2 research outputs found
Quantum channels from association schemes
We propose in this note the study of quantum channels from association
schemes. This is done by interpreting the -matrices of a scheme as the
Kraus operators of a channel. Working in the framework of one-shot zero-error
information theory, we give bounds and closed formulas for various independence
numbers of the relative non-commutative (confusability) graphs, or,
equivalently, graphical operator systems. We use pseudocyclic association
schemes as an example. In this case, we show that the unitary
entanglement-assisted independence number grows at least quadratically faster,
with respect to matrix size, than the independence number. The latter parameter
was introduced by Beigi and Shor as a generalization of the one-shot Shannon
capacity, in analogy with the corresponding graph-theoretic notion.Comment: 6 page
Quantum source-channel coding and non-commutative graph theory
Alice and Bob receive a bipartite state (possibly entangled) from some finite
collection or from some subspace. Alice sends a message to Bob through a noisy
quantum channel such that Bob may determine the initial state, with zero chance
of error. This framework encompasses, for example, teleportation, dense coding,
entanglement assisted quantum channel capacity, and one-way communication
complexity of function evaluation.
With classical sources and channels, this problem can be analyzed using graph
homomorphisms. We show this quantum version can be analyzed using homomorphisms
on non-commutative graphs (an operator space generalization of graphs).
Previously the Lov\'{a}sz number has been generalized to
non-commutative graphs; we show this to be a homomorphism monotone, thus
providing bounds on quantum source-channel coding. We generalize the Schrijver
and Szegedy numbers, and show these to be monotones as well. As an application
we construct a quantum channel whose entanglement assisted zero-error one-shot
capacity can only be unlocked by using a non-maximally entangled state.
These homomorphisms allow definition of a chromatic number for
non-commutative graphs. Many open questions are presented regarding the
possibility of a more fully developed theory.Comment: 24 page