Informational power of the Hoggar SIC-POVM

We compute the informational power for the Hoggar SIC-POVM in dimension 8, i.e. the classical capacity of a quantum-classical channel generated by this measurement. We show that the states constituting a maximally informative ensemble form a twin Hoggar SIC-POVM being the image of the original one under a conjugation.Comment: 6 double column page

Orthogonal Projections on Hyperplanes Intertwined With Unitaries

Consider a sequence in a finite-dimensional complex (resp. real) vector space arising as the iterates of an arbitrary point under the composition of a unitary (resp. orthogonal) map with the orthogonal projection on the hyperplane orthogonal to the starting point. We show that, generically, the series of the squared norms of those points sums to the dimension of the underlying space. The exact formula for this series in non-generic cases is provided as well, along with its application to determining the number of quantum degrees of freedom.Comment: 14 pages, 4 figure

Localization of Eigenstates & Mean Wehrl Entropy

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 quantitatively characterized by the eigenvector entropy, the inverse participation ratio or the eigenvector statistics. Since these quantities depend on the choice of the eigenbasis, we suggest to use the overcomplete basis of coherent states, uniquely determined by the classical phase space. In this way we define the mean Wehrl entropy of eigenvectors of the Floquet operator and demonstrate that this quantity is useful to describe quantum chaotic systems.Comment: 7 pages in Latex with 4 pictures in .ps (included), submitted to Physica

Tadeusz Krauze (1934–2019)

W dniu 8 stycznia 2019 roku (w wieku 84 lat) zmarł w Warszawie Tadeusz Krauze, emerytowany profesor Hofstra University w Hempstead, NY, USA, matematyk i socjolog. Członek Polskiego Towarzystwa Matematycznego od 2005 roku.Tadeusz Krauze studia matematyczne ukończył na Uniwersytecie Łódzkim w 1955 r. Doktorat z socjologii uzyskał w 1974 r. w USA na New York University. Przez wiele lat był profesorem socjologii na Hofstra University w Hempstead (USA), a w latach 1985-88 kierował tamtejszym wydziałem socjologii.Jako naukowiec zajmował się problemami stratyfikacji społecznej i socjologii nauki, a także metodami matematycznymi socjologii.Przez ponad 20 lat był redaktorem „International Journal of Sociology”. Członek międzynarodowych, amerykańskich i polskich towarzystw naukowych

Average Weights and Power in Weighted Voting Games

We investigate a class of weighted voting games for which weights are randomly distributed over the standard probability simplex. We provide close-formed formulae for the expectation and density of the distribution of weight of the $k$-th largest player under the uniform distribution. We analyze the average voting power of the $k$-th largest player and its dependence on the quota, obtaining analytical and numerical results for small values of $n$ and a general theorem about the functional form of the relation between the average Penrose--Banzhaf power index and the quota for the uniform measure on the simplex. We also analyze the power of a collectivity to act (Coleman efficiency index) of random weighted voting games, obtaining analytical upper bounds therefor.Comment: 12 pages, 7 figure

Morphophoric POVMs, generalised qplexes, and 2-designs

We study the class of quantum measurements with the property that the image of the set of quantum states under the measurement map transforming states into probability distributions is similar to this set and call such measurements morphophoric. This leads to the generalisation of the notion of a qplex, where SIC-POVMs are replaced by the elements of the much larger class of morphophoric POVMs, containing in particular 2-design (rank-1 and equal-trace) POVMs. The intrinsic geometry of a generalised qplex is the same as that of the set of quantum states, so we explore its external geometry, investigating, inter alia, the algebraic and geometric form of the inner (basis) and the outer (primal) polytopes between which the generalised qplex is sandwiched. In particular, we examine generalised qplexes generated by MUB-like 2-design POVMs utilising their graph-theoretical properties. Moreover, we show how to extend the primal equation of QBism designed for SIC-POVMs to the morphophoric case.Comment: 27 pages, 5 figure

Different Traces of Quantum Systems Having the Same Classical Limit

Many quantum systems may have the same classical limit. We argue that in the classical limit their traces do not necessarily converge one to another. The trace formula allows to express quantum traces by means of classical quantities as sums over periodic orbits of the classical system. To explain the lack of convergence of the traces we need the quantum corrections to the classical actions of periodic orbits. The four versions of the quantum baker map on the sphere serve as an illustration of this problem.Comment: LaTeX 4 pages, 2 figures included, final published versio