Canonical decompositions of n-qubit quantum computations and concurrence

Stephen S. Bullock*, Gavin K. Brennen

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

35 Citations (Scopus)


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 ε 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 1ak 2 for such an a ε 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.

Original languageEnglish
Pages (from-to)2447-2467
Number of pages21
JournalJournal of Mathematical Physics
Issue number6
Publication statusPublished - Jun 2004
Externally publishedYes


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