The complete finite interface and error conventions

The word “complete” is a modeling requirement, not a consequence of naming one environmental marginal. It is essential to Theorem thm:exact and Example ex:purifier.

One way to obtain (eq:interface) is to specify an encoding $J:C\to H_{\mathrm{payload}}$, a fixed normalized ancilla $|\Omega ⟩$ containing any necessary purifiers, and a complete unitary $U$, and then set

The ancilla is fixed independently of the unknown logical input. The construction is mathematically isometric; whether it describes a gravitational preparation and evolution is a separate obligation.

For a linear map $\Lambda$ between matrix spaces, its diamond norm is

For channels with a common input and output,

For channel differences, the diamond norm can be evaluated on density operators with a reference of dimension equal to the input dimension. Thus the errors used here are worst-case reference-preserving errors, not errors for one selected input.

The finite stabilized norm facts in this convention are standard channel-norm facts; see the preliminaries of KSW. All distances between channels in this paper use (eq:distances).