Manuscript Em05-B@1 · Em05-B-v1.tex · theorem · 6c050f5b

Theorem: Exact specified-decoder performance

The specified decoder and its flagged instrument · lines 1004–1078 · proof included or linked

Em05-B@1#thm:specified, Em05-B@1#eq:specifiedChannel, Em05-B@1#eq:specifiedError, Em05-B@1#eq:specifiedBell, Em05-B@1#eq:specifiedWorst

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The map $D_{⋆}$ is CPTP and

Λ_{η} := D_{⋆} N_{η} = η id_{M} + t R_{∣0 ⟩ ⟨ 0∣} .

Its performance is

d_{⋄} (Λ_{η}, id_{M}) F_{e} (Λ_{η}) F_{wc} (Λ_{η}) = t, = \frac{1 + 3 η}{4}, = η .

Its decoded Bell state

ω_{Q M^{'}} = η ∣Φ ⟩ ⟨ Φ∣ + t \frac{I _{Q}}{2} \otimes ∣0 ⟩ ⟨ 0∣

has concurrence $C (Q : M^{'}) = η$ .

Proof. A Kraus family is $W^{†}$ , together with

\frac{1}{2} ∣ i ⟩ ⟨ a ∣, i = 0, 1, a = 01, 10.

Its effects sum to $I_{R}$ , proving CPTP completeness.

The odd occupations contribute $p η t I_{M^{'}}$ . The decoded lower-right entry is $p η^{2} + p η t = p η$ , and its coherence is $η z$ . This proves (eq:specifiedChannel).

The identity and replacement by $∣0 ⟩$ have half-diamond distance one, attained on input $∣1 ⟩$ . Homogeneity proves (eq:specifiedError). The replacement term has Bell overlap $1/4$ , proving (eq:specifiedBell). Every purified target overlap is at least $η$ , and input $∣1 ⟩$ attains $η$ ; this proves (eq:specifiedWorst).

For concurrence, use the spin-flip definition Wootters, with

ω = (Y \otimes Y) ω (Y \otimes Y) .

In the basis $00, 01, 10, 11$ ,

ω = 1/2 00 η /2 0000 00 t /2 0 η /2 00 η /2 .

The square-root eigenvalues of $ω ω$ are

\frac{η + η}{2}, \frac{η - η}{2}, 0, 0.

The positive part of the largest minus the other three is $η$ .

Displayed formulas in this passage

Em05-B@1#eq:specifiedChannel · line 1007
$Λ_{η} := D_{⋆} N_{η} = η id_{M} + t R_{∣0 ⟩ ⟨ 0∣} .$
Em05-B@1#eq:specifiedError · line 1014
$d_{⋄} (Λ_{η}, id_{M}) F_{e} (Λ_{η}) F_{wc} (Λ_{η}) = t, = \frac{1 + 3 η}{4}, = η .$
Em05-B@1#eq:specifiedBell · line 1014
$d_{⋄} (Λ_{η}, id_{M}) F_{e} (Λ_{η}) F_{wc} (Λ_{η}) = t, = \frac{1 + 3 η}{4}, = η .$
Em05-B@1#eq:specifiedWorst · line 1014
$d_{⋄} (Λ_{η}, id_{M}) F_{e} (Λ_{η}) F_{wc} (Λ_{η}) = t, = \frac{1 + 3 η}{4}, = η .$
unlabeled · line 1023
$ω_{Q M^{'}} = η ∣Φ ⟩ ⟨ Φ∣ + t \frac{I _{Q}}{2} \otimes ∣0 ⟩ ⟨ 0∣$
unlabeled · line 1034
$\frac{1}{2} ∣ i ⟩ ⟨ a ∣, i = 0, 1, a = 01, 10.$
unlabeled · line 1055
$ω = (Y \otimes Y) ω (Y \otimes Y) .$
unlabeled · line 1061
$ω = 1/2 00 η /2 0000 00 t /2 0 η /2 00 η /2 .$
unlabeled · line 1072
$\frac{η + η}{2}, \frac{η - η}{2}, 0, 0.$

Theorem: Exact specified-decoder performance

Displayed formulas in this passage

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