A record of primes is a classical output

For a measured register $Q$, define its destructive record channel by

Expansion in the measured basis gives

The same identity holds with any untouched reference.

This is a complete readout-and-drop identity. It is not probability-one factoring. To obtain a prime list, the preceding circuit must first satisfy the arithmetic support condition.

If only a classical record is available for recovering an unknown quantum state, any subsequent quantum-output decoder is measure-and-prepare. On half of an entangled input it produces a separable state, whereas the identity preserves the entanglement. Thus record-only quantum recovery and computation of a classical function are different tasks Watrous,CheungVIIIB,CheungHawk04B.

Landmark I concerns the declared destructive-readout semantics CheungI. The coherent simulation in Lemma lem:records-local is not an invocation of Enter in the middle of an interference computation.