Manuscript En09-A@1 · En09-A.tex · section · eafec9de

The instrument and the conditioning target

The complete relational Enter theorem › The instrument and the conditioning target · lines 398–449

En09-A@1#eq:readout, En09-A@1#eq:computational-readout, En09-A@1#eq:pw-subnormalized, En09-A@1#eq:pw-probability, En09-A@1#eq:pw-normalized, En09-A@1#eq:ideal-relational-instrument

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Define the complete destructive clock-readout channel by

M_{C} (X) = k \in Z \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} X D_{k}^{†} .

Equivalently, in the computational presentation,

M_{C} (X) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes (⟨ k ∣_{C} \otimes I_{S}) (B_{C} \otimes I_{S}) X (B_{C}^{†} \otimes I_{S}) (∣ k ⟩_{C} \otimes I_{S}) .

For a preparation $ρ_{C S}$ , define

σ_{S, k} p_{k} ρ_{S} (t_{k}) = Tr_{C} [(P_{k} \otimes I_{S}) ρ_{C S} (P_{k} \otimes I_{S})], = Tr σ_{S, k}, = \frac{σ _{S, k}}{p _{k}} when p_{k} > 0.

Equation (eq:pw-normalized) is the clock-conditioned Page–Wootters state for this finite observable.

The appropriate channel target is not the normalized quotient in (eq:pw-normalized). It is the linear recorded map

J_{C} (X) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I_{S}) X (P_{k} \otimes I_{S})] .

Keeping subnormalized blocks prevents postselection from being mistaken for a deterministic channel.

Displayed formulas in this passage

En09-A@1#eq:readout · line 401
$M_{C} (X) = k \in Z \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes D_{k} X D_{k}^{†} .$
En09-A@1#eq:computational-readout · line 409
$M_{C} (X) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes (⟨ k ∣_{C} \otimes I_{S}) (B_{C} \otimes I_{S}) X (B_{C}^{†} \otimes I_{S}) (∣ k ⟩_{C} \otimes I_{S}) .$
En09-A@1#eq:pw-subnormalized · line 420
$σ_{S, k} p_{k} ρ_{S} (t_{k}) = Tr_{C} [(P_{k} \otimes I_{S}) ρ_{C S} (P_{k} \otimes I_{S})], = Tr σ_{S, k}, = \frac{σ _{S, k}}{p _{k}} when p_{k} > 0.$
En09-A@1#eq:pw-probability · line 420
$σ_{S, k} p_{k} ρ_{S} (t_{k}) = Tr_{C} [(P_{k} \otimes I_{S}) ρ_{C S} (P_{k} \otimes I_{S})], = Tr σ_{S, k}, = \frac{σ _{S, k}}{p _{k}} when p_{k} > 0.$
En09-A@1#eq:pw-normalized · line 420
$σ_{S, k} p_{k} ρ_{S} (t_{k}) = Tr_{C} [(P_{k} \otimes I_{S}) ρ_{C S} (P_{k} \otimes I_{S})], = Tr σ_{S, k}, = \frac{σ _{S, k}}{p _{k}} when p_{k} > 0.$
En09-A@1#eq:ideal-relational-instrument · line 438
$J_{C} (X) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes Tr_{C} [(P_{k} \otimes I_{S}) X (P_{k} \otimes I_{S})] .$

The instrument and the conditioning target

Displayed formulas in this passage

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