Manuscript Oai272-A@1 · Oai272-A.tex · lemma · a40828a0

Lemma: Positive block estimate

Stage 1: the operational certificate › A four-effect inequality · lines 459–483 · proof included or linked

Oai272-A@1#lem:block

(H Z^{†} Z K) ⪰ 0,

then

∥ Z ∥_{1} \leq Tr H Tr K .

Proof. Write the block matrix as $\sum_{α} (v _{α} u _{α}) (v _{α} u _{α})^{†}$ . Then

∥ Z ∥_{1} \leq α \sum ∥ u_{α} ∥∥ v_{α} ∥ \leq (α \sum ∥ u_{α} ∥^{2})^{1/2} (α \sum ∥ v_{α} ∥^{2})^{1/2} .

Displayed formulas in this passage

unlabeled · line 462
$(H Z^{†} Z K) ⪰ 0,$
unlabeled · line 466
$∥ Z ∥_{1} \leq Tr H Tr K .$
unlabeled · line 476
$∥ Z ∥_{1} \leq α \sum ∥ u_{α} ∥∥ v_{α} ∥ \leq (α \sum ∥ u_{α} ∥^{2})^{1/2} (α \sum ∥ v_{α} ∥^{2})^{1/2} .$