207,815 research outputs found
Information and communication in polygon theories
Generalized probabilistic theories (GPT) provide a framework in which one can
formulate physical theories that includes classical and quantum theories, but
also many other alternative theories. In order to compare different GPTs, we
advocate an approach in which one views a state in a GPT as a resource, and
quantifies the cost of interconverting between different such resources. We
illustrate this approach on polygon theories (Janotta et al. New J. Phys 13,
063024, 2011) that interpolate (as the number n of edges of the polygon
increases) between a classical trit (when n=3) and a real quantum bit (when
n=infinity). Our main results are that simulating the transmission of a single
n-gon state requires more than one qubit, or more than log(log(n)) bits, and
that n-gon states with n odd cannot be simulated by n'-gon states with n' even
(for all n,n'). These results are obtained by showing that the classical
capacity of a single n-gon state with n even is 1 bit, whereas it is larger
than 1 bit when n is odd; by showing that transmitting a single n-gon state
with n even violates information causality; and by showing studying the
communication complexity cost of the nondeterministic not equal function using
n-gon states.Comment: 18 page
On the Areas of Cyclic and Semicyclic Polygons
We investigate the ``generalized Heron polynomial'' that relates the squared
area of an n-gon inscribed in a circle to the squares of its side lengths. For
a (2m+1)-gon or (2m+2)-gon, we express it as the defining polynomial of a
certain variety derived from the variety of binary (2m-1)-forms having m-1
double roots. Thus we obtain explicit formulas for the areas of cyclic
heptagons and octagons, and illuminate some mysterious features of Robbins'
formulas for the areas of cyclic pentagons and hexagons. We also introduce a
companion family of polynomials that relate the squared area of an n-gon
inscribed in a circle, one of whose sides is a diameter, to the squared lengths
of the other sides. By similar algebraic techniques we obtain explicit formulas
for these polynomials for all n <= 7.Comment: 22 page
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