TY - JOUR
T1 - Canonical decompositions of n-qubit quantum computations and concurrence
AU - Bullock, Stephen S.
AU - Brennen, Gavin K.
PY - 2004/6
Y1 - 2004/6
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=2942675041&partnerID=8YFLogxK
U2 - 10.1063/1.1723701
DO - 10.1063/1.1723701
M3 - Article
AN - SCOPUS:2942675041
SN - 0022-2488
VL - 45
SP - 2447
EP - 2467
JO - Journal of Mathematical Physics
JF - Journal of Mathematical Physics
IS - 6
ER -