42,064 research outputs found
Violating the Shannon capacity of metric graphs with entanglement
The Shannon capacity of a graph G is the maximum asymptotic rate at which
messages can be sent with zero probability of error through a noisy channel
with confusability graph G. This extensively studied graph parameter disregards
the fact that on atomic scales, Nature behaves in line with quantum mechanics.
Entanglement, arguably the most counterintuitive feature of the theory, turns
out to be a useful resource for communication across noisy channels. Recently,
Leung, Mancinska, Matthews, Ozols and Roy [Comm. Math. Phys. 311, 2012]
presented two examples of graphs whose Shannon capacity is strictly less than
the capacity attainable if the sender and receiver have entangled quantum
systems. Here we give new, possibly infinite, families of graphs for which the
entangled capacity exceeds the Shannon capacity.Comment: 15 pages, 2 figure
Shannon capacity and the Lovász Theta function
This thesis focuses on the study of two graph parameters known as the Shannon capacity and the Lovász number. The first was introduced by Claude Elwood Shannon on \cite{shannon} and describes the maximum rate at which information can be transmitted through a noisy channel of comunication, where the noise of the channel is encoded by a graph. The second one was introduced by László Lovász as an attempt to determine the first. We start by introducing all the required concepts in order to formally state the problem by Shannon. We then move on to develop some theory regarding the spectrum of a set of matrices associated with graphs, providing an algebraic aproach to the matter in question. Then we prove some results due to Lovász on \cite{Lovasz} that allow us to effectively compute the Shannon capacity for specific families of graphs, including the Kneser graphs and the perfect graphs. We end up by studying further properties of the Lovász number, of particular interest for the case of perfect graphs
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.
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