Quantitative preparation-tail and waiting-time condition

The normalization lemma defines $\mathrm{\Gamma }>0$ and $\tau \ge 0$. ULE kernel adjoint symmetry and rate identities

Under the convolution premise, the nonnegative measurable function $u\mapsto \mathrm{\parallel }C(u){\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t}}$ satisfies $4\gamma {\int }_0^{\mathrm{\infty }}u\mathrm{\parallel }C(u){\mathrm{\parallel }}_{\mathrm{e}\mathrm{n}\mathrm{t}}du\le \mathrm{\Gamma }\tau$. A first-moment bound for the bath correlation convolution

A real or complex normed space with an ordered finite basis is Banach. The space $E$ above has the $d^4$ coordinate maps that send one standard matrix unit to one standard matrix unit and all other matrix units to zero as an ordered basis. Every finite-dimensional normed space is Banach

Strong measurability means almost-everywhere norm convergence of measurable simple functions. A Bochner integral is the norm limit of the integrals of integrable simple approximants converging in $L^1$. Strongly measurable Banach-valued function Bochner-integrable function

For a strongly measurable Banach-valued function $f$, finiteness of $\int \mathrm{\parallel }f\mathrm{\parallel }$ is equivalent to Bochner integrability, and $\mathrm{\parallel }\int f\mathrm{\parallel }\le \int \mathrm{\parallel }f\mathrm{\parallel }$. Bochner integrability criterion Bochner integral norm inequality

A density operator on a nonzero finite-dimensional complex inner-product space is a non-negative endomorphism of trace one; in an orthonormal basis its matrix is Hermitian positive semidefinite. Density operator

The nonnegative integral is monotone and positively homogeneous. Under countable choice, a $C^1$ diffeomorphism $T:U\to W$ between open subsets of Euclidean space satisfies ${\int }_{W}f(u)du={\int }_{U}f(T(s))\mathrm{\mid }\det DT(s)\mathrm{\mid }ds$ for nonnegative Lebesgue measurable $f$. Monotonicity and nonnegative homogeneity of the nonnegative integral A C¹ diffeomorphism satisfies the change-of-variables formula for nonnegative Lebesgue measurable functions

A normal endomorphism of a finite-dimensional complex inner-product space admits an orthonormal eigenbasis. Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely

Assume countable choice: every sequence of nonempty sets has a choice function. It is used here in the scalar change-of-variables supplier in [F7]. The Axiom of Countable Choice (AC_ω)