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

• URL: http://arxiv.org/abs/2203.09610v2
• Date: Sun, 15 Jan 2023 02:45:50 GMT
• Title: A relation among tangle, 3-tangle, and von Neumann entropy of entanglement for three qubits
• Authors: Dafa Li, Maggie Cheng, Xiangrong Li, and Shuwang Li
• Abstract summary: We derive a formula of the tangle for pure states of three qubits, and present three explicit local unitary (LU) invariants. Our result goes beyond the classical work of tangle, 3-tangle and von Neumann entropy of entanglement. We obtain all the states of three qubits of which tangles, concurrence, 3-tangle and von Neumann entropy don't vanish.
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• License: http://creativecommons.org/licenses/by/4.0/
• 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\'{\i}n et al.' Schmidt decomposition (ASD) of three qubits by connecting the tangle, 3-tangle, and von Neumann entropy for ASD with Ac\'{\i}n 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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