We present entanglement witness operators for detecting genuine multipartite entanglement. These witnesses are robust against noise and require only two local measurement settings when used in an ...experiment, independent of the number of qubits. This allows detection of entanglement for an increasing number of parties without a corresponding increase in effort. The witnesses presented detect states close to Greenberger-Horne-Zeilinger, cluster, and graph states. Connections to Bell inequalities are also discussed.
The crossing number of a graph
G
is the minimum number of edge crossings over all drawings of
G
in the plane. A graph
G
is
k
-crossing-critical if its crossing number is at least
k
, but if we ...remove any edge of
G
, its crossing number drops below
k
. There are examples of
k
-crossing-critical graphs that do not have drawings with exactly
k
crossings. Richter and Thomassen proved in 1993 that if
G
is
k
-crossing-critical, then its crossing number is at most
2.5
k
+
16
. We improve this bound to
2
k
+
8
k
+
47
.
A Crossing Lemma for Multigraphs Pach, János; Tóth, Géza
Discrete & computational geometry,
06/2020, Letnik:
63, Številka:
4
Journal Article
Recenzirano
Odprti dostop
Let
G
be a drawing of a graph with
n
vertices and
e
>
4
n
edges, in which no two adjacent edges cross and any pair of independent edges cross at most once. According to the celebrated Crossing Lemma ...of Ajtai, Chvátal, Newborn, Szemerédi and Leighton, the number of crossings in
G
is at least
c
e
3
n
2
, for a suitable constant
c
>
0
. In a seminal paper, Székely generalized this result to multigraphs, establishing the lower bound
c
e
3
m
n
2
, where
m
denotes the maximum multiplicity of an edge in
G
. We get rid of the dependence on
m
by showing that, as in the original Crossing Lemma, the number of crossings is at least
c
′
e
3
n
2
for some
c
′
>
0
, provided that the “lens” enclosed by every pair of parallel edges in
G
contains at least one vertex. This settles a conjecture of Bekos, Kaufmann, and Raftopoulou.
Let
H
be a complete
r
-uniform hypergraph such that two vertices are marked in each edge as its ‘boundary’ vertices. A linear ordering of the vertex set of
H
is called an
agreeing linear order
, ...provided all vertices of each edge of
H
lie between its two boundary vertices. We prove the following Helly-type theorem: if there is an agreeing linear order on the vertex set of every subhypergraph of
H
with at most 2
r
− 2 vertices, then there is an agreeing linear order on the vertex set of
H
. We also show that the constant 2
r
− 2 cannot be reduced in the theorem. The case
r
= 3 of the theorem has particular interest in the axiomatic theory of betweenness. Similar results are obtained for further
r
-uniform hypergraphs (
r
≥ 3), where one or two vertices are marked in each edge, and the linear orders need to satisfy various rules of agreement. In one of the cases we prove that no such Helly-type statement holds.
A planar point set of
n
points is called
γ
-
dense
if the ratio of the largest and smallest distances among the points is at most
γ
n
. We construct a dense set of
n
points in the plane with
n
e
Ω
(
...log
n
)
halving lines. This improves the bound
Ω
(
n
log
n
)
of Edelsbrunner et al. (Discrete Comput Geom 17(3):243–255,
1997
). Our construction can be generalized to higher dimensions, for any
d
we construct a dense point set of
n
points in
R
d
with
n
d
-
1
e
Ω
(
log
n
)
halving hyperplanes. Our lower bounds are asymptotically the same as the best known lower bounds for general point sets.
Linear-optics quantum logic operations enabled the observation of a four-photon cluster state. We prove genuine four-partite entanglement and study its persistency, demonstrating remarkable ...differences from the usual Greenberger-Horne-Zeilinger (GHZ) state. Efficient analysis tools are introduced in the experiment, which will be of great importance in further studies on multiparticle entangled states.
We report the synthesis of N-doped TiO2 nanofibers and high photocatalytic efficiency in generating hydrogen from ethanol–water mixtures under UV-A and UV-B irradiation. Titanate nanofibers ...synthesized by hydrothermal method are annealed in air and/or ammonia to achieve N-doped anatase fibers. Depending on the synthesis route, either interstitial N atoms or new N–Ti bonds appear in the lattice, resulting in slight lattice expansion as shown by XPS and HR-TEM analysis, respectively. These nanofibers were then used as support for Pd and Pt nanoparticles deposited with wet impregnation followed by calcination and reduction. In the hydrogen generation tests, the N-doped samples were clearly outperforming their undoped counterparts, showing remarkable efficiency not only under UV-B but also with UV-A illumination. When 100 mg of catalyst (N-doped TiO2 nanofiber decorated with Pt nanoparticles) was applied to 1 L of water–ethanol mixture, the H2 evolution rates were as high as 700 μmol/h (UV-A) and 2250 μmol/h (UV-B) corresponding to photo energy conversion percentages of ∼3.6 and ∼12.3%, respectively.
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