Manuscript En10-A@1 · En10-A.tex · section · 0a1d7743

Record and residual are specified together

The destructive instrument and its complete identity › Record and residual are specified together · lines 395–428

En10-A@1#eq:enter, En10-A@1#eq:target, En10-A@1#eq:instrument

For any retained factor $T$ , including $T = S \otimes Q$ , define

D_{r} M_{B}^{T} (X) J_{B}^{T} (X) = ⟨ r ∣_{B} \otimes I_{T}, = r \in R \sum ∣ r ⟩ ⟨ r ∣_{C} \otimes D_{r} X D_{r}^{†}, = r \in R \sum ∣ r ⟩ ⟨ r ∣_{C} \otimes Tr_{B} [(P_{r} \otimes I_{T}) X (P_{r} \otimes I_{T})] .

When $T$ is one-dimensional, the $T$ -blocks are scalars and the output is a classical record state.

The complete one-shot instrument is

I^{T} = M_{B}^{T} \circ (Φ \otimes id_{T}) .

Its selective $r$ -branch is the corresponding subnormalized block. All four outcomes are retained. There is no postselection.

Displayed formulas in this passage

En10-A@1#eq:enter · line 398
$D_{r} M_{B}^{T} (X) J_{B}^{T} (X) = ⟨ r ∣_{B} \otimes I_{T}, = r \in R \sum ∣ r ⟩ ⟨ r ∣_{C} \otimes D_{r} X D_{r}^{†}, = r \in R \sum ∣ r ⟩ ⟨ r ∣_{C} \otimes Tr_{B} [(P_{r} \otimes I_{T}) X (P_{r} \otimes I_{T})] .$
En10-A@1#eq:target · line 398
$D_{r} M_{B}^{T} (X) J_{B}^{T} (X) = ⟨ r ∣_{B} \otimes I_{T}, = r \in R \sum ∣ r ⟩ ⟨ r ∣_{C} \otimes D_{r} X D_{r}^{†}, = r \in R \sum ∣ r ⟩ ⟨ r ∣_{C} \otimes Tr_{B} [(P_{r} \otimes I_{T}) X (P_{r} \otimes I_{T})] .$
En10-A@1#eq:instrument · line 420
$I^{T} = M_{B}^{T} \circ (Φ \otimes id_{T}) .$