Manuscript Hawk04-B@2 · Hawk04-B.tex · proposition · ba3b0713

Proposition: Finite-dimensional semantic realization

A finite typed circuit semantics › Destructive partial readout · lines 741–768 · proof included or linked

Hawk04-B@2#prop:finite-realization

A finite-dimensional isometry has a qubit encoding and a unitary extension after pure allocation.

A finite-dimensional CPTP channel can be represented by pure allocation, a specified unitary, destructive readout of its unused work register, discard of those work outcomes, and return of its quantum output.

Proof. Choose orthonormal basis dictionaries into sufficiently large qubit registers. Complete the columns of the encoded isometry to an orthonormal basis, obtaining a unitary extension.

For Kraus operators $D_{j}$ , the map

L ∣ ψ ⟩ = j \sum D_{j} ∣ ψ ⟩ ∣ j ⟩_{G}

is an isometry because $\sum_{j} D_{j}^{†} D_{j} = I$ . A unitary extension implements this coherent map. Readout of $G$ and discard of the outcome give

X ⟼ j \sum D_{j} X D_{j}^{†} .

The designated logical output remains quantum.

Displayed formulas in this passage

unlabeled · line 758
$L ∣ ψ ⟩ = j \sum D_{j} ∣ ψ ⟩ ∣ j ⟩_{G}$
unlabeled · line 764
$X ⟼ j \sum D_{j} X D_{j}^{†} .$

Proposition: Finite-dimensional semantic realization

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