Prepared control and expectation convention

On a finite-dimensional subsystem with unperturbed $H=H^{\ast }$ in J, bounded self-adjoint control observable $B$, $\mathrm{\hslash }>0$ in J s, and finite interval $[0,T]$ with $T>0$ in s, adopt the idealized controlled Hamiltonian $H_{ϵ}(t)=H-ϵh(t)B,$ where $h:[0,T]\to \mathbb{R}$ is continuous, $ϵ\in \mathbb{R}$ is dimensionless, and $h(t)B$ has unit J. The initial prepared state is held fixed at $t=0$ as the Gibbs matrix $\sigma =Z_{\beta }^{-1}{e}^{-\beta H}$, $\beta >0$ in J${}^{-1}$; the model does not re-prepare a Gibbs state for $H_{ϵ}(t)$. Expectations use the adopted normal-state Born convention of Normal-state quantum measurement interpretation. The full Axiom of Choice convention, stated in The Axiom of Choice, of that wider measurement framework is retained. This is an adopted control model; it makes no claim about a particular device or data.

For the controlled propagator $U_{ϵ}$ satisfying $i\mathrm{\hslash }{U}_{ϵ}^{\mathrm{\prime }}(t)=H_{ϵ}(t)U_{ϵ}(t)$ and $U_{ϵ}(0)=I$, define ${\rho }_{ϵ}(t)=U_{ϵ}(t)\sigma U_{ϵ}(t{)}^{\ast }$ and ${\mathbb{E}}_{ϵ,t}[A]=\mathrm{Tr}({\rho }_{ϵ}(t)A)$ for bounded self-adjoint matrices $A$. Existence and uniqueness of this finite-dimensional propagator are supplied by Finite-time Kubo formula with norm remainder.

The finite-time response theorem Finite-time Kubo formula with norm remainder supplies the exact mathematical response interface used to formulate this postulate. It does not establish the physical applicability of the control model.