# Particle Trajectories for Quantum Maps

Yonah Borns-Weil<EMAIL_ADDRESS>Department of Mathematics,
University of California, Berkeley, CA 94720  and  Izak Oltman
<EMAIL_ADDRESS>Department of Mathematics, University of California,
Berkeley, CA 94720

###### Abstract.

We study the trajectories of a semiclassical quantum particle under repeated
indirect measurement by Kraus operators, in the setting of the quantized
torus. In between measurements, the system evolves via either Hamiltonian
propagators or metaplectic operators. We show in both cases the convergence in
total variation of the quantum trajectory to its corresponding classical
trajectory, as defined by propagation of a semiclassical defect measure. This
convergence holds up to the Ehrenfest time of the classical system, which is
larger when the system is “less chaotic”. In addition, we present numerical
simulations of these effects.

In proving this result, we provide a characterization of a type of semi-
classical defect measure we call uniform defect measures. We also prove
derivative estimates of a function composed with a flow on the torus.

##  1\. Introduction

Figure 1. A numerical simulation demonstrating this paper’s main result. Each
grey curve represents the trajectory of a quantum particle (under the
evolution procedure discussed in this paper) on a torus with identical initial
conditions and a fixed Hamiltonian. Initially, all particles roughly follow
the same trajectory (and agree with the corresponding classical trajectory),
but after some time become decoherent. More numerics are provided in §4. This
simulation is discussed at the end of §4

The framework of “quantum trajectories” has gotten increasing attention in
recent years. Instead of treating measurement as a one-time event, researchers
consider many measurements obtained over time and study their effects on the
particle being measured, described mathematically via quantum instruments. The
sequence of measured values is called the _quantum trajectory_ , and is a
random variable worthy of study.

In this paper, we take quantum trajectories into the setting of the
semiclassical quantized torus, as studied in Bouzouina–De Biévre [BD96],
Schenck [Sch09], Dyatlov–Jezequel [DJ21], and others. The spaces involved are
all finite-dimensional, which makes numerical simulations easier than in PDE-
based models. Our result is inspired by the recent work of Benoist, Fraas, and
Fröhlich [BFF22], who prove in the PDE setting that the trajectory of a
repeatedly-observed quantum particle in the semiclassical limit approaches the
natural notion of a classical trajectory. See Figure 1 for a simulation of
this quantum-classical correspondence of particle trajectories on the
quantized torus and §4 for more numerics. We strengthen the result in [BFF22]
in our setting by allowing the time to depend on the semiclassical parameter,
and we relate the rate of convergence to the “chaotic behavior” of the
underlying classical system. This matches the intuition that a less chaotic
transformation should be approximable for longer by its classical counterpart.
Additionally, we illustrate the results numerically to demonstrate in the
chaotic case that our time restriction is essentially optimal.

###  1.1. Model

In our setting, a quantum particles is modeled by a density operator (see
Definition 2.13) $\rho_{N}$ on the quantized $2d-$dimensional torus $H_{N}$
(see (2.8)) where $N\in\mathbb{N}$ is proportional to the reciprocal of the
semi-classical parameter $h$. This particle is “indirectly measured” by
conjugating by Kraus operators (see Definition 2.14)
$\operatorname{Op}_{N}(f_{q})$ where $\\{f_{q}\\}_{q\in\Omega}$ are fixed
elements of $C^{\infty}(\mathbb{T}^{2d})$ with uniformly bounded derivatives
(such that $\operatorname{Op}_{N}(f_{q})$ provide a resolution of the
identity), $\Omega$ is a compact metric space with finite Borel measure $\nu$,
and $\operatorname{Op}_{N}$ is the quantization of functions on the torus as
described in §2.4. The measurement procedure is further described in §2.5.

After observing the particle, we evolve it for one unit of time by conjugating
$\rho_{N}$ by a unitary operator $U$, which is either a quantization of a
fixed symplectic matrix $M$ (defined in §2.2) or Schrödinger evolution
$e^{-\frac{i}{h}P}$ for a fixed Hamiltonian $P$. If we repeat this process $n$
times, we get a resulting measure on the space of trajectories. Suppose
$F\subset\Omega^{n+1}$, then the probability of obtaining the set of
trajectories $F$ is given by:

$\displaystyle\int_{F}\operatorname{tr}_{H_{N}}\left(\Phi_{N,{q_{n}}}^{*}\circ\dots\circ\Phi_{N,{q_{0}}}^{*}(\rho_{N})\right)\mathop{}\\!\mathrm{d}\nu(q_{0})\cdots\mathop{}\\!\mathrm{d}\nu(q_{n})$

where

$\displaystyle\Phi_{N,{q_{i}}}^{*}(\rho):=U\operatorname{Op}_{N}(f_{q_{i}})\rho_{N}\operatorname{Op}_{N}(f_{q_{i}})^{*}U^{*}.$

Therefore, we have a “quantum probability measure” given by

$\mathbb{P}^{(n)}_{N,\rho}:=\operatorname{tr}_{H_{N}}\left(\Phi_{N,{q_{n}}}^{*}\circ\dots\circ\Phi_{N,{q_{0}}}^{*}\rho\right)\mathop{}\\!\mathrm{d}\nu(q_{0})\cdots\mathop{}\\!\mathrm{d}\nu(q_{n})$
(1.1)

We now describe the trajectory of a corresponding classical particle. Let
$\mu$ be the defect measure (defined in Definition 2.5) of $\rho_{N}$. We
interpret $\mu$ as the probability distribution of the initial position and
momentum of a classical particle. We apply an “approximate measurement,” which
computes an observable quantity $q$ to be in $E_{0}\subseteq\Omega$ with
probability
$\int_{E_{0}}|f_{q_{0}}(\zeta)|^{2}\mathop{}\\!\mathrm{d}\nu(q_{0}).$ The
particle is then allowed to classically evolve for one unit of time by a flow
$\phi^{t}$ on $\mathbb{T}^{2d}$ which is either multiplication by the
symplectic matrix $M$ or else the nonlinear Hamiltonian evolution
$\text{exp}(H_{p})$. We again repeat the process of alternating measurement
and evolution, though we must remark that unlike in the quantum case, the
measurement does not affect the location of the particle. Conditioned on
initially having position/momentum $\zeta$, we have the probability of
classically measuring $(q_{0},\dots q_{n})$ in a measurable set
$F\subset\Omega^{n+1}$ to be

$\int_{F}|f_{q_{n}}(\phi^{n}(\zeta))|^{2}\cdots|f_{q_{0}}(\zeta)|^{2}\mathop{}\\!\mathrm{d}\nu(q_{0})\cdots\mathop{}\\!\mathrm{d}\nu(q_{n}).$

One can easily check that $(q_{n})_{n}$ is a process of independent random
variables (although not identically distributed). As the initial value of
$\zeta$ was taken to be randomly chosen from the distribution $\mu$, we take
the classical probability of measuring $(q_{0},\dots,q_{n})\in F$ to be

$\int_{\mathbb{T}^{2d}}\int_{F}|f_{q_{n}}(\phi^{n}(\zeta))|^{2}\cdots|f_{q_{0}}(\zeta)|^{2}\mathop{}\\!\mathrm{d}\nu(q_{0})\cdots\mathop{}\\!\mathrm{d}\nu(q_{n})\mathop{}\\!\mathrm{d}\mu(\zeta).$

We therefore have a “classical probability measure” given by

$P^{(n)}_{\mu}:=\int_{\mathbb{T}^{2d}}|f_{q_{n}}(\phi^{n}(\zeta))|^{2}\cdots|f_{q_{0}}(\zeta)|^{2}\mathop{}\\!\mathrm{d}\mu(\zeta)\mathop{}\\!\mathrm{d}\nu(q_{0})\cdots\mathop{}\\!\mathrm{d}\nu(q_{n}).$
(1.2)

Note that the functions $f_{q}$ are allowed to overlap so that $P^{(n)}_{\mu}$
is a probability of trajectories in $\Omega^{n+1}$ while $\phi^{t}$ is the
unique trajectory on the torus.

The goal of this paper is to describe in what sense, and at what quantitative
rate, the measure $\mathbb{P}^{(n)}_{N,\rho}$ approaches $P^{(n)}$ as $N$
grows large.

We can state our main result.

######  Theorem (Main result).

Suppose $\rho=\rho_{N}:H_{N}\to H_{N}$ is a density operator with
semiclassical defect measure $\mu$. Fix $n\in\mathbb{N}$, and let
$(q_{N,0},q_{N,1},\dots,q_{N,n})$ be random variables with law
$\mathbb{P}_{N,\rho}^{(n)}$ given by (1.1) and let $(q_{0},q_{1},\dots,q_{n})$
be random variables with law $P_{\mu}^{(n)}$ given by (1.2). Then as
$N\to\infty$, $(q_{N,0},q_{N,1},\dots,q_{N,n})$ converges weakly to
$(q_{0},q_{1},\dots,q_{n})$.

In proving this, we show uniform convergence of $\mathbb{P}_{N,\rho}^{(n)}$ to
$P_{\mu}^{(n)}$ and hence

$\mathbb{P}_{N,\rho}^{(n)}\xrightarrow{N\to\infty}P_{\mu}^{(n)}$

in total variation. For a stronger version, see Theorem 1, in which we provide
a quantitative rate of convergence in the case where the number of steps $n$
approaches the Ehrenfest time (defined in 2.4).

Figure 1 numerically samples 200 quantum trajectories from such an evolution
procedure (see the end of §4 for more details) for $1000$ time steps of $0.01$
units of time. Observe that up $t=0.4$, all trajectories roughly agree, while
after this time, they disagree. This paper’s main result estimates how long
they agree, and to what distribution they agree with.

###  1.2. Previous Work

The understanding that a measurement affects a quantum state goes back to the
early days of quantum mechanics. In his paper on the uncertainty principle,
Heisenberg [Hei27] (see [Hei83] for an English translation) discussed the so-
called “collapse” of the wavefunction, which was later rephrased in
mathematical terms by von Neumann [Von13]. When a system is only partially or
indirectly measured, the necessary framework is that of positive operator-
valued measures, which were first studied by Naimark [Neu43]. The analogue of
wave-function collapse was introduced by Davies and Lewis [DL70] with the
development of quantum instruments.

Further research focused on multiple measurements as a probabilistic process,
which could be modeled in either discrete or continuous time. The first
analysis of continuous-time measurements was due to Davies [Dav69], and
required extensive machinery from stochastic calculus. (See [BG09] or [Hol03]
for a brief introduction.) Meanwhile, the discrete-time model consisting of
repeated applications of a quantum instrument, while requiring substantially
fewer technicalities, has nonetheless demonstrated a rich variety of behavior
and is a subject of current study. Kümmerer and Maassen [KM03] demonstrated
ergodicity of a repeated measurement process and showed that the states
approach a pure state provided the measurement operators do not have a “dark”
subspace [KM06]. Ballesteros et al. [Bal+18] show that when the dynamic is
trivial, the state of the system localizes in space.

If the state is allowed to evolve in between measurements, the situation
becomes more complex. The mathematical analysis of such a quantum trajectory,
which approximates a classical one, was done by Ballesteros et al. [Bal+21]
following extensive physical evidence of the phenomenon, see the exposition by
Figari [FT13]. That paper studied particles, initially in the set of normal
states, evolving under a quadratic Hamiltonian. The motivation for our work
came from the study of the semiclassical case under a more general Hamiltonian
by Benoist, Fraas, and Fröhlich in [BFF22]. We adapt their framework to the
setting of quantum maps. This has two advantages: we can improve on the number
of measurements exhibiting classical/quantum correspondence and we can provide
(more easily than in the PDE setting) numerical simulations. In particular,
the numerics indicate the accuracy of the dynamical bound on the number of
measurements.

##  2\. Background

In this section we provide a background on the measurement procedure as well
as the semiclassical analysis tools required in proving our main result.
Throughout this paper we consider $h=(2\pi N)^{-1}>0$ as the semiclassical
parameter where $N\in\mathbb{N}$.

###  2.1. Semiclassical Analysis on the Real Line

Physically, we are often concerned with how our quantum mechanical world
approximately produces classical mechanics. This limit is achieved by taking a
semiclassical parameter $h\in(0,1)$ to be very small, and the math involved is
called _semiclassical analysis_.

######  Definition 2.1 (symbol class).

For each $\delta\in[0,1/2)$, the _symbol class_ $S_{\delta}$ is defined as

$S_{\delta}:=\\{a\in
C^{\infty}(T^{*}\mathbb{R}^{d}):\forall\alpha,\beta\in\mathbb{N}^{d},\,|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\leqslant
C_{\alpha,\beta}h^{-\delta|\alpha+\beta|}\\}.$

Of most importance is the case where $\delta=0$, though technical steps of the
proof of Theorem 1 will require us to consider general $\delta$. Nevertheless,
if $\delta$ is omitted it will be taken to be $0$. Elements of $S_{\delta}$
(called _symbols_) may depend on $h$, though we require that the constants
$C_{\alpha,\beta}$ in the definition be uniform in $h$. Real-valued symbols
correspond to classical observables, which can be “quantized” to get quantum
observables as follows.

######  Definition 2.2 (Weyl quantization).

For $\delta\in[0,1/2)$, the _Weyl quantization_ of a symbol $a\in S_{\delta}$
is an operator
$\operatorname{Op}_{h}a:\mathcal{S}(\mathbb{R}^{d})\to\mathcal{S}^{\prime}(\mathbb{R}^{d})$
(sometimes written $a(x,hD)$) given by

$(\operatorname{Op}_{h}(a)u)(x):=\frac{1}{(2\pi h)^{d}}\iint
e^{\frac{i}{h}\langle
x-y,\xi\rangle}a\left(\frac{x+y}{2},\xi\right)u(y)\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}\xi,$

for $u\in\mathcal{S}(\mathbb{R}^{d})$.

A theorem of Calderón and Vaillancourt states that if $\delta\in[0,1/2)$ and
$a\in S_{\delta}$, then $\operatorname{Op}_{h}(a)$ is a bounded operator on
$L^{2}(\mathbb{R}^{d})$, with bound

$\|\operatorname{Op}_{h}(a)\|_{L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})}\leqslant
C\|a\|_{L^{\infty}(\mathbb{R}^{d})}+C\sum_{|\alpha|+|\beta|\leqslant
K}h^{\frac{|\alpha|+|\beta|}{2}}\|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a\|_{L^{\infty}(\mathbb{R}^{d})}$
(2.1)

where $C>0$ and $K\in\mathbb{N}$ depends only on $d$. (In fact, we remark that
(2.1) even holds for the threshold scaling $\delta=\frac{1}{2}$, though in
this case all derivatives of $a$ in the sum will contribute equal orders of
magnitude.)

If $a\in S$ is real-valued, $\operatorname{Op}_{h}(a)$ is a self-adjoint
operator on $L^{2}(\mathbb{R}^{d})$, and more generally we have

$\operatorname{Op}_{h}(a)^{*}=\operatorname{Op}_{h}(\overline{a}).$

Pseudodifferential operators enjoy good properties under composition. In
particular, if $\delta\in[0,\frac{1}{2})$ and $a,b\in S_{\delta}$, then

$\operatorname{Op}_{h}(a)\operatorname{Op}_{h}(b)=\operatorname{Op}_{h}(a\\#b)$

where $a\\#b\in S_{\delta}$ is the _Moyal product_ given explicitly by

$a\\#b(x,\xi):=e^{ih\sigma(D_{x},D_{\xi},D_{y},D_{\eta})}(a(x,\xi)b(y,\eta))\Big{|}_{\begin{subarray}{c}y=x\\\
\eta=\xi\end{subarray}}$

from which one may derive the asymptotic summation formula

$a\\#b(x,\xi)\sim\sum_{k=0}^{\infty}\frac{i^{k}h^{k}}{k!}\sigma(D_{x},D_{\xi},D_{y},D_{\eta})^{k}(a(x,\xi)b(y,\eta))\Big{|}_{\begin{subarray}{c}y=x\\\
\eta=\xi\end{subarray}}$ (2.2)

where in these formulas $\sigma(x,\xi,y,\eta):=\langle\xi,y\rangle-\langle
x,\eta\rangle$, and $a\sim\sum_{k=0}^{\infty}h^{k}a_{k}$ means that for each
$K\in\mathbb{N}$, $(a-\sum_{k=0}^{K-1}h^{k}a_{k})\in h^{K}S_{\delta}.$

In particular, one has that if $a,b\in S_{\delta}$, then
$(\operatorname{Op}_{h}a)(\operatorname{Op}_{h}b)-\operatorname{Op}_{h}(ab)=\operatorname{Op}_{h}(r)$
where $r$ has $S_{\delta}$ seminorms at most $O(h^{1-2\delta})$, with
constants depending only on finitely many derivatives of $a$ and $b$. More
generally, given $a_{1},\dots,a_{n}\in S_{\delta}$, we have by an easy
induction together with $(\ref{eq:CV})$ that

$\left\|\operatorname{Op}_{h}(a_{1})\cdots\operatorname{Op}_{h}(a_{n})-\operatorname{Op}_{h}(a_{1}\cdots
a_{n})\right\|_{L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})}=\operatorname{poly}(n)O(h^{1-2\delta}).$ (2.3)

Finally, we will need a theorem of Egorov, which gives a sense in which
quantum operators evolve in time analogously to their classical counterparts.
We provide the quantitative version due to Bouzouina and Robert [BR02], see
also Zworski [Zwo12, Chapter 11.4]. We first require the following definition.

######  Definition 2.3 (Lyapunov exponent).

For any dynamical system given by a flow $\phi_{t}$ on $T^{*}\mathbb{R}^{d}$
or $\mathbb{T}^{2d}$ (smooth in time), define the _Lyapunov exponent_ $\Gamma$
to be

$\Gamma=\lim_{t\to\infty}\sup_{y}\frac{1}{t}\log|\partial_{y}\phi_{t}(y)|$

and analogously for iterative maps.

Given $p\in S$, let $\phi_{t}$ be evolution by the classical Hamiltonian flow
of $p$, induced by the Hamiltonian vector field
$X_{p}=\partial_{\xi}p\partial_{x}-\partial_{x}p\partial_{\xi}$. On the
quantum side, a particle evolves by the Schrödinger equation

$hD_{t}\psi=P\psi,$

so $\psi(t)=e^{-\frac{i}{h}tP}\psi(0)$ and observables $A$ evolve in the
Heisenberg picture to $A(t)=e^{\frac{i}{h}tP}Ae^{-\frac{i}{h}tP}$. Egorov’s
theorem states that this evolved operator is equal, up to order $h$, to the
quantization of the classically observed symbol of $A$, for $t$ less than
_Ehrenfest time_ $t_{E}\sim\log\frac{1}{h}$. Specifically, we have that if
$\Gamma$ is the Lyapunov exponent of $\phi_{t}$ and

$|t|\leqslant T+\frac{\delta}{\Gamma+\varepsilon}\log\frac{1}{h}$ (2.4)

for some $\delta<\frac{1}{2}$ and $\varepsilon>0$, then

$e^{\frac{i}{h}tP}\operatorname{Op}_{h}(a)e^{-\frac{i}{h}tP}-\operatorname{Op}_{h}(\phi_{t}^{*}a)=\operatorname{Op}_{h}(r)$
(2.5)

for $a\in S$, and $r\in S_{\delta}$ has all seminorms of order
$O(h^{2-3\delta})$. In cases where $\Gamma=0$, such as a completely integrable
classical system, some stronger results are available, see [BR02].

###  2.2. Fourier Integral Operators

As a generalization of the operators $e^{-\frac{i}{h}tP}$ in Egorov’s theorem,
we briefly discuss Fourier integral operators. One may view the operator
$e^{-\frac{i}{h}P}$ as quantizing a transformation (or equivalently, a “change
of variables”) on phase space; namely the symplectomorphism on
$T^{*}\mathbb{R}^{d}$ given by $(x,\xi)\mapsto\phi_{t}(x,\xi)$. It is then
natural to generalize the quantization to arbitrary symplectomorphisms. This
abstraction will be rewarded when we work on the quantized torus in §2.4, on
which the most natural classical transformations do not arise from a flow at
all. Let $\phi:T^{*}\mathbb{R}^{d}\to T^{*}\mathbb{R}^{d}$ be any
symplectomorphism. We consider only unitary Fourier integral operators here,
and only consider their action in the case $\delta=0$. For the purposes of
this paper, define a _Fourier integral operator quantizing $\phi$_ in the
following way.

######  Definition 2.4 (quantized flow).

For $\phi:T^{*}\mathbb{R}^{d}\to T^{*}\mathbb{R}^{d}$, the Fourier integral
operator quantizing $\phi$ is the operator $U:L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})$ such that the Egorov-type relation

$U^{-1}\operatorname{Op}_{h}(a)U=\operatorname{Op}_{h}(b)$

holds for any $a\in S$ with $b=\phi^{*}a+O_{S}(h)$.

In particular (using these definitions), Egorov’s theorem is precisely the
statement that $e^{\frac{i}{h}tP}$ is a one-parameter family of Fourier
integral operators, which quantize the classical Hamiltonian flow $\phi_{t}$.

Perhaps the most important Fourier integral operators are the _metaplectic
operators_ [Zwo12, Chapter 11.3], which are the special case when $\phi$ is a
linear symplectomorphism. In this case for
$M\in\operatorname{Sp}(2d,\mathbb{R})$, we write the corresponding metaplectic
operator as $\widehat{M}$. The metaplectic operators are unitary, and benefit
from an exact version of (2.4), ie they satisfy

$\widehat{M}^{-1}\operatorname{Op}_{h}(a)\widehat{M}=\operatorname{Op}_{h}(M^{*}a)$

for all $a\in S$, with no error. We may remark that while the metaplectic
operators define a projective unitary representation of
$\operatorname{Sp}(2d,\mathbb{R})$, they are in fact a unitary representation
of its double cover, which is known as the _metaplectic group_. This is
analogous to projective representations of rotation operators on spinor
spaces, and leads similarly to a non-canonical choice of phase for a
metaplectic operator. As we will only encounter metaplectic operators when we
are conjugating by them, this phase vanishes and we need not concern ourselves
with it.

###  2.3. Semiclassical Defect Measures

We now discuss semiclassical defect measures, which give a quantitative answer
to where in phase space a quantum particle “is” in the semiclassical limit as
$h\to 0$.

######  Definition 2.5 (defect measure).

We say an $h$-dependent set of density operators $\rho_{h}$ on
$L^{2}(\mathbb{R}^{d})$, has _defect measure_ $\mu$ if for all $a\in
C^{\infty}_{0}(T^{*}\mathbb{R}^{d})$,

$\lim_{h\to
0}\operatorname{tr}\left(\rho_{h}\operatorname{Op}_{h}(a)\right)=\int_{T^{*}\mathbb{R}^{d}}a\mathop{}\\!\mathrm{d}\mu.$
(2.6)

If $\psi_{h}\in L^{2}(\mathbb{R}^{d})$, we say it has defect measure $\mu$ if
the density operator
$\left\lvert\psi_{h}\right\rangle\left\langle\psi_{h}\right\rvert$ has defect
measure $\mu$.

Having a semiclassical defect measure is a very special property of a
sequence, corresponding to being “essentially classical” as $h\to 0$.
Nevertheless, such a limit necessarily exists along a subsequence of $h$’s, as
the following proposition states.

######  Proposition 2.6.

Let $\rho_{h}$ be an $h$-dependent family of density operators on
$L^{2}(\mathbb{R}^{d})$. Then there is a nonnegative Radon measure $\mu$ on
$T^{*}\mathbb{R}^{d}$ and a sequence $h_{j}\to 0$ such that

$\lim_{j\to\infty}\operatorname{tr}\left(\rho_{h_{j}}\operatorname{Op}_{h_{j}}(a)\right)=\int_{T^{*}\mathbb{R}^{d}}a\mathop{}\\!\mathrm{d}\mu$

for all $a\in C_{0}^{\infty}(T^{*}\mathbb{R}^{d})$.

The proof of Proposition 2.6 is essentially the same as [Zwo12, Theorem 5.2],
which proves it in the special case of pure states.

In many natural cases, we may take the limit in (2.6) to be uniform in the
symbol $a$, in which case we refer to $\mu$ as a _uniform defect measure_. For
a precise characterization of aforesaid measures, see Proposition A.1. In
particular, in this paper we will work on the quantized torus $H_{N}$ (see
§2.4) where all defect measures are uniform.

For semiclassical pseudodifferential operators on the real line, there are
standard classes of states with known defect measures [BFF22, Proposition 3.3]
restated in Proposition 2.7. For a full proof (in French) see [LP93, §3] and a
proof for less general coherent states see [Zwo12, Chapter 5.1].

######  Proposition 2.7.

Fix $g\in\mathcal{S}(\mathbb{R}^{d})$ with $\|g\|_{L^{2}}=1$, define

$\psi_{h}(x):=h^{-\frac{d\beta}{2}}g\left(\frac{x-x_{0}}{h^{\beta}}\right)\exp\left(i\dfrac{x\cdot\xi_{0}}{h}\right),$

and let
$\rho_{h}=\left\lvert\psi_{h}\right\rangle\left\langle\psi_{h}\right\rvert.$
Then $\rho_{h}$ has defect measure $\mu$, given as follows:

  * •

If $\beta=0$,
$\mu=\delta_{\xi=\xi_{0}}|g(x-x_{0})|^{2}\mathop{}\\!\mathrm{d}x$,

  * •

If $\beta=1$,
$\mu=\delta_{x=x_{0}}|\hat{g}(\xi-\xi_{0})|^{2}\mathop{}\\!\mathrm{d}\xi$,

  * •

If $\beta\in(0,1)$, $\mu=\delta_{x=x_{0},\xi=\xi_{0}}$.

Here $\hat{g}$ refers to the usual (non-semiclassical) Fourier transform.

In the last case of Proposition 2.7, the states $\psi_{h}$ are called
_generalized coherent states_.

In fact, as we will be working with $S_{\delta}$ classes and must control the
error of our estimates, we will need a slightly stronger definition to
accommodate $h$-dependent symbols.

######  Definition 2.8 (uniform defect measure).

If $\rho_{h}$ has defect measure $\mu$, $\delta\in[0,\frac{1}{2})$, and
$\theta>0$, then we say $\mu$ is a _uniform $(\delta,\theta)$-defect measure_
provided

$\left|\operatorname{tr}(\rho_{h}\operatorname{Op}_{h}(a))-\int_{T^{*}\mathbb{R}^{d}}a\mathop{}\\!\mathrm{d}\mu\right|=O(h^{\theta})$
(2.7)

holds for all $a=a_{h}\in C^{\infty}_{0}(T^{*}\mathbb{R}^{d})\cap S_{\delta}$
with constant depending only on the symbol norms of $a$.

We easily see that if $\mu$ is a $(\delta,\theta)$-defect measure, it is
necessarily a $(\delta^{\prime},\theta^{\prime})$-defect measure for all
$\delta^{\prime}\leqslant\delta$ and $\theta^{\prime}\leqslant\theta$. To
address a potential point of confusion, note that in (2.7) we are enlarging
the space of symbols used in (2.6). These symbols are allowed to depend on $h$
with derivatives growing in $h$ provided they do not grow too fast. And their
support may not be uniformly bounded in $h$.

We can now state a refinement of Proposition 2.7 using this new terminology.

######  Proposition 2.9.

Let $\rho_{h}$ be as in Proposition 2.7. Then the defect measures $\mu$ given
in Proposition 2.7 are uniform $(\delta,\theta)$-defect measures with
$\delta,\theta$ given as follows:

  * •

If $\beta\in\\{0,1\\}$, then $\delta<\frac{1}{2}$ and $\theta<1-2\delta$,

  * •

If $\beta\in(0,1)$, then $\delta<\min(\beta,1-\beta)$ and
$\theta<\min(\beta-\delta,1-\beta-\delta)$.

The proof of Proposition 2.9 is straightforward and given in Appendix B.1.

###  2.4. The Quantized Torus

We now give an introduction to the quantized torus, and the pseudodifferential
calculus on the Hilbert space of corresponding quantum states.

For each $N\in\mathbb{N}$, define

$\displaystyle
H_{N}:=\mathrm{span}\left\\{\frac{1}{N^{\frac{d}{2}}}\sum_{k\in\mathbb{Z}^{d}}\delta_{x=k+\frac{n}{N}}:n\in\\{0,1,\dots,N-1\\}^{d}\right\\}.$
(2.8)

We may remark that $H_{N}$ consists precisely of the elements of
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$ which are periodic in physical and
Fourier space under the semiclassical Fourier transform $\mathcal{F}_{h}$,
given by

$\mathcal{F}_{h}u(\xi)=\frac{1}{(2\pi h)^{\frac{d}{2}}}\int
e^{\frac{i}{h}x\cdot\xi}u(x)\mathop{}\\!\mathrm{d}x$

where $h=(2\pi N)^{-1}$. For simplicity define

$\displaystyle
Q_{n}=\frac{1}{N^{\frac{d}{2}}}\sum_{k\in\mathbb{Z}^{d}}\delta_{x=k+\frac{n}{N}},$
(2.9)

and define a Hilbert space structure on $H_{N}$ such that $(Q_{n})$ is an
orthonormal basis of $H_{N}$. Though it is not necessary here, we may note
that this is an elementary example of the general geometric Toeplitz
quantization studied in [Del19].

We now give an overview of the pseudodifferential calculus on $H_{N}$, which
also may be found in [CZ10], [DJ21], [Sch09]. Let $a\in
C^{\infty}(\mathbb{T}^{2d})$. Intuitively, $a$ corresponds to a classical
observable, with $\mathbb{T}^{2d}$ playing the role of phase space. We see
that $a$ lifts to a doubly-periodic function $\tilde{a}$ on
$T^{*}\mathbb{R}^{d}$, which is in the symbol class $S$. By [CZ10, §2.3]
$\operatorname{Op}_{N}a$ maps $H_{N}$ (as a subset of
$\mathcal{S}^{\prime}(\mathbb{R}^{d})$) to itself, so we may define

$\operatorname{Op}_{N}a:=\operatorname{Op}_{h}\tilde{a}|_{H_{N}}:H_{N}\to
H_{N}$ (2.10)

which is also given in coordinates [CZ10, Lemmma 2.4] by

$\operatorname{Op}_{N}(a)Q_{j}=\sum_{m=0}^{N-1}A_{mj}Q_{m}$

with

$A_{mj}=\sum_{k,l\in\mathbb{Z}}\hat{a}(k,j-m-lN)(-1)^{kl}e^{\pi
i\frac{(j+m)k}{N}},$

where $\hat{a}$ is the (non-semiclassical) Fourier series in both variables.
Furthermore, we may define an analogue of $S_{\delta}$ symbols as

$S_{\delta}(\mathbb{T}^{2d}):=\\{a\in
C^{\infty}(\mathbb{T}^{2d}):\forall\alpha,\beta\in\mathbb{Z}^{d},\,|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}|\leqslant
C_{\alpha\beta}N^{\delta|\alpha+\beta|}\\}$

from which the obvious generalization of (2.10) can be defined.

Most results from pseudodifferential calculus carry over to $H_{N}$, see
[CZ10], [DJ21], for details. In particular, the formulas for compositions and
Egorov’s theorem are the same and follow immediately, and we have a version of
the Calderón-Vaillancourt theorem [CZ10, Proposition 2.7] that says

$\|\operatorname{Op}_{N}(a)\|_{H_{N}}\leqslant\|a\|_{L^{\infty}(\mathbb{T}^{2d})}+o(1),$

where $\|\cdot\|_{H_{N}}$ is the norm given by the Hilbert space structure. In
fact, one can directly bound $\operatorname{Op}_{N}(a)$ by the corresponding
map on $L^{2}(\mathbb{R}^{d})$:

$\|\operatorname{Op}_{N}(a)\|_{H_{N}}\leqslant\|\operatorname{Op}_{h}(\tilde{a})\|_{L^{2}(\mathbb{R}^{d})},$
(2.11)

see [DJ21, §2.2.3] for an explanation using the direct integral decomposition
of $L^{2}(\mathbb{R}^{d})$.

A particular class of metaplectic operators can also be defined on $H_{N}$.
Let $M\in\operatorname{Sp}(2d,\mathbb{Z})$ be an integer symplectic matrix,
and let $N$ be even. Define $\widehat{M}_{N}=\widehat{M}|_{H_{N}}$, which can
be shown [DJ21, §2.2.4] to be a unitary map on $H_{N}$. We then have an
analogous exact Egorov’s theorem:

$\operatorname{Op}_{N}(a)\widehat{M}_{N}=\widehat{M}_{N}\operatorname{Op}_{N}(a\circ
M).$ (2.12)

The definition of a defect measure on $H_{N}$ is also identical. Given an
$N$-dependent sequence of density operators on $H_{N}$, we say they have
_defect measure_ $\mu$ if for every $a\in C^{\infty}(\mathbb{T}^{2d})$,

$\lim_{N\to\infty}\operatorname{tr}\left(\rho_{N}\operatorname{Op}_{N}(a)\right)=\int_{T^{*}\mathbb{R}^{d}}a\mathop{}\\!\mathrm{d}\mu.$

The proof of Proposition A.1 gives that every defect measure on the torus is
uniform. In particular, we have the following result to be an immediate
consequence.

######  Proposition 2.10.

If $\rho_{N}$ are density operators on $H_{N}$ with defect measure $\mu$, then
$\mu$ is a probability measure, and for every $\varepsilon>0$ there is an
$h_{0}$ such that if $h<h_{0}$ and $\|a^{(\alpha)}\|\leqslant 1$ for all
$|\alpha|<K(d)$, then

$\left|\operatorname{tr}_{H_{N}}(\rho_{N}\operatorname{Op}_{N}(a))-\int_{\mathbb{T}^{2d}}a\mathop{}\\!\mathrm{d}\mu\right|\leqslant\varepsilon.$

The notion of a $(\delta,\theta)$-defect measure (2.7) also carries over to
the quantized torus case, where the symbols in this case are in
$S_{\delta}(\mathbb{T}^{2d})$.

Proposition 2.7, which gives the defect measures of generalized coherent
states, has a natural analogue on the quantized torus. We do not claim these
values of $\delta,\theta$ to be sharp.

######  Proposition 2.11.

Given $g\in C^{\infty}(\mathbb{T}^{d};\mathbb{C})$ such that
$\|g\|_{L^{2}(\mathbb{T}^{d})}=1$, and $\xi_{0}\in\mathbb{T}^{d}$, define

$\displaystyle\psi_{N}:=\sum_{k\in(\mathbb{Z}/N\mathbb{Z})^{d}}N^{-\frac{d}{2}}g\left(\frac{k}{N}\right)e^{2\pi
ik\cdot\xi_{0}}Q_{k},$

and set
$\rho_{N}:=C_{N}\left\lvert\psi_{N}\right\rangle\left\langle\psi_{N}\right\rvert$
with $C_{N}:=\|\psi_{N}\|_{H_{N}}^{-2}$ (for $Q_{k}$ defined in (2.9)). Then
$\rho_{N}$ has the semiclassical defect measure
$\mu=\delta_{\xi=\xi_{0}}|g(x)|^{2}\mathop{}\\!\mathrm{d}x$. In particular,
$\mu$ is a $(\delta,\theta)$-defect measure for $\delta<(d+2)^{-1}$,
$\theta<1-(d+2)\delta.$

To prepare for the next proposition, we define the map
$\Pi_{N}:\mathcal{S}(\mathbb{R}^{d})\to H_{N}$ by

$(\Pi_{N}u)_{n}=\frac{1}{N^{\frac{d}{2}}}\sum_{k\in\mathbb{Z}^{d}}u\left(k+\frac{n}{N}\right)Q_{n}.$
(2.13)

######  Proposition 2.12.

Given $\beta\in(0,1)$ and $g\in\mathcal{S}(\mathbb{R}^{d})$, define

$\displaystyle g_{N}(x):=N^{\frac{d\beta}{2}}g(N^{\beta}(x-x_{0}))e^{2\pi
iNx\cdot\xi_{0}}.$

Next, define $\psi_{N}:=\Pi_{N}g_{N}$, and
$\rho_{N}:=C_{N}^{\prime}\left\lvert\psi_{N}\right\rangle\left\langle\psi_{N}\right\rvert$,
where $C_{N}^{\prime}:=\|\psi_{N}\|_{H_{N}}^{-2}$. Then $\rho_{N}$ has the
semiclassical defect measure $\mu=\delta_{x=x_{0},\xi=\xi_{0}}$, which is a
$(\delta,\theta)$-defect measure for $\delta<\min(\beta,\frac{1-\beta}{d+1})$,
$\theta<\min(\beta-\delta,1-\beta-(d+1)\delta)$.

######  Remark 1.

We can easily see that $C_{N},C_{N}^{\prime}=1+o(1)$. Indeed one has by
Riemann sums:

$C_{N}^{-1}=N^{-d}\sum_{k\in(\mathbb{Z}/N\mathbb{Z})^{d}}\left|g\left(\frac{k}{N}\right)\right|^{2}=1+O(N^{-1}),$

so $C_{N}=1+O(N^{-1})$. Similarly,

$\begin{gathered}(C_{N}^{\prime})^{-1}=N^{-d(\beta-1)}\sum_{m\in(\mathbb{Z}/N\mathbb{Z})^{d}}\left|g\left(N^{\beta}\left(\frac{m}{N}-x_{0}\right)\right)\right|^{2}\\\
=N^{-d}\sum_{m\in(\mathbb{Z}/N\mathbb{Z})^{d}}N^{d\beta}\left|g\left(N^{\beta}\left(\frac{m}{N}-x_{0}\right)\right)\right|^{2}=1+O(N^{-(1-\beta)}),\end{gathered}$

so $C_{N}^{\prime}=1+O(N^{-(1-\beta)})$.

In particular, this shows that the constants $C_{N}$, $C_{N}^{\prime}$ in
Propositions 2.11 and 2.12 can actually be taken to be $1$ without affecting
the value of the defect measure, or even the possible values of
$(\delta,\theta)$. They are only included so the density operators match our
physical motivation of states with norm exactly $1$.

The proofs of Propositions 2.11 and 2.12 are given in Appendix B.2.

###  2.5. Measurement Procedure

We provide a brief background on the quantum mechanics used in this paper. For
a more comprehensive overview, see [BG09] [Hol03]. To model particles in
quantum mechanics, we begin by fixing a Hilbert space $H$, known as the _state
space_. Particles in our model are described by _density operators_.

######  Definition 2.13 (density operator).

A density operator on $H$ is a symmetric, positive semidefinite, trace-class
operator on $H$ with trace $1$.

Quantum particles in this paper are measured to have some value in $\Omega$ (a
locally compact metric space) by a type of quantum instrument using _Kraus
operators_.

######  Definition 2.14 (Kraus operators).

Suppose $\Omega$ is a locally compact metric space with Borel measure $\nu$.
And suppose for $q\in\Omega$, $A_{q}$ is a bounded operator on $H$, $q\mapsto
A_{q}$ is measurable, and

$\displaystyle\int_{\Omega}A_{q}^{*}A_{q}\mathop{}\\!\mathrm{d}\nu(q)=I.$

Then we call $A_{q}$’s _Kraus operators_.

Given a quantum particle described by a density operator $\rho$ and Kraus
operators $A_{q}$ on $\Omega$, the probability of measuring $\rho$ in
$F\subset\Omega$ (a measurable set) is:

$\displaystyle\operatorname{tr}\left(\int_{F}A_{q}\rho
A_{q}^{*}\mathop{}\\!\mathrm{d}\nu(q)\right).$

After measurement within $F$, the state changes to:

$\displaystyle\rho(F):=\frac{\int_{F}A_{q}\rho
A_{q}^{*}\mathop{}\\!\mathrm{d}\nu(q)}{\operatorname{tr}\left(\int_{F}A_{q}\rho
A_{q}^{*}\mathop{}\\!\mathrm{d}\nu(q)\right)}.$

For each measurable set $F\subset\Omega$, we can define
$\mathcal{I}(F):=\rho\mapsto\int_{F}A_{q}\rho
A_{q}^{*}\mathop{}\\!\mathrm{d}\nu(q)$. In this case, $\mathcal{I}$ is an
example of a quantum instrument.

This paper considers a specific class of Kraus operators. The Hilbert space we
work on is the quantized torus $H_{N}$ (recall §2.4). We fix a compact metric
space $\Omega$ with finite Borel measure $\nu(q)$, and let $f_{q}\in
C^{\infty}(\mathbb{T}^{2d})$ satisfy

$\int_{\Omega}\operatorname{Op}_{N}(f_{q})^{*}\operatorname{Op}_{N}(f_{q})\mathop{}\\!\mathrm{d}\nu(q)=Id,$
(2.14)

where $\operatorname{Op}_{N}$ is defined in (2.10). Additionally, assume that
$f_{q}$ are in $S$ (recall Definition 2.1) uniformly in $q$, meaning
$\|\partial^{\alpha}f_{q}\|_{L^{\infty}}\leqslant C_{\alpha}$ with constant
independent of $q$. These $\operatorname{Op}_{N}(f_{q})$ are the Kraus
operators we will consider and we will refer to $f_{q}$ as the symbols of the
Kraus operators.

As a natural example of the above construction, the reader may consider the
example where $\Omega=\mathbb{T}^{d}$, $d\nu(q)=dq$, and
$f_{q}(x,\xi)=f(x-q)$, where
$\int_{\mathbb{T}^{d}}|f(x)|^{2}\mathop{}\\!\mathrm{d}x=1$. Then conjugation
by $\operatorname{Op}_{N}(f_{q})$ corresponds intuitively to an “approximate
position measurement” (see also [Bal+18]), where the approximation approaches
a (nonexistent) “exact” position measurement as $f$ approaches a delta
function.

###  2.6. Evolution Procedure

This paper models quantum particles which are repeatedly measured and evolved.
The previous section discussed the measurement procedure, here we discuss the
evolution procedure.

We consider two different evolution procdures. Particles will either be
evolved by (1) a metaplectic operator or (2) exponentiation of a quantization
of a Hamiltonian on the torus. In the first case, we fix
$M\in\operatorname{Sp}(2d,\mathbb{Z})$ and let $\widehat{M}_{N}$ be the
corresponding metaplectic operator acting on $H_{N}$ (recall §2.2). In the
second case, we fix $p\in C^{\infty}(\mathbb{T}^{2d})$, and let
$P=\operatorname{Op}_{N}(p)$ (recall (2.10)).

Define an operator $\Phi_{N,q}^{*}$ on density operators given by

$\Phi_{N,q}^{*}(\rho):=U\operatorname{Op}_{N}(f_{q})\rho\operatorname{Op}_{N}(f_{q})^{*}U^{-1}$
(2.15)

where $U$ is a unitary operator given either by $U=\widehat{M}_{N}$ or
$U=e^{-2\pi iNP}$. We remark that $\Phi^{*}_{N,q}(\rho)$ is
$U\tilde{\rho}(q)U^{-1}$ where
$\tilde{\rho}(q)=\operatorname{Op}_{N}(f_{q})\rho\operatorname{Op}_{N}(f_{q})^{*}$
is an un-normalized a posteriori state of the instrument determined by Kraus
operators $\operatorname{Op}_{N}(f_{q})$. We stress again that
$\Phi^{*}_{N,q}(\rho)$ is not a density operator, as it is not normalized; the
trace will be taken at the end and interpreted as the probability density of
the trajectory $(q_{0},\dots,q_{n})$.

The evolution of the quantum system by $U$ corresponds to a classical
dynamical system $\phi^{t}:\mathbb{T}^{2d}\to\mathbb{T}^{2d}$. In the case of
evolving by the metaplectic operator $\widehat{M_{N}}$, $\phi^{t}:=M^{t}$
(where $t$ takes integer values). In the case of evolving by $e^{-2\pi
iNt\operatorname{Op}_{N}(p)}$, the corresponding classical dynamical system is
$\phi^{t}:=\exp(tH_{p})$ where $H_{p}$ is the Hamiltonian vector field
generated by $p$.

The quantitative convergence rates of the measures on the quantum and
classical trajectories depend on the Lyapunov exponents (recall Definition
2.3) of these corresponding classical dynamical systems. A system with a
larger Lyapunov exponent is more exponentially sensitive to initial
conditions, and hence seen as “more chaotic.” In particular, completely
integrable systems have Lyapunov exponents equal to zero.

Lyapunov exponents appear crucially in the following bound on symbol seminorms
of a time-evolved symbol.

######  Proposition 2.15.

Suppose $\phi_{t}$ is a smooth dynamical system on $\mathbb{T}^{2d}$ where $t$
either takes values in $\mathbb{Z}_{\geqslant 0}$ or $\mathbb{R}_{\geqslant
0}$ (in which case we assume $\phi_{t}$ is smooth in time), with Lyapunov
exponent $\Gamma$. Supose $a\in C^{\infty}(\mathbb{T}^{2d})$, and define
$a_{t}=a\circ\phi_{t}$. Then for $\alpha\in\mathbb{N}^{2d}$, $\varepsilon>0$,

$\|\partial^{\alpha}a_{t}\|_{L^{\infty}}\leqslant
C_{\alpha,\varepsilon}e^{(\Gamma+\varepsilon)|\alpha|t}\sup_{|\beta|\leqslant|\alpha|}\|\partial^{\beta}a\|_{L^{\infty}}.$

In particular if $a\in S$ and
$t\leqslant\frac{\delta}{\Gamma+\varepsilon}\log\frac{1}{h}$, then $a_{t}\in
S_{\delta}$ with seminorms independent of $t$.

A version of Proposition 2.15 on manifolds is discussed in [AN07, §5.2],
[DG14, Appendix C], and [DJN22, §5.2]. We present a proof of Proposition 2.15
in Appendix C, adapting the arguments of the mentioned references to the
simpler case of the torus.

##  3\. Main Result and Proof

We are now able to provide a quantitative proof of our result.

######  Theorem 1.

Suppose $H_{N}$ is the quantized torus of dimension $2d$,
$\rho=\rho_{N}:H_{N}\to H_{N}$ is a set of density operators depending on $N$
with semiclassical defect measure $\mu$ on $\mathbb{T}^{2d}$, $\Omega$ is a
compact metric space with finite Borel measure $\nu$, $f_{q}\in
C^{\infty}(\mathbb{T}^{2d})$ satisfy (2.14), $P_{\mu}^{(n)}$ is defined by
(1.2), and $\mathbb{P}_{N,\rho}^{(n)}$ is defined by (1.1), with
$\Phi_{N,q}^{*}$ defined by $(\ref{eq:defPhi})$. Then for every
$n\in\mathbb{N}$ we have as $N\to\infty$ the convergence in total variation

$\mathbb{P}_{N,\rho}^{(n)}\xrightarrow{N\to\infty}P_{\mu}^{(n)}.$

Furthermore, we have the quantitative estimate that if $\mu$ is a
$(\delta,\theta)$-defect measure as defined in (2.7) and $\varepsilon>0$, then
there are $C_{0},C_{1}$ independent of $n$ such that if

$n<\frac{\delta\log(N/C_{0})}{\Gamma+\varepsilon},$ (3.1)

then

$\displaystyle\left\|\mathbb{P}_{N,\rho}^{(n)}-P_{\mu}^{(n)}\right\|_{TV}\leqslant
C_{1}N^{-\min(1-2\delta,\theta)}$ (3.2)

where $\Gamma$ is the Lyapunov exponent of $M$ (metaplectic case) or
$\phi_{t}$ (Hamiltonian evolution case).

######  Remark 2.

The condition (3.1) can be recognized as the _Ehrenfest time_ given in (2.4).

###### Proof of Theorem 1.

This proof goes along the lines of [BFF22, Theorem 3.1], but gives
quantitative bounds. The version for $U=e^{-2\pi iNP}$ requires the
quantitative version of Egorov’s theorem given in (2.5).

We first consider when $U=\widehat{M}_{N}.$ We have

$\displaystyle\begin{split}&d\mathbb{P}^{(n)}_{N,\rho}(q_{0},\dots,q_{n})=\operatorname{tr}_{H_{N}}\left(\prod_{i=n}^{0}\left(\widehat{M}_{N}\operatorname{Op}_{N}(f_{q_{i}})\right)\rho\prod_{i=0}^{n}\left(\operatorname{Op}_{N}(f_{q_{i}})^{*}\widehat{M}_{N}^{-1}\right)\right)\mathop{}\\!\mathrm{d}\nu(q)\\\
&=\operatorname{tr}_{H_{N}}\left(\prod_{i=0}^{n}\left(\operatorname{Op}_{N}(f_{q_{i}})^{*}\widehat{M}_{N}^{-1}\right)\prod_{i=n}^{0}\left(\widehat{M}_{N}\operatorname{Op}_{N}(f_{q_{i}})\right)\rho\right)\mathop{}\\!\mathrm{d}\nu(q)\\\
&=\operatorname{tr}_{H_{N}}\left(\prod_{i=0}^{n}\left(\operatorname{Op}_{N}(f_{q_{i}}\circ
M^{i})^{*}\right)\prod_{i=n}^{0}\left(\operatorname{Op}_{N}(f_{q_{i}}\circ
M^{i})\right)\rho\right)\mathop{}\\!\mathrm{d}\nu(q)\\\
&=\left(\operatorname{tr}_{H_{N}}\left(\operatorname{Op}_{N}(|f_{q_{0}}|^{2}\cdots|f_{q_{n}}\circ
M^{n}|^{2})\rho\right)+\operatorname{poly}(n)O(N^{-(1-2\delta)})\right)\mathop{}\\!\mathrm{d}\nu(q)\\\
&=\left(\int_{\mathbb{T}^{2d}}|f_{q_{0}}(\zeta)|^{2}\cdots|f_{q_{n}}(M^{n}\zeta)|^{2}\mathop{}\\!\mathrm{d}\mu+\operatorname{poly}\log(N)(O(N^{-(1-2\delta)})+O(N^{-\theta}))\right)\mathop{}\\!\mathrm{d}\nu(q).\end{split}$
(3.3)

The first three equalities follow from the definition of $\Phi_{N,q_{j}}^{*}$,
commutativity of the trace, the exact Egorov property (2.12) of
$\widehat{M}_{N}$, and the unitarity of $\widehat{M}_{N}^{*}$. We then used
Proposition 2.15 to see that for each $i$, $f\circ M^{i}\in S_{\delta}$, for

$\displaystyle\delta\geqslant\frac{(\Gamma+\varepsilon)n}{\log(N)}$

so that we could use the composition rule for pseudodifferential operators
(2.3) together with (2.11) to get the fourth equality. We lastly used that
$\rho$ has $(\delta,\theta)$-defect measure $\mu$ to get the last equality.
Note that we may remove the $\operatorname{poly}\log(N)$ term by adjusting
$\varepsilon$ and replacing $\delta$ with a slightly smaller
$\delta^{\prime}$.

On the other-hand,

$\displaystyle\mathop{}\\!\mathrm{d}P^{(n)}_{\mu}=\int_{\mathbb{T}^{2d}}|f_{q_{0}}(\zeta)|^{2}\cdots|f_{q_{n}}(M^{n}\zeta)|^{2}\mathop{}\\!\mathrm{d}\mu\mathop{}\\!\mathrm{d}\nu(q).$

We may now integrate in $q$ and apply the dominated convergence theorem (using
that $\|f_{q}\|$ is uniformly bounded in $q$) to get the theorem for the case
that $U=\widehat{M}_{N}$.

When $U=e^{-2\pi iNP}$, the proof is the same, but uses quantitative Egorov’s
theorem (2.5) on the second line of (3.3), to give

$\displaystyle\operatorname{tr}_{H_{N}}\left(\prod_{i=0}^{n}\left(\operatorname{Op}_{N}(f_{q_{i}})^{*}e^{2\pi
iNP}\right)\prod_{i=n}^{0}\left(e^{-2\pi
iNP}\operatorname{Op}_{N}(f_{q_{i}})\right)\rho\right)\mathop{}\\!\mathrm{d}\nu(q)$
$\displaystyle\quad=\left(\operatorname{tr}_{H_{N}}\left(\prod_{i=0}^{n}\left(\operatorname{Op}_{N}(f_{q_{i}}\circ\phi_{i})^{*}\right)\prod_{i=n}^{0}\left(\operatorname{Op}_{N}(f_{q_{i}}\circ\phi_{i})\right)\rho\right)\right)\mathop{}\\!\mathrm{d}\nu(q)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+\left(\operatorname{poly}\log(N)O(N^{-(2-3\delta)})\right)\mathop{}\\!\mathrm{d}\nu(q).$

The $O(N^{-(2-3\delta)})$ error is dominated by the $O(N^{-(1-2\delta)})$
error later on in (3.3), and the remainder of the proof is the same. ∎

##  4\. Numerical Illustrations

Here we present numerical simulations to illustrate the results of this paper.
In these numerics, we simulate the evolution of a state on the quantization of
the 2-dimensional torus. The measurement procedure will use Kraus operators
$f_{q}(x,\xi)=f(x-q)$, with

$\displaystyle f(x)=c\left(\sum_{k\in\mathbb{Z}}\exp\left(\frac{-(x-2\pi
k)^{2}}{2\sigma^{2}}\right)\right)^{1/2}$ (4.1)

with $\sigma>0$ a parameter to be chosen111$\sigma$ can be viewed as the
measurement precision. The smaller the value of $\sigma$, the more precise.
and $c>0$ such that $\int_{\mathbb{T}}|f|^{2}=1$. At each time step, after
observation, we evolve the particle with the metaplectic operator
$\widehat{M}_{N}$ (as defined in §2.4). For now we let $M$ be Arnold’s cat
map,

$\displaystyle M=\begin{pmatrix}2&1\\\ 1&1\end{pmatrix}.$ (4.2)

We also choose the initial state to be
$\rho=\rho(N)=|\psi_{N}\rangle\langle\psi_{N}|$ with

$\displaystyle\psi_{N}=\Pi_{N}\left(\frac{N^{\frac{\beta}{2}}}{\sigma^{\prime}\sqrt{2\pi}}\exp\left(\frac{-((x-x_{0})N^{\beta})^{2}}{2(\sigma^{\prime})^{2}}+2\pi
iNxp_{0}\right)\right)$ (4.3)

with $\Pi_{N}$ defined in (2.13) and some fixed $x_{0},p_{0}\in[0,1]$ and
$\sigma^{\prime}>0$. By Proposition 2.12, $\rho$ has semiclassical defect
measure $\delta_{x=x_{0},\xi=p_{0}}$ when $\beta\in(0,1)$.

Let $Q_{i}$ be the observed location of the state at time $i$. These $Q_{i}$’s
are random variables, and their joint distribution is described by (1.1). To
numerically simulate this distribution, we note that formally
$\mathbb{P}\left(\bigcap_{i=0}^{n}\left\\{Q_{i}=q_{i}\right\\}\right)$ can be
written

$\displaystyle\mathbb{P}\left(Q_{n}=q_{n}\;\middle|\;\bigcap_{i=0}^{n-1}\left\\{Q_{i}=q_{i}\right\\}\right)\mathbb{P}\left(Q_{n-1}=q_{n-1}\;\middle|\;\bigcap_{i=0}^{n-2}\left\\{Q_{i}=q_{i}\right\\}\right)\cdots\mathbb{P}(Q_{0}=q_{0}).$

Therefore to sample $Q_{i}$ from this distribution, we first sample $Q_{0}$
with law $\mathbb{P}(Q_{0}=q_{0})\mathop{}\\!\mathrm{d}\nu(q_{0})$, and get a
value, which we call $\bar{q}_{0}$. Then we sample $Q_{1}$ with law

$\mathbb{P}\left(Q_{1}=q_{1}\;\middle|\;Q_{0}=\bar{q}_{0}\right)\mathop{}\\!\mathrm{d}\nu(q_{1}).$

We continue this process to get values of $\bar{q}_{i}$.

For our choice of initial state and Kraus operators, the probability density
functions of these distributions greatly simplify. Working in the basis of
$H_{N}$ described in §2.4, we write $|\psi_{N}\rangle$ as a vector
$v(0)\in\mathbb{C}^{N}$ (the argument of $v$ will denote the time at which the
state is to be observed). It is a straightforward computation that, with
respect to this basis,

$\operatorname{Op}_{N}(f_{q})=\operatorname{diag}(f(x_{1}-q),f(x_{2}-q),\dots,f(x_{N}-q))$

where $x_{i}=i/N$. To ease notation, let $\widehat{M}_{N}:=U$ and
$\operatorname{Op}_{N}(f_{q}):=A_{q}$. Then, by (1.1),

$\displaystyle\mathbb{P}(Q_{0}=q_{0})$
$\displaystyle=\operatorname{tr}(UA_{q_{0}}v(0)\bar{v}(0)^{t}A_{q_{0}}U^{*})$
$\displaystyle=\sum_{k}(A_{q_{0}})_{k,k}^{2}v(0)_{k}^{2}=\sum_{k}f(x_{k}-q_{0})^{2}v(0)_{k}^{2}.$

Observe that this is the convolution of the probability density functions
$f(x)^{2}$ and $\sum_{k}1_{x_{k}}v(0)_{k}^{2}$ (call random variables with
these distributions $F$ and $V(0)$ respectively). Here $1_{x_{k}}$ is $1$ at
$x_{k}$ and zero everywhere else. Therefore to sample $Q_{0}$, we sample from
the random variable $F+V(0)$ to get $\bar{q}_{0}$.

To sample $Q_{i}$, for $i>1$, we proceed in a similar manner. Assuming we have
already computed that $Q_{j}=\bar{q}_{j}$ for $j<i$, we must sample a random
variable with distribution

$\begin{split}\mathbb{P}&\left(Q_{i}=q_{i}\;\middle|\;\bigcap_{j=1}^{i-1}Q_{j}=\bar{q}_{j}\right)d\nu(q_{i})\\\
&=\mathbb{P}\left(\bigcap_{j=1}^{i-1}Q_{j}=\bar{q}_{j}\right)^{-1}\operatorname{tr}\left(A_{q_{i}}\prod_{j=1}^{i}(UA_{\bar{q}_{i-j}})v(0)v(0)^{t}\prod_{j=0}^{i-1}(A_{\bar{q}_{j}}U^{*})A_{q_{i}}\right)d\nu(q_{i}).\end{split}$
(4.4)

Let:

$\displaystyle v(i)=\prod_{j=1}^{i}(UA_{\bar{q}_{i-j}})v(0),$

so that (4.4) is $c\sum_{k}f(x_{k}-q_{i})^{2}v(i)_{j}^{2}$, with
$c^{-1}=\sum_{k}v(i)_{k}^{2}$. Like before, let $V(i)$ be a random variable
with probability density function $\sum_{k}v(i)_{k}^{2}1_{x_{k}}$. Therefore,
to sample $Q_{i}$, sample from $V(i)+F$. This proves the following
proposition.

######  Proposition 4.1.

Suppose $\rho=|\psi_{N}\rangle\langle\psi_{N}|$ is an initial state (as
defined by (4.3)), $f(x-q)$ are the symbols of Kraus operators (given by
(4.1)), and $n>0$. Then a trajectory of a quantum particle
$\mathbb{P}^{(n)}_{N,\rho}$, described by (1.1), can be sampled by applying
Algorithm 1 by storing the values $q_{0},q_{1},\dots,q_{n}$ (representing the
observed locations of the particle at each instance of time).

Algorithm 1 Quantum Particle Evolution

1:$\textbf{v}\leftarrow|\psi_{N}\rangle$

2:while $i\leqslant n$ do

3: $q_{i}\leftarrow$ sampled random number from distribution
$\sum_{k}1_{x_{k}}\textbf{v}_{k}^{2}$

4: $g\leftarrow$ sampled random number from Gaussian distribution of mean $0$,
variance $\sigma^{2}$

5: $q_{i}\leftarrow q_{i}+g\pmod{1}$

6:
$\textbf{v}\leftarrow\operatorname{diag}(f(x_{1}-q_{i}),\dots,f(x_{N}-q_{i}))\textbf{v}$

7: $\textbf{v}\leftarrow\hat{M}_{N}\textbf{v}$

8: $\textbf{v}\leftarrow\left\|\textbf{v}\right\|^{-1}\textbf{v}$

9: $i\leftarrow i+1$

10:end while

In this algorithm, we use the fact that in the fixed basis of $H_{N}$, the
components of $\widehat{M}_{N}$ are given by

$\displaystyle(\widehat{M})_{j,k}=\frac{1}{\sqrt{N}}\exp\left(\pi
i(2k^{2}-2jk+j^{2})/N\right),$ (4.5)

as computed in [DJ21].

Theorem 1 states that this distribution, $\mathbb{P}^{(n)}_{N,\rho}$, will
converge (as $N\to\infty$) to $P_{\mu}^{(n)}$. By Proposition 2.12, the semi-
classical defect measure $\mu$ for our chosen $\rho$ is
$\delta_{x=x_{0},\xi=p_{0}}$. We can then compute

$\displaystyle
P_{\mu}^{(n)}(q_{0},\dots,q_{n})=\prod_{k=0}^{n}\left|f\left(M^{k}\begin{pmatrix}x_{0}\\\
p_{0}\end{pmatrix}-q_{0}\right)\right|^{2}d(q_{0}).$

Therefore the distribution of $q_{k}$ according to $P_{\mu}^{(n)}$ is a
Gaussian centered at the position component of $M^{k}(x_{0}\ \xi_{0})^{t}$,
with variance $\sigma^{2}$.

Now we present several numerical simulations.

First, Figure 2 demonstrates the evolution procedure. An initial state is
given by (4.3) with $N=100$, $\beta=0.5$, $\sigma^{\prime}=0.1$, $x_{0}=0.5$,
and $p_{0}=0$. The symbols of the Kraus operators are given by $f(x-q)$ where
$f$ is given by (4.1), with $\sigma=0.1$. The absolute value squared of the
components of $\psi$ (with respect to the basis of $H_{N}$ previously
described) is plotted in blue. We then apply Algorithm 1 to get a value
$q_{1}$ which is plotted as an orange dot. The state is then changed by
observation (by step 6 of Algorithm 1), which is plotted in the next plot.
Then the particle is evolved by applying $\widehat{M}$, where $M$ is the cat
map (4.2) to get a new states. The absolute value squared of the components of
the new state are plotted in the next plot, and the process is repeated for 3
time steps.

Figure 2. Evolution Procedure Here we simulate the trajectory of a qunatum
particle for three time steps under the discussed evolution procedure. The
specifics of this simulation are discussed in §4.

Figure 3 compares an approximation of the marginal distributions of
$\mathbb{P}_{N,\rho}^{(n)}$ against $P_{\mu}^{(n)}$ for $n=4$. For this
simulation, the initial state is given by (4.3) with $N=2000$, $\beta=0.5$,
$\sigma^{\prime}=0.1$, $x_{0}=0.1$, and $p_{0}=0.1$. The symbols of the Kraus
operators are given by $f(x-q)$ where $f$ is given by (4.1), with
$\sigma=0.1$. We then run the evolution procedure 2000 times (we call this the
number of trials), saving the observed positions for each time step. The
relative frequency of the recorded positions are plotted as a histrogram in
blue. As the number of trials goes to infinity, these histograms should
converge to the marginal distributions of $\mathbb{P}^{(n)}_{N,\rho}$ at each
time step. For each instance in time, we plot the marginal distribution of
$P_{\mu}^{(n)}$ as an orange curve. This paper’s main result states that the
two distributions should agree until the Ehrenfest time. For the cat map, the
Lyapunov exponent is $\Gamma\approx.96$, so the Ehrenfest time is
approximately $\log(N)/(2\Gamma)\approx 3.95$.

Figure 3. Marginal distributions of $\bm{\mathbb{P}_{N,\rho}^{(n)}}$ vs
$\bm{P_{\mu}^{(n)}}$: We preform Algorithm 1 (observation then evolution by
the quantum cat map). We plot the relative frequencies of the observed
positions (approximating the marginal distribution of
${\mathbb{P}_{N,\rho}^{(n)}}$ against the marginal distributions of
${P_{\mu}^{(n)}}$). See §4 for more details.

It is apparent that the convergence of $\mathbb{P}_{N,\rho}^{(n)}$ to
$P_{\mu}^{(n)}$ as $N\to\infty$ is relatively slow. As seen in Figure 3, after
only $4$ time steps, the distribution of observed positions looks uniform.
This should be expected, as Theorem 1 only guarantees (3.2) for
$N\gtrsim\exp((\Gamma+\varepsilon)n/\delta)$. For the cat map,
$\Gamma=\log((3+\sqrt{5})/2)$, and in this case $\delta<1/4$. Then, (3.2)
requires (roughly) $N\gtrsim 50^{n}$. These estimates are far from sharp, but
give some sense of the exponentially slow rate of convergence as seen in
Figure 3.

In Figure 4, we run the same simulation as in Figure 3 but vary $N$ from $100$
to $4000$. For each value of $N$, we compute the total variation of the
(approximate) marginal distribution of $\mathbb{P}_{N,\rho}^{(time)}$ against
$P_{\mu}^{(time)}$ for times between $1$ and $6$. Observe that increasing $N$
leads to smaller total variation at an exponentially slow rate.

Figure 4. Convergence of marginal distributions of
$\bm{\mathbb{P}_{N,\rho}^{(n)}\to P_{\mu}^{(n)}}$: We run the same simulation
as in Figure 3, but vary $N$ from $100$ to $4000$ and compare
$\mathbb{P}_{N,\rho}^{(n)}$ to $P_{\mu}^{(n)}$. See §4 for more details.

Next we show how changing the Lyapunov exponent changes the rate of
convergence of $\mathbb{P}_{N,\rho}^{(n)}$ to $P_{\mu}^{(n)}$. In Figure 5,
the same evolution procedure is preformed for $4$ time steps but with $3$
different quantized symplectic matrices. In each case we plot the relative
frequencies of observed positions (approximating the marginal distributions of
$\mathbb{P}^{(n)}_{N,\rho}$) with a plot of the distribution of
$P_{\mu}^{(n)}$. In all cases, we chose $N=3000$, $\beta=0.5$,
$\sigma^{\prime}=\sigma=0.1$, $x_{0}=0.3$, $p_{0}=0.9$, and simulated 500
trajectories. The symplectic matrices chosen were:

$\displaystyle M_{1}=\begin{pmatrix}2&1\\\ 1&1\end{pmatrix},$ $\displaystyle
M_{2}=\begin{pmatrix}2&1\\\ 3&2\end{pmatrix},$ $\displaystyle
M_{3}=\begin{pmatrix}4&1\\\ 15&4\end{pmatrix},$ (4.6)

which have Lyapunov exponent roughly $0.962,1.317$, and $2.063$ respectively.
Computing the Ehrenfest time as $\log(N)/(2\Gamma)$, then the three evolution
procedures have Ehrenfest time approximately $4.16$, $3.04$, and $1.94$
respectively. This roughly agrees with the numerics provided. After only a few
time steps, $\mathbb{P}_{N,\rho}^{(n)}$ and $P_{\mu}^{(n)}$ disagree. Because
the Ehrenfest time grows logarithmically with $N$, and the computations grow
like $N^{2}$, it is not feasible to simulate numerics for which the two
measures agree for many more time steps.

Figure 5. Changing the Lyapunov exponent: We present the same numerics as in
Figure 3 but evolve by quantizations of three different matrices given by
(4.6) with 3 different Lyapunov exponents. See §4 for more details.

In Figure 6 we replace the evolution of the particle by $\widehat{M}_{N}$ at
each time step, with evolution by $\exp(-2\pi itN\operatorname{Op}_{N}(a))$,
where $a=\cos(2\pi x)+\cos(2\pi\xi)$, and $t=0.05$ – which should be thought
of as the interval of time between observations. In this case, $P_{\mu}^{(n)}$
is a Gaussian whose mean is computed by evolving $(x_{0},p_{0})$ along the
Hamiltonian flow generated by $a(x,\xi)$ for time $tn$. Note that the
convergence of $\mathbb{P}_{N,\rho}^{(n)}$ to $P_{\mu}^{(n)}$ is much more
rapid in terms of $N$ (as long as $t$ is small). We chose the following
parameters: $N=1000$, $\beta=0.5$, $\sigma^{\prime}=\sigma=0.1$, $x_{0}=0.4$,
$p_{0}=0.7$, with $1000$ trials.

Figure 6. Evolution of $\bm{\cos(2\pi x)+\cos(2\pi\xi)}$: We present the same
numerics as in Figure 3 but instead of evolving the quantum state by applying
the quantum cat map, we multiply our state by $\exp(-2\pi
itN\operatorname{Op}_{N}(\cos(2\pi x)+\cos(2\pi\xi)))$. See §4 for more
details.

Lastly, in Figure 7, we present numerics for the evolution of quantum
particles, where the cat map is replaced by the matrix:

$\displaystyle E:=\begin{pmatrix}-1&1\\\ -1&0\end{pmatrix}$ (4.7)

which is an example of an elliptic matrix, having Lyapunov exponent equal to
zero (in fact, $E^{3}=I$). The distribution of observed locations of evolved
particles is presented, and stays in agreement with the classical distribution
for significantly more steps. The parameters chosen here are $N=1000$,
$\beta=0.5$, $\sigma=\sigma^{\prime}=0.1$, $x_{0}=0.1$, $p_{0}=0.9$, with
$1000$ trials. In this case $\mathbb{P}_{N,\rho}^{(n)}$ and $P_{\mu}^{(n)}$
agree for more time steps than the evolution of other symplectic maps. The
Ehrenfest time will subtlety depend on the $\varepsilon$ in (3.1).

Figure 7. Evolution by Elliptic Matrix: We present the same numerics as in
Figure 3 but instead of evolving by the quantum cat map, we evolve by the
quantization of the elliptic matrix $E$ which has Lyapunov exponent zero. See
§4 for more details.

The figure on the first page (Figure 1) is simulated in the following way. We
chose an inital state $\rho$ given in (4.3) with $N=500$,
$\sigma^{\prime}=0.1$, $\beta=0.1$, $x_{0}=0.4$, $p_{0}=0.7$. We indirectly
measured the particle location using Kraus operators with symbol given by
(4.1) with $\sigma=0.1$. Between each measurement, the particle is evolved by
$\exp(-2\pi itNP)$ with $P=\operatorname{Op}_{N}(\cos(2\pi x)+\cos(2\pi\xi))$
and $t=0.01$. This procedure is done for $100$ time steps to sample a single
trajectory with law $\mathbb{P}^{(100)}_{N,\rho}$. We then repeat this $200$
times to get $200$ different trajectories, which are plotted on top of each
other.

##  5\. Acknowledgements

The authors are grateful to Maciej Zworski and Martin Fraas for helpful
discussions as well as two anonymous referees for numerous helpful corrections
and suggestions. The authors also thank Semyon Dyatlov for providing a
reference to proving Proposition 2.15 and to Ian Gleason and Oliver Edtmair
for fruitful discussions in writing up the proof in the appendix. This paper
is based upon work supported by National Science Foundation grant DMS-1952939.
The second author gratefully acknowledges support from the National Science
Foundation Graduate Research Fellowship under grant DGE-1650114.

##  Appendix A Uniformity of Defect Measures

Proposition 2.6 motivates the definition of defect measures, but this
definition allows for some undesirable consequences, which show up in the
following pathological example. Let $f\in C^{\infty}_{0}(\mathbb{R}^{d})$ with
$\|f\|_{L^{2}(\mathbb{R}^{d})}=1$, and define
$\psi_{h}(x)=f\left(x+h^{-1}\right).$ It is easy to show that $\psi_{h}$ has
defect measure equal to the zero measure, corresponding intuitively to the
mass having “escaped to infinity.” Indeed, one has for any $a\in
C^{\infty}_{0}(T^{*}\mathbb{R}^{d})$ supported in a ball of radius $R$ in
$\mathbb{R}^{2d}\cong T^{*}\mathbb{R}^{d}$, $h<(2R)^{-1}$, and $\chi\in
C^{\infty}(\mathbb{R}^{d})$ such that $\chi(x)=1$ on $|x|\geqslant 2R$,
$\chi=0$ on $|x|\leqslant R$ that

$\operatorname{Op}_{h}(a)\psi_{h}=\operatorname{Op}_{h}(a)\operatorname{Op}_{h}(\chi)\psi_{h}=O(h^{\infty})$
(A.1)

which gives

$\lim_{h\to 0}\langle\operatorname{Op}_{h}(a)\psi_{h},\psi_{h}\rangle=0.$
(A.2)

Unfortunately, there is no way to quantify the rate of decay of (A.2), as it
depends on not just the symbol norms of $a$ but also the location of its
support. In addition, the resulting measure is not a probability measure,
which defies our intuition of the defect measure representing the classical
location of the particle in phase space. We are therefore motivated to make a
more restrictive definition of defect measures to prohibit examples such as
(A.1). Call $\mu$ a _uniform semiclassical defect measure_ of $\rho_{h}$ if
for any $a\in S$,

$\lim_{h\to
0}\operatorname{tr}\left(\rho_{h}\operatorname{Op}_{h}(a)\right)=\int_{T^{*}\mathbb{R}^{d}}a\mathop{}\\!\mathrm{d}\mu.$
(A.3)

The reason for the name is given in the following proposition. We use the
standard multi-index notation. For $a\in C^{\infty}(T^{*}\mathbb{R}^{d})$ and
$\alpha\in\mathbb{N}^{2d}$, we let

$\displaystyle
a^{(\alpha)}(x)=\left(\prod_{j=1}^{2d}\partial_{j}^{\alpha_{j}}\right)a(x).$

######  Proposition A.1.

Let $\rho_{h}$ be a set of density operators with defect measure $\mu$. The
following are equivalent.

  1. (1)

$\mu$ is a uniform defect measure.

  2. (2)

For all $\varepsilon>0$, there is an $h_{0}(\varepsilon)>0$ such that if
$0<h<h_{0}$ and $a\in C^{\infty}_{0}(T^{*}\mathbb{R}^{d})$ with
$\|a^{(\alpha)}\|_{L^{\infty}}\leqslant 1$ for all $|\alpha|<K(d)$ then

$\left|\operatorname{tr}\left(\rho_{h}\operatorname{Op}_{h}(a)\right)-\int_{T^{*}\mathbb{R}^{d}}a\mathop{}\\!\mathrm{d}\mu\right|\leqslant\varepsilon.$

  3. (3)

For all $\varepsilon>0$, there is an $h_{0}(\varepsilon)>0$ such that if
$0<h<h_{0}$ and $a\in S$ with $\|a^{(\alpha)}\|_{L^{\infty}}\leqslant 1$ for
all $|\alpha|<K(d)$ then

$\left|\operatorname{tr}\left(\rho_{h}\operatorname{Op}_{h}(a)\right)-\int_{T^{*}\mathbb{R}^{d}}a\mathop{}\\!\mathrm{d}\mu\right|\leqslant\varepsilon.$

  4. (4)

$\mu$ is a probability measure.

###### Proof.

We trivially see that $(3)\implies(2)$ and $(3)\implies(1)$.

To show $(2)\implies(3)$, let $a\in S$, and let $\chi_{k}\in
C^{\infty}_{0}(T^{*}\mathbb{R}^{d})$ be smooth partition of unity of
$T^{*}\mathbb{R}^{d}$, chosen as in [Zwo12, Theorem 4.23], with the first
$K(d)$ derivatives chosen small enough. By the Cotlar-Stein-Knapp lemma, we
have the limit in the strong operator topology:

$\lim_{K\to\infty}\left(\sum_{k=0}^{K}\operatorname{Op}_{h}(\chi_{k}a)\right)\psi=\operatorname{Op}_{h}(a)\psi$

for every $\psi\in L^{2}(\mathbb{R}^{d})$. We need the following brief lemma.

######  Lemma A.2.

Let $A_{K}$ be a sequence of uniformly bounded operators converging to $A$ in
the strong topology, and let $\rho$ be self-adjoint and trace class. Then

$\lim_{K\to\infty}\operatorname{tr}(\rho A_{K})=\operatorname{tr}(\rho A).$

###### Proof of Lemma A.2.

By subtracting we may assume $A=0$, and without loss of generality let
$\sup_{K}\|A_{K}\|=1$. Let $\psi_{j}$ be an orthonormal basis of eigenvectors
of the compact operator $\rho$, so

$\rho=\sum_{j}\lambda_{j}\left\lvert\psi_{j}\right\rangle\left\langle\psi_{j}\right\rvert$

with $\lambda_{j}\to 0$. as $j\to\infty$. Let $\varepsilon>0$, and choose $J$
such that $\lambda_{j}<\varepsilon$ for $j>J$. Then

$|\operatorname{tr}(\rho A_{K})|=\left|\sum_{j\leqslant J}\lambda_{j}\langle
A_{K}\psi_{j},\psi_{j}\rangle+\sum_{j\leqslant J}\lambda_{j}\langle
A_{K}\psi_{j},\psi_{j}\rangle\right|\leqslant\sum_{j\leqslant
J}\left|\lambda_{j}\langle
A_{K}\psi_{j},\psi_{j}\rangle\right|+\varepsilon\xrightarrow{K\to\infty}\varepsilon$

which shows $\operatorname{tr}(\rho A_{K})\xrightarrow{K\to\infty}0$ as
desired. ∎

By Lemma A.2, we see that for fixed $h$,

$\lim_{K\to\infty}\operatorname{tr}\left(\rho_{h}\sum_{k=0}^{K}\operatorname{Op}_{h}(\chi_{k}a)\right)=\operatorname{tr}(\rho_{h}\operatorname{Op}_{h}(a))$

and similarly as $\mu$ is a finite measure

$\lim_{K\to\infty}\int_{T^{*}\mathbb{R}^{d}}\sum_{k=0}^{K}\chi_{k}a\mathop{}\\!\mathrm{d}\mu=\int_{T^{*}\mathbb{R}^{d}}a\mathop{}\\!\mathrm{d}\mu.$

Then letting $\varepsilon>0$ and picking the $h_{0}(\varepsilon)$ from part
$(2)$ we get

$\left|\operatorname{tr}\left(\rho_{h}\operatorname{Op}_{h}(a)\right)-\int_{T^{*}\mathbb{R}^{d}}a\mathop{}\\!\mathrm{d}\mu\right|=\lim_{K\to\infty}\left|\operatorname{tr}\left(\rho_{h}\sum_{k=0}^{K}\operatorname{Op}_{h}(\chi_{k}a)\right)-\int_{T^{*}\mathbb{R}^{d}}\sum_{k=0}^{K}\chi_{k}a\mathop{}\\!\mathrm{d}\mu\right|\leqslant\varepsilon$

To show $(1)\implies(3)$, let $a\in S$, and let $\chi\in
C^{\infty}_{0}(T^{*}\mathbb{R}^{d})$ be supported in a ball of radius $R$,
such that $\mu(T^{*}\mathbb{R}^{d}\setminus B_{R}(0))\leqslant\varepsilon/8$.
We also note that $\chi$ may be chosen so its first $K(d)$ derivatives are
small (possibly making $R$ larger). Then we have that

$\operatorname{tr}(\operatorname{Op}_{h}(a)\operatorname{Op}_{h}(1-\chi)\rho_{h})\leqslant\|\operatorname{Op}_{h}(a)\|\operatorname{tr}(\operatorname{Op}_{h}(1-\chi)\rho_{h})\leqslant
C\operatorname{tr}(\operatorname{Op}_{h}(1-\chi)\rho_{h})$

for $C$ depending on the first $K(d)$ derivatives of $a$. Next we apply (A.3)
with the symbol $1-\chi$ (which is independent of $a$) to get that

$\displaystyle
C\operatorname{tr}(\operatorname{Op}_{h}(1-\chi)\rho_{h})=C\varepsilon/8+o(1)$

as $h\to 0$. So that there exists $h_{0}>0$ such that for $0<h<h_{0}$ we have
$C\operatorname{tr}(\operatorname{Op}_{h}(1-\chi)\rho_{h})<\varepsilon/4$.
With this, the triangle inequality, and symbol calculus, we have for
$h<h_{0}(\varepsilon)$

$\displaystyle\begin{split}\left|\operatorname{tr}\left(\rho_{h}\operatorname{Op}_{h}(a)\right)-\int_{T^{*}\mathbb{R}^{d}}a\mathop{}\\!\mathrm{d}\mu\right|&\leqslant\left|\operatorname{tr}\left(\rho_{h}\operatorname{Op}_{h}(a)\operatorname{Op}_{h}(\chi)\right)-\int_{T^{*}\mathbb{R}^{d}}a\chi\mathop{}\\!\mathrm{d}\mu\right|+\frac{\varepsilon}{2}\\\
&\leqslant\left|\operatorname{tr}\left(\rho_{h}\operatorname{Op}_{h}(a\chi)\right)-\int_{T^{*}\mathbb{R}^{d}}a\chi\mathop{}\\!\mathrm{d}\mu\right|+\frac{3\varepsilon}{4}\end{split}$
(A.4)

where for the last inequality we have chosen possibly smaller
$h_{0}(\varepsilon)$ and used the fact that $a$ has bounded derivatives.

We now prove $(3)$ using a compactness argument. Observe that by Calderón-
Vaillancourt, if (A.3) holds for $a\in S$, it must also hold for any $a\in
C^{K_{0}(d)}(T^{*}\mathbb{R}^{d})$ for some sufficiently large $K_{0}$. Given
$h_{j}$, let $X_{j}$ be the set of $b\in C^{K_{0}(d)}(B_{R}(0))$ such that

$\left|\operatorname{tr}(\rho_{h}\operatorname{Op}_{h}(b\chi))-\int_{B_{R}(0)}b\chi\mathop{}\\!\mathrm{d}\mu\right|<\frac{\varepsilon}{4}$
(A.5)

for all $h<h_{j}$. Then for $K_{0}(d)$ large enough, condition (A.5) makes
$X_{j}$ to be an open set in $C^{K_{0}(d)}(B_{R}(0))$, so letting $h_{j}\to
0$, gives $(X_{j})_{j\in\mathbb{N}}$ to be an open cover of
$C^{K_{0}(d)}(B_{R}(0))$. Let

$Y=\\{b\in
C^{\infty}(B_{R}(0)):\|b^{(\alpha)}\|_{L^{\infty}(B_{R}(0))}\leqslant
1\quad\forall\,\,|\alpha|\leqslant K_{0}(d)+1\\},$

and let $\overline{Y}$ denote the closure of $Y$ in $C^{K_{0}(d)}(B_{R}(0))$.
By the Arzela-Ascoli theorem, $\overline{Y}$ is compact in the topology of
$C^{K_{0}(d)}(B_{R}(0))$, so there is a finite subset of the $X_{j}^{\prime}s$
that covers $\overline{Y}$, and hence an $h_{0}$ such that for $h<h_{0}$

$\left|\operatorname{tr}(\rho_{h}\operatorname{Op}_{h}(b))-\int_{B_{R}(0)}b\mathop{}\\!\mathrm{d}\mu\right|<\frac{\varepsilon}{4}$

holds for all $b$ with $\|b^{(\alpha)}\|_{L^{\infty}}\leqslant 1$ for
$|\alpha|\leqslant K_{0}(d)+1$. Letting $K(d)=K_{0}(d)+1$, and
$b=a|_{B_{R}(0)}$ in (A.4) completes the proof of $(3)$.

To show $(3)\implies(4)$, we simply see that

$\int_{T^{*}\mathbb{R}^{d}}\mathop{}\\!\mathrm{d}\mu=\lim_{h\to
0}\operatorname{tr}(\rho_{h}\operatorname{Op}_{h}(1))=\lim_{h\to
0}\operatorname{tr}\rho_{h}=1.$

Here we use that the function $f(x)=1$ is in the symbol class $S$.

Finally, we show $(4)\implies(3)$. Let $\chi\in
C^{\infty}_{0}(T^{*}\mathbb{R}^{d})$, so

$\operatorname{tr}\left(\rho_{h}\operatorname{Op}_{h}(1-\chi)\right)-\int_{T^{*}\mathbb{R}^{d}}1-\chi\mathop{}\\!\mathrm{d}\mu=-\left(\operatorname{tr}\left(\rho_{h}\operatorname{Op}_{h}(\chi)\right)-\int_{T^{*}\mathbb{R}^{d}}\chi\mathop{}\\!\mathrm{d}\mu\right)\to
0$

as $h\to 0$, so in particular (A.3) holds in the special case of the
noncompact symbol $1-\chi$. But that was the only symbol used in the proof of
$(1)\implies(3)$, so in particular shows $(4)\implies(3)$ as well. This shows
all equivalences, and hence completes the proof. ∎

##  Appendix B Coherent States

In this appendix, we prove Propositions 2.9, 2.11, and 2.12, which give
examples of $(\delta,\theta)$-defect measures.

###  B.1. Coherent States on $\mathbb{R}^{d}$

We prove Proposition 2.9. The proof is essentially the same as [Zwo12, §5.1
Examples 1 and 2], but is quantitative and keeps track of errors.

###### Proof of Proposition 2.9.

We first consider the case when $\beta=0$. In this case, we can without loss
of generality assume $x_{0}=0$, by absorbing it into the definition of $g$.
Then

$\psi_{h}(x)=g(x)e^{\frac{i}{h}\langle x,\xi_{0}\rangle},$

so for $a\in S_{\delta}\cap C_{0}^{\infty}(T^{*}\mathbb{R}^{d})$:

$\displaystyle\operatorname{tr}(\rho_{h}\operatorname{Op}_{h}(a))$
$\displaystyle=\langle\operatorname{Op}_{h}(a)\psi_{h},\psi_{h}\rangle$
$\displaystyle=\int\frac{1}{(2\pi h)^{d}}\left(\iint e^{\frac{i}{h}\langle
x-y,\xi-\xi_{0}\rangle}g(y)\,a\left(\frac{x+y}{2},\xi\right)\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}\xi\right)\overline{g(x)}\mathop{}\\!\mathrm{d}x$
$\displaystyle=\int\frac{1}{(2\pi h)^{d}}\left(\iint e^{\frac{i}{h}\langle
y,\xi\rangle}g(x-y)\,a\left(x-\frac{y}{2},\xi+\xi_{0}\right)\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}\xi\right)\overline{g(x)}\mathop{}\\!\mathrm{d}x.$

We evaluate the inner double integral using the explicit form of stationary
phase given in [Zwo12, Theorem 3.17].

$\begin{gathered}\iint e^{\frac{i}{h}\langle
y,\xi\rangle}g(x-y)\,a\left(x-\frac{y}{2},\xi+\xi_{0}\right)\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}\xi\\\
\sim(2\pi h)^{d}\left(\sum_{k=0}^{\infty}\frac{h^{k}}{i^{k}k!}\langle
D_{y},D_{\xi}\rangle^{k}\left.g(x-\cdot_{y})a\left(x-\frac{\cdot_{y}}{2},\cdot_{\xi}+\xi_{0}\right)\right|_{y=0,\xi=0}\right)\\\
=(2\pi h)^{d}\left(g(x)a(x,\xi_{0})+O(h^{1-2\delta})\right)\end{gathered}$

with $\sim$ referring to asymptotic summation and the last equality coming
from $a$ being an element of $S_{\delta}$. Then

$\left|\operatorname{tr}(\rho_{h}\operatorname{Op}_{h}(a))-\int|g(x)|^{2}a(x,\xi_{0})\mathop{}\\!\mathrm{d}x\right|\leqslant
O(h^{1-2\delta})\int|g(x)|\mathop{}\\!\mathrm{d}x=O(h^{1-2\delta})$

which gives the result for $\beta=0$. The case of $\beta=1$ follows from
taking the semiclassical Fourier transform.

We now treat the case when $0<\beta<1$. Here we have

$\psi_{h}(x)=h^{-\frac{d\beta}{2}}g\left(\frac{x-x_{0}}{h^{\beta}}\right)e^{\frac{i}{h}\langle
x,\xi_{0}\rangle}.$

We again have that
$\operatorname{tr}(\rho_{h}\operatorname{Op}_{h}(a))=\langle\operatorname{Op}_{h}(a)\psi_{h},\psi_{h}\rangle$.
We now approximate $\operatorname{Op}_{h}(a)$ by its left quantization defined
as:

$\operatorname{Op}_{h}^{L}(a)u(x):=\frac{1}{(2\pi h)^{d}}\iint
e^{\frac{i}{h}\langle
x-y,\xi\rangle}a(x,\xi)u(y)\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}\xi.$

By [Zwo12, Theorem 4.13],
$\operatorname{Op}_{h}(a)=\operatorname{Op}_{h}^{L}(a)+O(h^{1-2\delta})_{L^{2}(\mathbb{R}^{d})\to
L^{2}(\mathbb{R}^{d})}$. Therefore

$\displaystyle\operatorname{tr}(\rho_{h}\operatorname{Op}_{h}(a))=\langle\operatorname{Op}_{h}^{L}(a)\psi_{h},\psi_{h}\rangle+O(h^{1-2\delta})$
$\displaystyle=\frac{1}{(2\pi
h)^{d}}\int\overline{g\left(\frac{x-x_{0}}{h^{\beta}}\right)}\left(\iint
h^{-d\beta}g\left(\frac{y-x_{0}}{h^{\beta}}\right)e^{\frac{i}{h}\langle
x-y,\xi-\xi_{0}\rangle}a(x,\xi)\mathop{}\\!\mathrm{d}y\mathop{}\\!\mathrm{d}\xi\right)\mathop{}\\!\mathrm{d}x$
$\displaystyle\quad+O(h^{1-2\delta})$ $\displaystyle=\frac{1}{(2\pi
h)^{d}}\int\overline{g\left(\frac{x-x_{0}}{h^{\beta}}\right)}\int
e^{\frac{i}{h}\langle x-x_{0},\xi\rangle}a(x,\xi+\xi_{0})\left(\int
h^{-d\beta}g\left(\frac{y}{h^{\beta}}\right)e^{-\frac{i}{h}\langle
y,\xi\rangle}\mathop{}\\!\mathrm{d}y\right)\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}x$
$\displaystyle\quad+O(h^{1-2\delta})$ $\displaystyle=\frac{1}{(2\pi
h)^{d}}\iint\overline{g\left(\frac{x}{h^{\beta}}\right)}e^{\frac{i}{h}\langle
x,\xi\rangle}a(x+x_{0},\xi+\xi_{0})\hat{g}\left(\frac{\xi}{h^{1-\beta}}\right)\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}x+O(h^{1-2\delta})$
$\displaystyle=\frac{1}{(2\pi)^{d}}\iint\overline{g(x)}\hat{g}(\xi)e^{i\langle
x,\xi\rangle}a\left(h^{\beta}x+x_{0},h^{1-\beta}\xi+\xi_{0}\right)\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}x+O(h^{1-2\delta}).$

Because $\|g\|_{L^{2}(\mathbb{R}^{d})}=1$, we have by the Fourier inversion
formula that:

$\displaystyle\frac{1}{(2\pi)^{d}}\iint\overline{g(x)}\hat{g}(\xi)e^{i\langle
x,\xi\rangle}\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}x=1$

so that

$\displaystyle|\operatorname{tr}(\rho_{h}\operatorname{Op}_{h}(a))-a(x_{0},\xi_{0})|$
$\displaystyle\leqslant\left|\frac{1}{(2\pi)^{d}}\iint\overline{g(x)}\hat{g}(\xi)e^{i\langle
x,\xi\rangle}(a\left(h^{\beta}x+x_{0},h^{1-\beta}\xi+\xi_{0}\right)-a(x_{0},\xi_{0}))\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}x\right|+O(h^{1-2\delta}).$

By the mean value theorem and the fact that $a\in S_{\delta}\cap
C_{0}^{\infty}(T^{*}\mathbb{R}^{d})$, we have for $x,\xi\in\mathbb{R}^{d}$,

$\displaystyle|a\left(h^{\beta}x+x_{0},h^{1-\beta}\xi+\xi_{0}\right)-a(x_{0},\xi_{0})|$
$\displaystyle\leqslant(h^{\beta}|x|+h^{1-\beta}|\xi|)\|\nabla
a\|_{L^{\infty}}$ $\displaystyle\leqslant
C(h^{\beta}|x|+h^{1-\beta}|\xi|)h^{-\delta}.$

So, using that $g$ is Schwartz,

$\displaystyle|\operatorname{tr}(\rho_{h}\operatorname{Op}_{h}(a))-a(x_{0},\xi_{0})|$
$\displaystyle\leqslant
C\iint|g(x)||\hat{g}(\xi)|x||\xi|((h^{\beta}|x|+h^{1-\beta}|\xi|)h^{-\delta})\mathop{}\\!\mathrm{d}\xi\mathop{}\\!\mathrm{d}x$
$\displaystyle=O(h^{\beta-\delta})+O(h^{1-\beta-\delta})+O(h^{1-2\delta}).$

Noting that $1-2\delta=\beta-\delta+1-\beta-\delta$ shows the
$O(h^{1-2\delta})$ term to be superfluous and gives the desired result. ∎

###  B.2. Coherent States on the Quantized Torus

We provide full proofs of Propositions 2.11 and 2.12, though we do not claim
our parameters to be optimal.

###### Proof of Proposition 2.11.

For this proof, we work in the orthonormal basis $(Q_{n})$. Then for $a\in
S_{\delta}(\mathbb{T}^{2d})$, $\operatorname{Op}_{N}(a)$ has a representation
as a matrix $A$ with

$A_{mj}=\sum_{l,r\in\mathbb{Z}^{d}}\hat{a}(l,j-m-rN)(-1)^{\langle
r,l\rangle}e^{\pi i\frac{\langle j+m,l\rangle}{N}}.$ (B.1)

Let $M=\lfloor N^{\gamma}\rfloor$ for $\gamma\in(\delta,1)$. We see that if
$|j-m|>M$ (in distance $\operatorname{mod}N$), then for any
$k\in\mathbb{N}_{>d}$

$\displaystyle\begin{split}|A_{mj}|&\leqslant\sum_{l,r\in\mathbb{Z}^{d}}|\hat{a}(l,j-m-
rN)|\\\ &\leqslant C\sum_{l,r\in\mathbb{Z}^{d}}\langle
l\rangle^{-(d+1)}\|(\partial_{x}^{d+1}+1)\partial_{\xi}^{k}a\|_{L^{\infty}}|j-m-
rN|^{-k}\\\ &\leqslant CN^{(d+1+k)\delta}\sum_{r\in\mathbb{Z}^{d}}|j-m-
rN|^{-k}.\end{split}$ (B.2)

Note that there exists a large constant $M_{0}>0$ (independent of $N$) such
that

$\displaystyle\sum_{\begin{subarray}{c}r\in\mathbb{Z}^{d}\\\
|r|>M_{0}\end{subarray}}|j-m-rN|^{-k}\leqslant 1.$

Therefore:

$\displaystyle\sum_{r\in\mathbb{Z}^{d}}|j-m-rN|^{-k}\leqslant
M_{0}M^{-k}+1\leqslant CN^{-\gamma k}$

for sufficiently large $N$. Therefore

$\displaystyle\begin{split}|A_{mj}|&\leqslant
CN^{(d+1)\delta}N^{k\delta}N^{-k\gamma}=CN^{-k(\gamma-\delta)+(d+1)\delta}\end{split}$

where $C=C(a,\delta,k)$ is independent of $N$. Taking $k$ arbitrarily large
gives that $|A_{mj}|=O(N^{-\infty})$, so (as with pseudodifferential operators
on $\mathbb{R}^{d}$), the mass of the kernel is concentrated on the diagonal
and falls off superpolynomially away from it. We also note that even when $j$
and $m$ are close, we may remove one of the sums in (B.1) up to an
$O(N^{-\infty})$ error. Specifically we see that when $|j-m|\leqslant M$,
$k\in\mathbb{N}$

$\left|A_{mj}-\sum_{l\in\mathbb{Z}^{d}}\hat{a}(l,j-m)e^{\pi i\frac{\langle
j+m,l\rangle}{N}}\right|=O(N^{-\infty}).$

Indeed

$\displaystyle\left|A_{mj}-\sum_{l\in\mathbb{Z}^{d}}\hat{a}(l,j-m)e^{\pi
i\frac{\langle j+m,l\rangle}{N}}\right|$
$\displaystyle=\left|\sum_{\begin{subarray}{c}l,r\in\mathbb{Z}^{d}\\\ r\neq
0\end{subarray}}\hat{a}(l,j-m-rN)(-1)^{\langle r,l\rangle}e^{\pi
i\frac{\langle j+m,l\rangle}{N}}\right|$
$\displaystyle\leqslant\sum_{\begin{subarray}{c}l,r\in\mathbb{Z}^{d}\\\ r\neq
0\end{subarray}}|\hat{a}(l,j-m-rN)|.$

Note that there exists $c_{d}>0$ such that for all $r\neq 0$,

$\displaystyle|j-m-rN|\geqslant c_{d}N^{\gamma}|r|$

therefore for each $k\in\mathbb{Z}_{>d}$, using that $a\in
S_{\delta}(\mathbb{T}^{2d})$,

$\displaystyle\sum_{\begin{subarray}{c}l,r\in\mathbb{Z}^{d}\\\ r\neq
0\end{subarray}}|\hat{a}(l,j-m-rN)|$ $\displaystyle\leqslant
C\sum_{\begin{subarray}{c}l,r\in\mathbb{Z}^{d}\\\ r\neq
0\end{subarray}}\langle l\rangle^{-(d+1)}|j-m-
rN|^{-k}\|(\partial_{x}^{d+1}+1)\partial_{\xi}^{k}a\|_{L^{\infty}}$
$\displaystyle\leqslant CN^{\delta(d+1+k)}\sum_{r\in\mathbb{Z}^{d}_{\neq
0}}c_{d}N^{-k\gamma}\langle r\rangle^{-k}$

by the same reasoning as in (B.2). By Fourier inversion in the first variable,
we have in that case

$A_{mj}=\mathscr{F}_{2}a\left(\frac{j+m}{2N},j-m\right)+O(N^{-\infty})$ (B.3)

where $\mathscr{F}_{2}$ is the Fourier coefficient in the second component (so
$\mathscr{F}_{2}a$ acts on $\mathbb{T}^{d}\times\mathbb{Z}^{d}$).

Then we compute

$\displaystyle\begin{split}&\operatorname{tr}_{H_{N}}(\rho_{N}\operatorname{Op}_{N}(a))=C_{N}\left\langle\psi(N)\right\rvert\operatorname{Op}_{N}(a)\left\lvert\psi(N)\right\rangle\\\
&=C_{N}\sum_{m,j\in(\mathbb{Z}/N\mathbb{Z})^{d}}\frac{1}{N^{d}}\overline{g(m/N)}A_{mj}g(j/N)e^{2\pi
i(j-m)\xi_{0}}\\\ &=C_{N}\sum_{|m-j|\leqslant
M}\frac{1}{N^{d}}\overline{g(m/N)}g(j/N)\int_{\mathbb{T}^{d}}a\left(\frac{j+m}{2N},\xi\right)e^{2\pi
i(m-j)(\xi-\xi_{0})}\mathop{}\\!\mathrm{d}\xi+O(N^{-\infty})\\\
&=C_{N}\sum_{|m-j|\leqslant
M}\frac{1}{N^{d}}\overline{g(m/N)}g(j/N)\int_{\mathbb{T}^{d}}a\left(\frac{j+m}{2N},\xi+\xi_{0}\right)e^{2\pi
i(m-j)\xi}\mathop{}\\!\mathrm{d}\xi+O(N^{-\infty})\end{split}$ (B.4)

But when $|j-m|\leqslant M$, we have

$\left|a\left(\frac{j+m}{2N},\xi+\xi_{0}\right)-a\left(\frac{m}{N},\xi+\xi_{0}\right)\right|\leqslant
MN^{-1}\|\partial a\|_{L^{\infty}}\leqslant CMN^{-(1-\delta)}$

and

$\left|g\left(\frac{m}{N}\right)-g\left(\frac{j}{N}\right)\right|\leqslant
CMN^{-1},$

so

$\begin{gathered}\overline{g(m/N)}g(j/N)\int_{\mathbb{T}^{d}}a\left(\frac{j+m}{2N},\xi+\xi_{0}\right)e^{2\pi
i(m-j)\xi}\mathop{}\\!\mathrm{d}\xi\\\
=|g(m/N)|^{2}\int_{\mathbb{T}^{d}}a\left(\frac{m}{N},\xi+\xi_{0}\right)e^{2\pi
i(m-j)\xi}\mathop{}\\!\mathrm{d}\xi+O(MN^{-(1-\delta)})\end{gathered}$ (B.5)

As there are $O(N^{d}M^{d})$ pairs $(j,m)$ with $|m-j|\leqslant M$, the errors
(B.5) sum in (B.4) to give

$\displaystyle\begin{split}&\operatorname{tr}_{H_{N}}(\rho_{N}\operatorname{Op}_{N}(a))\\\
&=\sum_{m\in(\mathbb{Z}/N\mathbb{Z})^{d}}\frac{1}{N^{d}}|g(m/N)|^{2}\sum_{j:|m-j|\leqslant
M}\int_{\mathbb{T}^{d}}a\left(\frac{m}{N},\xi+\xi_{0}\right)e^{2\pi
i(m-j)\xi}\mathop{}\\!\mathrm{d}\xi+O(M^{d+1}N^{-(1-\delta)})\end{split}$
(B.6)

where we have used Remark 1 to remove the $C_{N}$. The inner sum in (B.6) is
just a partial Fourier series, and as $\gamma>\delta$, it differs from the
full Fourier series by order $O(N^{-\infty})$. This gives

$\operatorname{tr}_{H_{N}}(\rho_{N}\operatorname{Op}_{N}(a))=\sum_{m\in(\mathbb{Z}/N\mathbb{Z})^{d}}\frac{1}{N^{d}}|g(m/N)|^{2}a\left(\frac{m}{N},\xi_{0}\right)+O(M^{d+1}N^{-(1-\delta)}).$
(B.7)

But (B.7) is just a Riemann sum, and we have the estimate

$\left|\sum_{m\in(\mathbb{Z}/N\mathbb{Z})^{d}}\frac{1}{N^{d}}|g(m/N)|^{2}a\left(\frac{m}{N},\xi_{0}\right)-\int_{\mathbb{T}^{d}}a(x,\xi_{0})|g(x)|^{2}\mathop{}\\!\mathrm{d}x\right|=O(N^{-(1-\delta)}).$

Therefore

$\left|\operatorname{tr}_{H_{N}}(\rho_{N}\operatorname{Op}_{N}(a))-\int_{\mathbb{T}^{d}}a(x,\xi_{0})|g(x)|^{2}\mathop{}\\!\mathrm{d}x\right|\leqslant
O(M^{d+1}N^{-(1-\delta)})=(N^{(d+1)\gamma-(1-\delta)})$

for all $\gamma>\delta$. ∎

###### Proof of Proposition 2.12.

Let $a\in S_{\delta}(\mathbb{T}^{2d})$ and let $A=\operatorname{Op}_{N}(a)$
given in the basis $(Q_{n})$. Once again choose $M=\lfloor N^{\gamma}\rfloor$
for $\gamma\in(\delta,1)$. We have by (B.2) and (B.3) that (writing
$\psi:=\psi_{N}$ for convenience and letting subscripts refer to the indices)

$\displaystyle\begin{split}\operatorname{tr}(\rho_{N}\operatorname{Op}_{N}(a))&=C_{N}^{\prime}\sum_{m,j\in(\mathbb{Z}/N\mathbb{Z})^{d}}\overline{\psi_{m}}A_{mj}\psi_{j}\\\
&=C_{N}^{\prime}\sum_{|m-j|\leqslant
M}\overline{\psi_{m}}\int_{\mathbb{T}^{d}}a\left(\frac{j+m}{2N},\xi\right)e^{2\pi
i(m-j)\xi}\mathop{}\\!\mathrm{d}\xi\psi_{j}+O(N^{-\infty}).\end{split}$ (B.8)

But as $g\in\mathcal{S}(\mathbb{R}^{d})$, it has superpolynomial decay, and we
get

$\displaystyle\begin{split}\psi_{m}&=N^{-\frac{d}{2}}\sum_{k\in\mathbb{Z}^{d}}g_{N}\left(\frac{m}{N}+k\right)\\\
&=N^{\frac{d(\beta-1)}{2}}\sum_{k\in\mathbb{Z}^{d}}g\left(N^{\beta}\left(\frac{m}{N}+k-x_{0}\right)\right)e^{2\pi
i(m+kN)\cdot\xi_{0}}\\\
&=N^{\frac{d(\beta-1)}{2}}g\left(N^{\beta}\left(\frac{m}{N}-x_{0}\right)\right)e^{2\pi
im\cdot\xi_{0}}+O(N^{-\infty})\end{split}$ (B.9)

by throwing away all but the first term in the sum. In fact, for many $m$ we
may even throw out the first term. Given any $\varepsilon>0$, let
$R_{\varepsilon}=\lfloor N^{1-\beta+\varepsilon}\rfloor$. Then if
$|m-Nx_{0}|\leqslant R_{\varepsilon}$, we have by (B.9) that

$|\psi_{m}|=O(N^{-\infty}).$ (B.10)

We then approximate terms in the summand as in Proposition 2.11, and get when
$|j-m|\leqslant M$ that

$\left|a\left(\frac{j+m}{2N},\xi+\xi_{0}\right)-a\left(\frac{m}{N},\xi+\xi_{0}\right)\right|\leqslant
MN^{-1}\|\partial a\|_{L^{\infty}}\leqslant CMN^{-(1-\delta)}$

and

$\left|g\left(N^{\beta}\left(\frac{m}{N}-x_{0}\right)\right)-g\left(N^{\beta}\left(\frac{j}{N}-x_{0}\right)\right)\right|\leqslant
CMN^{-(1-\beta)}$

so in that case (using $\delta\leqslant\beta$)

$\begin{gathered}\overline{g\left(N^{\beta}\left(\frac{m}{N}-x_{0}\right)\right)}g\left(N^{\beta}\left(\frac{j}{N}-x_{0}\right)\right)\int_{\mathbb{T}^{d}}a\left(\frac{j+m}{2N},\xi+\xi_{0}\right)e^{2\pi
i(m-j)\xi}\mathop{}\\!\mathrm{d}\xi\\\
=\left|g\left(N^{\beta}\left(\frac{m}{N}-x_{0}\right)\right)\right|^{2}\int_{\mathbb{T}^{d}}a\left(\frac{m}{N},\xi-\xi_{0}\right)e^{2\pi
i(j-m)\xi}\mathop{}\\!\mathrm{d}\xi+O(MN^{1-\beta})\end{gathered}$ (B.11)

But by (B.8), (B.9), and (B.10) we have
$\operatorname{tr}(\rho_{N}\operatorname{Op}_{N}(a))$ to be equal to

$\begin{gathered}C_{N}^{\prime}N^{d(\beta-1)}\sum_{m:\left|m-Nx_{0}\right|\leqslant
R_{\varepsilon}}\sum_{j:|m-j|\leqslant
M}\overline{g\left(N^{\beta}\left(\frac{m}{N}-x_{0}\right)\right)}g\left(N^{\beta}\left(\frac{j}{N}-x_{0}\right)\right)\cdots\\\
\cdots\int_{\mathbb{T}^{d}}a\left(\frac{j+m}{2N},\xi\right)e^{2\pi
i(m-j)(\xi-\xi_{0})}\mathop{}\\!\mathrm{d}\xi+O(N^{-\infty})\end{gathered}$
(B.12)

The sum in (B.12) has $O(M^{d}R_{\varepsilon}^{d})$ terms, so adding up all
the errors given in (B.11) gives

$\displaystyle\operatorname{tr}(\rho_{N}\operatorname{Op}_{N}(a))$
$\displaystyle=N^{d(\beta-1)}\sum_{m:\left|m-Nx_{0}\right|\leqslant
R_{\varepsilon}}\left|g\left(N^{\beta}\left(\frac{m}{N}-x_{0}\right)\right)\right|^{2}\sum_{j:|m-j|\leqslant
M}\int_{\mathbb{T}^{d}}a\left(\frac{m}{N},\xi+\xi_{0}\right)e^{2\pi
i(m-j)\xi}\mathop{}\\!\mathrm{d}\xi$
$\displaystyle\quad+O(M^{d+1}N^{(d+1)(\beta-1)}R_{\varepsilon}^{d})$
$\displaystyle=N^{d(\beta-1)}\sum_{m:\left|m-Nx_{0}\right|\leqslant
R_{\varepsilon}}\left|g\left(N^{\beta}\left(\frac{m}{N}-x_{0}\right)\right)\right|^{2}a\left(\frac{m}{N},\xi_{0}\right)+O(M^{d+1}N^{\beta-1}N^{d\varepsilon})$

where in the last line we approximated the inner sum with a Fourier series as
in (B.7), obtaining an $O(N^{-\infty})$ error, and applied Remark 1 to ignore
the $C_{N}^{\prime}$. At this point, we (up to $O(N^{-\infty})$ error) add
back in the remaining $m$ terms and compute the Riemann sum to get

$\displaystyle\operatorname{tr}(\rho_{N}\operatorname{Op}_{N}(a))$
$\displaystyle=N^{d(\beta-1)}\sum_{m\in(\mathbb{Z}/N\mathbb{Z})^{d}}\left|g\left(N^{\beta}\left(\frac{m}{N}-x_{0}\right)\right)\right|^{2}a\left(\frac{m}{N},\xi_{0}\right)+O(M^{d+1}N^{\beta-1}N^{d\varepsilon})$
$\displaystyle=\int
N^{d\beta}|g(N^{\beta}(x-x_{0}))|^{2}a(x,\xi_{0})\mathop{}\\!\mathrm{d}x+O(N^{\beta-1})+O(N^{\delta-1})+O(M^{d+1}N^{\beta-1}N^{d\varepsilon})$
$\displaystyle=\int
N^{d\beta}|g(N^{\beta}(x-x_{0}))|^{2}a(x,\xi_{0})\mathop{}\\!\mathrm{d}x+O(M^{d+1}N^{\beta-1}N^{d\varepsilon})$

with the last line using that $\delta\leqslant\beta$. We substitute and use
$\int|g|^{2}=1$ to obtain

$\begin{gathered}\left|\int_{\mathbb{T}^{d}}N^{d\beta}|g(N^{\beta}(x-x_{0}))|^{2}a(x,\xi_{0})\mathop{}\\!\mathrm{d}x-a(x_{0},\xi_{0})\right|\\\
\leqslant\left|\int_{\mathbb{R}^{d}}|g(y)|^{2}\left(a(N^{-\beta}y+x_{0},\xi_{0})-a(x_{0},\xi_{0})\right)dy\right|+\left|\int_{\mathbb{R}^{d}\setminus
N^{\beta}\mathbb{T}^{d}}|g(y)|^{2}\mathop{}\\!\mathrm{d}y\right|\\\ \leqslant
C\left|\int_{\mathbb{R}^{d}}|g(y)|^{2}N^{-\beta}N^{\delta}|y|\mathop{}\\!\mathrm{d}y\right|+O(N^{-\infty})\leqslant
O(N^{\delta-\beta}).\end{gathered}$

This gives

$|\operatorname{tr}(\rho_{N}\operatorname{Op}_{N}(a))-a(x_{0},\xi_{0})|\leqslant
O(N^{\delta-\beta})+O(M^{d+1}N^{\beta-1}N^{d\varepsilon})$

which completes the proof by taking $\varepsilon$ and $\gamma-\delta$ small
enough. ∎

##  Appendix C Estimating Derivatives of the Flow

Here we present a proof of Proposition 2.15 which we restate here for
convenience.

######  Proposition C.1.

Suppose $\phi^{t}:\mathbb{T}^{2d}\to\mathbb{T}^{d}$ is a smooth dynamical
system on $\mathbb{T}^{2d}$ with Lyapunov exponent $\Lambda$ where $t$ either
takes integer values, or values in $\mathbb{R}_{\geqslant 0}$. Let $a\in
C^{\infty}(\mathbb{T}^{d})$. Then for all $\varepsilon>0$,
$\alpha\in\mathbb{N}^{d}$, there exists $C=C(\varepsilon,\alpha)$ such that:

$\displaystyle|\partial^{\alpha}(a\circ\phi^{t})(x)|\leqslant
Ce^{(\Lambda+\varepsilon)t|\alpha|}\|a\|_{C^{|\alpha|}}.$

Here $C^{k}$ is the norm defined as

$\displaystyle\|a\|_{C^{K}}=\sum_{|\alpha|\leqslant
k}\|\partial^{\alpha}a\|_{L^{\infty}(\mathbb{T}^{d})}.$

This proof essentially follows [DJN22, §5.2] but in the simpler setting of the
torus. To our knowledge, such a proof in this setting is not present in the
current literature. In an effort to improve readability, we include examples
throughout the proof, at the cost of being rather voluminous.

###### Proof.

We prove this by strong induction on the order of $\alpha$. We trivially have
the claim when $|\alpha|=0$. If $|\alpha|=1$, suppose
$\partial^{\alpha}=\partial_{i}$, then we have:

$\displaystyle\partial^{\alpha}_{x}(a\circ\phi^{t})(x)=\sum_{j=1}^{d}(\partial_{j}a)(\phi^{t}(x))\partial_{i}(\phi_{t}(x))_{j}$

where $(\phi^{t}(x))_{j}$ is the $j$th component of $\phi^{t}(x)$. By the
definition of Lyapunov exponent, for any $\varepsilon>0$, there exists
$C=C(\varepsilon)>0$ such that:

$\displaystyle|\partial_{i}\phi^{t}(x)|\leqslant Ce^{(\Lambda+\varepsilon)t},$

so that

$\displaystyle|\partial^{\alpha}_{x}(a\circ\phi^{t})(x)|\leqslant
Ce^{(\Lambda+\varepsilon)t}\|a\|_{C^{1}}.$

To ease notation in the remained of the proof, define
$\lambda:=\Lambda+\varepsilon$.

Before proving the inductive step, we show how to go from $\alpha$ of order
$1$ to order $2$. We will use the following notation for Jacobians and
tensors. If $f:\mathbb{R}\to\mathbb{R}^{d}$, then $Df$ will be the vector with
$i$th component $\partial_{i}f$. Note that $Df$ will be a $(0,1)$ tensor. We
define $D^{2}f$ as the $(0,2)$ tensor with $i,j$ entry
$(D^{2}f)_{i,j}=\partial_{i}\partial_{j}f$. If
$\phi:\mathbb{R}^{d}\to\mathbb{R}^{d}$, then we let $D\phi$ be the $(1,1)$
tensor with $i,j$ entry $(D\phi)_{j}^{i}=\partial_{j}\phi_{i}$. We let
$D^{2}\phi$ denote the $(1,2)$ tensor with $i,j,k$ entry
$(D^{2}\phi)_{j,k}^{i}=(\partial_{j}\partial_{k}\phi_{i})$.

Observe that:

$\displaystyle(D(a\circ\phi^{n}))(x)_{i}=\partial_{i}(a\circ\phi^{n})=\sum_{j}\partial_{j}((a\circ\phi^{n-1})\circ\phi^{1}(x))(\partial_{i}\phi_{j}^{1})(x)$

so that

$\displaystyle
D(a\circ\phi^{n})(x)=(D(a\circ\phi^{n-1})\circ\phi^{1}(x))D\phi^{1}(x)$

where the product of two tensors is the contraction of the upper and lower
components. That is, $(D(a\circ\phi^{n-1})\circ\phi^{1})$ is a $(0,1)$ tensor
which can be written $A_{\mu}$, and $D\phi^{1}$ is a $(1,1)$ tensor which can
be written $B_{\gamma}^{\nu}$. In this case $AB$ is the $(0,1)$ tensor
contracting the lower component of $A$ with the upper component of $B$:
$(AB)_{\gamma}=\sum_{\nu}A_{\nu}B^{\nu}_{\gamma}$.

Similarly

$\displaystyle(D^{2}(a\circ\phi^{n})(x))_{i,j}$
$\displaystyle=\partial_{i}\partial_{j}(a\circ\phi^{n})(x)$
$\displaystyle=\sum_{k,\ell}((\partial_{k}\partial_{\ell}(a\circ\phi^{n-1}))\circ\phi^{1}(x))\partial_{j}\phi^{1}_{\ell}(x)\partial_{i}\phi^{1}_{k}(x)$
$\displaystyle\quad+\sum_{k}(\partial_{k}(a\circ\phi^{n-1})\circ\phi^{1}(x))\partial_{j}\partial_{i}\phi^{1}_{k}(x)$

so that

$\displaystyle D^{2}(a\circ\phi^{n})(x)$
$\displaystyle\quad=(D^{2}(a\circ\phi^{n-1})\circ\phi^{1}(x))(D\phi^{1}(x)\otimes
D\phi^{1}(x))+(D(a\circ\phi^{n-1})\circ\phi^{1}(x))D^{2}\phi^{1}(x)$

Therefore

$\displaystyle\underbrace{\begin{pmatrix}D(a\circ\phi^{n+1})(x)\\\
D^{2}(a\circ\phi^{n+1})(x)\end{pmatrix}^{t}}_{:=A_{n+1}(x)^{t}}=\begin{pmatrix}D(a\circ\phi^{n})\circ\phi^{1}(x)\\\
D^{2}(a\circ\phi^{n})\circ\phi^{1}(x)\end{pmatrix}^{t}\underbrace{\begin{pmatrix}D\phi^{1}(x)&D^{2}\phi^{1}(x)\\\
0&D\phi^{1}(x)\otimes D\phi^{1}(x)\end{pmatrix}}_{:=M(x)},$

so that

$\displaystyle\begin{split}A_{n}(x)&=A_{n-1}(\phi^{1}(x))M(x)=A_{n-2}(\phi^{2}(x))M(\phi^{1}(x))M(x))\\\
&=A_{0}(\phi^{n}(x))M(\phi^{n-1}(x))M(\phi^{n-2}(x))\cdots M(x).\end{split}$
(C.1)

By the upper triangular structure of $M$, it is easy to check that

$\displaystyle M(\phi^{n-1}(x))M(\phi^{n-2}(x))\cdots
M(x)=\begin{pmatrix}M_{1}(x)&M_{2}(x)\\\ 0&M_{3}(x)\end{pmatrix}$ (C.2)

where:

$\displaystyle M_{1}(x)$
$\displaystyle:=\prod_{j=1}^{n}((D\phi^{1})\circ(\phi^{n-j}(x))),$
$\displaystyle M_{2}(x):=\sum_{j=1}^{n}(M_{2})_{j}(x),$ $\displaystyle
M_{3}(x)$
$\displaystyle=\prod_{j=1}^{n}(((D\phi^{1})\circ\phi^{n-j}(x))\otimes((D\phi^{1})\circ\phi^{n-j}(x)))$

with $(M_{2})_{j}(x)$ defined as

$\displaystyle\left(\prod_{i=1}^{j-1}((D\phi^{1})\circ(\phi^{n-i}(x)))\right)((D^{2}\phi^{1})\circ\phi^{n-j}(x))\left(\prod_{i=j+1}^{n}((D\phi^{1})\circ(\phi^{n-i}(x)))^{\otimes
2}\right)$ (C.3)

where $a^{\otimes 2}:=a\otimes a$.

We then have, by (C.1) and (C.2),

$\displaystyle\begin{split}|D^{2}(a\circ\phi^{n})(x)|&=|(Da\circ\phi^{n}(x))M_{2}(x)+(D^{2}a\circ\phi^{n}(x))M_{3}(x)|\\\
&\leqslant C\|a\|_{C^{1}}|M_{2}(x)|+C\|a\|_{C^{2}}|M_{3}(x)|\end{split}$ (C.4)

where the absolute value of a tensor denotes the maximum of the absolute value
of all entries.

But now observe that

$\displaystyle\prod_{i=1}^{j-1}((D\phi^{1})\circ(\phi^{n-i}(x))$
$\displaystyle=D(\phi^{j-1})\circ\phi^{n-j+1}(x),$
$\displaystyle\prod_{i=j+1}^{n}((D\phi^{1})\circ(\phi^{n-i}(x)))^{\otimes 2}$
$\displaystyle=(D(\phi^{n-j})(x))^{\otimes 2}$

so that

$\displaystyle|M_{3}(x)|$ $\displaystyle=|D(\phi^{n})(x)|^{2}\leqslant
Ce^{2\lambda n}$ (C.5)

by the inductive hypothesis and setting $a$ to be coordinate functions. And we
also have, by (C.3), that

$\displaystyle|(M_{2})_{j}(x)|$ $\displaystyle\leqslant
Ce^{\lambda(j-1)}e^{2\lambda(n-j)}\|D^{2}\phi^{1}\|_{L^{\infty}}=C^{\prime}e^{2\lambda
n-\lambda j-\lambda}$

where we use that $\phi^{1}$ is a smooth function on a compact set so that
$\|D^{2}\phi^{1}\|_{L^{\infty}}<\infty$. But we stress that this norm only
comes up once in each $(M_{2})_{j}$ term, so that the constant at the end is
independent of $n$. We therefore have that:

$\displaystyle|M_{2}(x)|\leqslant\sum_{j=1}^{n}|(M_{2})_{j}(x)|\leqslant
Ce^{2\lambda n-\lambda}\sum_{j=1}^{n}e^{-\lambda j}\leqslant Ce^{2\lambda
n}(1-e^{-\lambda})^{-1}$ (C.6)

We can absorb the $(1-e^{-\lambda})^{-1}$ term into the constant $C$ to get,
using (C.4), (C.5), and (C.6), that

$\displaystyle|D^{2}(a\circ\phi^{n})(x)|\leqslant C\|a\|_{C^{2}}e^{2\lambda
n}.$

Now if the flow is continuous in time and $t\in\mathbb{R}_{\geqslant 0}$, we
let $n\in\mathbb{N}$ be such that $n\leqslant t\leqslant n+1$ and observe
that:

$\displaystyle|D^{2}(a\circ\phi^{t})(x)|=|D^{2}(a\circ\phi^{t-n}\circ\phi^{n})(x)|\leqslant
C\|a\circ\phi^{t-n}\|_{C^{2}}e^{2\lambda n}.$

Note that $s:=t-n\in[0,1]$ which is notably a compact set, so that:

$\displaystyle\|a\circ\phi^{t-n}\|_{C^{2}}\leqslant\|a\|_{C^{2}}\max_{s\in[0,1]}\|\phi^{s}\|_{C^{2}}.$
(C.7)

We can then absorb $\max_{s\in[0,1]}\|\phi^{s}\|_{C^{2}}$ into the constant
$C$ to get the result for continuous time.

Now we present the inductive step, but the reader should keep in mind that
morally the idea is the same as above, but with more notation. Suppose we have
for all $k^{\prime}\leqslant k-1$ and $|\alpha|=k^{\prime}$:

$\displaystyle|\partial^{\alpha}(a\circ\phi^{t})(x)|\leqslant
C\|a\|_{C^{k^{\prime}}}e^{k^{\prime}\lambda t}.$

We then fix $n\in\mathbb{N}$, and aim to bound $D^{k}(a\circ\phi^{n})(x)$.
Define $A_{n}(x)$ as the $1\times k$ vector of tensors whose $i$th entry is
$D^{i}(a\circ\phi^{n})(x)$. By writing

$\displaystyle
D^{i}(a\circ\phi^{n})=D^{i-1}((D(a\circ\phi^{n-1})\circ\phi^{1})D\phi^{1})(x),$
(C.8)

and repeatedly applying the product rule, we can find a $k\times k$ matrix of
tensors, $M(x)$ such that $A_{n}(x)=A_{n-1}(\phi^{1}(x))M(x)$. Note that
$M(x)$ will be upper triangular as for any $i\leqslant k$,
$D^{i}(a\circ\phi^{n})$ is a linear combination of
$D^{j}(a\circ\phi^{n-1})\circ\phi^{1}$, for $j\leqslant i$. Also, for each
$i,j=1,\dots,k$, the $i,j$ entry of $M(x)$ must be a $(i,j)$ tensor because it
will be paired with at $(0,i)$ tensor to get a $(0,j)$ tensor. Each component
of $(M(x))_{i,j}$ will contain $j$ derivatives of the components of
$\phi^{1}(x)$ with a coefficient depending on $k$. Lastly note that for each
$i=1,\dots,k$, $(M(x))_{i,i}=(D\phi^{1}(x))^{\otimes i}$ as this is the term
in (C.8) when all derivatives fall on $D(a\circ\phi^{n-1}\circ\phi^{1})$.

We then recursively use the relation of $A_{n}$ to $A_{n-1}$ to get that:

$\displaystyle
A_{n}(x)=A_{0}(\phi^{n}(x))\left(\prod_{j=1}^{n}M(\phi^{n-j}(x))\right).$

For $i=1,\dots,k$, let $M_{i}$ be the $(i,k)$ entry of
$\prod_{j=1}^{n}M(\phi^{n-j})$, so that:

$\displaystyle
D^{k}(a\circ\phi^{n})(x)=\sum_{j=1}^{k}(D^{j}(a)\circ\phi^{n}(x))M_{j}(x).$
(C.9)

It now suffices to provide appropriate estimates of $M_{j}(x)$. For
$i=1,\dots,n$ let $B^{i}:=M(\phi^{n-i})(x)$. Then for each fixed
$j=1,\dots,k$:

$\displaystyle
M_{j}=\sum_{\ell_{1},\ell_{2},\dots,\ell_{n-1}}B^{1}_{j,\ell_{1}}B^{2}_{\ell_{1},\ell_{2}}\cdots
B^{n}_{\ell_{n-1},k}.$ (C.10)

Because $B^{i}$ are upper triangular, this sum is nonzero only if
$\ell_{m}\leqslant\ell_{m+1}$ for each $m=1,\dots,n-1$, and
$j\leqslant\ell_{1}$. Each term in the sum can be expressed as a path on a
grid of $k\times(n+1)$ nodes. Labeling the bottom left node $(1,1)$, every
path must begin at $(j,1)$ and end at $(k,n+1)$. The path can only go to the
right or up. That is $(m,n)$ can only go to $(m^{\prime},n+1)$ for
$m^{\prime}\geqslant m$ (provided that the path terminates at $(k,n+1)$). The
edges of the path give the index of $B^{i}$ in the sum. That is, if the nodes
$(m,i),(n,i+1)$ are on the path, then we include the term $B^{i}_{m,n}$ in the
sum.

For example, if $k=4$ and $n=5$, then the following path represents the term
$B^{1}_{1,2}B^{2}_{2,2}B_{2,2}^{3}B^{4}_{2,4}B^{5}_{4,4}$.

Each path can be represented as a vector whose $i$th component is $m$ where
$(m,i)$ is on the path. So that the example path is written as
$(1,2,2,2,4,4)$.

Each path will consist of flat sections (where the $i$ and $i-1$ component of
the vector representation are the same) and jumps (where the component of the
vector representation is not flat). In the above example, the jumps are at
indices $2$ and $5$. The corresponding entry of $B^{i}$ for the jumps will be
individually norm bounded. Recall that $B^{i}_{j,\ell}$ will be a $(j,\ell)$
tensor where all terms are derivatives of $\phi^{1}$. As long as
$(j,\ell)\neq(1,k)$, at most $k-1$ derivatives are taken of $\phi^{1}$. Note
that $B_{1,k}$ can only appear once in any term in the sum (C.10), and this is
bounded by the $C^{k}$ norm of $\phi^{1}$. We therefore can apply the
inductive hypothesis to get that $|B^{i}_{j,\ell}(x)|\leqslant
Ce^{\lambda\ell}$. Next we bound the flat sequences. Suppose such a sequence
is $\prod_{i=j_{1}}^{j_{2}}B_{\ell,\ell}^{i}$. Recall that
$B_{\ell,\ell}^{i}=(D\phi^{1})^{\otimes\ell}\circ(\phi^{n-i}(x))$, so that

$\displaystyle\left|\prod_{i=j_{1}}^{j_{2}}B_{\ell,\ell}^{i}(x)\right|=|(D(\phi^{j_{2}-j_{1}+1})\circ\phi^{n-j_{2}+1}(x))^{\otimes\ell}|\leqslant
Ce^{\ell(j_{2}-j_{1}+1)}.$ (C.11)

This follows by the inductive hypothesis by setting $a$ to be coordinate
functions. Now suppose we have a term of the sum (C.10) whose path has a
vector representation $(a_{1},a_{2},\dots,a_{n+1})$ with jumps at indices
$\ell_{1},\ell_{2},\dots,\ell_{s}$. We assume that there is a jump at index
$2$ and the path ends in flat section (as in the example), but all cases can
be treated similarly. We can then bound this term of the sum (using (C.11))
by:

$\displaystyle|B_{a_{\ell_{1}-1},a_{\ell_{1}}}^{\ell_{1}}|\cdots|B_{a_{\ell_{s}-1},a_{\ell_{s}}}^{\ell_{s}}|\left|\prod_{i=\ell_{1}+1}^{\ell_{2}-1}B^{i}_{a_{\ell_{1}},a_{\ell_{1}}}\right|\cdots\left|\prod_{i=\ell_{s}+1}^{n+1}B^{i}_{a_{\ell_{s}},a_{\ell_{s}}}\right|$
$\displaystyle\leqslant
Ce^{\lambda(a_{\ell_{1}}+\cdots+a_{\ell_{s}})}e^{\lambda
a_{\ell_{1}}(\ell_{2}-\ell_{1}-1)}\cdots e^{\lambda
a_{\ell_{s}}(n+1-\ell_{s}-1)}$
$\displaystyle=Ce^{\lambda(a_{\ell_{1}}(\ell_{2}-\ell_{1})+\cdots+a_{\ell_{s-1}}(\ell_{s}-\ell_{s-1})+a_{\ell_{s}}(n+1-\ell_{s}))}$
$\displaystyle=Ce^{\lambda(a_{2}+\cdots+a_{n+1})}.$

The last equality follows by noting that $\ell_{i+1}-\ell_{i}$ is the length
of each flat section and is where the sequence has values $a_{\ell_{i}}$.
Therefore:

$\displaystyle|M_{j}(x)|\leqslant
C\left|\sum_{(a_{n})\in\mathcal{A}_{j,k}}e^{\lambda(a_{2}+\dots+a_{n+1})}\right|$
(C.12)

where $\mathcal{A}_{j,k}$ is the set of all sequences $(a_{n})$ of length
$n+1$ such that $a_{i}\in\mathbb{Z}_{>0}$, $a_{1}=j$, $a_{n+1}=k$, and
$a_{i}\leqslant a_{i+1}$. Let $V$ be the $k\times k$ upper triangular matrix
such that $V_{\ell,m}=e^{m\lambda}$ for $\ell\leqslant m$ and zero otherwise.
Then by the exact same argument for computing products of upper triangular
matrices, we see that

$\displaystyle(V^{n})_{j,k}=\sum_{(a_{n})\in\mathcal{A}_{j,k}}e^{\lambda(a_{2}+\dots+a_{n+1})}.$
(C.13)

Because the eigenvalues of $V$ are distinct, $V$ is diagonalizable. We can
then write the entries of $V^{n}$ as linear combinations (with coefficients
independent of $n$) of the eigenvalues of $V$ raised to the $n$th power. That
is there exist $c_{j,\ell,k}\in\mathbb{R}$ independent of $n$ such that

$\displaystyle(V^{n})_{j,k}=\sum_{\ell=1}^{k}e^{\lambda\ell n}c_{j,\ell,k}.$

But then we immediately see that there exists a $C=C(k)>0$ such that
$|(V^{n})_{j,k}|\leqslant Ce^{\lambda kn}$ for $j=1,\dots,k$. We therefore
have that $|M_{j}(x)|\leqslant Ce^{n\lambda k}$, so that (C.9) becomes:

$\displaystyle|D^{k}(a\circ\phi^{n})(x)|\leqslant C\|a\|_{C^{k}}e^{n\lambda
k}.$

Applying the exact same argument at the end of the case $k=2$ to go from
discrete time to continuous time completes the proof.

∎

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