Canonical decompositions of n -qubit quantum computations and concurrence

• Published in: Journal of mathematical physics 2004-06, Vol.45 (6), p.2447-2467
• Main Authors: Bullock, Stephen S., Brennen, Gavin K.
• Format: Article
• Language: English
• Subjects: Information systems; Physics; Quantum theory
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.1723701

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(2 n )=KAK. The group K fixes a bilinear form related to the concurrence, and in particular any unitary in K preserves the tangle | 〈φ| ¯ (−iσ 1 y )⋯(−iσ n y )|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(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 ak 2 for such an a∈A has the same property. Finally, although | 〈φ| ¯ (−iσ 1 y )⋯(−iσ n y )|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.