Manuscript Psy02-B@2 · Psy02-B-v2.tex · proposition · cfd06e2e

Proposition: Dynamical residual bounds

Track A2: observation, channel, and instrument comparisons · lines 540–576 · proof included or linked

Psy02-B@2#prop:dynamic

For common bounded Hamiltonians and independent bath states,

d_{⋄} (Λ_{t}, Λ_{t}^{'}) \leq d_{tr} (τ, τ^{'}) + ℏ^{- 1} \int_{0}^{t} \norm H (s) - H^{'} (s) d s .

For bounded generators and a common encoding, if

\frac{1}{2} \norm L_{b, s} E - E L_{m, s}_{⋄} \leq r (s),

then

d_{⋄} (Λ_{b, t} E, E Λ_{m, t}) \leq \int_{0}^{t} r (s) d s .

Preparation errors are added separately.

Proof. Duhamel's formula bounds the unitary difference by the displayed Hamiltonian integral. Unitary-output trace distance is at most that operator norm. Bath-state and partial-trace contraction give the first assertion. Variation of constants expresses the second defect as

\int_{0}^{t} Λ_{b} (t, s) (L_{b, s} E - E L_{m, s}) Λ_{m} (s, 0) d s .

Channel composition contracts the diamond norm.

Displayed formulas in this passage

unlabeled · line 543
$d_{⋄} (Λ_{t}, Λ_{t}^{'}) \leq d_{tr} (τ, τ^{'}) + ℏ^{- 1} \int_{0}^{t} \norm H (s) - H^{'} (s) d s .$
unlabeled · line 549
$\frac{1}{2} \norm L_{b, s} E - E L_{m, s}_{⋄} \leq r (s),$
unlabeled · line 555
$d_{⋄} (Λ_{b, t} E, E Λ_{m, t}) \leq \int_{0}^{t} r (s) d s .$
unlabeled · line 569
$\int_{0}^{t} Λ_{b} (t, s) (L_{b, s} E - E L_{m, s}) Λ_{m} (s, 0) d s .$

Proposition: Dynamical residual bounds

Displayed formulas in this passage

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