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

Theorem: Probability comparison with arithmetic error

Compressed oscillator position measurements › Separate preparation, tail, quadrature, and arithmetic errors · lines 1288–1338 · proof included or linked

Em04-A@2#thm:bosonerror, Em04-A@2#eq:bosonbudget, Em04-A@2#eq:tail

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Let $ρ$ be an oscillator state, $σ_{d}$ a density matrix on $H_{d}$ , and $A_{d} (B; L)$ a Hermitian numerical matrix with a certified bound

A_{d} (B; L) - Q_{d}^{trap} (B; L)_{\infty} \leq α_{d} (B; L) .

Then

∣ Tr (ρE (B)) - Tr (σ_{d} A_{d} (B; L)) ∣ \leq preparation d_{tr} (ρ, σ_{d}) + domain tail Tr (σ_{d} E_{d} (R ∖ [- L, L])) + quadrature η_{d}^{trap} (B; L) + arithmetic α_{d} (B; L) .

The tail term is bounded by

min {1, \frac{Tr ( σ _{d} X _{osc}^{2} )}{L ^{2}}} \leq min {1, \frac{2 d - 1}{L ^{2}}} .

The preparation term can use (eq:stateerror) and (eq:dispbound). A separately rounded scalar evaluation of the trace requires its own additional certified error.

Proof. Insert the exact probabilities for $σ_{d}$ on $B$ , then on $B \cap [- L, L]$ , followed by the exact trapezoidal value. The effect inequality in Lemma lem:projection, the outside-domain probability, Theorem thm:trap, and the arithmetic norm bound control the four differences.

The pointwise inequality $1_{{∣ x ∣ > L}} \leq x^{2} / L^{2}$ proves the first tail bound. For a vector $ψ \in H_{d}$ ,

∥ X_{osc} ψ ∥^{2} \leq ∥ a ψ ∥^{2} + a^{†} ψ^{2} = ⟨ ψ ∣ (2 a^{†} a + I) ∣ ψ ⟩ \leq (2 d - 1) ∥ ψ ∥^{2} .

A convex decomposition proves the mixed-state bound.

Displayed formulas in this passage

Em04-A@2#eq:bosonbudget · line 1298
$∣ Tr (ρE (B)) - Tr (σ_{d} A_{d} (B; L)) ∣ \leq preparation d_{tr} (ρ, σ_{d}) + domain tail Tr (σ_{d} E_{d} (R ∖ [- L, L])) + quadrature η_{d}^{trap} (B; L) + arithmetic α_{d} (B; L) .$
Em04-A@2#eq:tail · line 1311
$min {1, \frac{Tr ( σ _{d} X _{osc}^{2} )}{L ^{2}}} \leq min {1, \frac{2 d - 1}{L ^{2}}} .$
unlabeled · line 1293
$A_{d} (B; L) - Q_{d}^{trap} (B; L)_{\infty} \leq α_{d} (B; L) .$
unlabeled · line 1331
$∥ X_{osc} ψ ∥^{2} \leq ∥ a ψ ∥^{2} + a^{†} ψ^{2} = ⟨ ψ ∣ (2 a^{†} a + I) ∣ ψ ⟩ \leq (2 d - 1) ∥ ψ ∥^{2} .$

Nearby in this section

remark: Remark: Raw numerical values need not be probabilities