The E-characteristic polynomial of an even order supersymmetric tensor is a useful tool in determining the positive definiteness of an even degree multivariate form. In this paper, for an even order tensor, we first establish the formula of its E-characteristic polynomial by using the classical Macaulay formula of resultants, then give an upper bound for the degree of that E-characteristic polynomial. Examples illustrate that this bound is attainable in some low order and dimensional cases.

We study tensor analysis problems motivated by the geometric measure of quantum entanglement. We define the concept of the unitary eigenvalue (U-eigenvalue) of a complex tensor, the unitary symmetric eigenvalue (US-eigenvalue) of a symmetric complex tensor, and the best complex rank-one approximation. We obtain an upper bound on the number of distinct US-eigenvalues of symmetric tensors and count all US-eigenpairs with nonzero eigenvalues of symmetric tensors. We convert the geometric measure of the entanglement problem to an algebraic equation system problem. A numerical example shows that a symmetric real tensor may have a best complex rankone approximation that is better than its best real rank-one approximation, which implies that the absolute-value largest Z-eigenvalue is not always the geometric measure of entanglement.Key words. unitary eigenvalue (U-eigenvalue), Z-eigenvalue, symmetric real tensor, geometric measure of entanglement, the best rank-one approximation 1. Introduction. Entanglement of compound systems is a key resource in quantum information processing. In many practical applications it is of fundamental importance to know whether a state is entangled or not. However, this information is often not sufficient, and it is also required to know how much a state is entangled. A useful tool for quantifying the amount of entanglement of a state is given by the so-called entanglement measures [1, 2].A widely used entanglement measure is provided by the geometric measure of entanglement [3,4,5] that is defined for a pure state. The geometric measure of entanglement was first proposed by Shimony [6] and extended to multipartite systems by Wei and Goldbart [7]. It has applications in various different topics, including many body physics [4], local discrimination, quantum computation, condensed matter systems, entanglement witnesses, and the study of quantum channel capacities. The geometric measure of entanglement is nothing but the injective tensor norm itself [8], which appears in the theory of operator algebra [9] and has now become increasingly important in theoretical physics-particularly in quantum channel capacities [10,11,12,13,14,15,16,17,18].A tensor is a multidimensional array [19]. Tensor decompositions originated with Hitchcock in 1927 [20], and the idea of a multiway model is attributed to Cattell in 1944 [21]. Recently, interest in tensor decompositions has expanded to other fields. Examples include signal processing [22,23], numerical linear algebra [24,25], computer

An order 2m complex tensor H is said to be Hermitian ifIt can be regarded as an extension of Hermitian matrix to higher order. A Hermitian tensor is also seen as a representation of a quantum mixed state. Motivated by the separability discrimination of quantum states, we investigate properties of Hermitian tensors including: unitary similarity relation, partial traces, nonnegative Hermitian tensors, Hermitian eigenvalues, rank-one Hermitian decomposition and positive Hermitian decomposition, and their applications to quantum states.

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