Manuscript XXII-C@1 · XXII-C.tex · lemma · b9a05a24

Lemma: Compactness and a base witness

The frozen process class › Fixed parties, dimensions, and Choi convention · lines 268–298 · proof included or linked

XXII-C@1#lem:base, XXII-C@1#eq:base-process

The set $W_{3}$ is nonempty and compact. Every $W \in W_{3}$ satisfies $Tr W = 8$ .

Proof. The completely depolarizing channel has Choi matrix $I_{a_{I} a_{O}} /2$ . Substituting these three channels into (eq:normalization) gives

1 = Tr (W I_{H} /8) = Tr W /8.

The normalization equations and positivity define a closed subset of the finite-dimensional positive matrices with trace eight, which is bounded and hence compact.

The matrix

W_{0} = I_{H} /8

is positive. Since every $M_{a} \in C_{a}$ has trace two,

Tr [W_{0} (M_{A} \otimes M_{B} \otimes M_{C})] = \frac{1}{8} a \in J \prod Tr M_{a} = 1.

Thus $W_{0} \in W_{3}$ .

Displayed formulas in this passage

unlabeled · line 279
$1 = Tr (W I_{H} /8) = Tr W /8.$
XXII-C@1#eq:base-process · line 287
$W_{0} = I_{H} /8$
unlabeled · line 292
$Tr [W_{0} (M_{A} \otimes M_{B} \otimes M_{C})] = \frac{1}{8} a \in J \prod Tr M_{a} = 1.$

Lemma: Compactness and a base witness

Displayed formulas in this passage

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