Introduction and scope

A spectrum does not specify a quantum channel. Occupation statistics leave phases, temporal correlations, remnant states, and logical encoding undetermined. An entropy curve likewise does not determine a decoder or its execution resources.

Hayden–Preskill retrieval assumes rapidly mixing unitary dynamics and extensive access to radiation HP. Random-unitary evaporation models already distinguish internal and environment-coupling timescales and investigate information retrieval Piroli. Memory minimization is a global multistep optimization problem, even when classical memory is free Bisio.

Our mathematical contributions are:

1. exact transmission-dependent recovery and atomic-delivery memory optima for a finite benchmark;

2. realizability constraints for prescribed emission probabilities;

3. loss-compatible repeated emission, quantitative absorption, and explicit complete recovery programmes.

The underlying benchmark state is locally Clifford-equivalent to Smolin's unlockable state Smolin. We therefore claim neither a new unlockable state nor a new joint-measurement unlocking mechanism. Our additional statements concern complete-channel error, propagation loss, and a declared delivery interface.

Operator-spreading diagnostics have a separate role Nahum; they are not recovery certificates. Clifford components admit stabilizer simulation AaronsonGottesman, but arbitrary emission amplitudes and non-Clifford internal gates do not make the complete process a stabilizer circuit. Pure-loss purification provides relevant propagation and error-detection context Winnel, without establishing the finite deterministic recovery results proved here.