Special core tensors of multi-qubit states and the concurrency of three lines

• Published in: Quantum information processing Vol. 22; no. 5
• Main Authors: Choong, Pak Shen, Zainuddin, Hishamuddin, Chan, Kar Tim, Said Husain, Sharifah Kartini
• Format: Journal Article
• Language: English
• Published: New York Springer US 28.04.2023
• Subjects: Data Structures and Information Theory; Mathematical Physics; Physics; Physics and Astronomy; Quantum Computing; Quantum Information Technology; Quantum Physics; Spintronics; Higher order singular value decomposition; Multi-qubit states; Quantum entanglement; Local unitary classification
• ISSN: 1573-1332; 1573-1332
• DOI: 10.1007/s11128-023-03939-w

In this work, we propose a computationally simple approach to identify the local unitary (LU) entanglement classes of multi-qubit states by higher-order singular value decomposition (HOSVD). For multipartite states, HOSVD simultaneously diagonalizes their one-body reduced density matrices (RDM) by LU actions. Therefore, the zeros of the all-orthogonality conditions due to HOSVD, also known as the core tensors, are the pure-state representations of such simultaneously diagonalized one-body RDM for a given multipartite state. By using the concurrency of three lines, we simplified the calculations and coarse-grained the classification into a finite number of families of states based on the square of their first n -mode singular values, σ 1 ( n ) 2 . These special core tensors are genuinely entangled by default. For three and four qubits, we identified two and four families of states respectively. A generalization of the algorithm to multi-qubit states is provided.