\section{Result}

Here, we show that in fact the opposite is true if the environment is sufficiently large. Almost all states of the system and the environment are ``pretty lazy'', that is the entropy rate
on the system is very low for any coupling Hamiltonian. With low we thereby mean that the entropy rate scales as some vanishing parameter $\varepsilon$ times the interaction strength.
Note that in contrast to the study of zero entropy rates, 
this is all one could hope for when talking about low entropy rates -- a stronger interaction strength will necessarily increase any non-zero rate. 

Our main result that almost all states have low entropy rates can now be stated slightly more formally.
In particular, we will show that the probability that a randomly chosen state $\rho_{SE}$ has large entropy rate is very small. 
That is,
\begin{align}\label{eq:mainResult}
	\Pr_{\rho_{SE}}\left[\left|\frac{d}{dt} H(S)_{\rho}\right| \geq \|H_{\rm int}\|_{\infty} \varepsilon\right] \leq \delta\ ,
\end{align}
where 
\begin{align}
	\varepsilon = 2^{-\frac{1}{2}\left(\log d_{{\ensuremath{ E } }} - 3 \log d_{{\ensuremath{ S } }} - 4\right)},\ \delta = 2e^{-d_{{\ensuremath{ S } }}^2/16}\ , 
\end{align}
and the distribution over the set of possible states on $\mathcal{H}_S \otimes \mathcal{H}_E$ can be any unitarily invariant measure.
If the environment is sufficiently large ($\log d_{{\ensuremath{ E } }} > 3 \log d_{{\ensuremath{ S } }}$) and the system itself is not too small ($\log d_S > 2$), then
we obtain a strong statement. We will furthermore show a similar bound that is also interesting for extremely small systems $\log d_S \leq 2$
as long as $\log d_{{\ensuremath{ E } }} > (9/2) \log d_{{\ensuremath{ S } }}$. In this case, we have 
\begin{align}
\varepsilon = 2^{-\frac{1}{2} \left(\log {d_{\ensuremath{ E } }} - \frac{9}{2}\log d_{{\ensuremath{ S } }} - 5\right)},\ \delta = 2e^{-d_{\ensuremath{ S } } d^{1/3}_{\ensuremath{ E } }/16}\ .
\end{align}

Since the Hilbert space dimension grows exponentially with the number of constituent particles of a physical system and since we usually assume the environment ${\ensuremath{ E } }$ to consist of a large number of particles, at least one of the dimensional constraints will be fulfilled in typical situations of physical interest.

It is important to note that while the entropy rate in general depends on the relation between the Hamiltonian and the state (see \eqref{eq:vonNeumanRate}), the condition for a state being lazy expressed in \eqref{eq:lazyCondition} describes a property of the state alone. Similarly, given the discussed dimensional constraints, the very structure of most states $\rho_{SE}$ is such that they do not allow a fast change of the entropy in $S$ -- even for ``unphysical'' Hamiltonians $H_{SE}$.

In the appendix, we show that analogous results can be obtained for the linear entropy or purity, which has been studied in the context of entropy rates
in ~\cite{lazyPaper,hayashi:purity,gogolin:mthesis}. In this case, we even obtain slightly more favourable parameters.



\section{Proof}

Let us now see how we can prove said results. Our proof thereby proceeds in two steps. First of all, we recall 
that for a randomly chosen pure state from $\mathcal{H}_{\ensuremath{ S } } \otimes \mathcal{H}_{\ensuremath{ E } }$ the state will almost certainly be close to
fully mixed on $\mathcal{H}_{\ensuremath{ S } }$, if the environment is significantly larger than the system~\cite{popescu:entanglement}. For completeness, we provide a simpler proof of this claim in the appendix. Second, we show that if a state is close to fully mixed on the system $\mathcal{H}_{\ensuremath{ S } }$ then it is indeed pretty lazy.

\smallskip
\noindent
{\bf Fully mixed on $\mathcal{H}_{\ensuremath{ S } }$:\ } 
Let us first consider only pure states on $\mathcal{H}_{\ensuremath{ S } } \otimes \mathcal{H}_{\ensuremath{ E } }$. Note that chosing a random pure state according to the Haar measure is equivalent to applying a randomly chosen unitary $U$ to a fixed starting state, say, $\ket{0}_{{\ensuremath{ S } }{\ensuremath{ E } }}$.
In contrast to~\cite{popescu:entanglement} our proof (see appendix) that such a random pure state is fully mixed on the system follows by an easy application of the decoupling theorem~\cite{dupuis:diss,dupuis:decoupling}. Furthermore, if we apply the decoupling theorem we do not have to restrict to pure states as in~\cite{popescu:entanglement}. That is, our statement does not only hold for most states of the form $U\proj{0}_{{\ensuremath{ S } }{\ensuremath{ E } }}U^{\dag}$ but more generally for most states of the form $U\sigma_{{\ensuremath{ S } }{\ensuremath{ E } }}U^{\dag}$ where $\sigma_{{\ensuremath{ S } }{\ensuremath{ E } }}$ is an arbitrary state (pure or mixed) on $\mathcal{H}_{\ensuremath{ S } } \otimes \mathcal{H}_{\ensuremath{ E } }$. Equivalently we may state that most states $\rho_{\sys\env}$ with given eigenvalues and randomly chosen eigenstates are close to fully mixed on the system. ``Randomly chosen'' here means that the eigenbasis of $\rho_{\sys\env}$ is chosen from the Haar measure, which by definition is unitarily invariant. Since our assertion holds for any fixed set of eigenvalues, it also holds if we pick $\rho_{\sys\env}$ from any unitarily invariant measure on $\ensuremath{\mathcal{S}}\left(\mathcal{H}_{\ensuremath{ S } } \otimes \mathcal{H}_{\ensuremath{ E } }\right)$, the set of density operators on $\mathcal{H}_{\ensuremath{ S } } \otimes \mathcal{H}_{\ensuremath{ E } }$.
Summarizing, we obtain the following little lemma, which is proven in the appendix.\\
\begin{lemma}\label{lem:closetoMixed}
For a bipartite system $\mathcal{H}_{\ensuremath{ S } } \otimes \mathcal{H}_{\ensuremath{ E } }$ 
\begin{align}
\Pr_{\rho_{{\ensuremath{ S } }{\ensuremath{ E } }}} \left\{\left\|\rho_{{\ensuremath{ S } }}-\frac{\id\ssys}{\dS}\right\|_1 \geq \chi\right\} 
\leq \delta ,\label{eq:prob}
\end{align}
where the probability is computed over the choice of $\rho_{\sys\env}$ from any unitarily invariant measure on $\ensuremath{\mathcal{S}}\left(\mathcal{H}_{\ensuremath{ S } } \otimes \mathcal{H}_{\ensuremath{ E } }\right)$,
and where we may choose either
\begin{align}\label{eq:parameters1}
\chi = 2^{-\frac{1}{2}\left(\log d\senv - \log d\ssys - 4\right)} , \delta = 2e^{-d_{{\ensuremath{ S } }}^2/16}
\end{align}
or 
\begin{align}\label{eq:parameters2}
\chi = 2^{-\frac{1}{3}\left(\log d\senv - \frac{3}{2} \log d\ssys - 5\right)} , \delta = 2e^{-d\ssysd\senv^{1/3}/16}\ .
\end{align}
\end{lemma}


\smallskip
\noindent
{\bf Pretty lazy for the von Neumann entropy:\ }
Let us now turn to the main part of our proof. A small calculation~\cite{lazyPaper} shows that the rate of change of 
the von Neumann entropy is given by
\begin{align}\label{eq:vonNeumanRate}
	\frac{d{\ensuremath{ { \rm H} } }({\ensuremath{ S } })}{dt} = -i \mathop{\mathrm{tr}}\nolimits\left(H_{\rm int}\left[\log(\rho_{\ensuremath{ S } }(t)) \otimes \mathbb{I}_{\ensuremath{ E } },\rho_{\sys\env}(t) \right]\right)\ .
\end{align}
Note that $[\log(\rho_{{\ensuremath{ S } }})\otimes \mathbb{I}_{\ensuremath{ E } },\rho_{\sys\env}] = 0$ if and only if~\eqref{eq:lazyCondition} holds, and thus
the latter is a sufficient condition for a state $\rho_{{\ensuremath{ S } }{\ensuremath{ E } }}$ to be lazy~\cite{lazyPaper}. Consider now a state $\rho_{\sys\env}$ such that 
its reduced state $\rho_{\ensuremath{ S } } = \mathop{\mathrm{tr}}\nolimits_{\ensuremath{ E } }(\rho_{\sys\env}) = \mathbb{I}_{\ensuremath{ S } }/d_{\ensuremath{ S } }$ is fully mixed. Clearly, any such state satisfies~\eqref{eq:lazyCondition}
and is a lazy state. 

How about states which are merely close to being fully mixed on $\mathcal{H}_{{\ensuremath{ S } }}$? The following lemma captures our
intuition that states which are close to lazy states on $\mathcal{H}_{{\ensuremath{ S } }}$ are in fact pretty lazy themselves.
Closeness it thereby measured in terms of the trace distance~\cite{nielsen&chuang:qc} 
which is the relevant quantity for distinguishing to quantum states~\cite{helstrom:detection}.

\begin{lemma}\label{lem:closeLazy}
	Consider a Hamiltonian with interaction strength $\left\|H_{\rm int}\right\|_\infty$. 
	For any quantum state $\rho_{\sys\env}$ on $H_{\sys\env}$ such that its reduced state is $\chi$-close to fully mixed, i.e.,
	$\chi = \|\rho_{\ensuremath{ S } }- \mathbb{I}_{\ensuremath{ S } }/d_{\ensuremath{ S } }\|_1$ where $\chi \leq 1/d_{\ensuremath{ S } }$ with $d_{\ensuremath{ S } } \geq 2$,  
	its entropy rate is bounded by
	\begin{align}
		\left|\frac{d{\ensuremath{ { \rm H} } }({\ensuremath{ S } })}{dt}\right| \leq \left\|H_{\rm int}\right\|_\infty  2 d_{\ensuremath{ S } } \chi\ .
	\end{align}
\end{lemma}
\begin{proof}
	Using~\eqref{eq:vonNeumanRate} we can upper bound the entropy rate by
\begin{align}
&\left|\frac{d{\ensuremath{ { \rm H} } }({\ensuremath{ S } })}{dt}\right| 
\leq \left\|H_{\rm int}\right\|_\infty\left\|\left[\log(\rho_S)\otimes\mathbb{I}_E,\rho_{SE}\right]\right\|_1 \label{eq:step1Bound}
\\&= \left\|H_{\rm int}\right\|_\infty\left\|\left[\left(\log(\rho_S)-\log(\frac{\id\ssys}{\dS})\right)\otimes\mathbb{I}_E,\rho_{SE}\right]\right\|_1 \label{eq:step2Bound}
\\&\leq 2\left\|H_{\rm int}\right\|_\infty\left\|\left(\log(\rho_S)-\log(\frac{\id\ssys}{\dS})\right)\otimes\mathbb{I}_E\right\|_\infty\left\|\rho_{SE}\right\|_1 \label{eq:step3Bound}
\\&= 2\left\|H_{\rm int}\right\|_\infty\left\|\log(\rho_S)-\log(\frac{\id\ssys}{\dS})\right\|_\infty\ , \label{eq:stepLastBound}
\end{align}
where~\eqref{eq:step1Bound} follows from the fact that for any bounded operators $A$ and $B$
\begin{align}\label{normineq}
\left|\mathop{\mathrm{tr}}\nolimits(AB)\right| \leq \mathop{\mathrm{tr}}\nolimits\left|AB\right| = \left\|AB\right\|_1 \leq \left\|A\right\|_1 \left\|B\right\|_\infty\ \ ,
\end{align}
\eqref{eq:step3Bound} follows from the convexity of the L1-norm, and~\eqref{eq:stepLastBound} follows from the definition
of the L1-norm $\|A\|_1 = \mathop{\mathrm{tr}}\nolimits\sqrt{A^\dagger A}$.
Now let $\left\{p_i\right\}_{i=1}^{d_S}$ denote the eigenvalues of $\rho_S$, so 
\begin{align}\label{wantmax}
\left|\frac{d{\ensuremath{ { \rm H} } }({\ensuremath{ S } })}{dt}\right|\leq2\left\|H_{\rm int}\right\|_\infty\cdot\max_{i=1}^{d_S}\left|\log(p_id_S)\right|\ . 
\end{align}
We want to maximize the r.h.s.\ of (\ref{wantmax}) for fixed 
\begin{align}\label{chidef}
\chi=\left\|\rho_S - \frac{\id\ssys}{\dS}\right\|_1=\sum_{i=1}^{d_S}\left|p_i-\frac{1}{d_S}\right|\ .
\end{align} 
Without loss of generality, let $p_1$ denote the smallest eigenvalue and $p_2$ the largest, so $p_1 \leq \frac{1}{d_S} \leq p_2$. The quantity $\left|\log(p_id_S)\right|$ in \eqref{wantmax} is monotously decreasing in $p_i$ if $0\leq p_i \leq \frac{1}{d_S}$ and monotously increasing if $\frac{1}{d_S} \leq p_i \leq 1$. The following procedure therefore allows to increase the r.h.s.\ of (\ref{wantmax}) while keeping $\chi$ constant: For all $3 \leq i \leq d_S$, if $p_i < \frac{1}{d_S}$ replace $p_1 \mapsto p_1 + p_i - \frac{1}{d_S}$ and $p_i \mapsto \frac{1}{d_S}$. For all $3 \leq i \leq d_S$, if $p_i > \frac{1}{d_S}$ replace $p_2 \mapsto p_2 + p_i - \frac{1}{d_S}$ and $p_i \mapsto \frac{1}{d_S}$. We end up with $p_1=\frac{1}{d_S}-\frac{\chi}{2}$, $p_2=\frac{1}{d_S}+\frac{\chi}{2}$, $p_i=\frac{1}{d_S}$ for $3\leq i\leq d_S$. 
For $\chi\geq0$ we have 
\begin{align}
\left|\log(p_1d_S)\right|\geq\left|\log(p_2d_S)\right|
\end{align}
so that
\begin{align}
\left|\frac{d{\ensuremath{ { \rm H} } }({\ensuremath{ S } })}{dt}\right|
&\leq2\left\|H_{\rm int}\right\|_\infty\cdot\left|\log(p_1d_S)\right|
\\&=2\left\|H_{\rm int}\right\|_\infty\cdot\left|\log\left(\left(\frac{1}{d_S}-\frac{\chi}{2}\right)d_S\right)\right|
\\&=2\left\|H_{\rm int}\right\|_\infty\cdot\left(-\log\left(1-\frac{1}{2} d_S\chi\right)\right)\ .\label{eq:boundEnd}
\end{align}
Let us now upper bound the term on the r.h.s. Note that for $0 \leq x \leq \frac{1}{2}$ the function $f(x):=-\log\left(1-x\right)$ is well defined
and convex.
By convexity we thus have $f(x)\leq2f(\frac{1}{2})x=2 x$ on the interval, and hence for $x = (1/2)d_{\ensuremath{ S } }\chi \leq 1/2$ we have
\begin{align}\label{eq:chiUpper}
-\log\left(1-\frac{1}{2} d_{\ensuremath{ S } }\chi\right) \leq d_{\ensuremath{ S } } \chi\ .
\end{align}
Upper bounding~\eqref{eq:boundEnd} using~\eqref{eq:chiUpper} now leads to the claimed result.
\end{proof}

Our claim that almost all states are pretty lazy now follows immediately by combining the two lemmas.
Lemma~\ref{lem:closetoMixed} tells us that the probability that a randomly chosen state $\rho_{\sys\env}$ is $\chi$-close to maximally mixed on $\mathcal{H}_{\ensuremath{ S } }$ 
is extremely high, where $\chi = 2\sqrt{d_{\ensuremath{ S } }/d_{\ensuremath{ E } }}$ and $\chi = 2\sqrt{d_{\ensuremath{ S } }}/\sqrt[3]{d_{\ensuremath{ E } }}$ respectively.
Lemma~\ref{lem:closeLazy} now tells us that for sufficiently large $d_{\ensuremath{ E } }$ such states are indeed pretty lazy. The values for $\varepsilon$ in~\eqref{eq:mainResult} are $d_{\ensuremath{ S } } \chi$.

{\bf Interaction strengths.}
For completeness, we discuss how our bounds can be improved by a more refined measure of interaction strength.
First of all, note that the operators $H_{{\ensuremath{ S } }}$ and $H_{{\ensuremath{ E } }}$ in~\eqref{eq:decomposition} are not unique and freedom in their choice may be used to minimize $\|H_{\rm int}\|_\infty$. Finding the optimal operators $H_{{\ensuremath{ S } }}$ and $H_{{\ensuremath{ E } }}$ forms in general a non-trivial endeavour. Here, we do not deal with this general task but ask how to best define the ``interaction strength'' for a given operator $H_{\rm int}$. Indeed, the decomposition~\eqref{eq:decomposition} does become unique 
if we require
$H_{{\ensuremath{ S } }}$ and $H_{{\ensuremath{ E } }}$ to be traceless and $H_{\rm int}$ to have vanishing partial traces on both ${\ensuremath{ S } }$ and ${\ensuremath{ E } }$ (e.g.\ \cite{linden:fluctuations,lazyPaper}).
We can now measure the interaction strength
as
\begin{align}\label{eq:deltaDef}
	\Delta(H_{\rm int}) := 2 \min_{\lambda \in \mathbb{R}}\|H_{\rm int} - \lambda \mathbb{I}_{SE}\|_{\infty}\ .
\end{align}
First of all, note that shifting all energy levels of a certain system by a constant amount does not affect the dynamics of that system. These only depend on the \emph{differences} between the energy eigenvalues. The quantity $H_{\rm int}$ as defined in the decomposition \eqref{eq:decomposition} is indeed invariant under addition of a multiple of $\mathbb{I}_{{\ensuremath{ S } }{\ensuremath{ E } }}$ to $H_{\sys\env}$. Similarly, we can see from \eqref{eq:vonNeumanRate} that adding a multiple of $\mathbb{I}_{{\ensuremath{ S } }{\ensuremath{ E } }}$ to $H_{\rm int}$ alone does not affect the rate of change of the local entropy. For this reason, the quantity $\Delta(H_{\rm int})$ defined in~\eqref{eq:deltaDef}
provides a more robust measure of the ``interaction strength'' of $H_{\sys\env}$ than $\left\|H_{\rm int}\right\|_{\infty}$, as it is already invariant under a shift of eigenvalues in $H_{\rm int}$. 
From~\eqref{eq:deltaDef} we furthermore see that this quantity can easily be computed using a semidefinite program (SDP)~\cite{boyd:convex} since we may also write
$\Delta(H_{\rm int}) = 2\gamma$ where $\gamma$ is the solution of the following SDP
\begin{sdp}{minimize}{$\gamma$}
	&$\gamma \mathbb{I} \geq H_{\rm int} - \lambda \mathbb{I} \geq -\gamma \mathbb{I}\ ,$
\end{sdp}
where the minimization is taken over variables $\gamma$ and $\lambda$.
Since $\Delta(H_{\rm int})$ equals the difference between the smallest and largest eigenvalue of $H_{\rm int}$ we have $\Delta(H_{\rm int}) \leq 2 \left\|H_{\rm int}\right\|_{\infty}$. An upper bound on the entangling rate which is proportional to $\left\|H_{\rm int}\right\|_{\infty}$ may therefore be strengthened by noting that we may replace $H_{\rm int}$ by $H_{\rm int} - \lambda\mathbb{I}$ without affecting time scales. This allows us to replace $\left\|H_{\rm int}\right\|_{\infty}$ by $\frac{1}{2}\Delta(H_{\rm int})$ in all the bounds if desired.

\section{Discussion}

We have shown that almost all states of the system and the environment are in fact pretty lazy. If the environment $E$ is sufficiently larger than our system $S$ -- which we assume to be the case in physical scenarios -- the vast majority of bipartite states is such that their entropy in $S$ can only be changed at a vanishing rate. The relevant timescale is thereby given by the inverse of the interaction strength $\left\|H_{\rm int}\right\|_\infty$.
Our results should be compared to~\cite{linden:evolution,linden:fluctuations}  in which it
was shown that 
\emph{equilibration} is a generic property of pure states on $\mathcal{H}_{\ensuremath{ S } } \otimes \mathcal{H}_{\ensuremath{ E } }$ if {\ensuremath{ E } } is sufficiently larger than ${\ensuremath{ S } }$. That is, under this conditions almost all joint initial states will lead to the state of {\ensuremath{ S } } being close to its temporal average for most times. Furthermore it is shown in \cite{linden:fluctuations} that for almost all joint initial states, the rate of change of ${\ensuremath{ S } }$ (the speed of the fluctuations around the temporal average, that is)
will on average be small. The time scale that the speed of fluctuations is compared to is here given by $\left\|H_{\sys}\otimes\mathbb{I}_E + H_{\rm int}\right\|_\infty$. While only $H_{\rm int}$ is able to create entanglement between {\ensuremath{ S } } and ${\ensuremath{ E } }$, both $H_{\rm int}$ and $H_{\sys}\otimes\mathbb{I}_E$ are relevant for the evolution of the state of ${\ensuremath{ S } }$. If the rate of change of the state of {\ensuremath{ S } } is low, this implies by Fannes' inequality~\cite{nielsen&chuang:qc} that the rate of change of the von Neumann entropy is low as well. So while the results of \cite{linden:evolution,linden:fluctuations} imply that most initial states will lead to entropy rates on {\ensuremath{ S } } which in a long-time temporal average are low, we show that most bipartite states really \emph{are} such that the entropy rates on {\ensuremath{ S } } are low for any interaction Hamiltonian.


\acknowledgments

We thank Cesar Rodriguez-Rosario for an inspiring talk at CQT, and interesting discussions as well as comments on our draft. This research was supported by the National Research Foundation and the Ministry of Education, Singapore.


