Manuscript VIII-ST@1 · VIII-ST.tex · proposition · fe979f3b

Proposition: Informative records exclude full-input exact recovery

Named access, residual instruments, and recovery boundaries › Quantum recovery safeguards · lines 1232–1254 · proof included or linked

VIII-ST@1#prop:informative

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Let a complete recorded instrument have a CPTP left inverse on the full input matrix algebra. Every record effect is then a scalar multiple of the identity. Consequently an informative classical record cannot coexist with exact recovery of every unknown input from that complete recorded output.

Proof. Let $A_{i}$ be Kraus operators of the recorded channel and $B_{j}$ those of a left inverse. The products $B_{j} A_{i}$ are Kraus operators for the identity channel. Its Choi matrix has rank one, so $B_{j} A_{i} = c_{j i} I$ . Since $\sum_{j} B_{j}^{†} B_{j} = I$ ,

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

Sum the diagonal terms belonging to one record. Its effect is scalar.

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

Proposition: Informative records exclude full-input exact recovery

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