Manuscript Oai280-B@1 · Oai280-B.tex · section · 6d92ab75

The complete ideal region table

Privacy, perfectness, and exact access-dependent recovery › The complete ideal region table · lines 636–680

Strings such as $124$ denote the set ${1, 2, 4}$ . All subsets are covered below.

$m$	Regions $A$	Accessible channel	Complete complement	Optimal error
0	$\emptyset$	Replacement	Recoverable	$3/4$
1	$1, 2, 3, 4, 5$	Replacement	Recoverable	$3/4$
2	$12, 13, 14, 15, 23, 24, 25, 34, 35, 45$	Replacement	Recoverable	$3/4$
3	$123, 124, 125, 134, 135, 145, 234, 235, 245, 345$	Recoverable	$I_{4} /4$ replacement	$0$
4	$1234, 1235, 1245, 1345, 2345$	Recoverable	$I_{2} /2$ replacement	$0$
5	$12345$	Isometric	Scalar replacement	$0$

Here “recoverable” means the constructive channel (eq:DA); it is not just an existence label.

For each triple the decoder matrix is completely specified by (eq:TA). The corresponding omissions are:

$A$	$A ˉ$	$A$	$A ˉ$	$A$	$A ˉ$	$A$	$A ˉ$	$A$	$A ˉ$
123	45	124	35	125	34	134	25	135	24
145	23	234	15	235	14	245	13	345	12

For every triple, $T_{A}$ is an $8 \times 8$ unitary, and $D_{A} = Tr_{K_{A}} \circ Ad_{T_{A}^{†}}$ .