Three qubits can be entangled in two inequivalent ways

By Wolfgang Dür, Guifré Vidal, J. I. Cirac. 2000

2,983 citations recorded by the source corpus.

Invertible local transformations of a multipartite system are used to define equivalence classes in the set of entangled states. This classification concerns the entanglement properties of a single copy of the state. Accordingly, we say that two states have the same kind of entanglement if both of them can be obtained from the other by means of local operations and classical communication (LOCC) with nonzero probability. When applied to pure states of a three-qubit system, this approach reveals the existence of two inequivalent kinds of genuine tripartite entanglement, for which the Greenberger-Horne-Zeilinger state and a W state appear as remarkable representatives. In particular, we show that the W state retains maximally bipartite entanglement when any one of the three qubits is traced out. We generalize our results both to the case of higher-dimensional subsystems and also to more than three subsystems, for all of which we show that, typically, two randomly chosen pure states cannot be converted into each other by means of LOCC, not even with a small probability of success.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/503356faa033d7ab94ad28571f3985f2c0cc5638

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Citations in the corpus

Entanglement entropy: Holography and renormalization group (2018, neutral): “(28) Another type of entangled states is called the W state (Dur et al., 2000), |W 〉 = 1√ n (|10 · · · 00〉+ |010 · · · 0〉+ · · ·+ |00 · · · 01〉) .”
Colloquium: Trapping and manipulating photon states in atomic ensembles (2003, background): “Finally we point to the recent theoretical studies on decoherence properties of collective states (Dur et al., 2000).”
Entanglement in many-body systems (2007, background): “The latter is the invariance group relevant for qubit entanglement classification and quantification (Dür et al., 2000; Osterloh and Siewert, 2005; Verstraete et al., 2003).”
Quantum information (2005, positive): “Among them, three-particle W and GHZ states: |Ψ〉GHZ = (|000〉 + |111〉)/ √ 2 [80], |Ψ〉W = (|001〉 + |010〉 + |100〉)/ √ 3 [81], which represent two different types of entanglement that cannot be transformed into each other through local operations and classical communications (LOCC).”
Geometry of Quantum States: Index (2007, methodology): “The latter quantity, characterising the mean bipartite entanglement, is maximised by the W state [278],”
Macroscopic quantum states: Measures, fragility, and implementations (2017, background): “I.A.”
Multiboundary wormholes and holographic entanglement (2014, background): “Other forms of multiparty entanglement cannot be interconverted at all [35].”
Quantum Information with Continuous Variables (2004, background): “For the case of three qubits, any pure and fully entangled state can be transformed to either the GHZ state or the so-called W state Dür, Vidal, and Cirac, 2000 , W = 1 3 100 + 010 + 001 , 122 via stochastic local operations and classical communication stochastic LOCC, in which stochastic means…”
Quantum metrology with nonclassical states of atomic ensembles (2016, methodology): “While the W state is N -particle entangled (Dür et al., 2000), its quantum Fisher information only amounts to 3N − 2.”