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

Lemma: Transpose-adjoint Choi transfer

Choi transfer and product-free range · lines 1168–1200 · proof included or linked

Oai272-A@1#lem:transfer, Oai272-A@1#eq:transfer

For CP maps $F : M_{a} (C) \to M_{b} (C)$ and $G : M_{b} (C) \to M_{c} (C)$ , define

F^{♯} = T_{a} F^{†} T_{b} .

Then

J (GF) = (F^{♯} \otimes G) (∣ Ω_{b} ⟩ ⟨ Ω_{b} ∣) .

If both maps are PPT and $Z = J (GF) \neq = 0$ , then

\frac{Z}{Tr Z} = (F^{♯} \otimes G) (\frac{∣ Ω _{b} ⟩ ⟨ Ω _{b} ∣}{Tr Z}) \in C .

Proof. For Kraus matrices $A_{r}$ of $F$ and $B_{s}$ of $G$ , the Kraus matrices of $F^{♯}$ are $A_{r}^{T}$ , and

(A_{r}^{T} \otimes B_{s}) ∣ Ω_{b} ⟩ = (I_{a} \otimes B_{s} A_{r}) ∣ Ω_{a} ⟩ .

Sum their rank-one projectors to prove (eq:transfer). Lemma lem:stability supplies the four CP conditions. Finally $Tr Z > 0$ because $Z$ is positive and nonzero.

Displayed formulas in this passage

Oai272-A@1#eq:transfer · line 1176
$J (GF) = (F^{♯} \otimes G) (∣ Ω_{b} ⟩ ⟨ Ω_{b} ∣) .$
unlabeled · line 1172
$F^{♯} = T_{a} F^{†} T_{b} .$
unlabeled · line 1181
$\frac{Z}{Tr Z} = (F^{♯} \otimes G) (\frac{∣ Ω _{b} ⟩ ⟨ Ω _{b} ∣}{Tr Z}) \in C .$
unlabeled · line 1193
$(A_{r}^{T} \otimes B_{s}) ∣ Ω_{b} ⟩ = (I_{a} \otimes B_{s} A_{r}) ∣ Ω_{a} ⟩ .$

Lemma: Transpose-adjoint Choi transfer

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