1,481 research outputs found
Undirected Graphs of Entanglement Two
Entanglement is a complexity measure of directed graphs that origins in fixed
point theory. This measure has shown its use in designing efficient algorithms
to verify logical properties of transition systems. We are interested in the
problem of deciding whether a graph has entanglement at most k. As this measure
is defined by means of games, game theoretic ideas naturally lead to design
polynomial algorithms that, for fixed k, decide the problem. Known
characterizations of directed graphs of entanglement at most 1 lead, for k = 1,
to design even faster algorithms. In this paper we present an explicit
characterization of undirected graphs of entanglement at most 2. With such a
characterization at hand, we devise a linear time algorithm to decide whether
an undirected graph has this property
Quantum Experiments and Graphs: Multiparty States as coherent superpositions of Perfect Matchings
We show a surprising link between experimental setups to realize
high-dimensional multipartite quantum states and Graph Theory. In these setups,
the paths of photons are identified such that the photon-source information is
never created. We find that each of these setups corresponds to an undirected
graph, and every undirected graph corresponds to an experimental setup. Every
term in the emerging quantum superposition corresponds to a perfect matching in
the graph. Calculating the final quantum state is in the complexity class
#P-complete, thus cannot be done efficiently. To strengthen the link further,
theorems from Graph Theory -- such as Hall's marriage problem -- are rephrased
in the language of pair creation in quantum experiments. We show explicitly how
this link allows to answer questions about quantum experiments (such as which
classes of entangled states can be created) with graph theoretical methods, and
potentially simulate properties of Graphs and Networks with quantum experiments
(such as critical exponents and phase transitions).Comment: 6+5 pages, 4+7 figure
Quantum Capacities for Entanglement Networks
We discuss quantum capacities for two types of entanglement networks:
for the quantum repeater network with free classical
communication, and for the tensor network as the rank of the
linear operation represented by the tensor network. We find that
always equals in the regularized case for the samenetwork graph.
However, the relationships between the corresponding one-shot capacities
and are more complicated, and the min-cut upper
bound is in general not achievable. We show that the tensor network can be
viewed as a stochastic protocol with the quantum repeater network, such that
is a natural upper bound of . We analyze the
possible gap between and for certain networks,
and compare them with the one-shot classical capacity of the corresponding
classical network
Holographic entropy inequalities and gapped phases of matter
We extend our studies of holographic entropy inequalities to gapped phases of
matter. For any number of regions, we determine the linear entropy inequalities
satisfied by systems in which the entanglement entropy satisfies an exact area
law. In particular, we find that all holographic entropy inequalities are valid
in such systems. In gapped systems with topological order, the "cyclic
inequalities" derived recently for the holographic entanglement entropy
generalize the Kitaev-Preskill formula for the topological entanglement
entropy. Finally, we propose a candidate linear inequality for general 4-party
quantum states.Comment: 20 pages, 4 figures. v2: section 4 rewritten, where all linear
entropy (in)equalities satisfied by area-law systems are derived and an error
in their relations to graph theory is correcte
The Variable Hierarchy for the Games mu-Calculus
Parity games are combinatorial representations of closed Boolean mu-terms. By
adding to them draw positions, they have been organized by Arnold and one of
the authors into a mu-calculus. As done by Berwanger et al. for the
propositional modal mu-calculus, it is possible to classify parity games into
levels of a hierarchy according to the number of fixed-point variables. We ask
whether this hierarchy collapses w.r.t. the standard interpretation of the
games mu-calculus into the class of all complete lattices. We answer this
question negatively by providing, for each n >= 1, a parity game Gn with these
properties: it unravels to a mu-term built up with n fixed-point variables, it
is semantically equivalent to no game with strictly less than n-2 fixed-point
variables
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