### Abstract

An estimate is derived for the volume fraction of a subset C ^{P}_{ε} = {U : ∥ grad J(U) ∥ ≤ ε} ⊂ U(N) in the neighborhood of the critical set C^{P} ≃ U(n)PU(m) of the kinematic quantum ensemble control landscape J(U) = Tr(U _{ρ}U†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 language | English (US) |
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Article number | 255302 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 44 |

Issue number | 25 |

DOIs | |

State | Published - Jun 24 2011 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Modeling and Simulation
- Mathematical Physics
- Physics and Astronomy(all)