The Discretized Adiabatic Theorem

Abstract

A discretized version of the adiabatic theorem is described with the help of a rule relating a Hermitian operator to its expectation value and variance. The simple initial operator X with known ground state is transformed in a series of N small steps into a more complicated final operator Z with unknown ground state. Each operator along the discretised path in the space of Hermitian matrices is used to measure the state, initially the ground state of X. Measurements similar to the Zeno effect or Renninger's negative measurements modify the state incrementally. This process eventually leads to an eigenstate combination of Z. In the limit of vanishing step size the state stays with overwhelming probability in the ground state of each of the N observables.