Volume fractions of the kinematic 'near-critical' sets of the quantum ensemble control landscape

Jason Dominy, Herschel Rabitz

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


An estimate is derived for the volume fraction of a subset C Pε = {U : ∥ grad J(U) ∥ ≤ ε} ⊂ U(N) in the neighborhood of the critical set CP ≃ U(n)PU(m) of the kinematic quantum ensemble control landscape J(U) = Tr(U ρU†O), where U represents the unitary time evolution operator, ρ is the initial density matrix of the ensemble, and O is an observable operator. This estimate is based on the Hilbert.Schmidt geometry for the unitary group and a first-order approximation of ∥ grad J(U) ∥2. An upper bound on these near-critical volumes is conjectured and supported by numerical simulation, leading to an asymptotic analysis as the dimension N of the quantum system rises in which the volume fractions of these 'enear-critical' sets decrease to zero as N increases. This result helps explain the apparent lack of influence exerted by the many saddles of J over the gradient flow.

Original languageEnglish (US)
Article number255302
JournalJournal of Physics A: Mathematical and Theoretical
Issue number25
StatePublished - Jun 24 2011

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Modeling and Simulation
  • Mathematical Physics
  • General Physics and Astronomy


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