Canonical decompositions of n-qubit quantum computations and concurrence

The two-qubit canonical decomposition SU(4)=[SU(2)circle timesSU(2)]Delta[SU(2)circle timesSU(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 \(<phi vertical bar) over bar(-isigma(1)(y))...(-isigma(n)(y))\phi>\(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 ais an element ofAsubset ofSU(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 ais an element ofA has the same property. Finally, although \(<phi vertical bar) over bar(-isigma(1)(y))...(-isigma(n)(y))\phi>\(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. (C) 2004 American Institute of Physics.