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arXiv · 2203.09610

A relation among tangle, 3-tangle, and von Neumann entropy of entanglement for three qubits

Dafa Li·Maggie Cheng·Xiangrong Li·Shuwang Li
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Abstract

In this paper, we derive a general formula of the tangle for pure states of three qubits, and present three explicit local unitary (LU) polynomial invariants. Our result goes beyond the classical work of tangle, 3-tangle and von Neumann entropy of entanglement for Ac\'ın et al.' Schmidt decomposition (ASD) of three qubits by connecting the tangle, 3-tangle, and von Neumann entropy for ASD with Ac\'ın et al.'s LU invariants. In particular, our result reveals a general relation among tangle, 3-tangle, and von Neumann entropy, together with a relation among their averages. The relations can help us find the entangled states satisfying distinct requirements for tangle, 3-tangle, and von Neumann entropy. Moreover, we obtain all the states of three qubits of which tangles, concurrence, 3-tangle and von Neumann entropy don't vanish and these states are endurable when one of three qubits is traced out. We indicate that for the three-qubit W state, its average von Neumann entropy is maximal only within the W SLOCC class, and that under ASD the three-qubit GHZ state is the unique state of which the reduced density operator obtained by tracing any two qubits has the maximal von Neumann entropy.

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Dafa Li, Maggie Cheng, Xiangrong Li, Shuwang Li. 2023-01-15. A relation among tangle, 3-tangle, and von Neumann entropy of entanglement for three qubits. https://arxiv.org/abs/2203.09610

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