Manuscript IX-A-ST@1 · IX-A-ST.tex · proposition · 6e99d1b2

Proposition: Informative-record recovery obstruction

Residual compatibility, blindness, and recovery safeguards · lines 1736–1758 · proof included or linked

IX-A-ST@1#prop:recovery

If a complete recorded finite instrument has a CPTP left inverse on the full input matrix algebra, each record effect is a scalar multiple of the identity. An informative record is therefore incompatible with exact recovery of every unknown input from that complete recorded output.

Proof. Let $A_{i}$ be the recorded channel's Kraus operators and $B_{j}$ those of a left inverse. The operators $B_{j} A_{i}$ form a Kraus family for identity. Its Choi matrix has rank one, so $B_{j} A_{i} = c_{j i} I$ . Trace preservation of the inverse implies

A_{i}^{†} A_{k} = j \sum c_{j i} c_{j k} I .

Summing the diagonal terms associated with a record gives a scalar effect.

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unlabeled · line 1752
$A_{i}^{†} A_{k} = j \sum c_{j i} c_{j k} I .$

Proposition: Informative-record recovery obstruction

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