On orthogonal tensors and best rank-one approximation ratio

As is well known, the smallest possible ratio between the spectral norm and the Frobenius norm of an $m \times n$ matrix with $m \le n$ is $1/\sqrt{m}$ and is (up to scalar scaling) attained only by matrices having pairwise orthonormal rows. In the present paper, the smallest possible ratio between spectral and Frobenius norms of $n_1 \times \dots \times n_d$ tensors of order $d$, also called the best rank-one approximation ratio in the literature, is investigated. The exact value is not known for most configurations of $n_1 \le \dots \le n_d$. Using a natural definition of orthogonal tensors over the real field (resp., unitary tensors over the complex field), it is shown that the obvious lower bound $1/\sqrt{n_1 \cdots n_{d-1}}$ is attained if and only if a tensor is orthogonal (resp., unitary) up to scaling. Whether or not orthogonal or unitary tensors exist depends on the dimensions $n_1,\dots,n_d$ and the field. A connection between the (non)existence of real orthogonal tensors of order three and the classical Hurwitz problem on composition algebras can be established: existence of orthogonal tensors of size $\ell \times m \times n$ is equivalent to the admissibility of the triple $[\ell,m,n]$ to the Hurwitz problem. Some implications for higher-order tensors are then given. For instance, real orthogonal $n \times \dots \times n$ tensors of order $d \ge 3$ do exist, but only when $n = 1,2,4,8$. In the complex case, the situation is more drastic: unitary tensors of size $\ell \times m \times n$ with $\ell \le m \le n$ exist only when $\ell m \le n$. Finally, some numerical illustrations for spectral norm computation are presented

Tensor rank and entanglement of pure quantum states

The rank of a tensor is analyzed in context of quantum entanglement. A pure quantum state $\bf v$ of a composite system consisting of $d$ subsystems with $n$ levels each is viewed as a vector in the $d$-fold tensor product of $n$-dimensional Hilbert space and can be dentified with a tensor with $d$ indices, each running from $1$ to $n$. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state ${\bf v}$ is called {\sl entangled}, if it {\sl cannot} be written in the product form, ${\bf v} \ne {\bf v}_1 \otimes {\bf v}_2 \otimes \cdots \otimes {\bf v}_d$, what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed.Comment: 46 page

Perron-Frobenius theorem for multi-homogeneous mappings with applications to nonnegative tensors

The Perron-Frobenius theorem for nonnegative matrices has been generalized to order-preserving homogeneous mappings on a cone and more recently to nonnegative tensors. We unify both approaches by introducing the concept of order-preserving multi-homogeneous mappings defined on a product of cones and their associated eigenvectors. By considering a vector valued version of the Hilbert metric, we prove several Perron-Frobenius type results for these mappings. We discuss the existence, the uniqueness and the maximality of nonnegative and positive eigenvectors of multi-homogeneous mappings. We prove a Collatz-Wielandt formula and a multi-linear Birkhoff-Hopf theorem. We study the convergence of the normalized iterates of multi-homogeneous mappings and prove convergence rates. Applications of our main results include the study of the (p,q)-singular vectors of nonnegative matrices, the p-eigenvectors, rectangular (p,q)-singular vectors and (p_1,...,p_d)-singular vectors of nonnegative tensors, the generalized DAD problem and the discrete generalized Schrödinger equation arising in multi-marginal optimal transport. We recast these problems in the multi-homogeneous framework and explain how our theorems can be used to refine, improve and offer a new point of view on previous results of the literature.Das Perron-Frobenius Theorem für nichtnegative Matrizen wurde auf homogene, ordnungserhaltende Abbildungen auf einem Kegel erweitert und, in letzter Zeit, auf nichtnegative Tensoren. Wir vereinheitlichen beide Ansätze, indem wir das Konzept der ordnungserhaltenden, multi-homogenen Abbildungen, die auf einem Produkt von Kegeln definiert sind, sowie deren zugehörige Eigenvektoren einführen. Indem wir eine vektorisierte Version der Hilbert-Metrik in Betracht ziehen, beweisen wir für diese Abbildungen mehrere Perron-Frobenius-Typ Ergebnisse. Wir diskutieren die Existenz, die Einzigartigkeit und die Maximalität nichtnegativer und positiver Eigenvektoren multihomogener Abbildungen. Wir beweisen eine Collatz-Wielandt-Formel und einen multi-linearen Birkhoff-Hopf Satz. Wir untersuchen die Konvergenz der normierten Iterationen von multi-homogenen Abbildungen und beweisen Konvergenzraten. Anwendungen unserer Hauptergebnisse umfassen die Untersuchung der (p,q)-singulären Vektoren nichtnegativer Matrizen, der p-Eigenvektoren, rechteckiger (p,q)-singulärer Vektoren und (p_1,...,p_d)-singulärer Vektoren nichtnegativer Tensoren, das generalisierte DAD-Problem und die diskrete generalisierte Schrödinger Gleichung, die im Zusammenhang mit multi-marginalem optimalen Transport auftritt. Wir übertragen diese Probleme in den multi-homogenen Rahmen und erklären, wie unsere Theoreme verwendet werden können, um frühere Ergebnisse der Literatur zu verfeinern, zu verbessern und eine neue Sichtweise auf diese zu bieten