Manuscript XXII-F@1 · XXII-F.tex · lemma · 67821705

Lemma: Base-class nonemptiness

The named quantum data › Process matrices and access · lines 289–311 · proof included or linked

XXII-F@1#lem:base

The set $W_{3}$ is nonempty and compact, and every member has trace eight. In particular,

W_{0} = I_{H} /8 \in W_{3} .

Proof. The completely depolarizing local channel has Choi matrix $I /2$ . Substitution into (eq:validity) gives $1 = Tr W /8$ . Positivity and the normalization equations define a closed subset of the bounded finite-dimensional set of positive matrices of trace eight.

For $W_{0}$ , every deterministic local tester has trace two, so

Tr [W_{0} (M_{A} \otimes M_{B} \otimes M_{C})] = \frac{1}{8} (2) (2) (2) = 1.

Displayed formulas in this passage

unlabeled · line 293
$W_{0} = I_{H} /8 \in W_{3} .$
unlabeled · line 307
$Tr [W_{0} (M_{A} \otimes M_{B} \otimes M_{C})] = \frac{1}{8} (2) (2) (2) = 1.$

Nearby in this section

section: Process matrices and access
definition: Definition: Frozen one-use process class
section: Process matrices and access
section: Process matrices and access