3,836 research outputs found
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.
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
Canonical quantum gravity in the Vassiliev invariants arena: I. Kinematical structure
We generalize the idea of Vassiliev invariants to the spin network context,
with the aim of using these invariants as a kinematical arena for a canonical
quantization of gravity. This paper presents a detailed construction of these
invariants (both ambient and regular isotopic) requiring a significant
elaboration based on the use of Chern-Simons perturbation theory which extends
the work of Kauffman, Martin and Witten to four-valent networks. We show that
this space of knot invariants has the crucial property -from the point of view
of the quantization of gravity- of being loop differentiable in the sense of
distributions. This allows the definition of diffeomorphism and Hamiltonian
constraints. We show that the invariants are annihilated by the diffeomorphism
constraint. In a companion paper we elaborate on the definition of a
Hamiltonian constraint, discuss the constraint algebra, and show that the
construction leads to a consistent theory of canonical quantum gravity.Comment: 21 Pages, RevTex, many figures included with psfi
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