Manuscript Em03-B@2 · Em03-B-v2.tex · section · c1a0977d

Pointer interaction and complete instrument

B5: a complete readout apparatus and its errors › Pointer interaction and complete instrument · lines 1459–1497

Em03-B@2#eq:assignmentinstrument

Let $A$ consist of $b_{N}$ pointer qubits. With ideal preparation $∣ 0^{b_{N}} ⟩_{A}$ , the readout interaction is

U_{R} = j = 1 \prod^{b_{N}} CNOT_{Q_{j} \to A_{j}} .

One explicit bounded generator for duration $T_{R}$ is

H_{R} = j = 1 \sum^{b_{N}} \frac{π}{2 T _{R}} ∣ 1 ⟩ ⟨ 1 ∣_{Q_{j}} \otimes (I - X_{A_{j}}),

with identity placement on the other factors. Its summands commute, and its time- $T_{R}$ evolution is $U_{R}$ .

The pointer is read in its computational basis, both $Q$ and $A$ are consumed, and $S$ is returned. With no noise this induces exactly (eq:idealread), including correlated references.

For a stochastic assignment matrix $W (y ∣ j)$ , $y \in Y$ , define

P_{W}^{S} (X) = j, y \sum W (y ∣ j) ∣ y ⟩ ⟨ y ∣_{F} \otimes M_{j} X M_{j}^{†} .

This is a complete instrument, not a fitted probability rule for one input.

An explicit assignment dilation maps pointer word $j$ to

y \sum W (y ∣ j) ∣ y ⟩_{F} ∣ j, y ⟩_{E} .

The orthonormal environment labels make this an isometry and produce (eq:assignmentinstrument) after the discarded systems are traced out. Failure records remain in $Y$ .

Displayed formulas in this passage

Em03-B@2#eq:assignmentinstrument · line 1481
$P_{W}^{S} (X) = j, y \sum W (y ∣ j) ∣ y ⟩ ⟨ y ∣_{F} \otimes M_{j} X M_{j}^{†} .$
unlabeled · line 1463
$U_{R} = j = 1 \prod^{b_{N}} CNOT_{Q_{j} \to A_{j}} .$
unlabeled · line 1467
$H_{R} = j = 1 \sum^{b_{N}} \frac{π}{2 T _{R}} ∣ 1 ⟩ ⟨ 1 ∣_{Q_{j}} \otimes (I - X_{A_{j}}),$
unlabeled · line 1491
$y \sum W (y ∣ j) ∣ y ⟩_{F} ∣ j, y ⟩_{E} .$

Pointer interaction and complete instrument

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