## Abstract

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 _{1}ak _{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 language | English |
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Pages (from-to) | 2447-2467 |

Number of pages | 21 |

Journal | Journal of Mathematical Physics |

Volume | 45 |

Issue number | 6 |

DOIs | |

Publication status | Published - Jun 2004 |

Externally published | Yes |