($N=2$) all three quantities coincide. We demonstrate that sub-- and super--fidelity are concave functions. We also show that super--fidelity is super--multiplicative while sub--fidelity is sub--multiplicative and design feasible schemes to measure these quantities in an experiment. Super--fidelity can be used to define a distance between quantum states. With respect to this metric the set of quantum states forms a part of a $N^2-1$ dimensional hypersphere. http://arxiv.org/abs/0805.2037 0 Pucha{\l}a Z. Miszczak, J. A. 2009 Physical Review A 79 024302 We provide a bound for the trace distance between two quantum states. The lower bound is based on the superfidelity, which provides the upper bound on quantum fidelity. One of the advantages of the presented bound is that it can be estimated using a simple measurement procedure. We also compare this bound with the one provided in terms of fidelity. http://arxiv.org/abs/0811.2323 0 Pucha{\l}a, Z. Rolski, T. 2008 Probability Theory and Related Fields 3-4 595-617 11/2008 Collision time Brownian motion Probability Theory Stochastic Processes http://www.springerlink.com/content/x43570761868m431/ 0 Markham, D. Miszczak, J. A. Pucha{\l}a, Z. {\.Z}yczkowski, K. 2008 Physical Review A 77 042111 Quantum Information Geometry of Quantum States Discrimination of Quantum States We analyze the problem of finding sets of quantum states that can be deterministically discriminated. From a geometric point of view, this problem is equivalent to that of embedding a simplex of points whose distances are maximal with respect to the Bures distance (or trace distance). We derive upper and lower bounds for the trace distance and for the fidelity between two quantum states, which imply bounds for the Bures distance between the unitary orbits of both states. We thus show that, when analyzing minimal and maximal distances between states of fixed spectra, it is sufficient to consider diagonal states only. Hence when optimal discrimination is considered, given freedom up to unitary orbits, it is sufficient to consider diagonal states. This is illustrated geometrically in terms of Weyl chambers. http://link.aps.org/doi/10.1103/PhysRevA.77.042111 0 G{\l}omb, P. Pucha{\l}a, Z. Sochan, A. 2007 Theoretical and Applied Informatics Image Processing Image Compression 0 Pucha{\l}a, Z. Rolski, T. 2005 Electronic Journal of Probability 10 Probab 1359-1380 11 continuous time random walk Brownian motion Collision time skew Young tableaux tandem queue Probability Theory Stochastic Processes Abstract In this note we consider the time of the collision $tau$ for $n$ independent copies of Markov processes $X^1_t,. . .,X^n_t$, each starting from $x_i$,where $x_1 <. . .< x_n$. We show that for the continuous time random walk $P_x(tau > t) = t^-n(n-1)/4(Ch(x)+o(1)),$ where $C$ is known and $h(x)$ is the Vandermonde determinant. From the proof one can see that the result also holds for $X_t$ being the Brownian motion or the Poisson process. An application to skew standard Young tableaux is given. http://www.math.washington.edu/~ejpecp/viewarticle.php?id=1551&layout=abstract 0 Pucha{\l}a, Z. 2005 Probability and Mathematical Statistics 25 129-132 Brownian motion Collision time Probability Theory Stochastic Processes A detail proof of Grabiner's theorem on the exact asymptotics of the time to collision for n independent Brownian motions is given. http://www.math.uni.wroc.pl/~pms/publications.php?nr=25.1