ULE correction bound and validity condition

PDF p. 10: Paragraph after Eq. (S2.10)

Assume the Axiom of Choice and Countable Choice and all model and preparation hypotheses of the ULE equation and independent modified-state contract: a finite-dimensional system with a positive finite number $n$ of Hermitian coupling channels, system couplings normalized by $\mathrm{\parallel }{X}_{\alpha }\mathrm{\parallel }\le 1$, a centered stationary Gaussian bath, and the source's remote-past factorized preparation formulation. Adopt the system–bath postulate used by that equation. Let $r(t)$ be the exact interaction-picture reduced operator. Assume that $r(t)$ is differentiable and that the unitary system propagator $U(t)$ is differentiable with $\dot{U}(t)=-i{H}_S(t)U(t)$, in units with $\mathrm{\hslash }=1$. Let ${\tilde{\mathcal{D}}}_t$ be the interaction-picture ULE generator of that equation.

Adopt the trace and domain conditions of the jump-correlator interface: the bath products defining the two-time correlation are well defined on their required domains and trace class, with well-defined traces invariant under a common time translation. Write their stationary correlation matrix as $C(t-s)$ and require every entry of $C$ to be Lebesgue integrable on $\mathbb{R}$. With the spectral-density convention $J(\omega )=(2\pi {)}^{-1}{\int }_{\mathbb{R}}C(t)e^{i\omega t}dt$, require $J(\omega )$ to be Hermitian positive semidefinite for every real $\omega$. Require the entries of its positive square root $g(\omega )=[J(\omega )\mathrm{/}(2\pi ){]}^{1\mathrm{/}2}$ to be Lebesgue integrable, and define $g(t)={\int }_{\mathbb{R}}g(\omega )e^{-i\omega t}d\omega$ entrywise.

Let $I_0={\int }_{\mathbb{R}}\mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}dt$ and $I_1={\int }_{\mathbb{R}}\mathrm{\mid }t\mathrm{\mid }\mathrm{\parallel }g(t){\mathrm{\parallel }}_{2,1}dt$, with $0<I_0<\mathrm{\infty }$ and $I_1<\mathrm{\infty }$. For coupling parameter $\gamma >0$, set $\mathrm{\Gamma }=4\gamma I_0^2$ and ${\tau }_B=I_1\mathrm{/}{I}_0$. The mixed norm is $\mathrm{\parallel }M{\mathrm{\parallel }}_{2,1}={\sum }_b({\sum }_{\alpha }\mathrm{\mid }{M}_{\alpha b}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. Explicitly assume the independent contract's short-memory regime $\mathrm{\Gamma }{\tau }_B\ll 1$.

Under the independent modified-state contract there is a modified interaction-picture operator $\sigma (t)$ such that $\mathrm{\parallel }\sigma (t)-r(t)\mathrm{\parallel }\le \mathrm{\Gamma }{\tau }_B$ and, for its differential residual $\eta (t)=\dot{\sigma }(t)-{\tilde{\mathcal{D}}}_t(\sigma (t))$, $\mathrm{\parallel }\eta (t)\mathrm{\parallel }\le 2{\mathrm{\Gamma }}^2{\tau }_B$ at all times covered by that contract. Here $\mathrm{\parallel }A\mathrm{\parallel }$ is the spectral operator norm, the supremum of $\mathrm{\parallel }Av\mathrm{\parallel }$ over unit vectors $v$. Thus the differential correction is of order ${\mathrm{\Gamma }}^2{\tau }_B$ in the assumed approximation regime. These estimates alone do not give a uniform-in-time trajectory error after the residual is dropped.

The article states the same numerical residual bound for the correction $\xi$ in its exact reduced-state equation and describes the evolution as effectively Markovian when $\mathrm{\Gamma }{\tau }_B\ll 1$. The independent contract used here instead bounds $\eta$ for $\sigma$; it does not establish the stated bound for the exact-state correction $\xi$.

Universal Lindblad equation with correction

Jump correlators and bath-memory parameters

Stationary correlation spectra and ULE kernel regularity

ULE residual bound and the modified-state distinction

The Axiom of Choice

The Axiom of Countable Choice (AC_ω)

The article attribution records its exact-state assertion without using that assertion as a premise. The quantitative bound proved here is a conditional application of the independent modified-state contract; no unchanged remote-past constant is asserted for arbitrary finite preparation.