Classification of multi-qubit states with higher order singular value decompositon and concurrency of three lines - Choong, Pak Shen (2022)
This research studied the classification problem of multipartite states. Since the Hilbert space of a multipartite system has a tensor product structure, we made use of a tensor decomposition, called higher order singular value decomposition (HOSVD) to solve the problem. We focused on finding the solutions to the set of allorthogonality conditions from HOSVD to obtain a complete classification of three qubits. Based on the relationship between the set of first n-mode singular values, σ(n)2 1 , we identified three possible cases that contain all the entanglement classes of three qubits. An entanglement polytope was illustrated to demonstrate how the entanglement classes of three qubits change with respect to σ(n)2 1 , which is in accordance with the existing literature. The geometrical significance of our classification method was studied by finding the stabilizer dimensions of all the entanglement classes of three qubits. We found that different entanglement classes of three qubits have different stabilizer dimensions. Furthermore, by making use of the concurrency of three lines, we generalized our classification approach for multi-qubit states, which is computationally simple and yet capable of producing a finite number of family of states. As a demonstration, we classified four-qubit states with our proposal and found four possible cases that contain the genuinely entangled four-qubit states.