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arXiv:1402.2219 · Journal of Physics A Mathematical and Theoretical

The rebit three-tangle and its relation to two-qubit entanglement

William K. Wootters

Published Oct 9, 2014• 31 citations

Abstract

The three-tangle is a measure of three-way entanglement in a system of three qubits. For a pure state, it can be understood as the residual entanglement not accounted for by pairwise entanglements between individual qubits. Here we define and evaluate the analogous quantity for three rebits (that is, binary systems in the real-amplitude variant of quantum theory). We find that the resulting formula is the same as in the complex case, except that an overall absolute value sign is missing. As a result, the rebit three-tangle can be negative, expressing the possibility of non-monogamous entanglement in real-amplitude quantum theory (for entanglement based on the convex-roof construction). We then relate the entanglement among three rebits to the entanglement of two qubits, by re-expressing the two-qubit state as a three-rebit state in the ubit model. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

DOI: 10.1088/1751-8113/47/42/424037

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The three-tangle is a measure of three-way entanglement in a system of three qubits. For a pure state, it can be understood as the residual entanglement not accounted for by pairwise entanglements between individual qubits. Here we define and evaluate the analogous quantity for three rebits (that is, binary systems in the real-amplitude variant of quantum theory). We find that the resulting formula is the same as in the complex case, except that an overall absolute value sign is missing. As a result, the rebit three-tangle can be negative, expressing the possibility of non-monogamous entanglement in real-amplitude quantum theory (for entanglement based on the convex-roof construction). We then relate the entanglement among three rebits to the entanglement of two qubits, by re-expressing the two-qubit state as a three-rebit state in the ubit model. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.

We propose an entanglement measure to quantify three-qubit entanglement in terms of negativity. A monogamy inequality analogous to the Coffman-Kundu-Wootters inequality is established. This consequently leads to a definition of residual entanglement, which is referred to as the three-$\ensuremath{\pi}$ in order to distinguish it from the three-tangle. The three-$\ensuremath{\pi}$ is proved to be a natural entanglement measure. By contrast to the three-tangle, it is shown that the three-$\ensuremath{\pi}$ always gives greater than zero values for pure states belonging to the

W

and Greenberger-Horne-Zeilinger classes, implying that three-way entanglement always exists for them; the three-tangle generally underestimates the three-way entanglement of a given system. This investigation will offer an alternative tool to understand genuine multipartite entanglement.

We obtain, analytically, the global negativity, partial

K

-way negativities

(K=2,3)

, Wootter's tangle, and three-tangle for the generic three-qubit canonical state. It is found that the product of global negativity and partial three-way negativity is equal to the three-tangle, while the partial two-way negativity is related to the tangle of qubit pairs. We also calculate similar quantities for the state canonical to a single-parameter

(0<q<1)

pure state which is a linear combination of a Greenberger-Horne-Zeilinger state and a

W

state. In this case, for

q=0.626\phantom{\rule{0.2em}{0ex}}85

, the state has zero three-tangle and zero three-way negativity, having only

W

-like entanglement. The difference between the product of global and partial three-way negativities and the three-tangle for a given state is a quantitative measure of two-qubit coherences transformed by unitary transformations on the canonical state into three-qubit coherences. The global negativity and partial

K

-way negativities, obtained by selective partial transpositions on multiqubit state operators, satisfy inequalities which for three qubits are equivalent to the Coffman-Kundu-Wootter inequality.

Exploring an analytical expression for the convex roof of the pure state squared concurrence for rank 2 mixed states the entanglement of a system of three particles under decoherence is studied, using the monogamy inequality for mixed states and the residual entanglement obtained from it. The monogamy inequality is investigated both for the concurrence and the negativity in the case of local independent phase damping channel acting on generalized Greenberger-Horne-Zeilinger states of three particles and the local independent amplitude damping channel acting on generalized $\ensuremath{\mid}\mathrm{W}⟩=\ensuremath{\mid}001⟩+\ensuremath{\mid}010⟩+\ensuremath{\mid}100⟩$ state of three particles. It is shown that the bipartite entanglement between one qubit and the rest has a qualitative similar behavior to the entanglement between individual qubits, and that the residual entanglement in terms of the negativity cannot be a good entanglement measure for mixed states, since it can increase under local decoherence.

Quantum entanglement and non-locality are two special aspects of quantum correlations. The relationship between multipartite entanglement and non-locality is at the root of the foundations of quantum mechanics but there is still no general quantitative theory. In order to address this issue we analyze the relationship between tripartite non-locality and tripartite entanglement measure, called the three-tangle. We describe the states which give the extremal quantum values of a Bell-type inequality for a given value of the tripartite entanglement. Moreover, we show that such extremal states can be reached if one introduced an appropriate order induced by the three-π entanglement measure. Finally, we derive an analytical expression relating tripartite entanglement to the maximal violations of the Bell-type inequalities.