A constructive Hayden–Preskill decoder

We now change the initial model explicitly.

Let the message have $k$ qubits and dimension $m=2^k$. Let $B,E$ each have $b$ qubits, initially in $|{\Phi }_{\beta }{⟩}_{BE}$, where $\beta =2^b$. Let

Thus

The observer is given $ED$ in batch. It knows an explicit circuit for $U$, or is explicitly granted observer-register operations $U^{\ast }$ and, for amplification, $U^T$. It has no access to the actual remnant $C$.

The construction is the conjugate-evolution and Bell-projection architecture of Yoshida–Kitaev YK (Sections 4–5). We prove an exact specialization and its ledger below rather than importing an unquantified scrambling approximation.