Manuscript Em03-C@1 · Em03-C-v1.tex · section · 47dd71ec

Two cutoffs per packet

C1 interface: joint carriers and referenced records › Two cutoffs per packet · lines 606–643

Choose reported saturation levels $N_{i}^{cut} \geq 1$ and control-domain cutoffs $K_{i} \geq N_{i}^{cut}$ . To avoid confusing operators and cutoff values, abbreviate

a_{i} = N_{i}^{cut}, a = (a_{1}, a_{2}), K = (K_{1}, K_{2}) .

Let $P_{i, a_{i}} = \sum_{n = 0}^{a_{i}} ∣ n ⟩ ⟨ n ∣$ .

The one-mode carrier map is

C_{a_{i}} (X) = V_{i} P_{i, a_{i}} X P_{i, a_{i}} V_{i}^{†} + tr ((I - P_{i, a_{i}}) X) ∣ a_{i} ⟩ ⟨ a_{i} ∣ .

Its Kraus maps are $V_{i} P_{i, a_{i}}$ and $∣ a_{i} ⟩ ⟨ n ∣$ , $n > a_{i}$ , whose effects sum strongly to the identity.

Set

b_{i}^{reg} = ⌈ lo g_{2} (a_{i} + 1) ⌉, d_{i} = 2^{b_{i}^{reg}}, B_{reg} = b_{1}^{reg} + b_{2}^{reg}, d = d_{1} d_{2} .

The binary embeddings $J_{i}$ send carrier integer $k$ to its fixed binary word. Define

B_{a} = (J_{1} \otimes J_{2}) (C_{a_{1}} \otimes C_{a_{2}}) (\cdot) (J_{1} \otimes J_{2})^{†} .

Displayed formulas in this passage

unlabeled · line 611
$a_{i} = N_{i}^{cut}, a = (a_{1}, a_{2}), K = (K_{1}, K_{2}) .$
unlabeled · line 619
$C_{a_{i}} (X) = V_{i} P_{i, a_{i}} X P_{i, a_{i}} V_{i}^{†} + tr ((I - P_{i, a_{i}}) X) ∣ a_{i} ⟩ ⟨ a_{i} ∣ .$
unlabeled · line 629
$b_{i}^{reg} = ⌈ lo g_{2} (a_{i} + 1) ⌉, d_{i} = 2^{b_{i}^{reg}}, B_{reg} = b_{1}^{reg} + b_{2}^{reg}, d = d_{1} d_{2} .$
unlabeled · line 637
$B_{a} = (J_{1} \otimes J_{2}) (C_{a_{1}} \otimes C_{a_{2}}) (\cdot) (J_{1} \otimes J_{2})^{†} .$