Manuscript Hawk04-C@1 · Hawk04-C-v1.tex · proposition · 94d46352

Proposition: Readout probabilities, residuals, and references

Row 3: the radiation and readout analogy · lines 788–827 · proof included or linked

Hawk04-C@1#prop:instrument, Hawk04-C@1#eq:read-drop-general

The map (eq:enter) is CPTP. For a state $ρ$ , outcome $z$ has probability

p_{z} = Tr (M_{z} ρ M_{z}^{†}) .

If $p_{z} > 0$ , its surviving state is $M_{z} ρ M_{z}^{†} / p_{z}$ . For every untouched reference, the same formulas apply with $M_{z}$ tensored with the reference identity. Moreover,

dropBit_{K} M_{S} (X) = Tr_{S} X .

Proof. Kraus operators for the hybrid map are $∣ z ⟩_{K} \otimes M_{z}$ . They satisfy

z \sum M_{z}^{†} M_{z} = I_{S T},

which proves complete positivity and trace preservation. Taking the trace of one classical block gives $p_{z}$ . Positive-probability normalization gives the conditional state. No normalized conditional state is required for a zero-probability branch.

Finally,

z \sum (⟨ z ∣_{S} \otimes I_{T}) X (∣ z ⟩_{S} \otimes I_{T}) = Tr_{S} X

is the basis formula for partial trace. All these identities remain true after tensoring with $id_{Q}$ .

Displayed formulas in this passage

Hawk04-C@1#eq:read-drop-general · line 800
$dropBit_{K} M_{S} (X) = Tr_{S} X .$
unlabeled · line 792
$p_{z} = Tr (M_{z} ρ M_{z}^{†}) .$
unlabeled · line 810
$z \sum M_{z}^{†} M_{z} = I_{S T},$
unlabeled · line 820
$z \sum (⟨ z ∣_{S} \otimes I_{T}) X (∣ z ⟩_{S} \otimes I_{T}) = Tr_{S} X$

Proposition: Readout probabilities, residuals, and references

Displayed formulas in this passage

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