185 research outputs found
Linearly dependent vectorial decomposition of clutters
This paper deals with the question of completing a monotone increasing family of
subsets of a finite set
to obtain the linearly dependent subsets of a family of
vectors of a vector space. Specifically, we demonstrate that such vectorial completions
of the family of subsets Âż exist and, in addition, we show that the minimal
vectorial completions of the family Âż provide a decomposition of the clutter of the
inclusion-minimal elements of Âż. The computation of such vectorial decomposition
of clutters is also discussed in some cases.Peer ReviewedPostprint (authorâs final draft
Trade-offs in multi-party Bell inequality violations in qubit networks
Two overlapping bipartite binary input Bell inequalities cannot be
simultaneously violated as this would contradict the usual no-signalling
principle. This property is known as monogamy of Bell inequality violations and
generally Bell monogamy relations refer to trade-offs between simultaneous
violations of multiple inequalities. It turns out that multipartite Bell
inequalities admit weaker forms of monogamies that allow for violations of a
few inequalities at once. Here we systematically study monogamy relations
between correlation Bell inequalities both within quantum theory and under the
sole assumption of no signalling. We first investigate the trade-offs in Bell
violations arising from the uncertainty relation for complementary binary
observables, and exhibit several network configurations in which a tight
trade-off arises in this fashion. We then derive a tight trade-off relation
which cannot be obtained from the uncertainty relation showing that it does not
capture monogamy entirely. The results are extended to Bell inequalities
involving different number of parties and find applications in
device-independent secret sharing and device-independent randomness extraction.
Although two multipartite Bell inequalities may be violated simultaneously, we
show that genuine multi-party non-locality, as evidenced by a generalised
Svetlichny inequality, does exhibit monogamy property. Finally, using the
relations derived we reveal the existence of flat regions in the set of quantum
correlations.Comment: 15 pages, 5 figure
Multipartite entanglement detection for hypergraph states
We study the entanglement properties of quantum hypergraph states of
qubits, focusing on multipartite entanglement. We compute multipartite
entanglement for hypergraph states with a single hyperedge of maximum
cardinality, for hypergraph states endowed with all possible hyperedges of
cardinality equal to and for those hypergraph states with all possible
hyperedges of cardinality greater than or equal to . We then find a lower
bound to the multipartite entanglement of a generic quantum hypergraph state.
We finally apply the multipartite entanglement results to the construction of
entanglement witness operators, able to detect genuine multipartite
entanglement in the neighbourhood of a given hypergraph state. We first build
entanglement witnesses of the projective type, then propose a class of
witnesses based on the stabilizer formalism, hence called stabilizer witnesses,
able to reduce the experimental effort from an exponential to a linear growth
in the number of local measurement settings with the number of qubits
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