Manuscript Hawk02-E@1 · Hawk02-E.tex · section · 4841592b

Task, distances, and complete output spaces

Task, distances, and complete output spaces · lines 267–309

Hawk02-E@1#eq:rec-definition, Hawk02-E@1#eq:leak-definition

We use $ℏ = 1$ . The code $C$ has finite dimension $d$ . The main microscopic example has $d = 2$ .

For states and channels, write

d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .

Channel comparisons retain arbitrary external references. Finite-dimensional input permits a reference of dimension $d$ for channel discrimination Watrous.

For an isometry

V_{t} : C ⟶ R \otimes B,

define the accessible channel and complete complementary channel by

N_{t} (X) = Tr_{B} (V_{t} X V_{t}^{†}), C_{t} (X) = Tr_{R} (V_{t} X V_{t}^{†}) .

The replacement channel for a state $σ_{B}$ is

T_{σ_{B}} (X) = Tr (X) σ_{B} .

The two operational quantities are

ε_{rec} (t) δ_{full} (t) = D CPTP in f d_{⋄} (D N_{t}, id_{C}), = σ_{B} in f d_{⋄} (C_{t}, T_{σ_{B}}) .

Displayed formulas in this passage

Hawk02-E@1#eq:rec-definition · line 297
$ε_{rec} (t) δ_{full} (t) = D CPTP in f d_{⋄} (D N_{t}, id_{C}), = σ_{B} in f d_{⋄} (C_{t}, T_{σ_{B}}) .$
Hawk02-E@1#eq:leak-definition · line 297
$ε_{rec} (t) δ_{full} (t) = D CPTP in f d_{⋄} (D N_{t}, id_{C}), = σ_{B} in f d_{⋄} (C_{t}, T_{σ_{B}}) .$
unlabeled · line 273
$d_{tr} (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}, d_{⋄} (Φ, Ψ) = \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$
unlabeled · line 283
$V_{t} : C ⟶ R \otimes B,$
unlabeled · line 287
$N_{t} (X) = Tr_{B} (V_{t} X V_{t}^{†}), C_{t} (X) = Tr_{R} (V_{t} X V_{t}^{†}) .$
unlabeled · line 293
$T_{σ_{B}} (X) = Tr (X) σ_{B} .$

Task, distances, and complete output spaces

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