Fig. 2

Sketch of proof of the main result. We show how an operation of the form of Fig. 1b can be used to build an operation of the form Fig. 1a. This gives the direction ⇐ in (13) for the equivalence of Theorem 3 (the other direction is trivial, see Supplementary Methods 1). The construction has three sub-blocks: Box U₁ represents the fact that one can obtain the microstate γ _β (H_E) to arbitrary precision from many copies of the macrostate (e _β (H_E), H_E) using a macrostate operation (interestingly, this can be done with exact energy conservation). This result relies on a central limit theorem and typicality results for individual energy eigenspaces of many non-interacting systems. Box U₂ operates by choosing as H_E as a rescaled version of H and showing that one can then obtain the microstate γ _e (H) using a macrostate operation. Box U_mic exists by assumption: it uses the microstate operation to obtain ρ_f from γ _e (H) (it is the one represented in Fig. 1b))