arXiv subjects

Richard Stong

Publications and source records attributed to Richard Stong.

8 recordsLinked to original sources

arXiv2016

Quantum Max-flow/Min-cut

The classical max-flow min-cut theorem describes transport through certain idealized classical networks. We consider the quantum analog for tensor networks. By associating an integral capacity to each edge and a tensor to each vertex in a flow network, we can also interpret it as a tensor network, and more specifically, as a linear map from the input space to the output space. The quantum max flow is defined to be the maximal rank of this linear map over all choices of tensors. The quantum min cut is defined to be the minimum product of the capacities of edges over all cuts of the tensor network. We show that unlike the classical case, the quantum max-flow=min-cut conjecture is not true in general. Under certain conditions, e.g., when the capacity on each edge is some power of a fixed integer, the quantum max-flow is proved to equal the quantum min-cut. However, concrete examples are also provided where the equality does not hold. We also found connections of quantum max-flow/min-cut with entropy of entanglement and the quantum satisfiability problem. We speculate that the phenomena revealed may be of interest both in spin systems in condensed matter and in quantum gravity.

math.CO↗

arXiv2016

A lower bound for the dimension of a highest weight module

For each integer $t>0$ and each complex simple Lie algebra $\mathfrak{g}$, we determine the least dimension of an irreducible highest weight representation of $\mathfrak{g}$ whose highest weight has height $t$. As a corollary, we classify all irreducible modules whose dimension equals a product of two primes.

math.RT↗

arXiv2015

Partition and sum is fast

We consider the following "partition and sum" operation on a natural number: Treating the number as a long string of digits insert several plus signs in between some of the digits and carry out the indicated sum. This results in a smaller number and repeated application can always reduce the number to a single digit. We show that surprisingly few iterations of this operation are needed to get down to a single digit.

math.HO↗

arXiv2009

On classification of modular tensor categories

We classify all unitary modular tensor categories (UMTCs) of rank $\leq 4$. There are a total of 70 UMTCs of rank $\leq 4$ (Note that some authors would have counted as 35 MTCs.) In our convention there are two trivial unitary MTCs distinguished by the modular $S$ matrix $S=(\pm1)$. Each such UMTC can be obtained from 10 non-trivial prime UMTCs by direct product, and some symmetry operations. UMTCs encode topological properties of anyonic quantum systems and can be used to build fault-tolerant quantum computers. We conjecture that there are only finitely many equivalence classes of MTCs for any given rank, and a UMTC is universal for anyonic quantum computation if and only if its global quantum dimension $D^2$ is \emph{not} an integer. Discovery of non-abelain anyons in Nature will be a landmark in condensed matter physics. The non-abelian anyons in UMTCs of rank $\leq 4$ are the simplest, and, therefore, are most likely to be found. G. Moore and N. Read proposed that non-abelian statistics could occur in fractional quantum Hall (FQH) liquids. The Read-Rezayi conjecture predicts the existence of anyons related to $SU(2)_k$ in FQH liquids at filling fractions $ν=2+\frac{k}{k+2}$ for $k=1,2,3$. For $ν={5/2}$ and $SU(2)_2$, there is a numerical proof and experimental evidence for this conjecture. The Ising anyons might exist in chiral superconductors (strontium ruthenate). There are theoretical designs for the toric code MTC using Josephson junction array, for the $Fib\times Fib$ MTC using optical lattice, and for the $SU(2)_k$ using cold trapped bosonic atoms.

math.QA↗

arXiv2006

Comparing Corresponding Dihedral Angles on Classical Geometric Simplices

In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.

math.DG↗

arXiv2006

A Novel Symmetric Four Dimensional Polytope Found Using Optimization Strategies Inspired by Thomson's Problem of Charges on a Sphere

Inspired by, and using methods of optimization derived from classical three dimensional electrostatics, we note a novel beautiful symmetric four dimensional polytope we have found with 80 vertices. We also describe how the method used to find this symmetric polytope, and related methods can potentially be used to find good examples for the kissing and packing problems in D dimensions.

physics.comp-ph↗

arXiv2002

Lengths of simple loops on surfaces with hyperbolic metrics

Given a compact orientable surface of negative Euler characteristic, there exists a natural pairing between the Teichmueuller space of the surface and the set of homotopy classes of simple loops and arcs. The length pairing sends a hyperbolic metric and a homotopy class of a simple loop or arc to the length of geodesic in its homotopy class. We study this pairing function using the Fenchel-Nielsen coordinates on Teichmueller space and the Dehn-Thurston coordinates on the space of homotopy classes of curve systems. Our main result establishes Lipschitz type estimates for the length pairing expressed in terms of these coordinates. As a consequence, we reestablish a result of Thurston-Bonahon that the length pairing extends to a continuous map from the product of the Teichmueller space and the space of measured laminations.

math.GT↗

arXiv2000

Cauchy Inequality and the Space of Measured Laminations

We establish a Cauchy type inequality for the geometric intersection number between two 1-dimensional submanifolds in a surface. Some of the basic results in Thurston's theory of measured laminations on surfaces are derived from the Cauchy inequality.

math.GT↗