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Now showing items 1-10 of 10

#### On the minimum rank of a graph over finite fields

(Elsevier, 2012-03-15)

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

(Elsevier, 2011-06-23)

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

(Elsevier, 2012-01)

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

(Elsevier, 2017-08)

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 ...