Combining errors and binary embedding

For a positive approximation $L$, define ${\delta }_{K,L}=\min \{1,\sqrt{J{\Vert K-L\Vert }_{\infty }}\}$. In the common embedding,

Entrywise errors at most $\eta$ imply ${\delta }_{K,L}\le \min \{1,J\sqrt{\eta }\}$. A numerical convergence difference is not automatically a justified entrywise error bound.

Choose an ordered occupation basis, append the flag, and map these $d_N+1$ states to the first computational basis states of $q_{\mathrm{raw}}$ qubits. Unused binary states have zero prepared weight. A reverse representation map pinches valid versus invalid binary subspaces and sends invalid weight to the flag; it is CPTP by the same Kraus construction as the cutoff. Its action on valid encoded states is exact.

The representation does not realize bosonic CCR exactly in finite dimension. For bounded embedded observables the expectation error is at most $2t_N(K){\Vert A\Vert }_{\infty }$; number and energy require the separate moment estimates.