The bounded norm-continuous driven model: unique unitary propagator, Dyson truncation budget and outcome-law error

Adopt the framework data and the declared driven model of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models in the bounded norm-continuous class of clause (b)(i): $I=[a,b]$ is the declared compact time interval, $\mathfrak{H}$ the state space of the declared system, $\mathrm{\hslash }>0$, and $H:I\to \mathcal{B}(\mathfrak{H})$ is norm-continuous with self-adjoint values, $K:={\sup }_{t\in I}\mathrm{\parallel }H(t)\mathrm{\parallel }\mathrm{/}\mathrm{\hslash }<\mathrm{\infty }$. Let $U_0(t,s)=I$ and $U_{n+1}(t,s)={\int }_s^{t}A(r)U_n(r,s)dr$ with $A(r)=-\frac{i}{\mathrm{\hslash }}H(r)$ be the oriented Dyson iterates of Norm Dyson series and uniqueness, and let $U(t,s)={\sum }_{n\ge 0}{U}_n(t,s)$ be its norm-convergent propagator; for $N\in {\mathbb{N}}_0$ put $R_N(t,s):=U(t,s)-{\sum }_{n=0}^N{U}_n(t,s)$. 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), the latter for the outcome-law reading of P2. Then (all differential assertions below apply only when $a<b$, with one-sided derivatives at endpoints; for $a=b$ only $U(a,a)=I$ and the algebraic and norm assertions are claimed):

(a) Well-posedness of the declared model. The family $U$ is the unique family of bounded operators on $\mathfrak{H}$ that is jointly norm-continuous on $I\times I$, unitary-valued, satisfies $U(s,s)=I$ and the cocycle $U(t,r)=U(t,s)U(s,r)$, and solves $i\mathrm{\hslash }{\mathrm{\partial }}_{t}U(t,s)=H(t)U(t,s)$ together with its second-endpoint equation $i\mathrm{\hslash }{\mathrm{\partial }}_{s}U(t,s)=-U(t,s)H(s)$ in operator norm. In particular the model's clause (c)(i) declaration has exactly one solution: the driven model in the bounded norm-continuous class is well posed.

(b) Truncation budget. For all $s,t\in I$ and all $N\in {\mathbb{N}}_0$, with $\tau :=\mathrm{\mid }t-s\mathrm{\mid }$, $\parallel R_N(t,s)\parallel \le e^{K\tau }\frac{(K\tau {)}^{N+1}}{(N+1)!},\text{and}\mathrm{\parallel }U(t,s)-I\mathrm{\parallel }\le e^{K\tau }-1.$ In particular $\mathrm{\parallel }{R}_1(t,s)\mathrm{\parallel }\le e^{K\tau }(K\tau {)}^2\mathrm{/}2$ and $\mathrm{\parallel }{R}_2(t,s)\mathrm{\parallel }\le e^{K\tau }(K\tau {)}^3\mathrm{/}6$.

(c) Evolved states and outcome laws. For every preparation $\rho$ specified at time $s$ the operator $\rho (t):=U(t,s)\rho (s)U(t,s{)}^{\ast }$ is a preparation of the model with $\mathrm{tr}\rho (t)=1$ for every $t\in I$, and the specification is consistent with the cocycle: evolving first from $s$ to $r$ and then from $r$ to $t$ gives the same preparation. For a pure preparation of a unit vector $\psi (s)$ the vector $\psi (t):=U(t,s)\psi (s)$ is unit and spans a pure preparation for every $t$. For every assigned sharp observable $(A,u_A)$ (Assigned sharp observables and the outcome law of the framework) and every Borel set $B\subseteq \mathbb{R}$, the number ${\mu }_{\rho (t)}^A(B)$ is the model's outcome probability at time $t$.

(d) Error of the unnormalized truncated quadratic estimate. Let $\psi (s)$ be a unit vector, put ${\psi }^N(t):={\sum }_{n=0}^N{U}_n(t,s)\psi (s)$ and let $(A,u_A)$ be an assigned sharp observable with $A\in \mathcal{B}(\mathfrak{H})$ bounded. Then for every $t\in I$ and every Borel set $B\subseteq \mathbb{R}$, $\mid {\mu }_{\psi (t)}^A(B)-{\mu }_{{\psi }^N(t)}^A(B)\mid \le (1+\mathrm{\parallel }{\sum }_{n=0}^N{U}_n(t,s)\mathrm{\parallel })\mathrm{\parallel }{R}_N(t,s)\mathrm{\parallel }\le 2e^{K\tau }\frac{(K\tau {)}^{N+1}}{(N+1)!},$ where the truncated quantity is the unnormalized quadratic form ${\mu }_{{\psi }^N(t)}^A(B):=\mathrm{\parallel }{E}_A(B){\psi }^N(t){\mathrm{\parallel }}^2$ of the truncated vector, whose norm is at most $e^{K\tau }$. It is an approximation to the normalized exact Born law and need not itself be a probability law; for $N=1$ the bound also implies the looser estimate $2e^{K\tau }(K\tau {)}^2$.

(e) Scope. The theorem is conditional on the bounded norm-continuous class, on the compactness of $I$ and on $K<\mathrm{\infty }$. It asserts no existence or uniqueness statement for unbounded families (which are treated in their own classes), no uniform-in-time approximation, no rate or decay, no measurement or apparatus model, and no comparison with any recorded data.

• The stronger unitary bound. Because every $U(t,s)$ is unitary, the truncation lemma also records the bound $\mathrm{\parallel }{R}_N(t,s)\mathrm{\parallel }\le (K\tau {)}^{N+1}\mathrm{/}(N+1)!$, which does not grow exponentially; the theorem states the exponential version because it is the one used to convert to the outcome-law estimate with the same constant.
• From propagator error to probability error. Clause (d) is the mechanism by which the Dyson truncation of the page's amplitude theorems becomes a bound on an outcome probability; it applies to every Borel outcome set and every bounded assigned sharp observable, and it requires no additional physical assumption beyond P2.