Manuscript Psy03-A@1 · Psy03-A.tex · proposition · b30eb6e9

Proposition: Generator-to-channel budget

Complete instruments and operational intervention budgets › Reference-stable budgets · lines 1525–1567 · proof included or linked

Psy03-A@1#prop:dynamic, Psy03-A@1#eq:dynamic

For bounded time-dependent generators with CPTP propagators,

d_{⋄} (Λ_{T}, Λ_{T}^{'}) \leq \frac{1}{2} \int_{0}^{T} \norm L_{t} - L_{t}^{'}_{⋄} d t .

For generators (eq:qgenerator) in a common node basis,

D_{op} \leq \int_{0}^{T} \frac{\norm H _{t} - H _{t}^{'} _{op}}{ℏ} + i \neq = j \sum \abs k_{ij, t} - k_{ij, t}^{'} + i \sum \abs γ_{i, t} - γ_{i, t}^{'} d t .

Either bound may be truncated at one.

Proof. Variation of constants gives the difference as

\int_{0}^{T} Λ (T, s) (L_{s} - L_{s}^{'}) Λ^{'} (s, 0) d s .

Composition with channels contracts the diamond norm Watrous. This is the same bounded-generator interface recorded in Psy02-B CheungPsy02B.

For the component estimate,

\norm [A, \cdot]_{⋄} \leq 2 \norm A_{op}, \norm D_{L}_{⋄} \leq 2 \norm L_{op}^{2} .

Each displayed jump or projector has operator norm one. Substitute in (eq:dynamic).

Displayed formulas in this passage

Psy03-A@1#eq:dynamic · line 1528
$d_{⋄} (Λ_{T}, Λ_{T}^{'}) \leq \frac{1}{2} \int_{0}^{T} \norm L_{t} - L_{t}^{'}_{⋄} d t .$
unlabeled · line 1536
$D_{op} \leq \int_{0}^{T} \frac{\norm H _{t} - H _{t}^{'} _{op}}{ℏ} + i \neq = j \sum \abs k_{ij, t} - k_{ij, t}^{'} + i \sum \abs γ_{i, t} - γ_{i, t}^{'} d t .$
unlabeled · line 1549
$\int_{0}^{T} Λ (T, s) (L_{s} - L_{s}^{'}) Λ^{'} (s, 0) d s .$
unlabeled · line 1560
$\norm [A, \cdot]_{⋄} \leq 2 \norm A_{op}, \norm D_{L}_{⋄} \leq 2 \norm L_{op}^{2} .$