287 research outputs found
On the theta number of powers of cycle graphs
We give a closed formula for Lovasz theta number of the powers of cycle
graphs and of their complements, the circular complete graphs. As a
consequence, we establish that the circular chromatic number of a circular
perfect graph is computable in polynomial time. We also derive an asymptotic
estimate for this theta number.Comment: 17 page
Entanglement-assisted zero-error source-channel coding
We study the use of quantum entanglement in the zero-error source-channel
coding problem. Here, Alice and Bob are connected by a noisy classical one-way
channel, and are given correlated inputs from a random source. Their goal is
for Bob to learn Alice's input while using the channel as little as possible.
In the zero-error regime, the optimal rates of source codes and channel codes
are given by graph parameters known as the Witsenhausen rate and Shannon
capacity, respectively. The Lov\'asz theta number, a graph parameter defined by
a semidefinite program, gives the best efficiently-computable upper bound on
the Shannon capacity and it also upper bounds its entanglement-assisted
counterpart. At the same time it was recently shown that the Shannon capacity
can be increased if Alice and Bob may use entanglement.
Here we partially extend these results to the source-coding problem and to
the more general source-channel coding problem. We prove a lower bound on the
rate of entanglement-assisted source-codes in terms Szegedy's number (a
strengthening of the theta number). This result implies that the theta number
lower bounds the entangled variant of the Witsenhausen rate. We also show that
entanglement can allow for an unbounded improvement of the asymptotic rate of
both classical source codes and classical source-channel codes. Our separation
results use low-degree polynomials due to Barrington, Beigel and Rudich,
Hadamard matrices due to Xia and Liu and a new application of remote state
preparation.Comment: Title has been changed. Previous title was 'Zero-error source-channel
coding with entanglement'. Corrected an error in Lemma 1.
Fourier analysis on finite groups and the Lov\'asz theta-number of Cayley graphs
We apply Fourier analysis on finite groups to obtain simplified formulations
for the Lov\'asz theta-number of a Cayley graph. We put these formulations to
use by checking a few cases of a conjecture of Ellis, Friedgut, and Pilpel made
in a recent article proving a version of the Erd\H{o}s-Ko-Rado theorem for
-intersecting families of permutations. We also introduce a -analog of
the notion of -intersecting families of permutations, and we verify a few
cases of the corresponding Erd\H{o}s-Ko-Rado assertion by computer.Comment: 9 pages, 0 figure
Bounds on entanglement assisted source-channel coding via the Lovasz theta number and its variants
We study zero-error entanglement assisted source-channel coding
(communication in the presence of side information). Adapting a technique of
Beigi, we show that such coding requires existence of a set of vectors
satisfying orthogonality conditions related to suitably defined graphs and
. Such vectors exist if and only if where represents the Lov\'asz number. We
also obtain similar inequalities for the related Schrijver and
Szegedy numbers.
These inequalities reproduce several known bounds and also lead to new
results. We provide a lower bound on the entanglement assisted cost rate. We
show that the entanglement assisted independence number is bounded by the
Schrijver number: . Therefore, we are able to
disprove the conjecture that the one-shot entanglement-assisted zero-error
capacity is equal to the integer part of the Lov\'asz number. Beigi introduced
a quantity as an upper bound on and posed the question of
whether . We answer this in the
affirmative and show that a related quantity is equal to . We show that a quantity recently introduced
in the context of Tsirelson's conjecture is equal to .
In an appendix we investigate multiplicativity properties of Schrijver's and
Szegedy's numbers, as well as projective rank.Comment: Fixed proof of multiplicativity; more connections to prior work in
conclusion; many changes in expositio
The asymptotic spectrum of graphs and the Shannon capacity
We introduce the asymptotic spectrum of graphs and apply the theory of
asymptotic spectra of Strassen (J. Reine Angew. Math. 1988) to obtain a new
dual characterisation of the Shannon capacity of graphs. Elements in the
asymptotic spectrum of graphs include the Lov\'asz theta number, the fractional
clique cover number, the complement of the fractional orthogonal rank and the
fractional Haemers bounds
Graph-theoretical Bounds on the Entangled Value of Non-local Games
We introduce a novel technique to give bounds to the entangled value of
non-local games. The technique is based on a class of graphs used by Cabello,
Severini and Winter in 2010. The upper bound uses the famous Lov\'asz theta
number and is efficiently computable; the lower one is based on the quantum
independence number, which is a quantity used in the study of
entanglement-assisted channel capacities and graph homomorphism games.Comment: 10 pages, submission to the 9th Conference on the Theory of Quantum
Computation, Communication, and Cryptography (TQC 2014
A recursive Lov\'asz theta number for simplex-avoiding sets
We recursively extend the Lov\'asz theta number to geometric hypergraphs on
the unit sphere and on Euclidean space, obtaining an upper bound for the
independence ratio of these hypergraphs. As an application we reprove a result
in Euclidean Ramsey theory in the measurable setting, namely that every
-simplex is exponentially Ramsey, and we improve existing bounds for the
base of the exponential.Comment: 13 pages, 3 figure
- …