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*Value Added Tax charge for European Union residents. If you live in the European Union in one of the following member states (Belgium, France, Germany, Ireland, Italy, Portugal, Spain or Sweden) and are not registered for VAT we are required to charge VAT at the rate applicable in your country of residence. If you live in any other member state in the EU… (More)

- Wojciech Tadej, Karol Zyczkowski
- Open Syst. Inform. Dynam.
- 2006

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 and present a catalogue of inequivalent cases known for the dimensions N = 2, . . . , 16. In particular, we explicitly write down some families of complex… (More)

We analyse 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 we define ensembles of random matrices and study the statistical properties of the spectrum located inside the unit circle. In the limit of large matrices,… (More)

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 European Union, which was recently a subject of a vivid political discussion. The a priori voting power of a voter measures his potential influence over the decisions… (More)

- Karol Zyczkowski
- Open Syst. Inform. Dynam.
- 2003

Relations between Shannon entropy and Rényi entropies of integer order are discussed. For any N–point discrete probability distribution for which the Rényi entropies of order two and three are known, we provide an lower and an upper bound for the Shannon entropy. The average of both bounds provide an explicit extrapolation for this quantity. These results… (More)

We analyse 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 the set of all pure states of a N ×K composite system, induces a unique measure in the space of N × N mixed states (or in the space of K ×K mixed states, if the… (More)

- Karol Zyczkowski, Ingemar Bengtsson
- Open Syst. Inform. Dynam.
- 2004

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 certain class of unistochastic operations, determined by unitary matrices of extended dimensionality, is defined and analyzed. Using the concept of the dynamical… (More)

Erik M. Bollt,1 Theodore Stanford,2 Ying-Cheng Lai,3 Karol Życzkowski4,5 1Mathematics Department, 572 Holloway Road, U.S. Naval Academy, Annapolis, Maryland 21402-5002 2Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003-8001 3Departments of Mathematics, Electrical Engineering, and Physics, Center for Systems… (More)

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 of some composed ensembles and demonstrate their physical relevance. We discuss also the methods of generating random matrices distributed according to invariant… (More)

- Karol Zyczkowski
- 2007

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 the population he or she represents. For a generic population distribution the quota required for the qualified majority can be set in such a way that the… (More)