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

Theorem: Maximal decoded Bell fidelity

Improved intermediate recovery control › Exact optimal Bell fidelity · lines 1185–1264 · proof included or linked

Em05-B@1#thm:optimalBell, Em05-B@1#eq:optimalBellValue, Em05-B@1#eq:BellDecoder, Em05-B@1#eq:BellLower

For the declared extractor,

F_{e}^{opt} (N_{η}) = \frac{1 + 4 η - η ^{2}}{4} = \frac{2 - t ^{2} + 2 η}{4} .

It is attained by the complete decoder

D_{1} (Y) = W^{†} Y W + Tr ((I_{R} - P) Y) ∣1 ⟩ ⟨ 1∣.

Consequently,

ε_{rec} (N_{η}) \geq B (η) := \frac{t ( 2 + t )}{4} .

Proof. Let $D$ be any CPTP decoder and abbreviate

e_{0} = ∣00 ⟩_{R}, e_{1} = ∣11 ⟩_{R}, e_{2} = ∣01 ⟩_{R}, e_{3} = ∣10 ⟩_{R} .

Put

a = ⟨ 0∣ D (∣ e_{0} ⟩ ⟨ e_{0} ∣) ∣0 ⟩, b = ⟨ 1∣ D (∣ e_{1} ⟩ ⟨ e_{1} ∣) ∣1 ⟩, r_{j} = ⟨ 1∣ D (∣ e_{j} ⟩ ⟨ e_{j} ∣) ∣1 ⟩ .

All lie in $[0, 1]$ . If

k = ⟨ 0∣ D (∣ e_{0} ⟩ ⟨ e_{1} ∣) ∣1 ⟩,

a Kraus family ${K_{ℓ}}$ and Cauchy–Schwarz give

∣ k ∣ = ℓ \sum ⟨ 0∣ K_{ℓ} ∣ e_{0} ⟩ ⟨ 1∣ K_{ℓ} ∣ e_{1} ⟩ \leq ab .

The Bell overlap is therefore bounded by

4 F_{e} (D N_{η}) = a + t^{2} (1 - a) + η^{2} b + η t (r_{2} + r_{3}) + 2 η ℜ k \leq t^{2} + (1 - t^{2}) a + η^{2} b + 2 η t + 2 η ab \leq 1 + η^{2} + 2 η t + 2 η = 2 - t^{2} + 2 η .

Each step uses nonnegative coefficients and $a, b, r_{j} \leq 1$ .

The decoder $D_{1}$ is CPTP, with Kraus operators $W^{†}, ∣1 ⟩ ⟨ 01∣, ∣1 ⟩ ⟨ 10∣$ . It has $a = b = r_{2} = r_{3} = k = 1$ , attaining the upper bound. Equation (eq:BellWitness) gives

ε_{rec} (N_{η}) \geq 1 - F_{e}^{opt} (N_{η}) = \frac{t ( 2 + t )}{4} .

Displayed formulas in this passage

Em05-B@1#eq:optimalBellValue · line 1188
$F_{e}^{opt} (N_{η}) = \frac{1 + 4 η - η ^{2}}{4} = \frac{2 - t ^{2} + 2 η}{4} .$
Em05-B@1#eq:BellDecoder · line 1197
$D_{1} (Y) = W^{†} Y W + Tr ((I_{R} - P) Y) ∣1 ⟩ ⟨ 1∣.$
Em05-B@1#eq:BellLower · line 1205
$ε_{rec} (N_{η}) \geq B (η) := \frac{t ( 2 + t )}{4} .$
unlabeled · line 1242
$4 F_{e} (D N_{η}) = a + t^{2} (1 - a) + η^{2} b + η t (r_{2} + r_{3}) + 2 η ℜ k \leq t^{2} + (1 - t^{2}) a + η^{2} b + 2 η t + 2 η ab \leq 1 + η^{2} + 2 η t + 2 η = 2 - t^{2} + 2 η .$
unlabeled · line 1214
$e_{0} = ∣00 ⟩_{R}, e_{1} = ∣11 ⟩_{R}, e_{2} = ∣01 ⟩_{R}, e_{3} = ∣10 ⟩_{R} .$
unlabeled · line 1219
$a = ⟨ 0∣ D (∣ e_{0} ⟩ ⟨ e_{0} ∣) ∣0 ⟩, b = ⟨ 1∣ D (∣ e_{1} ⟩ ⟨ e_{1} ∣) ∣1 ⟩, r_{j} = ⟨ 1∣ D (∣ e_{j} ⟩ ⟨ e_{j} ∣) ∣1 ⟩ .$
unlabeled · line 1227
$k = ⟨ 0∣ D (∣ e_{0} ⟩ ⟨ e_{1} ∣) ∣1 ⟩,$
unlabeled · line 1231
$∣ k ∣ = ℓ \sum ⟨ 0∣ K_{ℓ} ∣ e_{0} ⟩ ⟨ 1∣ K_{ℓ} ∣ e_{1} ⟩ \leq ab .$
unlabeled · line 1260
$ε_{rec} (N_{η}) \geq 1 - F_{e}^{opt} (N_{η}) = \frac{t ( 2 + t )}{4} .$

Nearby in this section

remark: Remark: Bell optimality is not diamond optimality