The Quantum Clock and the Autonomization Theorem

The idea is to autonomize the dynamics: remove the explicit time dependence by promoting the controller to the status of a quantum system. To do so, we introduce a quantum clock C, coupled to the system S through a time-independent total Hamiltonian:

   Ĥ_tot = Ĥ_C + Ĥ_int                        (time-independent)
   Ĥ_int = ∫ dt'  Ĥ_S(t') ⊗ |t'⟩⟨t'|          (clock-system coupling)

The clock has a basis of states |t⟩ labeled by a "time". The interaction term reads this state: when the clock is in |t'⟩, the system feels the Hamiltonian Ĥ_S(t'). The clock, for its part, simply moves forward — its Hamiltonian Ĥ_C generates the translation e^(−i Ĥ_C s) |t⟩ = |t+s⟩. Nothing depends explicitly on time: the total energy is conserved.

The key result is the autonomization theorem. If we start from a product state (system and clock uncorrelated), the reduced dynamics of the system alone reproduces the original driven evolution exactly:

   initial state :  ρ̂_tot(0) = ρ̂_S ⊗ |t₀⟩⟨t₀|
   then          :  Tr_C[ ρ̂_tot(Δt) ] = Û(t,t₀) ρ̂_S Û†(t,t₀)     (with t = t₀ + Δt)

In other words: any dynamics generated by a time-dependent Hamiltonian admits an autonomous, energy-conserving realization on a larger system. There is no longer any need for an external agent — the "driving" emerges from the interaction with the clock.

This is stronger than the usual Stinespring dilation, which only guarantees that each evolution can be realized on an enlarged system. Here, it is the whole family of evolutions, at all times, that follows from a single time-independent Hamiltonian. Bonus: choosing a different initial clock state naturally models imperfect control, where time is ill-defined.