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# Communications in Mathematical Sciences

## Volume 15 (2017)

### Number 6

### Regularly decomposable tensors and classical spin states

Pages: 1651 – 1665

DOI: https://dx.doi.org/10.4310/CMS.2017.v15.n6.a8

#### Authors

#### Abstract

A spin-$j$ state can be represented by a symmetric tensor of order $N =2j$ and dimension $4$. Here, $j$ can be a positive integer, which corresponds to a boson; $j$ can also be a positive half-integer, which corresponds to a fermion. In this paper, we introduce regularly decomposable tensors and show that a spin-$j$ state is classical if and only if its representing tensor is a regularly decomposable tensor. In the even-order case, a regularly decomposable tensor is a completely decomposable tensor but not vice versa; a completely decomposable tensors is a sum-of-squares (SOS) tensor but not vice versa; an SOS tensor is a positive semi-definite (PSD) tensor but not vice versa. In the odd-order case, the first row tensor of a regularly decomposable tensor is regularly decomposable and its other row tensors are induced by the regular decomposition of its first row tensor. We also show that complete decomposability and regular decomposability are invariant under orthogonal transformations, and that the completely decomposable tensor cone and the regularly decomposable tensor cone are closed convex cones. Furthermore, in the even-order case, the completely decomposable tensor cone and the PSD tensor cone are dual to each other. The Hadamard product of two completely decomposable tensors is still a completely decomposable tensor. Since one may apply the positive semi-definite programming algorithm to detect whether a symmetric tensor is an SOS tensor or not, this gives a checkable necessary condition for classicality of a spin-$j$ state. Further research issues on regularly decomposable tensors are also raised.

#### Keywords

positive semi-definite tensors, sum-of-squares tensors, quantum entanglement, spin states, bosons, fermions, classicality

#### 2010 Mathematics Subject Classification

15A18, 15A69, 15B48

This research was supported by the Hong Kong Research Grant Council Grant No. PolyU 501212, 501913, 531213, 15302114, 15300715, 15206915, the National Natural Science Foundation of China (11401428,61374057), and the Deutsch-Französische Hochschule (Université franco-allemande), grant number CT-45-14-II/2015. Ce travail a bénéficié d’une aide Investissements d’Avenir du LabEx PALM (ANR-10-LABX-0039-PALM).

Received 14 December 2016

Published 27 June 2017