Assumption: Capacity and free-resource accounting on the cut

Conditional on an input-independent shared classical resource $\Lambda =\lambda$:

1. the wholly classical alternative has a complete input-bearing cut state $M$ with at most $K$ realized values;

2. any additional freely available post-cut resource has a specified input-independent law and does not carry additional input information jointly with $M$;

3. for the one-qubit comparison, the input-bearing cut state is a qubit, with no preshared entanglement or additional correlated decoder-accessible quantum system;

4. an uncounted auxiliary quantum system is free only when its joint cut state has the product form

   $${{\rho }_{x,\lambda }}^M\otimes {{\sigma }_{\lambda }}^F;$$

   merely requiring ${{\sigma }_{\lambda }}^F$ to be the same marginal for every $x$ is insufficient;

5. shared classical randomness is not mutated into an additional current-input-dependent storage variable;

6. the decoder receives realized classical memory values, not an exact observable description of their encoding probabilities.

For the main classical benchmark $K=2$.