|
The signaling dimension in generalized probabilistic theories (pp0411-0424)
Michele Dall'Arno, Alessandro Tosini, and
Francesco Buscemi
doi:
https://doi.org/10.26421/QIC24.5-6-2
Abstracts:
The signaling dimension
of a given physical system
quantifies the minimum dimension of a classical
system required to reproduce all input/output correlations of the
given system. Thus, unlike other dimension measures - such as the
dimension of the linear space or the maximum number of (jointly or
pairwise) perfectly
discriminable
states - which examine the correlation space only
along a single direction, the signaling dimension does
not depend on the arbitrary choice of a specific operational
task. In this sense, the signaling dimension summarizes the
structure of the entire set of input/output
correlations consistent with a given system in
a single scalar quantity. For quantum
theory, it was recently proved by
Frenkel
and
Weiner
in a seminal result that the signaling dimension
coincides with the Hilbert space dimension. Here, we derive
analytical and algorithmic
techniques to compute the signaling dimension for any
given system of any given generalized probabilistic theory.
We prove that it suffices to consider
extremal
measurements with ray-extremal
effects, and we bound the number of elements of
any such measurement in terms
of the linear dimension. For systems with
a finite number of
extremal
effects, we recast the problem of
characterizing the
extremal
measurements with ray-extremal
effects as the problem of deriving the vertex
description of a
polytope
given its face description, which can be
conveniently solved by standard techniques
such as the double description
algorithm. For each such measurement, we
recast the computation of the signaling dimension
as a linear program, and we propose a
combinatorial
branch and bound algorithm to reduce its size. We
apply our results to derive the
extremal
measurements with ray-extremal
effects of a composition of two square bits (or
squits)
and prove that their signaling dimension is
five, even though each
squit
has a signaling dimension equal to two.
Key Words:
signaling dimension,
generalized probabilistic theory,
GPT,
square bit,
squit,
extremal
measurements |
