Manuscript Em04-A@2 · Em04-A-v2.tex · proposition · f5b28abc

Proposition: Coherent-state truncation and displacement

Compressed oscillator position measurements › Exact objects and state approximation · lines 1110–1171 · proof included or linked

Em04-A@2#prop:coherent, Em04-A@2#eq:dispbound

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For finite $α \in C$ , let $μ = ∣ α ∣^{2}$ and

∣ α ⟩ = e^{- μ /2} n = 0 \sum^{\infty} \frac{α ^{n}}{n !} ∣ n ⟩ .

Its omitted probability is

δ_{d} = 1 - e^{- μ} n = 0 \sum^{d - 1} \frac{μ ^{n}}{n !} .

For $μ > 0$ and $d > μ$ ,

δ_{d} \leq e^{- μ} (e μ / d)^{d};

for $μ = 0$ , $δ_{d} = 0$ .

Define

a_{d} = P_{d} a P_{d} ∣_{H_{d}}, ∣ β_{d} ⟩ = e^{α a_{d}^{†} - α a_{d}} ∣0 ⟩, ∣ α_{d} ⟩ = \frac{P _{d} ∣ α ⟩}{1 - δ _{d}} .

These preparations need not agree, but

d_{tr} (∣ β_{d} ⟩ ⟨ β_{d} ∣, ∣ α_{d} ⟩ ⟨ α_{d} ∣) \leq min {1, I_{d} (α) + 1 - 1 - δ_{d}},

where

I_{d} (α) = ∣ α ∣ d \int_{0}^{1} e^{- s^{2} μ /2} \frac{( s ∣ α ∣ ) ^{d}}{d !} d s .

Proof. The squared coefficients are Poisson probabilities. For a Poisson variable $K$ , $Pr (K \geq d) \leq exp (μ (e^{s} - 1) - s d)$ . Choose $s = lo g (d / μ) > 0$ .

For displacement, write $A = α a^{†} - α a$ , $A_{d} = P_{d} A P_{d} ∣_{H_{d}}$ , and $u (s) = P_{d} ∣ s α ⟩$ . Differentiation gives

u^{'} = A_{d} u + P_{d} A (I - P_{d}) ∣ s α ⟩,

with boundary source

- α d e^{- s^{2} μ /2} \frac{( s α ) ^{d}}{d !} ∣ d - 1 ⟩ .

Variation of constants and unitarity of $e^{s A_{d}}$ give $∥ ∣ β_{d} ⟩ - P_{d} ∣ α ⟩ ∥ \leq I_{d} (α)$ . Normalization adds vector distance $1 - 1 - δ_{d}$ . Pure-state trace distance is at most the vector distance, proving (eq:dispbound).

Displayed formulas in this passage

Em04-A@2#eq:dispbound · line 1135
$d_{tr} (∣ β_{d} ⟩ ⟨ β_{d} ∣, ∣ α_{d} ⟩ ⟨ α_{d} ∣) \leq min {1, I_{d} (α) + 1 - 1 - δ_{d}},$
unlabeled · line 1113
$∣ α ⟩ = e^{- μ /2} n = 0 \sum^{\infty} \frac{α ^{n}}{n !} ∣ n ⟩ .$
unlabeled · line 1118
$δ_{d} = 1 - e^{- μ} n = 0 \sum^{d - 1} \frac{μ ^{n}}{n !} .$
unlabeled · line 1122
$δ_{d} \leq e^{- μ} (e μ / d)^{d};$
unlabeled · line 1128
$a_{d} = P_{d} a P_{d} ∣_{H_{d}}, ∣ β_{d} ⟩ = e^{α a_{d}^{†} - α a_{d}} ∣0 ⟩, ∣ α_{d} ⟩ = \frac{P _{d} ∣ α ⟩}{1 - δ _{d}} .$
unlabeled · line 1141
$I_{d} (α) = ∣ α ∣ d \int_{0}^{1} e^{- s^{2} μ /2} \frac{( s ∣ α ∣ ) ^{d}}{d !} d s .$
unlabeled · line 1158
$u^{'} = A_{d} u + P_{d} A (I - P_{d}) ∣ s α ⟩,$
unlabeled · line 1162
$- α d e^{- s^{2} μ /2} \frac{( s α ) ^{d}}{d !} ∣ d - 1 ⟩ .$

Proposition: Coherent-state truncation and displacement

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