Manuscript En06-B@1 · En06-B.tex · proposition · ee886ae8

Proposition: Exact finite dilation

An explicit local apparatus dilation · lines 1526–1555 · proof included or linked

En06-B@1#prop:dilation, En06-B@1#eq:H0, En06-B@1#eq:globalunitary

On the common finite Hilbert space

H_{0} = Q_{L} \otimes Q_{R} \otimes W \otimes S_{L} \otimes S_{R} \otimes R_{L} \otimes E_{L} \otimes F_{L} \otimes R_{R} \otimes E_{R} \otimes F_{R},

let

U = (U_{L} U_{R}) (U_{B} \otimes I_{rest}) .

Insert the six apparatus blank qubits, apply $U$ , and trace $E_{L} F_{L} E_{R} F_{R}$ . The resulting operational channel is exactly (eq:Bellchannel).

Proof. Equation (eq:dilationaction) copies the outcome into two orthogonal carriers $R, E$ and the setting into $S, F$ . Tracing $E$ removes cross-outcome terms; tracing $F$ removes off-diagonal setting terms. The remaining terms have precisely the Kraus expression in (eq:Bellchannel). The identity on $W$ is retained throughout.

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En06-B@1#eq:H0 · line 1529
$H_{0} = Q_{L} \otimes Q_{R} \otimes W \otimes S_{L} \otimes S_{R} \otimes R_{L} \otimes E_{L} \otimes F_{L} \otimes R_{R} \otimes E_{R} \otimes F_{R},$
En06-B@1#eq:globalunitary · line 1539
$U = (U_{L} U_{R}) (U_{B} \otimes I_{rest}) .$

Proposition: Exact finite dilation

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