5,580 research outputs found
Entanglement and quantum combinatorial designs
We introduce several classes of quantum combinatorial designs, namely quantum
Latin squares, cubes, hypercubes and a notion of orthogonality between them. A
further introduced notion, quantum orthogonal arrays, generalizes all previous
classes of designs. We show that mutually orthogonal quantum Latin arrangements
can be entangled in the same way than quantum states are entangled.
Furthermore, we show that such designs naturally define a remarkable class of
genuinely multipartite highly entangled states called -uniform, i.e.
multipartite pure states such that every reduction to parties is maximally
mixed. We derive infinitely many classes of mutually orthogonal quantum Latin
arrangements and quantum orthogonal arrays having an arbitrary large number of
columns. The corresponding multipartite -uniform states exhibit a high
persistency of entanglement, which makes them ideal candidates to develop
multipartite quantum information protocols.Comment: 14 pages, 3 figures. Comments are very welcome
Commutative association schemes
Association schemes were originally introduced by Bose and his co-workers in
the design of statistical experiments. Since that point of inception, the
concept has proved useful in the study of group actions, in algebraic graph
theory, in algebraic coding theory, and in areas as far afield as knot theory
and numerical integration. This branch of the theory, viewed in this collection
of surveys as the "commutative case," has seen significant activity in the last
few decades. The goal of the present survey is to discuss the most important
new developments in several directions, including Gelfand pairs, cometric
association schemes, Delsarte Theory, spin models and the semidefinite
programming technique. The narrative follows a thread through this list of
topics, this being the contrast between combinatorial symmetry and
group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes
(based on group actions) and its connection to the Terwilliger algebra (based
on combinatorial symmetry). We propose this new role of the Terwilliger algebra
in Delsarte Theory as a central topic for future work.Comment: 36 page
Computational complexity of reconstruction and isomorphism testing for designs and line graphs
Graphs with high symmetry or regularity are the main source for
experimentally hard instances of the notoriously difficult graph isomorphism
problem. In this paper, we study the computational complexity of isomorphism
testing for line graphs of - designs. For this class of
highly regular graphs, we obtain a worst-case running time of for bounded parameters . In a first step, our approach
makes use of the Babai--Luks algorithm to compute canonical forms of
-designs. In a second step, we show that -designs can be reconstructed
from their line graphs in polynomial-time. The first is algebraic in nature,
the second purely combinatorial. For both, profound structural knowledge in
design theory is required. Our results extend earlier complexity results about
isomorphism testing of graphs generated from Steiner triple systems and block
designs.Comment: 12 pages; to appear in: "Journal of Combinatorial Theory, Series A
Multiset Combinatorial Batch Codes
Batch codes, first introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai,
mimic a distributed storage of a set of data items on servers, in such
a way that any batch of data items can be retrieved by reading at most some
symbols from each server. Combinatorial batch codes, are replication-based
batch codes in which each server stores a subset of the data items.
In this paper, we propose a generalization of combinatorial batch codes,
called multiset combinatorial batch codes (MCBC), in which data items are
stored in servers, such that any multiset request of items, where any
item is requested at most times, can be retrieved by reading at most
items from each server. The setup of this new family of codes is motivated by
recent work on codes which enable high availability and parallel reads in
distributed storage systems. The main problem under this paradigm is to
minimize the number of items stored in the servers, given the values of
, which is denoted by . We first give a necessary and
sufficient condition for the existence of MCBCs. Then, we present several
bounds on and constructions of MCBCs. In particular, we
determine the value of for any , where
is the maximum size of a binary constant weight code of length
, distance four and weight . We also determine the exact value of
when or
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