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

Lemma: Elementary PPT stability and tensor closure

Stage 0: conventions and the operational package › The represented class · lines 331–361 · proof included or linked

Oai272-A@1#lem:stability

For a CP map $F : M_{a} (C) \to M_{b} (C)$ , its adjoint and $T_{b} F T_{a}$ are CP. Moreover,

F T_{a} is CP ⟺ T_{b} F is CP .

Adjoints and transpose-adjoints of PPT maps are PPT. Every state in $C$ is PPT, and $C$ is closed under finite tensor products after grouping the parties.

Proof. If $F (X) = \sum_{α} K_{α} X K_{α}^{†}$ , then

F^{†} (Y) = α \sum K_{α}^{†} Y K_{α}, T_{b} F T_{a} (X) = α \sum K_{α} X K_{α}^{†} .

These formulas prove the CP assertions and the equivalence. Taking adjoints proves the transpose-adjoint statements.

For (eq:C),

R^{T_{A}} = ((T_{A} L) \otimes M) (∣ χ ⟩ ⟨ χ ∣) ⪰ 0.

Tensor the representing maps and vectors. In product bases, the transpose on a tensor product is the tensor product of the transposes. The required CP conditions therefore persist.

Displayed formulas in this passage

unlabeled · line 335
$F T_{a} is CP ⟺ T_{b} F is CP .$
unlabeled · line 345
$F^{†} (Y) = α \sum K_{α}^{†} Y K_{α}, T_{b} F T_{a} (X) = α \sum K_{α} X K_{α}^{†} .$
unlabeled · line 355
$R^{T_{A}} = ((T_{A} L) \otimes M) (∣ χ ⟩ ⟨ χ ∣) ⪰ 0.$

Lemma: Elementary PPT stability and tensor closure

Displayed formulas in this passage

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