Manuscript Psy02-A@1 · Psy02-A.tex · proposition · d92ce402

Proposition: Trace distance alone does not bound field moments

Input, preparation, and measurement budgets › Preparation in a moment-compatible metric · lines 1333–1356 · proof included or linked

Psy02-A@1#prop:unbounded-prep

There are one-mode states $σ_{M}$ with

d_{tr} (σ_{M}, ∣ 0 ⟩ ⟨ 0 ∣) \to 0

but

Tr (σ_{M} Φ (e)^{2}) \to \infty.

Proof. Take

σ_{M} = (1 - M^{- 1/2}) ∣ 0 ⟩ ⟨ 0 ∣ + M^{- 1/2} ∣ M ⟩ ⟨ M ∣ .

The trace distance is $M^{- 1/2}$ . Since $⟨ M ∣ Φ (e)^{2} ∣ M ⟩ = 2 M + 1$ , the second moment is $1 + 2 M$ .

Displayed formulas in this passage

unlabeled · line 1336
$d_{tr} (σ_{M}, ∣ 0 ⟩ ⟨ 0 ∣) \to 0$
unlabeled · line 1340
$Tr (σ_{M} Φ (e)^{2}) \to \infty.$
unlabeled · line 1347
$σ_{M} = (1 - M^{- 1/2}) ∣ 0 ⟩ ⟨ 0 ∣ + M^{- 1/2} ∣ M ⟩ ⟨ M ∣ .$