Manuscript VII-G@1 · VII-G.tex · section · 54f561e8

An exact dilation and its resource meaning

Implementation equivalence and represented readout › An exact dilation and its resource meaning · lines 933–959

VII-G@1#eq:dilation, VII-G@1#eq:dilated-instrument

Fix an input label $g$ . Choose a Kraus representation for every nonzero branch and write $a = (h, α)$ . Define

V_{g} ψ = h, α \sum ∣ h, α ⟩_{A} \otimes K_{h ∣ g, α} ψ .

Completeness implies $V_{g}^{†} V_{g} = I$ .

Destructive computational readout of $A$ , followed by reporting only $h$ , gives

h \sum ∣ h ⟩ ⟨ h ∣_{R} \otimes α \sum K_{h ∣ g, α} X K_{h ∣ g, α}^{†} .

The matter register is returned. The carrier is consumed. This is a represented instrument, not destructive readout of the matter source itself.

The Born readout interface is the specified semantic primitive of the Enter calculus CheungEnter. No measurement-strategy uniqueness theorem is used as a premise here.

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VII-G@1#eq:dilation · line 937
$V_{g} ψ = h, α \sum ∣ h, α ⟩_{A} \otimes K_{h ∣ g, α} ψ .$
VII-G@1#eq:dilated-instrument · line 947
$h \sum ∣ h ⟩ ⟨ h ∣_{R} \otimes α \sum K_{h ∣ g, α} X K_{h ∣ g, α}^{†} .$

An exact dilation and its resource meaning

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