Manuscript Em05-A@1 · Em05-A-v1.tex · theorem · 9cac0def

Theorem: Exact specified-decoder performance

A complete specified decoder and optimal-recovery boundaries · lines 700–774 · proof included or linked

Em05-A@1#thm:specified, Em05-A@1#eq:decoded, Em05-A@1#eq:specifiedError, Em05-A@1#eq:BellFid, Em05-A@1#eq:WorstFid, Em05-A@1#eq:decodedBell, Em05-A@1#eq:decodedConc

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The decoded channel is

Λ_{η} := D_{⋆} N_{η} = η id + (1 - η) R_{∣0 ⟩ ⟨ 0∣} .

It satisfies

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

For the Bell-input decoded state,

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

the two-qubit concurrence is

C (Q : M^{'}) = η .

No optimality of $D_{⋆}$ is asserted for intermediate $η$ .

Proof. The odd occupations in (eq:rhoR) contribute $p η t I_{2}$ after decoding. The decoded lower-right entry is $p η^{2} + p η t = p η$ , and its off-diagonal entry is $η z$ . The upper-left entry is $1 - p η$ . This proves (eq:decoded).

A half-diamond distance between channels is at most one. The identity and replacement by $∣0 ⟩$ attain distance one on input $∣1 ⟩$ . Homogeneity therefore gives (eq:specifiedError).

The replacement term's Bell overlap is $1/4$ , proving (eq:BellFid). For any purified input, the target overlap of (eq:decoded) is at least $η$ , because the replacement contribution is nonnegative. Input $∣1 ⟩$ gives zero replacement contribution and attains $η$ , proving (eq:WorstFid).

For concurrence, use Wootters' definition Wootters: the concurrence is the positive part of the largest square-root eigenvalue of $ω ω$ minus the other three, where

ω = (Y \otimes Y) ω (Y \otimes Y) .

In the ordered basis $00, 01, 10, 11$ , (eq:decodedBell) is

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

The square-root eigenvalues are

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

Their difference is $η$ .

Displayed formulas in this passage

Em05-A@1#eq:decoded · line 703
$Λ_{η} := D_{⋆} N_{η} = η id + (1 - η) R_{∣0 ⟩ ⟨ 0∣} .$
Em05-A@1#eq:specifiedError · line 711
$d_{⋄} (Λ_{η}, id) F_{e} (Λ_{η}) F_{wc} (Λ_{η}) = 1 - η, = \frac{1 + 3 η}{4}, = η .$
Em05-A@1#eq:BellFid · line 711
$d_{⋄} (Λ_{η}, id) F_{e} (Λ_{η}) F_{wc} (Λ_{η}) = 1 - η, = \frac{1 + 3 η}{4}, = η .$
Em05-A@1#eq:WorstFid · line 711
$d_{⋄} (Λ_{η}, id) F_{e} (Λ_{η}) F_{wc} (Λ_{η}) = 1 - η, = \frac{1 + 3 η}{4}, = η .$
Em05-A@1#eq:decodedBell · line 717
$ω_{Q M^{'}} = η ∣Φ ⟩ ⟨ Φ∣ + t \frac{I _{Q}}{2} \otimes ∣0 ⟩ ⟨ 0∣,$
Em05-A@1#eq:decodedConc · line 724
$C (Q : M^{'}) = η .$
unlabeled · line 755
$ω = (Y \otimes Y) ω (Y \otimes Y) .$
unlabeled · line 759
$ω = 1/2 00 η /2 0000 00 t /2 0 η /2 00 η /2 .$
unlabeled · line 769
$\frac{η + η}{2}, \frac{η - η}{2}, 0, 0.$

Theorem: Exact specified-decoder performance

Displayed formulas in this passage

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