Trotter Product Formulae for $$*$$-Automorphisms of Quantum Lattice Systems

Abstract

AbstractWe consider the dynamics $$t\mapsto \tau _t$$ t ↦ τ t of an infinite quantum lattice system that is generated by a local interaction. If the interaction decomposes into a finite number of terms that are themselves local interactions, we show that $$\tau _t$$ τ t can be efficiently approximated by a product of n automorphisms, each of them being an alternating product generated by the individual terms. For any integer m, we construct a product formula (in the spirit of Trotter) such that the approximation error scales as $$n^{-m}$$ n - m . Our bounds hold in norm, pointwise for algebra elements that are sufficiently well approximated by finite volume observables.