arXiv subjects

Géza Tóth

Publications and source records attributed to Géza Tóth.

54 records · Page 3Linked to original sources

arXiv2021

On the hollow enclosed by convex sets

For $n\leq d$, a family ${\cal F}=\{C_0,C_1,\ldots, C_n\}$ of compact convex sets in $R^d$ is called an $n$-critical family provided any $n$ members of ${\cal F}$ have a non-empty intersection, but $\bigcap_{i=0}^n C_i=\varnothing$. If $n=d$ then a lemma on the intersection of convex sets due to Klee implies that the $d+1$ members of the $d$-critical family enclose a `hollow' in $R^d$, a bounded connected component of $R^d\setminus\bigcup_{i=0}^n C_i.$ Here we prove that the closure of the convex hull of a hollow in $R^d$ is a $d$-simplex.

math.CO↗

arXiv2021

Bound entangled singlet-like states for quantum metrology

Bipartite entangled quantum states with a positive partial transpose (PPT), i.e., PPT entangled states, are usually considered very weakly entangled. Since no pure entanglement can be distilled from them, they are also called bound entangled. In this paper we present two classes of ($2d\times 2d$)-dimensional PPT entangled states for any $d\ge 2$ which outperform all separable states in metrology significantly. We present strong evidence that our states provide the maximal metrological gain achievable by PPT states for a given system size. When the dimension $d$ goes to infinity, the metrological gain of these states becomes maximal and equals the metrological gain of a pair of maximally entangled qubits. Thus, we argue that our states could be called "PPT singlets."

quant-ph↗

arXiv2021

A slightly better bound on the crossing number in terms of the pair-crossing number

The crossing number of a graph $G$, ${\mbox{cr}}(G)$, is the minimum number of crossings, the pair-crossing number, ${\mbox{pcr}}(G)$, is the minimum number of pairs of crossing edges over all drawings of $G$. In this note we show that ${\mbox{cr}}(G)=O({\mbox{pcr}}(G)^{3/2}\log{\mbox{pcr}}(G))$, which is an improvement of the result of Matoušek, by a log factor.

math.CO↗

arXiv2021

Activating hidden metrological usefulness

We consider bipartite entangled states that cannot outperform separable states in any linear interferometer. Then, we show that these states can still be more useful metrologically than separable states if several copies of the state are provided or an ancilla is added to the quantum system. We present a general method to find the local Hamiltonian for which a given quantum state performs the best compared to separable states. We obtain analytically the optimal Hamiltonian for some quantum states with a high symmetry. We show that all bipartite entangled pure states outperform separable states in metrology. Some potential applications of the results are also suggested.

quant-ph↗

arXiv2020

Improvement on the crossing number of crossing-critical graphs

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 $2.5k+16$. We improve this bound to $2k+6\sqrt{k}+44$.

math.CO↗

arXiv2020

Crossings between non-homotopic edges

We call a multigraph {\em non-homotopic} if it can be drawn in the plane in such a way that no two edges connecting the same pair of vertices can be continuously transformed into each other without passing through a vertex, and no loop can be shrunk to its end-vertex in the same way. It is easy to see that a non-homotopic multigraph on $n>1$ vertices can have arbitrarily many edges. We prove that the number of crossings between the edges of a non-homotopic multigraph with $n$ vertices and $m>4n$ edges is larger than $c\frac{m^2}{n}$ for some constant $c>0$, and that this bound is tight up to a polylogarithmic factor. We also show that the lower bound is not asymptotically sharp as $n$ is fixed and $m$ tends to infinity.

math.CO↗

arXiv2019

Petruska's question on planar convex sets

Given $2k-1$ convex sets in $R^2$ such that no point of the plane is covered by more than $k$ of the sets, is it true that there are two among the convex sets whose union contains all $k$-covered points of the plane? This question due to Gy. Petruska has an obvious affirmative answer for $k=1,2,3$; we show here that the claim is also true for $k=4$, and we present a counterexample for $k=5$. We explain how Petruska's geometry question fits into the classical hypergraph extremal problems, called arrow problems, proposed by P. Erdős.

math.CO↗

arXiv2019

Dense point sets with many halving lines

A planar point set of $n$ points is called {\em $γ$-dense} if the ratio of the largest and smallest distances among the points is at most $γ\sqrt{n}$. We construct a dense set of $n$ points in the plane with $ne^{Ω\left({\sqrt{\log n}}\right)}$ halving lines. This improves the bound $Ω(n\log n)$ of Edelsbrunner, Valtr and Welzl from 1997. Our construction can be generalized to higher dimensions, for any $d$ we construct a dense point set of $n$ points in $\mathbb{R}^d$ with $n^{d-1}e^{Ω\left({\sqrt{\log n}}\right)}$ halving hyperplanes. Our lower bounds are asymptotically the same as the best known lower bounds for general point sets.

math.CO↗

arXiv2017

Entanglement and extreme planar spin squeezing

We introduce an entanglement-depth criterion optimized for planar quantum squeezed (PQS) states. It is connected with the sensitivity of such states for estimating an arbitrary, not necessarily small phase. We compare numerically our criterion with the well-known extreme spin squeezing condition of Sørensen and Mølmer [Phys. Rev. Lett. 86, 4431 (2001)] and show that our condition detects a higher depth of entanglement when both planar spin variances are squeezed below the standard quantum limit. We employ our theory to monitor the entanglement dynamics in a PQS state produced via quantum non-demolition (QND) measurements using data from a recent experiment [Phys. Rev. Lett. 118, 233603 (2017)].

quant-ph↗

arXiv2016

Note on k-planar crossing numbers

The crossing number $cr(G)$ of a graph $G=(V,E)$ is the smallest number of edge crossings over all drawings of $G$ in the plane. For any $k\ge 1$, the $k$-planar crossing number of $G$, $cr_k(G)$, is defined as the minimum of $cr(G_0)+cr(G_1)+\ldots+cr(G_{k-1})$ over all graphs $G_0, G_1,\ldots, G_{k-1}$ with $\cup_{i=0}^{k-1}G_i=G$. It is shown that for every $k\ge 1$, we have $cr_k(G)\le \left(\frac{2}{k^2}-\frac1{k^3}\right)cr(G)$. This bound does not remain true if we replace the constant $\frac{2}{k^2}-\frac1{k^3}$ by any number smaller than $\frac1{k^2}$. Some of the results extend to the rectilinear variants of the $k$-planar crossing number.

math.CO↗

arXiv2015

How long does it take to obtain a physical density matrix?

The statistical nature of measurements alone easily causes unphysical estimates in quantum state tomography. We show that multinomial or Poissonian noise results in eigenvalue distributions converging to the Wigner semicircle distribution for already a modest number of qubits. This enables to specify the number of measurements necessary to avoid unphysical solutions as well as a new approach to obtain physical estimates.

quant-ph↗

arXiv2015

Improvements on the density of maximal 1-planar graphs

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. A graph, together with a 1-planar drawing is called 1-plane. Brandenburg et al. showed that there are maximal 1-planar graphs with only $\frac{45}{17}n + O(1)\approx 2.647n$ edges and maximal 1-plane graphs with only $\frac{7}{3}n+O(1)\approx 2.33n$ edges. On the other hand, they showed that a maximal 1-planar graph has at least $\frac{28}{13}n-O(1)\approx 2.15n-O(1)$ edges, and a maximal 1-plane graph has at least $2.1n-O(1)$ edges. We improve both lower bounds to $\frac{20n}{9}\approx 2.22n$.

math.CO↗

arXiv2015

Separation with restricted families of sets

Given a finite $n$-element set $X$, a family of subsets ${\mathcal F}\subset 2^X$ is said to separate $X$ if any two elements of $X$ are separated by at least one member of $\mathcal F$. It is shown that if $|\mathcal F|>2^{n-1}$, then one can select $\lceil\log n\rceil+1$ members of $\mathcal F$ that separate $X$. If $|\mathcal F|\ge α2^n$ for some $0<α<1/2$, then $\log n+O(\log\frac1α\log\log\frac1α)$ members of $\mathcal F$ are always sufficient to separate all pairs of elements of $X$ that are separated by some member of $\mathcal F$. This result is generalized to simultaneous separation in several sets. Analogous questions on separation by families of bounded Vapnik-Chervonenkis dimension and separation of point sets in ${\mathbb{R}}^d$ by convex sets are also considered.

math.CO↗

arXiv2015

Erdős - Szekeres Theorem for Lines

According to the Erdős-Szekeres theorem, for every $n$, a sufficiently large set of points in general position in the plane contains $n$ in convex position. In this note we investigate the line version of this result, that is, we want to find $n$ lines in convex position in a sufficiently large set of lines that are in general position. We prove almost matching upper and lower bounds for the minimum size of the set of lines in general position that always contains $n$ in convex position. This is quite unexpected, since in the case of points, the best known bounds are very far from each other. We also establish the dual versions of many variants and generalizations of the Erd\H os-Szekeres theorem.

math.MG↗

arXiv2014

Saturated simple and $k$-simple topological graphs

A simple topological graph $G$ is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. $G$ is called saturated if no further edge can be added without violating this condition. We construct saturated simple topological graphs with $n$ vertices and $O(n)$ edges. For every $k>1$, we give similar constructions for $k$-simple topological graphs, that is, for graphs drawn in the plane so that any two edges have at most $k$ points in common. We show that in any $k$-simple topological graph, any two independent vertices can be connected by a curve that crosses each of the original edges at most $2k$ times. Another construction shows that the bound $2k$ cannot be improved. Several other related problems are also considered.

math.CO↗

arXiv2014

Multiple coverings with closed polygons

A planar set $P$ is said to be cover-decomposable if there is a constant $k=k(P)$ such that every $k$-fold covering of the plane with translates of $P$ can be decomposed into two coverings. It is known that open convex polygons are cover-decomposable. Here we show that closed, centrally symmetric convex polygons are also cover-decomposable. We also show that an infinite-fold covering of the plane with translates of $P$ can be decomposed into two infinite-fold coverings. Both results hold for coverings of any subset of the plane.

math.MG↗

arXiv2012

Improvement on the decay of crossing numbers

We prove that the crossing number of a graph decays in a continuous fashion in the following sense. For any epsilon>0 there is a delta>0 such that for a sufficiently large n, every graph G with n vertices and m > n^{1+epsilon} edges, has a subgraph G' of at most (1-delta)m edges and crossing number at least (1-epsilon)cr(G). This generalizes the result of J. Fox and Cs. Toth.

math.CO↗

arXiv2009

Towards The Albertson Conjecture

Albertson conjectured that if a graph $G$ has chromatic number $r$ then its crossing number is at least as much as the crossing number of $K_r$. Albertson, Cranston, and Fox verified the conjecture for $r\le 12$. We prove the statement for $r\le 16$.

math.CO↗