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arXiv:1803.09788 · Linear and Multilinear Algebra

Highly entangled tensors

Harm Derksen, Visu Makam

Published Feb 13, 2020• 7 citations
arXivHTMLPDF

Abstract

A geometric measure for the entanglement of a unit length tensor T∈(Cn)⊗k is given by −2log2⁡∥T∥σ, where ∥⋅∥σ denotes the spectral norm. A simple induction gives an upper bound of (k−1)log2⁡(n) for the entanglement. We show the existence of tensors with entanglement larger than klog2⁡(n)−log2⁡(k)−o(log2⁡(k)). Friedland and Kemp (Proc. AMS, 2018) have similar results in the case of symmetric tensors. Our techniques give improvements in this case.

DOI
10.1080/03081087.2020.1726276

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A geometric measure for the entanglement of a unit length tensor T∈(Cn)⊗k is given by −2log2⁡∥T∥σ, where ∥⋅∥σ denotes the spectral norm. A simple induction gives an upper bound of (k−1)log2⁡(n) for the entanglement. We show the existence of tensors with entanglement larger than klog2⁡(n)−log2⁡(k)−o(log2⁡(k)). Friedland and Kemp (Proc. AMS, 2018) have similar results in the case of symmetric tensors. Our techniques give improvements in this case.
We show that the two notions of entanglement: the maximum of the geometric measure of entanglement and the maximum of the nuclear norm is attained for the same states. We affirm the conjecture of Higuchi-Sudberry on the maximum entangled state of four qubits. We introduce the notion of d-density tensor for mixed d-partite states. We show that d-density tensor is separable if and only if its nuclear norm is 111. We suggest an alternating method for computing the nuclear norm of tensors. We apply the above results to symmetric tensors. We give many numerical examples.
We show that the spectral norm of a d d -mode real or complex symmetric tensor in n n variables can be computed by finding the fixed points of the corresponding polynomial map. For a generic complex symmetric tensor the number of fixed points is finite, and we give upper and lower bounds for the number of fixed points. For n = 2 n=2 we show that these fixed points are the roots of a corresponding univariate polynomial of degree at most ( d − 1 ) 2 + 1 (d-1)^2+1 , except certain cases, which are completely analyzed. In particular, for n = 2 n=2 the spectral norm of d d -symmetric tensor is polynomially computable in d d with a given relative precision. For a fixed n > 2 n>2 we show that the spectral norm of a d d -mode symmetric tensor is polynomially computable in d d with a given relative precision with respect to the Hilbert-Schmidt norm of the tensor. These results show that the geometric measure of entanglement of d d -mode symmetric qunits on C n \mathbb {C}^n are polynomially computable for a fixed n n .
This study focuses on tensor Z-eigenvalue localization and its application in the geometric measure of entanglement for multipartite quantum states. A new Z-eigenvalue localization theorem and the bounds for the Z-spectral radius are derived, which are more precise than some of the existing results. On the other hand, we present theoretical bounds of the geometric measure of entanglement for a weakly symmetric multipartite quantum state with non-negative amplitudes by virtue of different distance measures. Numerical examples show that these conclusions are superior to the existing results in quantum physics in some cases.
The main topic of this compilation thesis is the investigation of multipartite entanglement of finite dimensional systems. We developed a numerical algorithm that detects if a multipartite state is entangled or separable in a finite number of steps of a semi-definite optimization task. This method is an extension of previously known semi-definite methods, which are inconclusive when the state is separable. In our case, if the state is separable, an explicit decomposition into a mixture of separable states can be extracted. This was achieved by mapping the entanglement problem onto the mathematically well studied truncated moment problem. Additionally, a new way of writing the partially transposed state for symmetric multi-qubit states was developed which simplifies many results previously known in entanglement theory. This new way of writing the partial transpose criterion unifies different interpretations and alternative formulations of the partial transpose criterion and it is also a part in the aforementioned semi-definite algorithm. The geometric properties of entangled symmetric states of two qubits were studied in detail: We could answer the question of how far a given pure state is from the convex hull of symmetric separable states, as measured by the Hilbert-Schmidt distance, by giving an explicit formula for the closest separable symmetric state. For mixed states we could provide a numerical upper and analytical lower bound for this distance. For a larger number of qubits we investigated the ball of absolutely classical states, i.e. symmetric multi-qubit states that stay separable under any unitary transformation. We found an analytical lower bound for the radius of this ball around the maximally mixed symmetric state and gave a numerical upper bound on this radius, by searching for an entangled state as close as possible to the maximally mixed symmetric state. The tensor representation of a symmetric multi-qubit state, or spin-j state, allowed us to study entanglement properties based on the spectrum of the tensor via tensor eigenvalues. The definiteness of this tensor relates to the entanglement of the state it represents and, hence, the smallest tensor eigenvalue can be used to detect entanglement. However, the tensor eigenvalues are more difficult to determine than the familiar matrix eigenvalues which made the investigation computationally more challenging. The relationship between the value of the smallest tensor eigenvalue and the amount of entanglement in the state was also investigated. It turned out that they are strongly correlated for small system sizes, i.e. for up to six qubits. However, to investigate this correlation we needed an independent way to gauge the amount of entanglement of a state and in order to do so we improved existing numerical methods to determine the distance of a state to the set of separable symmetric states, using a combination of linear and quadratic programming. The tensor representation of symmetric multi-qubit states was also used to formally define a new tensor class of regularly decomposable tensors that corresponds to the set of separable symmetric multi-qubit states.
The geometric measure of entanglement for pure states has attracted much attention. On the other hand, the spectral theory of non-negative tensors (hypermatrices) has been developed rapidly. In this paper, we show how the spectral theory of non-negative tensors can be applied to the study of the geometric measure of entanglement for pure states. For symmetric pure multipartite qubit or qutrit states an elimination method is given. For symmetric pure multipartite qudit states, a numerical algorithm with randomization is presented. We also show that a nonsymmetric pure state can be augmented to a symmetric one whose amplitudes can be encoded in a non-negative symmetric tensor, so the geometric measure of entanglement can be calculated. Several examples, such as mGHZm\mathrm{GHZ}mGHZ states, WWW states, inverted WWW states, qudits, and nonsymmetric states, are used to demonstrate the power of the proposed methods. Given a pure state, one can always find a change of basis (a unitary transformation) so that all the probability amplitudes of the pure state are non-negative under the new basis. Therefore, the methods proposed here can be applied to a very wide class of multipartite pure states.