Canonical decompositions of n-qubit quantum computations and concurrence
Stephen S. Bullock, Gavin K. Brennen · Journal of Mathematical Physics · 2004
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.