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Multilinear singular value decomposition for two qubits

Pak Shen Choong, Hishamuddin B Zainuddin, I. S. Rakhimov

Published Jan 1, 2016• 1 citation
OpenAlex

Abstract

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.

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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.
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.
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.
Abstract The multipartite unitary gates are called genuine if they are not product unitary operators across any bipartition. We mainly investigate the classification of genuine multipartite unitary gates of Schmidt rank two (SR-2), by focusing on the multiqubit scenario. For genuine multipartite (excluding bipartite) unitary gates of SR-2, there is an essential fact that their Schmidt decompositions are unique. Based on this fact, we propose a key notion named as singular number (SN) to classify the unitary gates concerned. The SN is defined as the number of local singular operators in the Schmidt decomposition. We then determine the accurate range of SN. For each SN, we formulate the parametric Schmidt decompositions of genuine multiqubit unitary gates under local equivalence. Finally, we extend the study to three-qubit diagonal unitary gates due to the close relation between diagonal unitary gates and SR-2 unitaries. We start with discussing two typical examples of SR-2, one of which is a fundamental three-qubit unitary gate, i.e. the CCZ gate. Then we characterize the diagonal unitary gates of SR greater than two. We show that a three-qubit diagonal unitary gate has SR at most three, and present a necessary and sufficient condition for such a unitary gate of SR-3. This completes the characterization of all genuine three-qubit diagonal unitary gates.
In quantum information, pure states correspond to tensors. Two states equivalent under local unitary group or local general linear group can be used for the same quantum-information tasks, and it is thus an essential problem to determine whether two tensors are equivalent. In this paper, we determine the two sorts of equivalence for 2-tensors in terms of the Schmidt decomposition and singular value decomposition of matrices. We also extend the equivalence to some 3-tensors, proving that Schmidt decomposition is unachievable for 3-tensors. However, we present a method to express 3-tensors using the logic behind Schmidt decomposition. Furthermore, we generalized the scenario to higher dimensions and discussed the characteristic behaviors of different groups in tensor transformations.