Now showing items 1-10 of 10
On the minimum rank of a graph over finite fields
In this paper we deal with two aspects of the minimum rank of a simple undirected graph G on n vertices over a finite field Fq with q elements, which is denoted by mr(Fq,G). In the first part of this paper we show that ...
Best rank one approximation of real symmetric tensors can be chosen symmetric
(Springer Verlag, 2013-02)
We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Further- more, a symmetric best rank one approximation to a symmetric tensor is ...
An Asymptotic Expansion and Recursive Inequalities for the Monomer-Dimer Problem
(Springer Verlag, 2011-04)
Let lambda(d)(p) be the p monomer-dimer entropy on the d-dimensional integer lattice Z(d), where p is an element of [0, 1] is the dimer density. We give upper and lower bounds for lambda(d)(p) in terms of expressions ...
On tensors of border rank l in Cm×n×l
We study tensors in Cm×n×l whose border rank is l. We give a set-theoretic char- acterization of tensors in C3×3×4 and in C4×4×4 of border rank 4 at most.
Numerical Estimation of the Relative Entropy of Entanglement
(American Physical Society, 2010-11-29)
We propose a practical algorithm for the calculation of the relative entropy of entanglement (REE), defined as the minimum relative entropy between a state and the set of states with positive partial transpose. Our algorithm ...
A proof of the set-theoretic version of the salmon conjecture
We show that the irreducible variety of 4 4 4 complex valued tensors of border rank at most 4 is the zero set of polynomial equations of degree 5 (the Strassen commutative conditions), of degree 6 (the Landsberg-Manivel ...
A note on the nonzero spectra of irreducible matrices
(Taylor & Francis, 2011-09)
In this note we extend the necessary and sufficient conditions of Boyle-Handleman [M. Boyle and D. Handelman, The spectra of nonnegative matrices via symbolic dynamics, Ann. Math. 133 (1991), pp. 249-316] and Kim-Ormes-Roush ...
The tensor rank of tensor product of two three-qubit W states is eight
We show that the tensor rank of tensor product of two three-qubit W states is not less than eight. Combining this result with the recent result of M. Christandl, A. K. Jensen, and J. Zuiddam that the tensor rank of tensor ...
Nuclear norm of higher-order tensors
(American Mathematical Society, 2016-11)
We establish several mathematical and computational properties of the nuclear norm for higher-order tensors. We show that like tensor rank, tensor nuclear norm is dependent on the choice of base field --- the value of the ...
MOST BOSON QUANTUM STATES ARE ALMOST MAXIMALLY ENTANGLED
(American Mathematical Society, 2018-09-04)
The geometric measure E of entanglement of an m qubit quantum state takes maximal possible value m. In previous work of Gross, Flammia, and Eisert, it was shown that E ≥ m − O(log m) with high probability as m → ∞. They ...