40 research outputs found
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
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
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 -th largest player under the uniform distribution. We analyze
the average voting power of the -th largest player and its dependence on the
quota, obtaining analytical and numerical results for small values of 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
Birkhoff's polytope and unistochastic matrices, N=3 and N=4
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 matrices as a subset of Birkhoff's
polytope. For N=3 we present fairly complete results. For N=4 partial results
are obtained. An interesting difference between the two cases is that there is
a ball of unistochastic matrices around the van der Waerden matrix for N=3,
while this is not the case for N=4.Comment: 30 pages, 4 figure
Quantum Iterated Function Systems
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 functions system (QIFS), where
functions act randomly with prescribed probabilities in the Hilbert space. In a
more general setting a QIFS consists of completely positive maps acting in the
space of density operators. We present exemplary classical IFSs, the invariant
measure of which exhibits fractal structure, and study properties of the
corresponding QIFSs and their invariant states.Comment: 12 pages, 1 figure include
Wybrane aspekty funkcjonowania Sejmu w latach 1997–2007
Praca recenzowana / peer-reviewed paperPraca naukowa finansowana ze środków na naukę w latach 2006–2008 jako projekt
badawczy własny Nr 1 H02E 052 3
Highly symmetric POVMs and their informational power
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 two (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.Comment: 40 pages, 3 figure