Manuscript Hawk01-A@1 · Hawk01-A.tex · section · 11979653

Loss and complete number readout

Finite outgoing packets and ordinary detection › Loss and complete number readout · lines 1144–1185

Hawk01-A@1#eq:losskraus, Hawk01-A@1#eq:binomialloss, Hawk01-A@1#eq:backgroundoccupation, Hawk01-A@1#eq:numberinstrument

A single-mode pure-loss channel of transmissivity $η \in [0, 1]$ has Kraus operators

L_{r} = n = r \sum^{\infty} (r n) η^{(n - r) /2} (1 - η)^{r /2} ∣ n - r ⟩ ⟨ n ∣, r = 0, 1, \dots .

On a number input $n$ , the detected-number law is

Pr (k ∣ n) = (k n) η^{k} (1 - η)^{n - k} .

For a thermal independent environment of occupation $n_{env}$ , the beam-splitter relation gives

n_{det} = η n_{in} + (1 - η) n_{env} .

The complete destructive number instrument on $k$ modes, with spectator $S$ , is

M_{num} (ρ^{R S}) = n \sum ∣ n ⟩ ⟨ n ∣_{F} \otimes ⟨ n ∣ ρ^{R S} ∣ n ⟩ .

The positive block sum converges in trace norm. Clipping or saturation must retain an explicit overflow outcome.

Displayed formulas in this passage

Hawk01-A@1#eq:losskraus · line 1148
$L_{r} = n = r \sum^{\infty} (r n) η^{(n - r) /2} (1 - η)^{r /2} ∣ n - r ⟩ ⟨ n ∣, r = 0, 1, \dots .$
Hawk01-A@1#eq:binomialloss · line 1159
$Pr (k ∣ n) = (k n) η^{k} (1 - η)^{n - k} .$
Hawk01-A@1#eq:backgroundoccupation · line 1165
$n_{det} = η n_{in} + (1 - η) n_{env} .$
Hawk01-A@1#eq:numberinstrument · line 1174
$M_{num} (ρ^{R S}) = n \sum ∣ n ⟩ ⟨ n ∣_{F} \otimes ⟨ n ∣ ρ^{R S} ∣ n ⟩ .$