Entanglement Classification of Four-partite States

Ming-Yong Ye, Xiu‐Min Lin

이혁재

S. M. Zangi, Junli Li, Cong‐Feng Qiao

Abstract We present a practical classification scheme for the four-partite entangled states under stochastic local operations and classical communication (SLOCC). By transforming a four-partite state into a triple-state set composed of two tripartite states and a bipartite state, the entanglement classification is reduced to the classification of tripartite and bipartite entanglements. This reduction method has the merit of involving only the linear constrains, and meanwhile provides an insight into the entanglement character of the subsystems.

Oliver Viehmann, Christopher Eltschka, Jens Siewert

The number of entanglement classes in stochastic local operations and classical communication (SLOCC) classifications increases with the number of qubits and is already infinite for four qubits. Criteria for explicitly discriminating and classifying pure states of four and more qubits are highly desirable and therefore at the focus of intense theoretical research. We develop a general criterion for the discrimination of pure $N\text{\ensuremath{-}}$partite entangled states in terms of polynomial $\mathrm{SL}(d,\mathrm{C}){}^{\ensuremath{\bigotimes}N}$ invariants. By means of this criterion, existing SLOCC classifications of four-qubit entanglement are reproduced. Based on this we propose a polynomial classification scheme in which entanglement types are identified through ``tangle patterns.'' This scheme provides a practicable way to classify states of arbitrary multipartite systems. Moreover, the use of polynomials induces a corresponding quantification of the different types of entanglement.

Takahiro Matsuoka

Ming Li, Zong Wang, Jing Wang +2 more

We investigate the norms of the Bloch vectors for any quantum state with subsystems less than or equal to four. Tight upper bounds of the norms are obtained, which can be used to derive tight upper bounds for entanglement measures defined by the norms of Bloch vectors. By using these bounds a trade-off relation of the norms of Bloch vectors is discussed. These upper bounds are then applied on separability. Necessary conditions are presented for different kinds of separable states in four-partite quantum systems. We further present a complete classification of quantum states for four-qudits quantum systems.