Quantum Inf Process (2012) 11:481–492
The n-tangle of odd n qubits
Received: 7 February 2011 / Accepted: 5 July 2011 / Published online: 22 July 2011
© Springer Science+Business Media, LLC 2011
Abstract Coffman et al. presented the 3-tangle of three qubits in Phys Rev A 61,
052306 (2000). Wong and Christensen (Phys Rev A 63, 044301, 2001) extended the
standard form of the 3-tangle to even number of qubits, known as n-tangle. In this
paper, we propose a generalization of the standard form of the 3-tangle to any odd
n-qubit pure states and call it the n-tangle of odd n qubits. We show that the n-tangle
of odd n qubits is invariant under permutations of the qubits, and is an entanglement
monotone. The n-tangle of odd n qubits can be considered as a natural entanglement
measure of any odd n-qubit pure states, and used for stochastic local operations and
classical communication classiﬁcation.
Keywords 3-tangle · n-tangle · Concurrence · Residual entanglement
Quantum entanglement is a key quantum mechanical resource in quantum compu-
tation and information, such as quantum cryptography, quantum dense coding and
quantum teleportation . Entanglement measure, which characterizes the degree of
entanglement contained in a quantum state, has been a subject under intensive research.
The entanglement of bipartite systems is well understood. The concurrence is
a good entanglement measure for two-qubit states and is an entanglement monotone,
The paper was supported by NSFC(Grant No. 10875061) and Tsinghua National Laboratory for
Information Science and Technology.
D. Li (
Department of mathematical sciences, Tsinghua University, Beijing, 100084, China