3,121 research outputs found
Kochen-Specker Sets and the Rank-1 Quantum Chromatic Number
The quantum chromatic number of a graph is sandwiched between its
chromatic number and its clique number, which are well known NP-hard
quantities. We restrict our attention to the rank-1 quantum chromatic number
, which upper bounds the quantum chromatic number, but is
defined under stronger constraints. We study its relation with the chromatic
number and the minimum dimension of orthogonal representations
. It is known that . We
answer three open questions about these relations: we give a necessary and
sufficient condition to have , we exhibit a class of
graphs such that , and we give a necessary and
sufficient condition to have . Our main tools are
Kochen-Specker sets, collections of vectors with a traditionally important role
in the study of noncontextuality of physical theories, and more recently in the
quantification of quantum zero-error capacities. Finally, as a corollary of our
results and a result by Avis, Hasegawa, Kikuchi, and Sasaki on the quantum
chromatic number, we give a family of Kochen-Specker sets of growing dimension.Comment: 12 page
A Generalization of Kochen-Specker Sets Relates Quantum Coloring to Entanglement-Assisted Channel Capacity
We introduce two generalizations of Kochen-Specker (KS) sets: projective KS
sets and generalized KS sets. We then use projective KS sets to characterize
all graphs for which the chromatic number is strictly larger than the quantum
chromatic number. Here, the quantum chromatic number is defined via a nonlocal
game based on graph coloring. We further show that from any graph with
separation between these two quantities, one can construct a classical channel
for which entanglement assistance increases the one-shot zero-error capacity.
As an example, we exhibit a new family of classical channels with an
exponential increase.Comment: 16 page
Estimating quantum chromatic numbers
We develop further the new versions of quantum chromatic numbers of graphs
introduced by the first and fourth authors. We prove that the problem of
computation of the commuting quantum chromatic number of a graph is solvable by
an SDP algorithm and describe an hierarchy of variants of the commuting quantum
chromatic number which converge to it. We introduce the tracial rank of a
graph, a parameter that gives a lower bound for the commuting quantum chromatic
number and parallels the projective rank, and prove that it is multiplicative.
We describe the tracial rank, the projective rank and the fractional chromatic
numbers in a unified manner that clarifies their connection with the commuting
quantum chromatic number, the quantum chromatic number and the classical
chromatic number, respectively. Finally, we present a new SDP algorithm that
yields a parameter larger than the Lov\'asz number and is yet a lower bound for
the tracial rank of the graph. We determine the precise value of the tracial
rank of an odd cycle.Comment: 34 pages; v2 has improved presentation based after referees'
comments, published versio
The Game Chromatic Number of Complete Multipartite Graphs with No Singletons
In this paper we investigate the game chromatic number for complete
multipartite graphs. We devise several strategies for Alice, and one strategy
for Bob, and we prove their optimality in all complete multipartite graphs with
no singletons. All the strategies presented are computable in linear time, and
the values of the game chromatic number depend directly only on the number and
the sizes of sets in the partition
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