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Transactions on Computational and Applied Mathematics

Equivalence of 2-tensors and 3-tensors under Local Unitary and General Linear Groups

Nina Geng

Published Jan 1, 2023• 0 citations
Publisher

Abstract

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.

DOI
10.23977/tracam.2023.030107

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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.
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Since two quantum states that are local unitary (LU) equivalent have the same amount of entanglement, it is meaningful to find a practical method to determine the LU equivalence of given quantum states. In this paper, we present a valid process to find the unitary tensor product decomposition for an arbitrary unitary matrix. Then, by using this process, the conditions for determining the local unitary equivalence of quantum states are obtained. A numerical verification is carried out, which shows the practicability of our protocol. We also present a property of LU invariants by using the universality of quantum gates which can be used to construct the complete set of LU invariants.
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. 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.
We derive the necessary and sufficient conditions for the local unitary (LU) equivalence of two general (pure or mixed) nnn-qubit states and determine the local unitary operators connecting them. Almost all relevant information is contained in the one-qubit reduced matrices of the multiqubit states under investigation. Our technique relies on identifying ab initio all local symmetries and the corresponding local cyclic unitary operators. To derive the above conditions we use the reference forms of the multiqubit states whose definition requires the diagonalization of the one-qubit reduced matrices. Based on those conditions, we propose a straightforward protocol to decide whether or not two nnn-qubit states are LU equivalent.