The declared adiabatic protocol model: slow parameter, followed spectral subspace, uniform gap and leakage probability

(a) Framework data. Adopt the framework data and readings of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models clauses (a), (d) (P1, P2 and P4 and their state, observable and evolution readings; units of Dimensions, SI unit conventions and quantity types of the framework); assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice) as the choice hypotheses of the cited spectral and evolution suppliers. The autonomous-dynamics postulate P4 is adopted in the reading of Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution; the protocol below is the reading of its clause (iv) for slowly varying families.

(b) The declared slow family. As an additional model assumption of the pair, fix a protocol duration $T>0$ (seconds), the slow parameter $s\in [0,1]$, and a family $s\mapsto H(s)$ of self-adjoint operators on the declared state space $\mathfrak{H}$ (Symmetric, self-adjoint and essentially self-adjoint operators, Hilbert space), read as the model's energy at slow time $s$, in one of the two declared regularity branches:

• (f) finite branch. $\mathfrak{H}={\mathbb{C}}^d$ for some $d\ge 1$ and $H\in C^2([0,1];\mathrm{Herm}({\mathbb{C}}^d))$;
• (u) common-domain branch. $D(H(s))=Y$ for a dense subspace $Y\subseteq \mathfrak{H}$ that is a Hilbert space continuously embedded in $\mathfrak{H}$, every $H(s)$ is self-adjoint on $\mathfrak{H}$, and $s\mapsto H(s)$ is of class $C^1([0,1];\mathcal{B}(Y,\mathfrak{H}))$ (A bounded linear operator between normed spaces, The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

The family is prescribed model data: no laboratory ramp, field source or feedback is assumed. The protocol propagator is the unique unitary family $U_T(s,\sigma )$ on $\mathfrak{H}$ solving the slow-time Schroedinger equation $i\mathrm{\hslash }{\mathrm{\partial }}_s{U}_T(s,\sigma )=TH(s)U_T(s,\sigma ),U_T(\sigma ,\sigma )=I,0\le \sigma \le s\le 1,$ whose existence is cited from the mathematics: in branch (f) the family $s\mapsto TH(s)\mathrm{/}\mathrm{\hslash }$ is norm-continuous with bounded self-adjoint values and $U_T$ is its Dyson propagator (Norm Dyson series and uniqueness), and in branch (u) the family $s\mapsto TH(s)\mathrm{/}\mathrm{\hslash }$ lies in $C^1([0,1];\mathcal{B}(Y,\mathfrak{H}))$ and $U_T$ is its strong propagator (Common-domain nonautonomous evolution). The displayed differential equation holds in operator norm in branch (f); in branch (u) it is applied to each $y\in Y$, with derivative in $\mathfrak{H}$ and $U_T(s,\sigma )Y=Y$. Derivatives at endpoints are one-sided. The model's evolution in physical time is $\psi (t):=U_T(t\mathrm{/}T,0)\psi (0)$ for $t\in [0,T]$, and the evolved preparation is $\rho (t):=U_T(t\mathrm{/}T,0)\rho (0)U_T(t\mathrm{/}T,0{)}^{\ast }$, read as in clause (d) of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models.

(c) Followed spectral subspace and uniform gap. Write $E_{H(s)}$ for the spectral projection-valued measure of $H(s)$ in Assigned sharp observables and the outcome law of the framework clause (b), with the projection-valued-measure conventions of Projection valued measure. Fix a label: for branch (f) a spectral cluster, meaning a subset ${\sigma }_P(s)\subseteq \sigma (H(s))$ of the finite spectrum (Discrete and essential spectrum of a self-adjoint operator), with $P(s):=E_{H(s)}({\sigma }_P(s))$; for branch (u) a labelled eigenvalue $E(s)$ with $1\le \dim \ker (H(s)-E(s)I)<\mathrm{\infty }$, with ${\sigma }_P(s):=\{E(s)\}$ and $P(s):=E_{H(s)}(\{E(s)\})$. Here $\mathrm{dist}(A,B):=\inf \{\mathrm{\mid }a-b\mathrm{\mid }:a\in A,b\in B\}$, with $\inf \mathrm{\varnothing }=+\mathrm{\infty }$. The label is declared uniformly gapped when $\mathrm{dist}({\sigma }_P(s),\sigma (H(s))\setminus {\sigma }_P(s))\ge g>0(s\in [0,1])$ in branch (f), respectively $\mathrm{dist}(\{E(s)\},\sigma (H(s))\setminus \{E(s)\})\ge g>0$ in branch (u). For such a label, coherently followed means that $s\mapsto P(s)$ is continuous in operator norm and has the following local Riesz continuation property. For every $s_0\in [0,1]$ there is a relative neighborhood $J$ of $s_0$ and a fixed finite collection of circles ${\mathrm{\Gamma }}_j$ with pairwise disjoint closed discs, each centered at one of the distinct eigenvalues in ${\sigma }_P(s_0)$ and containing no other point of $\sigma (H(s_0))$. There is one circle for each such eigenvalue (one circle in branch (u)). Parametrize each circle counterclockwise by $z_j(\theta )=c_j+r_j{e}^{i\theta }$, $0\le \theta \le 2\pi$, with $r_j>0$. For every $s\in J$ and every $z$ on these circles, $zI-H(s):D(H(s))\to \mathfrak{H}$ is bijective with bounded inverse on $\mathfrak{H}$, and $P(s)=\frac{1}{2\pi i}{\sum }_j{\int }_0^{2\pi }(z_j(\theta )I-H(s){)}^{-1}{z}_j^{\mathrm{\prime }}(\theta )d\theta \mathrm{.}$ Each integral means the limit in $\mathcal{B}(\mathfrak{H})$ of Riemann sums of its norm-continuous integrand. For an empty cluster in branch (f) the circle collection and sum are empty and $P(s)=0$ on $J$; a full cluster is allowed. The local formulas agree on overlaps as the same prescribed $P(s)$ and thus specify continuation along $[0,1]$ from $P(0)$, without selecting instantaneous eigenvectors. This is a condition on the prescribed label, not a claim that an arbitrary choice of labels satisfies it. A preparation in the subspace is a preparation with $\rho (0)=P(0)\rho (0)P(0)$; for a unit vector $\psi (0)\in \mathrm{ran}P(0)$ the leakage probability at slow time $s$ is the model's outcome probability $p_{\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{k}}(s):={\mu }_{\psi (sT)}^{(I-P(s))}(\{1\})=\parallel (I-P(s))\psi (sT){\parallel }^2,\psi (sT)=U_T(s,0)\psi (0),$ that is, the Born probability in the reading of P2 that the evolved preparation is found outside the followed subspace at slow time $s$ (Assigned sharp observables and the outcome law of the framework, The state space: density operators, pure and mixed states, and purity; State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes).

(d) Status and scope. Clauses (b)-(c) adopt the protocol as an idealization and fix its vocabulary. The size of the leakage is not part of the definition: it is the content of the adiabatic theorems of the page, which are conditional on the uniform gap and on coherent following. The definition asserts no claim that a laboratory ramp realizes the family, no selection or relaxation of a branch by the protocol, no thermalization, no measurement record and no empirical result; whether a laboratory preparation realizes a preparation in the declared subspace is a tested identification in the sense of P7.

• Two branches, two theorems. The finite and common-domain branches are declared separately because their mathematics differ: the finite branch uses the norm-continuous Dyson propagator and a finite-dimensional contour construction, the common-domain branch uses the strong propagator and graph-norm regularity. The page's two adiabatic theorems prove the leakage bounds for the corresponding branches and add no assumption to this definition.
• The gap belongs to the family, not to the protocol. The uniform gap is a property of the prescribed family and its label, independent of $T$; the protocol duration enters only through the slow-time equation. This separation is what makes the $1\mathrm{/}T$ estimates of the page meaningful.