Partition Ghz Slocc Class of Three Qubits into Ten Families Under Lu

Dafa Li

Dafa Li

In [Science 340:1205, (2013)], via entanglement polytopes Michael Walter et al. obtained a finite yet systematic classification of multi-particle entanglement. It is well known that under SLOCC, pure states of three (four) qubits are partitioned into six (nine) families. In this paper,we present a LU invariant and an entanglement measures for the GHZ SLOCC class of three qubits, and partition states of the GHZ SLOCC class of three qubits into ten families and each family into two subfamilies under LU. We give a necessary and sufficient condition for the uniqueness of the generalized Schmidt decomposition for the GHZ SLOCC class.

Xin-Wei Zha, Ya-Jun Duan, Jianxia Qi +2 more

Dafa Li

In [Science 340:1205, (2013)], via entanglement polytopes Michael Walter et al. obtained a finite yet systematic classification of multi-particle entanglement. It is well known that under SLOCC, pure states of three (four) qubits are partitioned into six (nine) families. Ac\'{\i}n et al. proposed the generalized Schmidt decomposition for three qubits and partitioned pure states of three qubits into five types. In this paper,we present a LU invariant and an entanglement measures for the GHZ SLOCC class of three qubits, and partition states of the GHZ SLOCC class of three qubits into ten families and each family into two subfamilies under LU. We give a necessary and sufficient condition for the uniqueness of the generalized Schmidt decomposition for the GHZ SLOCC class.

Dafa Li, Yu Jing Guo

Abstract Classification of multipartite pure states via the local unitary (LU) equivalence or the stochastic local operations and classical communication (SLOCC) equivalence is a fundamental task in investigating multipartite entanglement. It is shown in the seminal paper (2000 Phys . Rev . A 62 062314) that pure states of three qubits can be partitioned into six SLOCC classes. We explore here a necessary and sufficient condition for LU equivalence of pure states for any SLOCC class via Acín et al ’s Schmidt coefficients (ASC) of three qubits in (2000 Phys. Rev. Lett. 85 1560). We show that the LU equivalent states in W SLOCC class have the same ASC irrespective of phases while there are generally two different ASCs for the LU equivalent states in GHZ SLOCC class (irrespective of phases for some subclass). Consequently, we obtain a complete LU classification of three qubit pure states in the sense of ASC. Moreover, we discuss the choice among the different ASCs and then propose an approach to find a unique ASC for any given three qubit pure state. At last we analyse the entanglement/separability of the marginal states of the two inequivalent genuine entangled SLOCC classes. We find that any bipartite marginal states of the W SLOCC class are entangled, and that of the GHZ SLOCC class has eight different cases which are depend on the ASCs.

Huan Sun, Huan Li, Huaixin Cao