Random Bures mixed states and the distribution of their purity (2009)
Osipov, Vladimir Al., Sommers, Hans-Juergen, Zyczkowski, Karol
Ensembles of random density matrices determined by various probability measures are analysed. A simple and efficient algorithm to generate at random density matrices distributed according to the...
Volume of the set of unistochastic matrices of order 3 and the mean Jarlskog invariant (2009)
Dunkl, Charles, Zyczkowski, Karol
A bistochastic matrix B of size N is called unistochastic if there exists a unitary U such that B_ij=|U_{ij}|^{2} for i,j=1,...,N. The set U_3 of all unistochastic matrices of order N=3 forms a...
Citation graph, weighted impact factors and performance indices (2009)
A scheme of evaluating an impact of a given scientific paper based on importance of papers quoting it is investigated. Introducing a weight of a given citation, dependent on the previous scientific...
Positive maps, positive polynomials and entanglement witnesses (2009)
Skowronek, Lukasz, Zyczkowski, Karol
We link the study of positive quantum maps, block positive operators, and entanglement witnesses with problems related to multivariate polynomials. For instance, we show how indecomposable block...
Cones of positive maps and their duality relations (2009)
Skowronek, Lukasz, Stormer, Erling, Zyczkowski, Karol
The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive...
Protected Subspaces in Quantum Information (2009)
Majgier, Krzysztof, Maassen, Hans, Zyczkowski, Karol
In certain situations the state of a quantum system, after transmission through a quantum channel, can be perfectly restored. This can be done by 'coding' the state space of the system before...
Entropy of a quantum error correction code (2008)
Kribs, David W., Pasieka, Aron, Zyczkowski, Karol
We define and investigate a notion of entropy for quantum error correcting codes. The entropy of a code for a given quantum channel has a number of equivalent realisations, such as through the...
Quartic quantum theory: an extension of the standard quantum mechanics (2008)
We propose an extended quantum theory, in which the number K of parameters necessary to characterize a quantum state behaves as fourth power of the number N of distinguishable states. As the simplex...
An Introduction to Quantum Entanglement (2008)
Ingemar Bengtsson, Karol Zyczkowski
[1] S. Abe. A note on the q-deformation-theoretic aspect of the
Mean Dynamical Entropy of Quantum System Tends to Infinity in the Semiclassical Limit (2007)
Wojciech Slomczynski, Karol Zyczkowski, Instytut Matematyki
We show that the mean dynamical entropy of a quantum map on the sphere tends logarithmically to infinity in the semiclassical limit. Consequences of this fact for classical dynamical systems are...
Szarek, Stanislaw J., Werner, Elisabeth, Zyczkowski, Karol
We investigate the set a) of positive, trace preserving maps acting on density matrices of size N, and a sequence of its nested subsets: the sets of maps which are b) decomposable, c) completely...
Composition of quantum states and dynamical subadditivity (2007)
Roga, Wojciech, Fannes, Mark, Zyczkowski, Karol
We introduce a composition of quantum states of a bipartite system which is based on the reshuffling of density matrices. This non-Abelian product is associative and stems from the composition of...
Defect of a unitary matrix (2007)
Tadej, Wojciech, Zyczkowski, Karol
We analyze properties of a map B = f(U) sending a unitary matrix U of size N into a doubly stochastic matrix defined by B_{i,j} = |U_{i,j}|^2. For any U we define its DEFECT, determined by the...
From a toy model to the double square root voting system (2007)
Slomczynski, Wojciech, Zyczkowski, Karol
We investigate systems of indirect voting based on the law of Penrose, in which each representative in the voting body receives the number of votes (voting weight) proportional to the square root of...
Subnormalized states and trace-nonincreasing maps (2007)
Cappellini, Valerio, Sommers, Hans-Juergen, Zyczkowski, Karol
We investigate the set of completely positive, trace-nonincreasing linear maps acting on the set M_N of mixed quantum states of size N. Extremal point of this set of maps are characterized and its...
Penrose voting system and optimal quota (2006)
Slomczynski, Wojciech, Zyczkowski, Karol
Systems of indirect voting based on the principle of qualified majority can be analysed using the methods of game theory. In particular, this applies to the voting system in the Council of the...
Mubs and Hadamards of Order Six (2006)
Bengtsson, Ingemar, Bruzda, Wojciech, Ericsson, Asa, Larsson, Jan-Ake, Tadej, Wojciech, Zyczkowski, Karol
We report on a search for mutually unbiased bases (MUBs) in 6 dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the...
Higher-Rank Numerical Ranges of Unitary and Normal Matrices (2006)
Choi, Man-Duen, Holbrook, John A., Kribs, David W., Zyczkowski, Karol
We verify a conjecture on the structure of higher-rank numerical ranges for a wide class of unitary and normal matrices. Using analytic and geometric techniques, we show precisely how the higher-rank...
An Introduction to Quantum Entanglement: a Geometric Approach (2006)
Zyczkowski, Karol, Bengtsson, Ingemar
We present a concise introduction to quantum entanglement. Concentrating on bipartite systems we review the separability criteria and measures of entanglement. We focus our attention on geometry of...
Distribution of G-concurrence of random pure states (2006)
Cappellini, Valerio, Sommers, Hans-Juergen, Zyczkowski, Karol
Average entanglement of random pure states of an N x N composite system is analyzed. We compute the average value of the determinant D of the reduced state, which forms an entanglement monotone....
2nd Polish Symposium on Econo- and Sociophysics (2006)
Burda, Zdzislaw, Holyst, Janusz, Jurkiewicz, Jerzy, Kulakowski, Krzysztof, Malarz, Krzysztof, Nowak, Maciej A, ...
Basis Markov Partitions and Transition Matrices for Stochastic Systems (2006)
Bollt, Erik, Gora, Pawel, Ostruszka, Andrzej, Zyczkowski, Karol
We analyze dynamical systems subjected to an additive noise and their deterministic limit. In this work, we will introduce a notion by which a stochastic system has something like a Markov partition...
Physical accessibility of non-completely positive maps (2005)
Carteret, Hilary, Terno, Daniel R., Zyczkowski, Karol
Maps that are not completely positive (CP) are often useful to describe the dynamics of open systems. An apparent violation of complete positivity can occur because there are prior correlations of...
A concise guide to complex Hadamard matrices (2005)
Tadej, Wojciech, Zyczkowski, Karol
Complex Hadamard matrices, consisting of unimodular entries with arbitrary phases, play an important role in the theory of quantum information. We review basic properties of complex Hadamard matrices...
Higher-Rank Numerical Ranges and Compression Problems (2005)
Choi, Man-Duen, Kribs, David W., Zyczkowski, Karol
We consider higher-rank versions of the standard numerical range for matrices. A central motivation for this investigation comes from quantum error correction. We develop the basic structure theory...
Quantum Error Correcting Codes From The Compression Formalism (2005)
Choi, Man-Duen, Kribs, David W., Zyczkowski, Karol
We solve the fundamental quantum error correction problem for bi-unitary channels on two-qubit Hilbert space. By solving an algebraic compression problem, we construct qubit codes for such channels...
On the structure of the body of states with positive partial transpose (2005)
Szarek, Stanislaw, Bengtsson, Ingemar, Zyczkowski, Karol
We show that the convex set of separable mixed states of the 2 x 2 system is a body of constant height. This fact is used to prove that the probability to find a random state to be separable equals 2...
Scalable Noise Estimation with Random Unitary Operators (2005)
Emerson, Joseph, Alicki, Robert, Zyczkowski, Karol
We describe a scalable stochastic method for the experimental measurement of generalized fidelities characterizing the accuracy of the implementation of a coherent quantum transformation. The method...
Negativity of the Wigner function as an indicator of nonclassicality (2004)
Kenfack, Anatole, Zyczkowski, Karol
A measure of nonclassicality of quantum states based on the volume of the negative part of the Wigner function is proposed. We analyze this quantity for Fock states, squeezed displaced Fock states...
Voting in the European Union: The square root system of Penrose and a critical point (2004)
Zyczkowski, Karol, Slomczynski, Wojciech
The notion of the voting power is illustrated by examples of the systems of voting in the European Council according to the Treaty of Nice and the more recent proposition of the European Convent. We...
Statistical properties of random density matrices (2004)
Sommers, Hans-Juergen, Zyczkowski, Karol
Statistical properties of ensembles of random density matrices are investigated. We compute traces and von Neumann entropies averaged over ensembles of random density matrices distributed according...
Experimental simulation of quantum graphs by microwave networks (2004)
Hul, Oleh, Bauch, Szymon, Pakonski, Prot, Savytskyy, Nazar, Zyczkowski, Karol, Sirko, Leszek
We present the results of experimental and theoretical study of irregular, tetrahedral microwave networks consisting of coaxial cables (annular waveguides) connected by T-joints. The spectra of the...
Birkhoff's polytope and unistochastic matrices, N=3 and N=4 (2004)
Bengtsson, Ingemar, Ericsson, Asa, Kus, Marek, Tadej, Wojciech, Zyczkowski, Karol
The set of bistochastic or doubly stochastic N by N matrices form a convex set called Birkhoff's polytope, that we describe in some detail. Our problem is to characterize the set of unistochastic...
On duality between quantum maps and quantum states (2004)
Zyczkowski, Karol, Bengtsson, Ingemar
We investigate the space of quantum operations, as well as the larger space of maps which are positive, but not completely positive. A constructive criterion for decomposability is presented. A...
Average fidelity between random quantum states (2003)
Zyczkowski, Karol, Sommers, Hans-Jurgen
We analyze mean fidelity between random density matrices of size N, generated with respect to various probability measures in the space of mixed quantum states: Hilbert-Schmidt measure, Bures...
Quantum dynamical entropy and decoherence rate (2003)
Alicki, Robert, Lozinski, Artur, Pakonski, Prot, Zyczkowski, Karol
We investigate quantum dynamical systems defined on a finite dimensional Hilbert space and subjected to an interaction with an environment. The rate of decoherence of initially pure states, measured...
Wehrl entropy, Lieb conjecture and entanglement monotones (2003)
Mintert, Florian, Zyczkowski, Karol
We propose to quantify the entanglement of pure states of $N \times N$ bipartite quantum system by defining its Husimi distribution with respect to $SU(N)\times SU(N)$ coherent states. The Wehrl...
Renyi extrapolation of Shannon entropy (2003)
Relations between Shannon entropy and Renyi entropies of integer order are discussed. For any N-point discrete probability distribution for which the Renyi entropies of order two and three are known,...
Bures volume of the set of mixed quantum states (2003)
Sommers, Hans-Juergen, Zyczkowski, Karol
We compute the volume of the N^2-1 dimensional set M_N of density matrices of size N with respect to the Bures measure and show that it is equal to that of a N^2-1 dimensional hyper-halfsphere of...
Hilbert--Schmidt volume of the set of mixed quantum states (2003)
Zyczkowski, Karol, Sommers, Hans-Juergen
We compute the volume of the convex N^2-1 dimensional set M_N of density matrices of size N with respect to the Hilbert-Schmidt measure. The hyper--area of the boundary of this set is also found and...
Entanglement of 2xK quantum systems (2003)
Lozinski, Artur, Buchleitner, Andreas, Zyczkowski, Karol, Wellens, Thomas
We derive an analytical expression for the lower bound of the concurrence of mixed quantum states of composite 2xK systems. In contrast to other, implicitly defined entanglement measures, the...
Quantization of Classical Maps with tunable Ruelle-Pollicott Resonances (2003)
Ostruszka, Andrzej, Manderfeld, Christopher, Zyczkowski, Karol, Haake, Fritz
We investigate the correspondence between the decay of correlation in classical system, governed by Ruelle--Pollicott resonances, and the properties of the corresponding quantum system. For this...
Euroattractor: a brief introduction to Iterated Function Systems (2002)
Zyczkowski, Karol, Lozinski, Artur
In this work we propose a definition of an Euroattractor: an attracting invariant measure of a certain iterated functions system (IFS). An IFS is defined by specifying a set of functions, defined in...
Quantum Iterated Function Systems (2002)
Lozinski, Artur, Zyczkowski, Karol, Slomczynski, Wojciech
Iterated functions system (IFS) is defined by specifying a set of functions in a classical phase space, which act randomly on an initial point. In an analogous way, we define a quantum iterated...
Bounds on entanglement in qudit subsystems (2002)
Kendon, Vivien M, Zyczkowski, Karol, Munro, William J
The entanglement in a pure state of N qudits (d-dimensional distinguishable quantum particles) can be characterised by specifying how entangled its subsystems are. A generally mixed subsystem of m...
Irreversible Quantum Baker Map (2002)
Lozinski, Artur, Pakonski, Prot, Zyczkowski, Karol
We propose a generalization of the model of classical baker map on the torus, in which the images of two parts of the phase space do overlap. This transformation is irreversible and cannot be...
Families of line-graphs and their quantization (2001)
Pakonski, Prot, Tanner, Gregor, Zyczkowski, Karol
Any directed graph G with N vertices and J edges has an associated line-graph L(G) where the J edges form the vertices of L(G). We show that the non-zero eigenvalues of the adjacency matrices are the...
Manifolds of interconvertible pure states (2001)
Sinolecka, Magdalena M., Zyczkowski, Karol, Kus, Marek
Local orbits of a pure state of an N x N bi-partite quantum system are analyzed. We compute their dimensions which depends on the degeneracy of the vector of coefficients arising by the Schmidt...
Fractality of certain quantum states (2001)
Wojcik, Daniel, Zyczkowski, Karol
We prove the theorem announced in Phys. Rev. Lett. {\bf 85}:5022, 2001 concerning the existence and properties of fractal states for the Schr\"odinger equation in the infinite one-dimensional well.
Renyi-Wehrl entropies as measures of localization in phase space (2001)
Gnutzmann, Sven, Zyczkowski, Karol
We generalize the concept of the Wehrl entropy of quantum states which gives a basis-independent measure of their localization in phase space. We discuss the minimal values and the typical values of...
Relativity of pure states entanglement (2001)
Zyczkowski, Karol, Bengtsson, Ingemar
Entanglement of any pure state of an N times N bi-partite quantum system may be characterized by the vector of coefficients arising by its Schmidt decomposition. We analyze various measures of...
Induced measures in the space of mixed quantum states (2000)
Zyczkowski, Karol, Sommers, Hans-Juergen
We analyze several product measures in the space of mixed quantum states. In particular we study measures induced by the operation of partial tracing. The natural, rotationally invariant measure on...
Classical 1D maps, quantum graphs and ensembles of unitary matrices (2000)
Pakonski, Prot, Zyczkowski, Karol, Kus, Marek
We study a certain class of classical one dimensional piecewise linear maps. For these systems we introduce an infinite family of Markov partitions into equal cells. The symbolic dynamics generated...
Dynamics of quantum entanglement (2000)
Zyczkowski, Karol, Horodecki, Pawel, Horodecki, Michal, Horodecki, Ryszard
A model of discrete dynamics of entanglement of bipartite quantum state is considered. It involves a global unitary dynamics of the system and periodic actions of local bistochastic or decaying...
Monge Metric on the Sphere and Geometry of Quantum States (2000)
Zyczkowski, Karol, Slomczynski, Wojciech
Topological and geometrical properties of the set of mixed quantum states in the N-dimensional Hilbert space are analysed. Assuming that the corresponding classical dynamics takes place on the sphere...
Level Dynamics and Universality of Spectral Fluctuations (2000)
Braun, Peter, Gnutzmann, Sven, Haake, Fritz, Kus, Marek, Zyczkowski, Karol
The spectral fluctuations of quantum (or wave) systems with a chaotic classical (or ray) limit are mostly universal and faithful to random-matrix theory. Taking up ideas of Pechukas and Yukawa we...
Geometry of entangled states (2000)
Geometric properties of the set of quantum entangled states are investigated. We propose an explicit method to compute the dimension of local orbits for any mixed state of the general K x M problem...
Validity of threshold-crossing analysis of symbolic dynamics from chaotic time series (2000)
Bollt, Erik M., Stanford, Theodore, Lai, Ying-Cheng, Zyczkowski, Karol
A practical and popular technique to extract the symbolic dynamics from experimentally measured chaotic time series is the threshold-crossing method, by which an arbitrary partition is utilized for...
Time Evolution of Quantum Fractals (2000)
Wojcik, Daniel, Bialynicki-Birula, Iwo, Zyczkowski, Karol
We propose a general construction of wave functions of arbitrary prescribed fractal dimension, for a wide class of quantum problems, including the infinite potential well, harmonic oscillator, linear...
Invariant sets for discontinuous parabolic area-preserving torus maps (2000)
Ashwin, Peter, Fu, Xin-Chu, Nishkawa, Takashi, Zyczkowski, Karol
We analyse a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in each component. We show that...
Truncations of random unitary matrices (1999)
Zyczkowski, Karol, Sommers, Hans-Juergen
We analyze properties of non-hermitian matrices of size M constructed as square submatrices of unitary (orthogonal) random matrices of size N>M, distributed according to the Haar measure. In this way...
Localization of Eigenstates & Mean Wehrl Entropy (1999)
Dynamics of a periodically time dependent quantum system is reflected in the features of the eigenstates of the Floquet operator. Of the special importance are their localization properties...
Invariant sets for discontinuous parabolic area-preserving torus maps (1999)
Ashwin, Peter, Fu, Xin-Chu, Nishikawa, Takashi, Zyczkowski, Karol
We analyze a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in each component. We show that...
Dynamical entropy for systems with stochastic perturbation (1999)
Ostruszka, Andrzej, Pakonski, Prot, Slomczynski, Wojciech, Zyczkowski, Karol
Dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively. Since for such systems the KS-entropy diverges we analyse the difference between the total...
Linear Parabolic Maps on the Torus (1999)
Zyczkowski, Karol, Nishikawa, Takashi
We investigate linear parabolic maps on the torus. In a generic case these maps are non-invertible and discontinuous. Although the metric entropy of these systems is equal to zero, their dynamics is...
On the volume of the set of mixed entangled states II (1999)
The problem of of how many entangled or, respectively, separable states there are in the set of all quantum states is investigated. We study to what extent the choice of a measure in the space of...
On the entropy devil's staircase in a family of gap-tent maps (1998)
Zyczkowski, Karol, Bollt, Erik M.
We analyze dynamical properties of a "gap-tent map" - a family of 1D maps with a symmetric gap, which mimics the presence of noise in physical realizations of chaotic systems. We demonstrate that the...
Quantum baker map on the sphere (1998)
Pakonski, Prot, Ostruszka, Andrzej, Zyczkowski, Karol
We define a class of dynamical systems on the sphere analogous to the baker map on the torus. The classical maps are characterized by dynamical entropy equal to ln 2. We construct and investigate a...
On the volume of the set of mixed entangled states (1998)
Zyczkowski, Karol, Horodecki, Pawel, Sanpera, Anna, Lewenstein, Maciej
A natural measure in the space of density matrices describing N-dimensional quantum systems is proposed. We study the probability P that a quantum state chosen randomly with respect to the natural...
Entropy computing via integration over fractal measures (1998)
Slomczynski, Wojciech, Kwapien, Jaroslaw, Zyczkowski, Karol
We discuss the properties of invariant measures corresponding to iterated function systems (IFSs) with place-dependent probabilities and compute their Renyi entropies, generalized dimensions, and...
Quantum scattering in the strip: from ballistic to localized regimes (1998)
Gebarowski, Robert, Seba, Petr, Zyczkowski, Karol, Zakrzewski, Jakub
Quantum scattering is studied in a system consisting of randomly distributed point scatterers in the strip. The model is continuous yet exactly solvable. Varying the number of scatterers (the sample...
Random matrix approach to `nonuniversal' conductance (1997)
Seba, Petr, Zyczkowski, Karol, Zakrzewski, Jakub
Recent experiments on the conductance of high quality quantum wires have revealed an unexpected feature: the quantization step of the conductance is apparently system dependent. We provide the...
Monge Distance between Quantum States (1997)
Zyczkowski, Karol, Slomczynski, Wojciech
We define a metric in the space of quantum states taking the Monge distance between corresponding Husimi distributions (Q--functions). This quantity fulfills the axioms of a metric and satisfies the...
Mean dynamical entropy of quantum system tends to infinity in the semiclassical limit (1997)
Slomczynski, Wojciech, Zyczkowski, Karol
We show that the mean dynamical entropy of a quantum map on the sphere is positive and tends logarithmically to infinity in the semiclassical limit. A link between chaotic dynamics of classical...
Composed ensembles of random unitary matrices (1997)
Pozniak, Marcin, Zyczkowski, Karol, Kus, Marek
Composed ensembles of random unitary matrices are defined via products of matrices, each pertaining to a given canonical circular ensemble of Dyson. We investigate statistical properties of spectra...
1 Artificial Economy Project, (1997)
Option Pricing, Marc Potters, Karol Zyczkowski
Abstract We introduce a prescription how to price options on a stock or commodity, in a situation where risk in options ' trading cannot be eliminated by hedging. First, a hedging strategy is...
Coherent States Measurement Entropy (1996)
Kwapien, Jaroslaw, Slomczynski, Wojciech, Zyczkowski, Karol
Coherent states (CS) quantum entropy can be split into two components. The dynamical entropy is linked with the dynamical properties of a quantum system. The measurement entropy, which tends to zero...
Statistical properties of random scattering matrices (1996)
Seba, Petr, Zyczkowski, Karol, Zakrzewski, Jakub
We discuss the properties of eigenphases of S--matrices in random models simulating classically chaotic scattering. The energy dependence of the eigenphases is investigated and the corresponding...
Secular determinants of random unitary matrices (1996)
Haake, Fritz, Kus, Marek, Sommers, Hans-Jurgen, Schomerus, Henning, Zyczkowski, Karol
We consider the characteristic polynomials of random unitary matrices U drawn from various circular ensembles. In particular, the statistics of the coefficients of these polynomials are studied. The...
Secular determinants of random unitary matrices (1996)
Haake, Fritz, Kus, Marek, Sommers, Hans-Jurgen, Schomerus, Henning, Zyczkowski, Karol
We consider the characteristic polynomials of random unitary matrices U drawn from various circular ensembles. In particular, the statistics of the coefficients of these polynomials are studied. The...
Secular determinants of random unitary matrices. (1996)
Haake, Fritz, Kus, Marek, Sommers, Hans-Jurgen, Schomerus, Henning, Zyczkowski, Karol
We consider the characteristic polynomials of random unitary matrices U drawn from various circular ensembles. In particular, the statistics of the coefficients of these polynomials are studied. The...
OPTION PRICING & PARTIAL HEDGING: THEORY OF POLISH OPTIONS
The twin problems of hedging and pricing of options in discrete-time markets are analyzed. We consider trading strategies consisting of one stock and one bond. The bond price rises deterministically...