Marek Kus

Quantum correlations in two-fermion systems

We characterize and classify quantum correlations in two-fermion systems having 2K single-particle states. For pure states we introduce the Slater decomposition and rank (in analogy to Schmidt decomposition and rank): i.e., we decompose the state into a combination of elementary Slater determinants formed by pairs of mutually orthogonal single-particle states. Mixed states can be characterized by their Slater number which is the minimal Slater rank required to generate them. For K = 2 we gi ve a necessary and sufficient condition for a state to have a Slater number 1. We introduce a correlation measure for mixed states which can be evaluated analytically for K = 2. For higher K, we provide a method of constructing and optimizing Slater number witnesses, i.e., operators that detect Slater numbers for some states.

Composed ensembles of random unitary matrices

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 also discuss the methods of generating random matrices distributed according to invariant Haar measure on the orthogonal and unitary group.

Random unistochastic matrices

An ensemble of random unistochastic (orthostochastic) matrices is defined by taking squared moduli of elements of random unitary (orthogonal) matrices distributed according to the Haar measure on U(N) (or O(N)). An ensemble of symmetric unistochastic matrices is obtained with use of unitary symmetric matrices pertaining to the circular orthogonal ensemble. We study the distribution of complex eigenvalues of bistochastic, unistochastic and orthostochastic matrices in the complex plane. We compute averages (entropy, traces) over the ensembles of unistochastic matrices and present inequalities concerning the entropies of products of bistochastic matrices.

Classical 1D maps, quantum graphs and ensembles of unitary matrices

We study a certain class of classical one-dimensional piecewise linear maps. For these systems we introduce an infinite family of Markov partitions in equal cells. The symbolic dynamics generated by these systems is described by bi-stochastic (doubly stochastic) matrices. We analyse the structure of graphs generated from the corresponding symbolic dynamics. We demonstrate that the spectra of quantized graphs corresponding to the regular classical systems have locally Poissonian statistics, while quantized graphs derived from classically chaotic systems display statistical properties characteristic of the circular unitary ensemble, even though the corresponding unitary matrices are sparse.

Coherent states and the classical limit on irreducible representations

We give an explicit construction of the coherent states for an arbitrary irreducible representation. We also construct the symplectic structure on the manifold of coherent states, find canonical variables and discuss various classical limits of quantum-mechanical systems with relevant observables that obey commutation relations.

Universality of eigenvector statistics of kicked tops of different symmetries

The authors show that the eigenvectors of the Floquet operators of periodically kicked tops with orthogonal, unitary and symplectic canonical transformations conform to the predictions of the respective circular ensembles of random matrices.

Separability, entanglement, and full families of commuting normal matrices

We reduce the question of whether a given quantum mixed state is separable or entangled to the problem of existence of a certain full family of commuting normal matrices whose matrix elements are partially determined by components of the pure states constituting a decomposition of the considered mixture. The method reproduces many known entanglement and/or separability criteria, and provides yet another geometrical characterization of mixed separable states.

Quantum effects of periodic orbits for the kicked top

We analyze traces of powers of the time evolution operator of a periodically kicked top. Semiclassically, such traces are related to periodic orbits of the classical map. We derive the semiclassical traces in a coherent state basis and show how the periodic orbits can be recovered via a Fourier transform. A breakdown of the stationary phase approximation is detected. The quasi energy spectrum remains elusive due to lack of knowledge of sufficiently many periodic orbits. Divergencies of periodic orbit formulas are avoided by appealing to the finiteness of the quantum mechanical Hilbert space. The traces also enter the coefficients of the characteristic polynominal of the Floquet operator. Statistical properties of these coefficients give rise to a new criterion for the distinction of chaos and regular motion.

A symplectic context for level dynamics

Abstract We consider a mathematical context which was suggested by quantum mechanical considerations of level dynamics. Although the situation is a general one, we restrict our attention to certain examples of physical relevance where explicit calculations are possible. Cases where M is the cotangent space of some Lie group or Lie algebra Q of operators on a finite-dimensional vector space are of particular interest.

Level dynamics for conservative and dissipative quantum systems

We establish level dynamics for finite matrices, employing a unified treatment of real symmetric, complex Hermitian, quaternion real, unitary, and arbitrary complex matrices. In all cases the level dynamics take the form of the classical Hamiltonian flow of some fictitious many-particle systems. Equilibrium statistical mechanics of the latter leads to the well known matrix ensembles of random-matrix theory. Ginibres ensemble, in particular, is thus associated with level dynamics of arbitrary complex matrices.