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Title: Regularly decomposable tensors and classical spin states
Authors: QI, LQ 
Zhang, GF 
Braun, D
Bohnet-Waldraff, F
Giraud, O
Keywords: Bosons
Positive semi-definite tensors
Quantum entanglement
Spin states
Sum-of-squares tensors
Issue Date: 2017
Publisher: International Press
Source: Communications in mathematical sciences, 2017, v. 15, no. 6, p. 1651-1665 How to cite?
Journal: Communications in mathematical sciences 
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.
ISSN: 1539-6746
EISSN: 1945-0796
DOI: 10.4310/CMS.2017.v15.n6.a8
Rights: © 2017 International Press
This paper was published in Communications in Mathematical Sciences and is made available as an electronic reprint with the permission of International Press of Boston.
The following publication Qi, L., Zhang, G., Braun, D., Bohnet-Waldraff, F., & Giraud, O. (2016). Regularly Decomposable Tensors and Classical Spin States. Communications in mathematical sciences, 15 (6), 1651-1665 is available at
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