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

Process matrices and access

The named quantum data › Process matrices and access · lines 205–234

XXII-F@1#eq:choi, XXII-F@1#eq:testers

Fix

J = {A, B, C}, H_{a_{I}} = H_{a_{O}} = C^{2}, H = a \in J ⨂ (H_{a_{I}} \otimes H_{a_{O}}) .

Thus $dim H = 64$ . The tensor-factor order is bookkeeping, not a temporal order.

For a local completely positive map $M$ , use

J (M) = i, j = 0 \sum^{1} ∣ i ⟩ ⟨ j ∣ \otimes M (∣ i ⟩ ⟨ j ∣), M = J (M)^{T},

with the full transpose in fixed bases. Define

C_{a} = {M \geq 0 : Tr_{a_{O}} M = I_{a_{I}}}, B_{a} = {B \geq 0 : Tr_{a_{O}} B \leq I_{a_{I}}} .

These represent deterministic channels and trace-nonincreasing instrument elements, respectively.

Displayed formulas in this passage

unlabeled · line 208
$J = {A, B, C}, H_{a_{I}} = H_{a_{O}} = C^{2}, H = a \in J ⨂ (H_{a_{I}} \otimes H_{a_{O}}) .$
XXII-F@1#eq:choi · line 217
$J (M) = i, j = 0 \sum^{1} ∣ i ⟩ ⟨ j ∣ \otimes M (∣ i ⟩ ⟨ j ∣), M = J (M)^{T},$
XXII-F@1#eq:testers · line 224
$C_{a} = {M \geq 0 : Tr_{a_{O}} M = I_{a_{I}}}, B_{a} = {B \geq 0 : Tr_{a_{O}} B \leq I_{a_{I}}} .$