Classification of Bipartite and Tripartite Qutrit Entanglement Under SLOCC

Yang, Xiaoping; Wang, Zhi‐Xi; Wang, Xiaohong; Fei, Shao-Ming

doi:10.1088/0253-6102/50/3/25

Cited by 5 publications

(4 citation statements)

References 23 publications

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“…In contrast, there is an infinite number of SLOCC classes of the tripartite qutrits in which we are interested here. A previous study used an inductive method [9] to identify some SLOCC classes of tripartite qutrits [20]. These classes are shown in Table 1 below.…”

Section: Definition 22 (Slocc-maximality)mentioning

confidence: 99%

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Graphical Classification of Entangled Qutrits

Honda¹

2012

Electron. Proc. Theor. Comput. Sci.

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A multipartite quantum state is entangled if it is not separable. Quantum entanglement plays a fundamental role in many applications of quantum information theory, such as quantum teleportation. Stochastic local quantum operations and classical communication (SLOCC) cannot essentially change quantum entanglement without destroying it. Therefore, entanglement can be classified by dividing quantum states into equivalence classes, where two states are equivalent if each can be converted into the other by SLOCC. Properties of this classification, especially in the case of non two-dimensional quantum systems, have not been well studied. Graphical representation is sometimes used to clarify the nature and structural features of entangled states. SLOCC equivalence of quantum bits (qubits) has been described graphically via a connection between tripartite entangled qubit states and commutative Frobenius algebras (CFAs) in monoidal categories. In this paper, we extend this method to qutrits, i.e., systems that have three basis states. We examine the correspondence between CFAs and tripartite entangled qutrits. Using the symmetry property, which is required by the definition of a CFA, we find that there are only three equivalence classes that correspond to CFAs. We represent qutrits graphically, using the connection to CFAs. We derive equations that characterize the three equivalence classes. Moreover, we show that any qutrit can be represented as a composite of three graphs that correspond to the three classes

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Section: Definition 22 (Slocc-maximality)mentioning

confidence: 99%

“…The entangled states are divided into equivalence classes by SLOCC-equivalence, which relates states that can be converted into each other by SLOCC. Several recent studies have investigated SLOCC-equivalence classes [8] [9][10] [12] [18] [20]. In this paper, we focus on tripartite qutrits.…”

Section: Introductionmentioning

confidence: 99%

Graphical Classification of Entangled Qutrits

Honda¹

2012

Electron. Proc. Theor. Comput. Sci.

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“…[1][2][3][4]. Consequently, a crucial question in studying entanglement is to explore the relation of two states under local operations , such as local operations and classical communication (LOCC) [6,22,39], local unitary (LU) operators [9, 18-21, 23-25, 29, 32, 36, 40-54] and stochastic local operations and classical communication (SLOCC) [7,8,12,23,[26][27][28][29][30][31][32][33][34][35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

“…Later, Verstraete et al in reference [8] partitioned the four qubits pure states into nine families under SLOCC. Afterwards, considerable efforts have been undertaken over the last two decades for the SLOCC classification of n-qubits, mixed states, two qutrits, and other multipartite states in higher dimensional systems [12,23,[26][27][28][29][30][31][32][33][34][35][36][37][38]. As a special case of SLOCC classification, LU equivalence is still a difficult issue for multipartite states since there does not exist canonical Schmidt decomposition (SD) in any multipartite space [55][56][57][58][59][60][61].…”

Section: Introductionmentioning

confidence: 99%

Local unitary equivalence of the SLOCC class of three qubits

Li¹,

Guo²

2020

J. Phys. A: Math. Theor.

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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.

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