Higher-order singular value decomposition and the reduced density matrices of three qubits
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Higher-order singular value decomposition and the reduced density matrices of three qubits Pak Shen Choong1 · Hishamuddin Zainuddin1,2 Sh. K. Said Husain1
· Kar Tim Chan1
·
Received: 24 March 2020 / Accepted: 25 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we demonstrate that higher-order singular value decomposition (HOSVD) can be used to identify special states in three qubits by local unitary (LU) operations. Since the matrix unfoldings of three qubits are related to their reduced density matrices, HOSVD simultaneously diagonalizes the one-body reduced density matrices of three qubits. From the all-orthogonality conditions of HOSVD, we computed the special states of three qubits. Furthermore, we showed that it is possible to construct a polytope that encapsulates all the special states of three qubits by LU operations with HOSVD. Keywords Quantum entanglement · Higher-order singular value decomposition · Local unitary equivalence · Three qubits Mathematics Subject Classification 15A69
B
Hishamuddin Zainuddin [email protected] https://profile.upm.edu.my/hisham Pak Shen Choong [email protected] Kar Tim Chan [email protected] Sh. K. Said Husain [email protected]
1
Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia
2
Malaysia-Italy Center for Mathematical Sciences (MICEMS), Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia 0123456789().: V,-vol
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C. P. Shen et al.
1 Introduction Being the central characteristic of composite quantum systems, entanglement has been studied extensively in the past from various perspectives, such as the classification of multipartite states [1,2,7,18,19,21], the geometry of quantum state space [5,15,25] and more recently, the resource-theoretic [4,9,11,13] and categorical approach [10,12]. Apart from the fact that entanglement connects deeply to the foundations of quantum theory, it can be utilized as a resource in quantum information processing. From this perspective, it is important to be able to quantify entanglement and classify entangled states based on the computational tasks they can perform. The Hilbert space of a composite quantum system is described by the tensor product of its subsystems’ Hilbert spaces. This tensor product structure naturally endows tensorial properties to the elements of multipartite states, thus allowing us to employ multilinear algebraic methods on them. As an example, we can apply singular value decomposition (SVD) on the elements of bipartite states and restate it as Schmidt decomposition [22], which is a widely used approach in the local unitary (LU) classification of bipartite states. It is also known that the Schmidt coefficients are LU invariants of the entanglement classes for bipartite states [7,25]. Given the successful precedence in bipartite states, it is natural to consider Schmidt decomposition in the LU classification of multipartite states. This idea turned out to be unsuccessful [23]
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