Canonical Decompositions of n-qubit Quantum Computations and Concurrence

Abstract

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.