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arXiv:2402.12542 · SIAM Journal on Applied Algebra and Geometry

Tensor Decompositions with Applications to Local Unitary and Stochastic Local Operations and Classical Communication Equivalence of Multipartite Pure States

Luke Oeding, Ian Tan

Published Jan 3, 2025• 0 citations
arXivHTMLPDF

Abstract

Abstract. We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors (L. De Lathauwer, B. De Moor, and J. Vandewalle, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1253–1278). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus’s algorithm (B. Kraus, Phys. Rev. Lett., 104 (2010), 020504) used the HOSVD to compute normal forms of almost all [Formula: see text]-qubit pure states under the action of the local unitary group. Taking advantage of the double cover [Formula: see text], we produce similar algorithms (distinguished by the parity of [Formula: see text]) that compute normal forms for almost all [Formula: see text]-qubit pure states under the action of the group of stochastic local operations and classical communication.

DOI
10.1137/24m1643451

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Abstract. We introduce a broad lemma, one consequence of which is the higher order singular value decomposition (HOSVD) of tensors (L. De Lathauwer, B. De Moor, and J. Vandewalle, SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1253–1278). By an analogous application of the lemma, we find a complex orthogonal version of the HOSVD. Kraus’s algorithm (B. Kraus, Phys. Rev. Lett., 104 (2010), 020504) used the HOSVD to compute normal forms of almost all [Formula: see text]-qubit pure states under the action of the local unitary group. Taking advantage of the double cover [Formula: see text], we produce similar algorithms (distinguished by the parity of [Formula: see text]) that compute normal forms for almost all [Formula: see text]-qubit pure states under the action of the group of stochastic local operations and classical communication.
In this study, local unitary (LU) properties of two- and three-qubit quantum systems are studied. Specifically, the Schmidt decomposition approach to LU classification for two qubits is re-examined using a more general approach, i.e. higher order singular value decomposition (HOSVD). Later, HOSVD is used to classify three-qubit pure states by LU operations. We found that due to HOSVD, it is possible to characterize the entanglement classes of three qubits by the eigenvalues distribution of its one-particle reduced density matrices. This finding generalized the similar classification results in the literature and it is hoped that this work will fill in the gap of LU classification from bipartite to multipartite quantum states by using HOSVD.
Schmidt decomposition has been used in the local unitary (LU) classification of bipartite quantum states for some time. In order to generalize the LU classification of bipartite quantum states into multipartite quantum states, higher order singular value decomposition (HOSVD) is introduced but specific examples have not been explicitly worked out. In this pedagogical paper, we would like to work out such details in two qubits since the LU classification of two qubits is well known. We first demonstrate the method of HOSVD in two-qubit systems and discuss its properties. In terms of the LU classification of two-qubit states, some subtle differences in the stabilizer groups of entanglement classes are noticed when Schmidt decomposition is substituted by HOSVD. To reconcile the differences between the two, further studies are needed.
Abstract Classification of multipartite pure states via the local unitary (LU) equivalence or the stochastic local operations and classical communication (SLOCC) equivalence is a fundamental task in investigating multipartite entanglement. It is shown in the seminal paper (2000 Phys . Rev . A 62 062314) that pure states of three qubits can be partitioned into six SLOCC classes. We explore here a necessary and sufficient condition for LU equivalence of pure states for any SLOCC class via Acín et al ’s Schmidt coefficients (ASC) of three qubits in (2000 Phys. Rev. Lett. 85 1560). We show that the LU equivalent states in W SLOCC class have the same ASC irrespective of phases while there are generally two different ASCs for the LU equivalent states in GHZ SLOCC class (irrespective of phases for some subclass). Consequently, we obtain a complete LU classification of three qubit pure states in the sense of ASC. Moreover, we discuss the choice among the different ASCs and then propose an approach to find a unique ASC for any given three qubit pure state. At last we analyse the entanglement/separability of the marginal states of the two inequivalent genuine entangled SLOCC classes. We find that any bipartite marginal states of the W SLOCC class are entangled, and that of the GHZ SLOCC class has eight different cases which are depend on the ASCs.
Abstract Tensors are increasingly ubiquitous in various areas of applied mathematics and computing, and tensor decompositions are of practical significance and benefit many applications in data completion, image processing, computer vision, collaborative filtering, etc. Recently, Kilmer and Martin propose a new tensor factorization strategy, tensor singular value decomposition (t-svd), which extends the matrix singular value decomposition to tensors. However, computing t-svd for high dimensional tensors costs much computation and thus cannot efficiently handle large scale datasets. Motivated by advantage of quantum computation, in this paper, we present a quantum algorithm of t-svd for third-order tensors and then extend it to order-p tensors. We prove that our quantum t-svd algorithm for a third-order N dimensional tensor runs in time  N polylog ( N ) if we do not recover classical information from the quantum output state. Moreover, we apply our quantum t-svd algorithm to context-aware multidimensional recommendation systems, where we just need to extract partial classical information from the quantum output state, thus achieving low time complexity.