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Quantum Information and Computation

Slocc classification for nine families of four-qubits

LiDafa, LiXiangrong, HuangHongtao, LiXinxin

Published Sep 1, 2009• 0 citations
Publisher

Abstract

In 2000, Dur, Vidal and Cirac indicated that there are infinitely many SLOCC classes for four qubits. Later, Verstraete, Dehaene, and Verschelde proposed nine families of states corresponding to ni...
DOI
10.5555/2011804.2011809

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In 2000, D\"{u}r, Vidal and Cirac indicated that there are infinitely many SLOCC classes for four qubits. Later, Verstraete, Dehaene, and Verschelde proposed nine families of states corresponding to nine different ways of entangling four qubits. And then in 2007 Lamata et al. reported that there are eight true SLOCC entanglement classes of four qubits up to permutations of the qubits. In this paper, we investigate SLOCC classification of the nine families proposed by Verstraete, Dehaene and Verschelde, and distinguish 49 true SLOCC entanglement classes from them.
It is a recent observation that entanglement classification for qubits is closely related to stochastic local operations and classical communication (SLOCC) invariants. Verstraete et al.[ Phys. Rev. A 65 (2002) 052112 ] showed that for pure states of four qubits there are nine different degenerate SLOCC entanglement classes. Li et al.[ Phys. Rev. A 76 (2007) 052311 ] showed that there are at feast 28 distinct true SLOCC entanglement classes for four qubits by means of the SLOCC invariant and semi-invariant. We give 16 different entanglement classes for four qubits by means of basic SLOCC invariants.
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.
Abstract We classify four qubit states under SLOCC operations, that is, we classify the orbits of the group S L ( 2 , C ) 4 on the Hilbert space H 4 = ( C 2 ) ⊗ 4 . We approach the classification by realising this representation as a symmetric space of maximal rank. We first describe general methods for classifying the orbits of such a space. We then apply these methods to obtain the orbits in our special case, resulting in a complete and irredundant classification of S L ( 2 , C ) 4 -orbits on H 4 . It follows that an element of ( C 2 ) ⊗ 4 is conjugate to an element of precisely 87 classes of elements. Each of these classes either consists of one element or of a parameterised family of elements, and the elements in the same class all have equal stabiliser in S L ( 2 , C ) 4 . We also present a complete and irredundant classification of elements and stabilisers up to the action of S y m 4 ⋉ S L ( 2 , C ) 4
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.