The n-tangle of odd n qubits

Xin-Wei Zha, Irfan Ahmed, Najeeb ur Rehman Lashari +1 more

Dafa Li, Xiangrong Li, Hongtao Huang +1 more

Coffman et al. [Phys. Rev. A 61, 052306 (2000)] introduced the residual entanglement for three qubits. In this paper, we present the entanglement measure τ(ψ) for even n qubits; for odd n qubits, we propose the residual entanglement τ(i)(ψ) with respect to qubit i and the odd n-tangle R(ψ) by averaging the residual entanglement with respect to each qubit. In this paper, we show that these measures are LU-invariant, entanglement monotones, invariant under permutations of the qubits, and multiplicative in some cases.

Stephen S. Bullock, Gavin K. Brennen

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.

Richard J. Fitzgerald

Bullock, S S, Gavin K. Brennen

The two-qubit canonical decomposition SU(4) = [SU(2) \\otimes SU(2)] Delta [SU(2) \\otimes SU(2)] writes any two-qubit quantum computation 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 (C.C.D.) SU(2^n)=KAK. The group K fixes a bilinear form related to the concurrence, and in particular any computation in K preserves the tangle |<phi^*|(-i sigma^y_1)...(-i sigma^y_n)|phi>|^2 for n even. Thus, the C.C.D. shows that any n-qubit quantum computation is a composition of a computation preserving this n-tangle, a computation in A which applies relative phases to a set of GHZ states, and a second computation which preserves it. 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 in A within SU(2^{2p}), 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=k_1 a k_2 for such an a \\in A has the same property. Finally, although |<phi^*|(-i sigma^y_1)...(-i sigma^y_n)|phi>|^2 vanishes identically when the number of qubits is odd, we show that a more complicated C.C.D. still exists in which K is a symplectic group.

Xu Xu