Tensor decompositions with applications to LU and SLOCC equivalence of multipartite pure states

Abstract

We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors defined by DeLathauwer, DeMoor and Vandewalle (2000). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus' (2010) algorithm used the HOSVD to compute normal forms of almost all $n$-qubit pure states under the action of the local unitary group. Taking advantage of the double cover ${\mathrm{SL}}_2(\mathbb{C})\times {\mathrm{SL}}_2(\mathbb{C})\to {\mathrm{SO}}_4(\mathbb{C})$, we produce similar algorithms (distinguished by the parity of $n$) that compute normal forms for almost all $n$-qubit pure states under the action of the SLOCC group.

arXiv comment23 pages, 2 figures

References in corpus (8)

• Three qubits can be entangled in two inequivalent ways
• Four qubits can be entangled in nine different ways
• Local unitary equivalence of multipartite pure states
• Almost all multipartite qubit quantum states have trivial stabilizer
• Complexity of matrix problems
• Necessary and sufficient conditions for local manipulation of multipartite pure quantum states
• Classification of four qubit states and their stabilisers under SLOCC operations
• Freudenthal ranks: GHZ vs. W