Manuscript Hawk04-D@1 · Hawk04-D-v1.tex · proposition · 39b16f67

Proposition: Readout, forgetting, and complete accounting

Complete interfaces and the finite operational core › Enter, discard, and finite soundness · lines 457–491 · proof included or linked

Hawk04-D@1#prop:drop, Hawk04-D@1#eq:drop

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The instrument is CPTP and

dropBit_{K} M_{S} = Tr_{S} .

For any CPTP operation on $B$ , with all its outputs ignored, the untouched $QR$ marginal is unchanged. Thus residual Enter followed by Drop leaves $N$ and $ε_{rec} (N)$ exactly unchanged.

A complete readout dilation is

W_{read} ∣ ψ ⟩ = x \sum ∣ x ⟩_{K} ∣ x ⟩_{E} \otimes M_{x} ∣ ψ ⟩ .

It is an isometry. Tracing $E$ gives (eq:Enter). Discarding $K$ may be represented by moving its carrier to another inaccessible store. Those outputs remain in a complete dilation.

Proof. The measurement bras satisfy $\sum_{x} M_{x}^{†} M_{x} = I$ . Summing the classical blocks gives the basis formula for partial trace. For a local channel $Λ_{B}$ , its adjoint is unital, so every untouched observable $A$ obeys

(id \otimes Λ_{B}^{*}) (A \otimes I) = A \otimes I .

All untouched expectations agree, including references. Orthogonality of the two outcome labels and measurement completeness give $W_{read}^{†} W_{read} = I$ . Tracing $E$ removes the cross-outcome terms.

Displayed formulas in this passage

Hawk04-D@1#eq:drop · line 460
$dropBit_{K} M_{S} = Tr_{S} .$
unlabeled · line 470
$W_{read} ∣ ψ ⟩ = x \sum ∣ x ⟩_{K} ∣ x ⟩_{E} \otimes M_{x} ∣ ψ ⟩ .$
unlabeled · line 484
$(id \otimes Λ_{B}^{*}) (A \otimes I) = A \otimes I .$

Proposition: Readout, forgetting, and complete accounting

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