Canonical Decompositions of n-qubit Quantum Computations and Concurrence

The two-qubit canonical decomposition SU(4) = [SU(2)⊗ SU(2)]∆[SU(2)⊗ 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(2n) = KAK. The group K fixes a bilinear form related to the concurrence, and in particular any computation in K preserves the tangle |〈φ|(−iσ1) · · · (−iσ y n)|φ〉| 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 ∈ 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|(−iσ1) · · · (−iσ y n)|φ〉| 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.