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Chinese Physics B

Relations between tangle and I concurrence for even n-qubit states

Xin-Wei Zha, Ning Miao, Ke Li

Published Oct 17, 2019• 0 citations
Publisher

Abstract

Gilad Gour and Nolan R Wallach [J. Math. Phys. 51 112201 (2010)] have proposed the 4-tangle and the square of the I concurrence. They also gave the relationship between the 4-tangle and the square of the I concurrence. In this paper, we give the expression of the square of the I concurrence and the n-tangle for six-qubit and eight-qubit by some local unitary transformation invariant. We prove that in six-qubit and eight-qubit states there exist strict monogamy laws for quantum correlations. We elucidate the relations between the square of the I concurrence and the n-tangle for six-qubit and eight-qubits. Especially, using this conclusion, we can show that 4-uniform states do not exist for eight-qubit states.

DOI
10.1088/1674-1056/ab4e7d

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Gilad Gour and Nolan R Wallach [J. Math. Phys. 51 112201 (2010)] have proposed the 4-tangle and the square of the I concurrence. They also gave the relationship between the 4-tangle and the square of the I concurrence. In this paper, we give the expression of the square of the I concurrence and the n-tangle for six-qubit and eight-qubit by some local unitary transformation invariant. We prove that in six-qubit and eight-qubit states there exist strict monogamy laws for quantum correlations. We elucidate the relations between the square of the I concurrence and the n-tangle for six-qubit and eight-qubits. Especially, using this conclusion, we can show that 4-uniform states do not exist for eight-qubit states.
We show that for even nnn qubits, n\ n n-tangle is the square of the SLOCC polynomial invariant of degree 2. We find that the nnn-tangle is not the residual entanglement for any even n≥4n\geq 4n≥4\ qubits. We give a necessary and sufficient condition for the vanishing of the concurrence C1(2...n)C_{1(2...n)}C1(2...n)​. The condition implies that the concurrence C1(2...n)C_{1(2...n)}C1(2...n)​ is always positive for any entangled states while the nnn-tangle vanishes for some entangled states. We argue that for even nnn\ qubits, the concurrence $ C_{1(2...n)}$\ is equal to or greater than the nnn-tangle.\ Further, we reveal that the residual entanglement is a partial measure for product states of any nnn qubits while the nnn-tangle is multiplicative for some product states.
The two-qubit canonical decomposition SU(4)=[SU(2)⊗SU(2)]Δ[SU(2)⊗SU(2)] writes any two-qubit unitary operator as a composition of a local unitary, a relative phasing of Bell states, and a second local unitary. Using Lie theory, we generalize this to an n-qubit decomposition, the concurrence canonical decomposition (CCD) SU(2n)=KAK. The group K fixes a bilinear form related to the concurrence, and in particular any unitary in K preserves the tangle |〈φ|¯(−iσ1y)⋯(−iσny)|φ〉|2 for n even. Thus, the CCD shows that any n-qubit unitary is a composition of a unitary operator preserving this n-tangle, a unitary operator in A which applies relative phases to a set of GHZ states, and a second unitary operator which preserves the tangle. As an application, we study the extent to which a large, random unitary may change concurrence. The result states that for a randomly chosen a∈A⊂SU(22p), the probability that a carries a state of tangle 0 to a state of maximum tangle approaches 1 as the even number of qubits approaches infinity. Any v=k1ak2 for such an a∈A has the same property. Finally, although |〈φ|¯(−iσ1y)⋯(−iσny)|φ〉|2 vanishes identically when the number of qubits is odd, we show that a more complicated CCD still exists in which K is a symplectic group.