7,991 research outputs found
A Combinatorial Approach to Nonlocality and Contextuality
So far, most of the literature on (quantum) contextuality and the
Kochen-Specker theorem seems either to concern particular examples of
contextuality, or be considered as quantum logic. Here, we develop a general
formalism for contextuality scenarios based on the combinatorics of hypergraphs
which significantly refines a similar recent approach by Cabello, Severini and
Winter (CSW). In contrast to CSW, we explicitly include the normalization of
probabilities, which gives us a much finer control over the various sets of
probabilistic models like classical, quantum and generalized probabilistic. In
particular, our framework specializes to (quantum) nonlocality in the case of
Bell scenarios, which arise very naturally from a certain product of
contextuality scenarios due to Foulis and Randall. In the spirit of CSW, we
find close relationships to several graph invariants. The recently proposed
Local Orthogonality principle turns out to be a special case of a general
principle for contextuality scenarios related to the Shannon capacity of
graphs. Our results imply that it is strictly dominated by a low level of the
Navascu\'es-Pironio-Ac\'in hierarchy of semidefinite programs, which we also
apply to contextuality scenarios.
We derive a wealth of results in our framework, many of these relating to
quantum and supraquantum contextuality and nonlocality, and state numerous open
problems. For example, we show that the set of quantum models on a
contextuality scenario can in general not be characterized in terms of a graph
invariant.
In terms of graph theory, our main result is this: there exist two graphs
and with the properties \begin{align*} \alpha(G_1) &= \Theta(G_1),
& \alpha(G_2) &= \vartheta(G_2), \\[6pt] \Theta(G_1\boxtimes G_2) & >
\Theta(G_1)\cdot \Theta(G_2),& \Theta(G_1 + G_2) & > \Theta(G_1) + \Theta(G_2).
\end{align*}Comment: minor revision, same results as in v2, to appear in Comm. Math. Phy
The Shannon capacity of a graph and the independence numbers of its powers
The independence numbers of powers of graphs have been long studied, under
several definitions of graph products, and in particular, under the strong
graph product. We show that the series of independence numbers in strong powers
of a fixed graph can exhibit a complex structure, implying that the Shannon
Capacity of a graph cannot be approximated (up to a sub-polynomial factor of
the number of vertices) by any arbitrarily large, yet fixed, prefix of the
series. This is true even if this prefix shows a significant increase of the
independence number at a given power, after which it stabilizes for a while
Simultaneous communication in noisy channels
A sender wishes to broadcast a message of length over an alphabet to
users, where each user , should be able to receive one of
possible messages. The broadcast channel has noise for each of the users
(possibly different noise for different users), who cannot distinguish between
some pairs of letters. The vector is said to be
feasible if length encoding and decoding schemes exist enabling every user
to decode his message. A rate vector is feasible if there
exists a sequence of feasible vectors such that
. We
determine the feasible rate vectors for several different scenarios and
investigate some of their properties. An interesting case discussed is when one
user can only distinguish between all the letters in a subset of the alphabet.
Tight restrictions on the feasible rate vectors for some specific noise types
for the other users are provided. The simplest non-trivial cases of two users
and alphabet of size three are fully characterized. To this end a more general
previously known result, to which we sketch an alternative proof, is used. This
problem generalizes the study of the Shannon capacity of a graph, by
considering more than a single user
Rainbow saturation and graph capacities
The -colored rainbow saturation number is the minimum size
of a -edge-colored graph on vertices that contains no rainbow copy of
, but the addition of any missing edge in any color creates such a rainbow
copy. Barrus, Ferrara, Vandenbussche and Wenger conjectured that for every and . In this short
note we prove the conjecture in a strong sense, asymptotically determining the
rainbow saturation number for triangles. Our lower bound is probabilistic in
spirit, the upper bound is based on the Shannon capacity of a certain family of
cliques.Comment: 5 pages, minor change
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