What reconstruction means

A channel acts on states, whereas an observable is an operator. Accordingly, the schematic expression

must be replaced by a statement with specified types.

For the present paper, the exact operator statement will be

where $U$ is the declared finite representation correspondence and ${\mathcal{O}}_B$ belongs to the algebra generated by boundary data on $B$. The exact operational statement will be

Here ${\mathcal{N}}_B$ supplies the declared boundary quantum data, ${\mathcal{R}}_B$ is a binary record channel, and ${\mathcal{M}}_{{\mathcal{O}}_{\mathrm{bulk}}}$ is the binary measurement channel associated with ${\mathcal{O}}_{\mathrm{bulk}}$.

Equation (eq:typed-channel-target) does not claim recovery of an unknown bulk quantum state. Nor does it identify the conditional postmeasurement bulk state with a boundary instrument output. Observable-algebra recovery and state recovery are distinct questions BKK2007,ADH2015.