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 ), respectively). 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.

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 we use the vector SU(2) coherent states and the generalised Husimi distributions to define the Monge distance between two arbitrary density matrices. The Monge metric has a simple semiclassical interpretation and induces a non-trivial geometry. Among all pure states the distance from the maximally mixed state ρ * , proportional to the identity matrix, admits the largest value for the coherent states, while the delocalized 'chaotic' states are close to ρ * . This contrasts the geometry induced by the standard (trace, Hilbert-Schmidt or Bures) metrics, for which the distance from ρ * is the same for all pure states. We discuss possible physical consequences including unitary time evolution and the process of decoherence. We introduce also a simplified Monge metric, defined in the space of pure quantum states and more suitable for numerical computation.

A new definition of the entropy of a given dynamical system and of an instrument describing the measurement process is proposed within the operational approach to quantum mechanics. It generalizes other definitions of entropy, in both the classical and quantum cases. The Kolmogorov-Sinai (KS) entropy is obtained for a classical system and the sharp measurement instrument. For a quantum system and a coherent states instrument, a new quantity, coherent states entropy, is defined. It may be used to measure chaos in quantum mechanics. The following correspondence principle is proved: the upper limit of the coherent states entropy of a quantum map as h-+0 is less than or equal to the KS-entropy of the corresponding classical map."Chaos umpire sits, And by decision more imbroils the fray By which he reigns: next him high arbiter Chance governs all. " John Milton, Paradise Lost, Book II 5674

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We discuss the dependence of the Shannon entropy of normalized finite rank-1 POVMs on the choice of the input state, looking for the states that minimize this quantity. To distinguish the class of measurements where the problem can be solved analytically, we introduce the notion of highly symmetric POVMs and classify them in dimension 2 (for qubits). In this case, we prove that the entropy is minimal, and hence, the relative entropy (informational power) is maximal, if and only if the input state is orthogonal to one of the states constituting a POVM. The method used in the proof, employing the Michel theory of critical points for group action, the Hermite interpolation, and the structure of invariant polynomials for unitary-antiunitary groups, can also be applied in higher dimensions and for other entropy-like functions. The links between entropy minimization and entropic uncertainty relations, the Wehrl entropy, and the quantum dynamical entropy are described.