2,629 research outputs found
On Hardness of the Joint Crossing Number
The Joint Crossing Number problem asks for a simultaneous embedding of two
disjoint graphs into one surface such that the number of edge crossings
(between the two graphs) is minimized. It was introduced by Negami in 2001 in
connection with diagonal flips in triangulations of surfaces, and subsequently
investigated in a general form for small-genus surfaces. We prove that all of
the commonly considered variants of this problem are NP-hard already in the
orientable surface of genus 6, by a reduction from a special variant of the
anchored crossing number problem of Cabello and Mohar
Proposed test of macroscopic quantum contextuality
We show that, for any system with a number of levels which can be identified
with n qubits, there is an inequality for the correlations between three
compatible dichotomic measurements which must be satisfied by any noncontextual
theory, but is violated by any quantum state. Remarkably, the violation grows
exponentially with n, and the tolerated error per correlation also increases
with n, showing that state-independent quantum contextuality is experimentally
observable in complex systems.Comment: REVTeX4, 5 pages, 1 figur
Kochen-Specker theorem and experimental test on hidden variables
A recent proposal to experimentally test quantum mechanics against
noncontextual hidden-variable theories [Phys. Rev. Lett. 80, 1797 (1998)] is
shown to be related with the smallest proof of the Kochen-Specker theorem
currently known [Phys. Lett. A 212, 183 (1996)]. This proof contains eighteen
yes-no questions about a four-dimensional physical system, combined in nine
mutually incompatible tests. When these tests are considered as tests about a
two-part two-state system, then quantum mechanics and non-contextual hidden
variables make the same predictions for eight of them, but make different
predictions for the ninth. Therefore, this ninth test would allow us to
discriminate between quantum mechanics and noncontextual hidden-variable
theories in a (gedanken) single run experiment.Comment: 4 pages, 1 figure. To appear in Int. J. Mod. Phys.
Twin inequality for fully contextual quantum correlations
Quantum mechanics exhibits a very peculiar form of contextuality. Identifying
and connecting the simplest scenarios in which more general theories can or
cannot be more contextual than quantum mechanics is a fundamental step in the
quest for the principle that singles out quantum contextuality. The former
scenario corresponds to the Klyachko-Can-Binicioglu-Shumovsky (KCBS)
inequality. Here we show that there is a simple tight inequality, twin to the
KCBS, for which quantum contextuality cannot be outperformed. In a sense, this
twin inequality is the simplest tool for recognizing fully contextual quantum
correlations.Comment: REVTeX4, 4 pages, 1 figur
Two Party Non-Local Games
In this work we have introduced two party games with respective winning
conditions. One cannot win these games deterministically in the classical world
if they are not allowed to communicate at any stage of the game. Interestingly
we find out that in quantum world, these winning conditions can be achieved if
the players share an entangled state. We also introduced a game which is
impossible to win if the players are not allowed to communicate in classical
world (both probabilistically and deterministically), yet there exists a
perfect quantum strategy by following which, one can attain the winning
condition of the game.Comment: Accepted in International Journal of Theoretical Physic
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Minimum Cell Connection in Line Segment Arrangements
We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points a and b in different cells of the induced arrangement:
[(i)] compute the minimum number of segments one needs to remove so that there is a path connecting a to b that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell.
We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting a
to b must stay inside a given polygon P with a constant number of holes, the segments are contained in P, and the endpoints of the segments are on the boundary of P. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution
Bell's theorem without inequalities and without alignments
A proof of Bell's theorem without inequalities is presented which exhibits
three remarkable properties: (a) reduced local states are immune to collective
decoherence; (b) distant local setups do not need to be aligned, since the
required perfect correlations are achieved for any local rotation of the local
setups; (c) local measurements require only individual measurements on the
qubits. Indeed, it is shown that this proof is essentially the only one which
fulfils (a), (b), and (c).Comment: REVTeX4, 4 page
Finite-precision measurement does not nullify the Kochen-Specker theorem
It is proven that any hidden variable theory of the type proposed by Meyer
[Phys. Rev. Lett. {\bf 83}, 3751 (1999)], Kent [{\em ibid.} {\bf 83}, 3755
(1999)], and Clifton and Kent [Proc. R. Soc. London, Ser. A {\bf 456}, 2101
(2000)] leads to experimentally testable predictions that are in contradiction
with those of quantum mechanics. Therefore, it is argued that the existence of
dense Kochen-Specker-colorable sets must not be interpreted as a nullification
of the physical impact of the Kochen-Specker theorem once the finite precision
of real measurements is taken into account.Comment: REVTeX4, 5 page
The complexity of separating points in the plane
We study the following separation problem: given n connected curves and two points s and t in the plane, compute the minimum number of curves one needs to retain so that any path connecting s to t intersects some of the retained curves. We give the first polynomial (O(n3)) time algorithm for the problem, assuming that the curves have reasonable computational properties. The algorithm is based on considering the intersection graph of the curves, defining an appropriate family of closed walks in the intersection graph that satisfies the 3-path-condition, and arguing that a shortest cycle in the family gives an optimal solution. The 3-path-condition has been used mainly in topological graph theory, and thus its use here makes the connection to topology clear. We also show that the generalized version, where several input points are to be separated, is NP-hard for natural families of curves, like segments in two directions or unit circles
The Complexity of Separating Points in the Plane
We study the following separation problem: given n connected curves and two points s and t in the plane, compute the minimum number of curves one needs to retain so that any path connecting s to t intersects some of the retained curves. We give the first polynomial (O(n3)) time algorithm for the problem, assuming that the curves have reasonable computational properties. The algorithm is based on considering the intersection graph of the curves, defining an appropriate family of closed walks in the intersection graph that satisfies the 3-path-condition, and arguing that a shortest cycle in the family gives an optimal solution. The 3-path-condition has been used mainly in topological graph theory, and thus its use here makes the connection to topology clear. We also show that the generalized version, where several input points are to be separated, is NP-hard for natural families of curves, like segments in two directions or unit circles
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