text:
\section{Introduction}

CAT(0) cube complexes are nice examples of CAT(0) spaces that share many similarities with trees. Their combinatoric and geometric properties provide useful tools to study the large class of groups which admit non-trivial actions on CAT(0) cube complexes. Group acting on CAT(0) cube complexes were shown to play an important role in low dimensional topology \cite{Haglund2008,Haglund2012,Wise2012,Agol2013} and in geometric group theory \cite{Niblo1997,Chatterji2010,Cherix2004,Sageev2005,Chatterji2005,Nica2004,Roller1998}.

In \cite{Grigorchuk1980}, Grigorchuk constructed an infinite finitely generated 2-group, now known as the \emph{first Grigorchuk group}, and showed that this group has a lot of exotic properties. Most notably, it is of intermediate growth and hence amenable, but not elementary amenable. This example was generalized in \cite{Grigorchuk1984} to an uncountable family of groups $\{G_\omega\}_{\omega \in \{0,1,2\}^\infty}$, the \emph{Grigorchuk groups}, that have since generated a lot of research. Grigorchuk groups are defined by their action by automorphisms on the infinite binary rooted tree. This action extends naturally to an action by homeomorphisms on the boundary of the tree. Schreier graphs of stabilizers of points in this action have linear structure and have proved an important tool in the study of these groups, see e.g. \cite{Grigorchuk2019,Bartholdi2000,Grigorchuk2018,Vorobets2012}

In \cite{Sageev1995}, Sageev showed that if a finitely generated group has a Schreier graph with more than one end, then one can construct a CAT(0) cube complex on which the group acts without bounded orbit. The way in which the complex and the action are constructed is explicitly described.

It is well known that Grigorchuk groups have Schreier graphs with 2 ends and hence it is tempting to apply Sageev's construction and to investigate the complex that it gives and the action of $G_\omega$ on it for different sequences $\omega$. Recall that the groups $G_\omega$ are not quasi-isometric and may have different algebraic and geometric properties, for example, being torsion's groups or not. Following Sageev's method we are able to construct a complex $\mathcal{X}_\omega$ on which the group $G_\omega$ acts for each sequence $\omega$. It turns out that these complexes are isometric for all $\omega$. In this paper we present a different approach by constructing an unique CAT(0) cube complex $\mathcal{X}$ (isometric to the $\mathcal{X}_\omega$) on which all $G_\omega$ act without bounded orbit, see Theorem~\ref{T:bounded_orbit}.

Moreover, we can strengthen this result for an uncountable subfamily of groups $G_\omega$, including the self-similar examples $G_{(012)^\infty}$ (the first Grigorchuk group which is an infinite 2-group) and $G_{(01)^\infty}$ (which has an element of infinite order), by proving that the actions of these groups on $\mathcal{X}$ are faithful and proper, see Theorem~\ref{T:Properness_action}. Here, an action is called proper if the cubes' stabilizers are all finite.

All Grigorchuk groups $G_\omega$ are amenable and therefore have the Haagerup property \cite{Bekka1995}. It is known that groups with the Haagerup property satisfy the Baum-Connes conjecture \cite{Higson1997}. Recall that this conjecture links the $K$-homology of the classifying space of proper actions of a group $G$ and the $K$-theory of its reduced $C^*$-algebra by conjecturing that the assembly map $\mu_i : RK_i^G( \underline{E}G) \rightarrow K_i (C^*_r(G))$ is an isomorphism for $i = 0,1$, see \cite{Valette2002}. Even if the conjecture is proved for a particular group, it is interesting to be able to explicit the isomorphisms $\mu_i$. For this, we need a model for the classifying space of proper actions $\underline{E}G$. The Theorem~\ref{T:model} shows that the complex $\mathcal{X}$ gives such a model for all the groups $G_\omega$ whose action on $\mathcal{X}$ is proper.

\subsection*{Acknowledgment}
The author is thankful to T. Nagnibeda for the idea of this project, the fruitful discussions and the careful rereading, to R. Grigorchuk, P-H. Leeman, D. Francoeur, P. Bagnoud, A. Vallette, A. Genevois and N. Matte Bon for helpful comments and ideas. The author acknowledge support of the Swiss NSF grants 200020-178828 and 200020-200400.

\section{Definitions}
\subsection{CAT(0) cube complexes}
A cube complex is a space obtained by gluing, via isometries, euclidean cubes with edges of length one. This space can be equiped with a metric induced by that of the cubes. A result of Gromov (see \cite[Theorem II.5.20]{Bridson1999} for the finite dimension and \cite[Theorem 40]{Leary2013} for the general case), ensures that if the complex is simply connected and if the link of each cube does not contain the boundary of a triangle which is not filled\footnote{Sometimes, this condition is formulated as: the link of each vertex does not contain the boundary of a simplex which is not filled. These two conditions are equivalent.}, then the metric verifies the CAT(0) condition. Recall that the link associated to an $n$ cube $C$ in a cube complex is the simplicial complex defined as follows. The vertices are the $n+1$ cubes whose boundary contains $C$. A pair of vertices in the link are joined by an edge if the two corresponding $n+1$ cubes are in the boundary of a common $n+2$ cube. If the resulting 1-complex contains a triangle appears, we fill it if the 3 vertices that form the triangle are in the boundary of a common $n+3$ cube. If the boundary of a $k$-simplex appears, we fill it if the $k+1$ vertices that form the simplex are in the boundary of a $n+k+1$ cube.

A key feature of CAT(0) cube complexes is the existence of hyperplanes. Two edges are said equivalent if they are the opposite sides of a square. An equivalent class of edges is a \emph{hyperplane}. It is also possible to define hyperplanes in a more geometrical way, as the codimension-1 subspace spanned by the midpoints of the edges. In the case of a CAT(0) cube complex, hyperplanes are also CAT(0) cube complexes which split the complex into exactly two connected components called \emph{half-spaces}, see \cite{Sageev1995} for details.
\subsection{Grigorchuk groups}
Let $\mathcal{T}_2$ denote the infinite binary rooted tree. The vertices of this tree can be described as (finite) binary sequences, see Figure~\ref{F:Tbin}.
\begin{figure}
\centering
\include{pictures/Tbin}
\caption{The first levels of $\mathcal{T}_2$}\label{F:Tbin}.
\end{figure}
\begin{defn}
Let $\omega = \omega_1\omega_2\ldots$ be an infinite sequence in $\{0,1,2\}^\infty$. Denote by $\sigma$ the shift on the space $\{0,1,2\}^\infty$. The group $G_\omega$ is the subgroup of the automorphisms group of $\mathcal{T}_2$ generated by the automorphisms $a,b_\omega,c_\omega,d_\omega$ where,
\begin{equation*}
a(0x) = 1x \quad \text{and} \quad a(1x) = 0x
\end{equation*}
for all binary sequences $x$ and
\begin{align*}
b_\omega(0x) &= \begin{cases} 0a(x)	& \text{if } \omega_1 \neq 2 \\ 0x		& \text{if } \omega_1 = 2 \end{cases} & b_\omega(1x) = 1 b_{\sigma \omega}(x) \\
c_\omega(0x) &= \begin{cases} 0a(x)	& \text{if } \omega_1 \neq 1 \\ 0x		& \text{if } \omega_1 = 1 \end{cases} & c_\omega(1x) = 1 c_{\sigma \omega}(x) \\
d_\omega(0x) &= \begin{cases} 0a(x)	& \text{if } \omega_1 \neq 0 \\ 0x		& \text{if } \omega_1 = 0 \end{cases} & d_\omega(1x) = 1 d_{\sigma \omega}(x).
\end{align*}
\end{defn}
The actions of these generators can be represented graphically, see Figure~\ref{F:def_grig_group}.
\begin{figure}
\centering
\include{pictures/def_grig_group}
\caption{The actions of the generators, where
$
I_\omega^i =
\begin{cases}
1	& \omega_1 = i \\
a	& \omega_1 \neq i
\end{cases}.
$}\label{F:def_grig_group}
\end{figure}

Let $v$ be a vertex of the $n$\th{} level of the tree, we can look at the action of an element $g \in G_\omega$ on the subtree rooted at $v$. This subtree is isometric to $\mathcal{T}_2$ and then, we can define a new automorphism, which is an element of $G_{\sigma^n \omega}$. This automorphism is called the \emph{restriction of $g$ on $v$} and is denoted by $g_v$.

An element $g$ \emph{stabilizes the $n$\th{} level} if $g(v) = v$ for every $v$ on the $n$\th{} level. The subgroup of all such elements in $G_\omega$ is $\stab_\omega(n)$, the \emph{stabilizer of the $n$\th{} level}. For an element $g$ in $\stab_\omega(1)$, we denote by $(g_0,g_1)$ the two restrictions of $g$ on the two subtrees rooted at the first level. If $g$ does not stabilize the first level, we can still write it as $a(g_0,g_1)$. We refer the reader to the section VII of \cite{DelaHarpe2000} and \cite{Grigorchuk2006} for a detailed general exposition of these concepts.
\section{Construction of the cube complex $\mathcal{X}$}
\subsection{The graph $\Gamma$}
The action of $G_\omega$ can be extended to $\partial \mathcal{T}_2$, the boundary of the tree. The points of the boundary $\partial \mathcal{T}_2$ of the tree can be seen as one-sided binary sequences. Consider the orbit of $0^\infty$ and define $\Gamma_\omega$ as the graph with $G_\omega 0^\infty$ as vertices and $\{(g0^\infty, sg0^\infty) : g \in G_\omega, s \in \{a, b_\omega, c_\omega, d_\omega \}\}$ as edges. This is the orbital graph of the action of $G_\omega$ on $0^\infty$. These graphs are not isomorphic if we look at them as labelled\footnote{By labelled graph we mean a graph with edges labelled by generators $\{a,b_\omega,c_\omega,d_\omega\}$.} graphs, but the underlying unlabelled graphs are the same.
\begin{prop}
The vertices of the graphs $\Gamma_\omega$ are exactly the infinite sequences with a finite number of $1$. Moreover all the $\Gamma_\omega$ are equal as unlabelled graphs.
\end{prop}
\begin{proof}
These facts are proved in \cite{MatteBon2015} for the orbital graph of $1^\infty$, but the proofs can be translated for our case directly.
\end{proof}
We will denote by $\Gamma$ the underlying unlabelled graph and by $V(\Gamma)$ its set of vertices.
\begin{figure}
\centering
\include{pictures/graph_gamma_label}
\caption{Some examples of graphs $\Gamma_\omega$ and the graph $\Gamma$. The red edge are labelled by $a$, the blue by $b_\omega$, the green by $c_\omega$ and the orange by $d_\omega$.}
\label{F:Graphe_gamma}
\end{figure}
\subsection{The cube complex $\overline{\mathcal{X}}$}
We begin by defining a huge cube complex $\overline{\mathcal{X}}$ with an action of $G_\omega$ on it. This space is not CAT(0) and the actions do not preserve the connected components in general. However, each connected component is a CAT(0) cube complex.

The set $\overline{\mathcal{V}}$ of vertices of $\overline{\mathcal{X}}$, is the set of the 2-coloring of vertices of $\Gamma$, i.e.
\begin{equation*}
\overline{\mathcal{V}}  = \{ v : V(\Gamma) \rightarrow \field{Z}/2\field{Z} \}= \left(\field{Z}/2\field{Z}\right)^{V(\Gamma)}.
\end{equation*}
A vertex $v \in \overline{\mathcal{V}}$ can be seen as a subset of vertices of $\Gamma$ whose image is equal to $1$. This set is called the \emph{support} of $v$.

We provide $\overline{\mathcal{V}}$ with a graph structure by adding an edge between two vertices $v$ and $w$  if and only if the symmetric difference of their support contains exactly one element, or in other words, if they are equal on all the vertices except one. Such an edge is labelled by the vertex of $\Gamma$ on which the two functions differ. The neighborhood of a vertex $v$ is all the vertices of the form $v+\delta_{x}$, for $x \in \Gamma$, where
\begin{equation*}
\delta_{x }(y) =
\begin{cases}
1 & x = y\\
0 & x \neq y
\end{cases}
\end{equation*}
We can further add the higher dimensional cubes inductively. Whenever the boundary of a square appears, we fill it. After filling all the squares, we do the same for the cubes, then for the cubes of dimension $4$, etc. In short, we add a $n$-cube at each appearance of its boundary. The resulting space is called $\overline{\mathcal{X}}$.

The combinatorial structure of the cubes of $\overline{\mathcal{X}}$ can be explicitly described. A subset $\{x_1, \ldots, x_n\}$ of $V(\Gamma)$ and a coloring $v$ of $V(\Gamma)$ define an unique $n$ cube in $\overline{\mathcal{X}}$ whose vertices are the colorings which differ from the coloring $v$ only on vertices in $\{x_1, \ldots, x_n\}$. We will denote this cube by $\mathscr{C}(v, x_1, x_2, \ldots,  x_n)$. Every cube of $\overline{\mathcal{X}}$ can be described in this way. The Figure~\ref{F:span} illustrates this characterization with an example.

\begin{figure}
\centering
\include{pictures/span_cubes}
\caption{The $3$-cube $\mathscr{C}(v, x_1, x_2, x_3)$.} \label{F:span} \label{F:description_cube}
\end{figure}

As $V(\Gamma)$ is infinite, the set $\overline{\mathcal{V}}$, which is equal to $(\field{Z}/2\field{Z})^{V(\Gamma)}$, is uncountable. Moreover, $\overline{\mathcal{X}}$ is of infinite dimension, is not locally finite and is not connected. Indeed, there is no finite path between two vertices (= colorings) which differ on an infinite number of vertices of $\Gamma$. However, it turns out that each connected component of $\overline{\mathcal{X}}$ is a CAT(0) cube complex.
\begin{prop}
Each connected component of $\overline{\mathcal{X}}$ is a CAT(0) cube complex.
\end{prop}
\begin{proof}
Let $Y$ be a connected component of $\overline{\mathcal{X}}$. To prove that this component is a CAT(0) complex, we will verify that it is simply connected and that the link of each cell does not contain the boundary of a triangle which is not filled.

Let $\gamma$ be a closed path passing through the vertices $v_1, v_2, \ldots, v_n$ of $Y$. By the description of a neighborhood, there exist $x_1, \ldots, x_n$, some vertices of $\Gamma$, such that $v_i = v_{i-1} + \delta_{x_i}$ for $i = 2, \ldots, n$ and $v_1 = v_n + \delta_{x_n}$. Using the description of cubes above, all these vertices are vertices of a cube of dimension at most $n$ and thus $\gamma$ is contractible.

Let $C$ be an $n$-cube of $Y$ and let $C_1,C_2$ and $C_3$ be $(n+1)$-cubes which form a triangle in the link of $C$. By definition of the link, $C$ is in the boundary of the cubes $C_1$, $C_2$ and $C_3$. Using the characterization of the cubes above, if $C = \mathscr{C}(v, x_1, \ldots, x_n)$, there exist $ x'_1, x'_2, x'_3$, some different vertices of $\Gamma$, such that $C_i = \mathscr{C}(v, x_1, \ldots, x_n,  x'_i)$. If we consider $\mathscr{C}(v, x_1, \ldots, x_n,  x'_1, x'_2, x'_3)$, we can see that it is an $(n+3)$-cube of $Y$ which contains $C_1,C_2$ and $C_3$ on its boundary. Then, the triangle in the link is filled.
\end{proof}
Using the action of a group $G_\omega$ on the orbit of $0^\infty$, we define an action of $G_\omega$ on $\overline{\mathcal{V}}$: %
\begin{equation*}
(gv)(x) = v(g^{-1}x )
\end{equation*}
for a vertex $v$ of $\overline{\mathcal{V}}$, an element $g$ of $G_\omega$ and a vertex $x$ of $\Gamma$. This action can be extended to an action on $\overline{\mathcal{X}}$ which preserves the structure of cube complex. Generally, it is not true that the connected components of $\overline{\mathcal{X}}$ are preserved by the action. For example, consider the vertex $v$ of $\overline{\mathcal{X}}$ which colors in black the set $L$ composed of the vertices of $\Gamma$ which are left ends of edges labelled by $a$ and only these ones. Then, $av$ will color in black the complement of $L$ in $V(\Gamma)$ and the distance between $v$ and $av$ will be infinite.
\subsection{The CAT(0) cube complex $\mathcal{X}$.}
We will now show that a particular connected component of $\overline{\mathcal{X}}$ is preserved under the action of $G_\omega$. Let $\Gamma_+$ be the set of vertices of $\Gamma$ including $0^\infty$ and all the vertices on the right side of it, see Figure~\ref{F:gamma_+}.
\begin{figure}
\centering
\include{pictures/graph_gamma}
\caption{The black vertices are elements of $\Gamma_+$} \label{F:gamma_+}
\end{figure}
The vertices of $\Gamma_+$ can be described in an explicit way as follows,
\begin{prop}\label{P:Gamma_description}
The set $\Gamma_+$ consists of $0^\infty$ and all the vertices of the form $x = x_1 x_2 \ldots x_n 0^\infty$ with $x_n = 1$ and $n$ odd.
\end{prop}
\begin{proof}
Let us begin by an observation. Given any element $x$ of $\{0,1\}^\infty$, a generator $s$ of $G_\omega$ can only acts non-trivially in two ways:
\begin{itemize
\item flip the first digit of $x$ (if $s = a$)
\item flip the digit after the first apparition of $0$ (if $s \in \{b_\omega,c_\omega,d_\omega\}$)
\end{itemize}
It is straightforward to see that these two moves do not change the parity of the position of the last $1$, except for $0^\infty$ and $10^\infty$. We constructed $\Gamma$ in a way to have $10^\infty$ on the right of $0^\infty$, and then, the parity of the position of the last $1$ is the same as for $10^\infty$ for all the other vertices on the right of $0^\infty$.
\end{proof}
\begin{defn}\label{D:v0}
The cube complex $\mathcal{X}$ is the connected component of $\overline{\mathcal{X}}$ which contains the vertex $v_0$:
\begin{equation*}
v_0(x)  = \chi_{\Gamma_+} (x) =
\begin{cases}
1 & x \in \Gamma_+ \\
0 & x \notin \Gamma_+
\end{cases}.
\end{equation*}
\end{defn}
We denote by $\mathcal{V}$ the set of vertices of $\mathcal{X}$. Let us show that the CAT(0) cube complex $\mathcal{X}$ is preserved by the action of every $G_\omega$.
\begin{prop}\label{P:Stab_connected_comp}
Let $v$ be a vertex of $\mathcal{X}$ and $g$ an element of $G_\omega$. Then $gv$ is a vertex of $\mathcal{X}$.
\end{prop}
\begin{proof}
By the construction of $\mathcal{V}$, it is sufficient to prove that $\Gamma_+ \Delta g\Omega$ is finite, where $\Omega$ is the support of $v$.

The first thing to show is that  $\Gamma_+ \Delta g\Gamma_+$ is finite for every $g$. Let $Y=\Gamma_+ \setminus g\Gamma_+= \{ x \in \Gamma : x \in \Gamma_+ \text{ and } gx \notin \Gamma_+ \}$. If $n$ is the length of $g$ with respect to the standard generating set, a vertex $x$ in $Y$ must be at distance at most $n$ of $\Gamma_+^c$ in the graph $\Gamma$. As this graph is locally finite, $Y$ is necessarily finite. Similarly $g\Gamma_+ \setminus \Gamma_+$ is finite and so $\Gamma_+ \Delta g\Gamma_+$ is also finite.

We need the following property of symmetric differences. Let $A,B,C$ be three sets. If $|A \Delta B|$ and $|B \Delta C|$ are finite, then $|A \Delta C|$ is finite.

The set $\Gamma_+ \Delta g\Gamma_+$ is finite by the first part of the proof. The action of $G_\omega$ preserves the size of the subsets of $\Gamma$, then $|g\Gamma_+ \Delta g\Omega| = |g(\Gamma_+ \Delta \Omega)| = |\Gamma_+ \Delta \Omega|$ which is finite because $v_0$ and $v$ are connected. Combining these equalities and using the property above, we prove that $|\Gamma \Delta g\Omega|$ is finite.
\end{proof}
\begin{cor}
The action $G_\omega \actsGroup \overline{\mathcal{X}}$ can be restricted to an action $G_\omega \actsGroup \mathcal{X}$, for every $\omega \in \{0,1,2\}^\infty$.
\end{cor}
\begin{rem}
To find this invariant connected component, it is crucial to have a Schreier graph that has at least 2 ends. Indeed, this allows us to construct this function whose support cannot be moved too much by the action of the group. Such a set is sometimes called a \emph{commensurated subset},\cite{Cornulier2013}.
\end{rem}
We next describe the hyperplanes of $\mathcal{X}$ and understand the induced action of the groups $G_\omega$ on the set $\mathfrak{H}(\mathcal{X})$ of hyperplanes of $\mathcal{X}$.
\begin{prop}
There exists a bijection $\phi : \Gamma \rightarrow \mathfrak{H}(\mathcal{X})$. More precisely, a hyperplane is composed of all the edges with the same label $x$, where $x$ is a vertex of $\Gamma$.
\end{prop}
\begin{proof}
We will show that for each hyperplane $\mathfrak{h}$ there exists a vertex $x$ in $\Gamma$ such that the edges forming $\mathfrak{h}$ are exactly those labelled by $x$. It follows directly from the description of the squares, see Figure~\ref{F:description_cube}, that two opposite sides have the same label. Now let $e$ and $f$ be two edges with the same label $x$. We denote by $v$ (resp. by $w$) the vertex of $\mathcal{X}$ which is the endpoint of $e$ (resp. of $f$) such that $v(x) = 0$ (resp. $w(x) =0$ ). The complex is connected, hence there exists a path from $v$ to $w$ and we can construct a sequence of adjacent squares, see Figure~\ref{F:proof_hyperplanes}, and then $v$ and $w$ are on the same hyperplane.
\begin{figure}
\centering
\include{pictures/proof_hyperplanes}
\caption{In bold, the sequence of equivalent edges between $e$ and $f$} \label{F:proof_hyperplanes}
\end{figure}
\end{proof
\begin{prop}
The action $G_\omega \actsGroup \mathcal{X}$ induces an action $ G_\omega \actsGroup \mathfrak{H}(\mathcal{X})$ which is transitive, for every $\omega$.
\end{prop}
\begin{proof}
Using the previous proposition, each hyperplane can be labelled by a vertex $x$ of $\Gamma$ and this is why we will denote them by $\mathfrak{h}_x$. A hyperplane $\mathfrak{h}_{x}$ splits $\mathcal{X}$ into 2 half-spaces $\mathfrak{h}_{x}^0$ and $\mathfrak{h}_{x}^1$:
\begin{align*}
\mathfrak{h}_{x}^0 &= \{v \in \mathcal{V} : v(x) = 0 \} \\
\mathfrak{h}_{x}^1 &= \{v \in \mathcal{V} : v(x) = 1 \}.
\end{align*}
Let $g$ be an element of $G_\omega$ and $v$ a vertex in $\mathfrak{h}_{x}^0$. As $(gv)(gx) = v(x) = 0$, then $gv$ is in $\mathfrak{h}_{gx}^0$. Similarly, $gv$ is in $ \mathfrak{h}_{g'x}^1$ for each $v$ in $\mathfrak{h}_{x}^1$. These images of half spaces lead us to define the induced action $G_\omega \actsGroup \mathfrak{H}(\mathcal{X})$ as
\begin{equation*}
g\mathfrak{h}_{x}=\mathfrak{h}_{gx}.
\end{equation*}
The proof of transitivity follows directly from the definition of the action and the transitivity of $G_\omega \actsGroup \Gamma$.
\end{proof}
\begin{rem}
Recall that hyperplanes split CAT(0) cube complexes into two connected components. The action of $G_\omega$ cannot swap these components delimited by an hyperplane. Indeed, for every $g$ in $G_\omega$ there exist vertices $v$ and $w$ in $\mathfrak{h}_{x}^0$ such that $v(gx) = 0$ and $w(gx) = 1$ and then $g\mathfrak{h}_{x}^0 \not\subset \mathfrak{h}_{x}^1$.\end{rem}

\section{Properties of the action of $G_\omega$ on $\mathcal{X}$}
In this section we will prove our main results. Namely, we will first show that every action $G_\omega \actsGroup \mathcal{X}$ has no bounded orbit. Then, we will prove that this action is also proper and faithful for a uncountable subfamily of groups $G_\omega$.
\subsection{Bounded orbits and fixed points}
The proofs of this section are inspired by \cite{Niblo1998}.
\begin{prop}
Let $\omega$ be a sequence in $\{0,1,2\}^\infty$. Then, the action $G_\omega \actsGroup \mathcal{X}$ does not fix any vertex.
\end{prop}
\begin{proof}
Suppose by contradiction that there exists a vertex $v$ of $\mathcal{X}$ which is fixed by the action of $G_\omega$. We will show that then $\Gamma_+$ or $\Gamma_+^c$ has to be finite which is obviously not the case.

Suppose that $v(0^\infty) = 0$. Then for every $g$ in $G_\omega$, we have
\begin{equation*}
gv = v \Rightarrow v(g^{-1}0^\infty) = v(0^\infty) = 0.
\end{equation*}
As the action of $G_\omega$ on the vertices of $\Gamma$ is transitive, $v(x)=0$ for every vertex $x$ of $\Gamma$. The distance between $v$ and $v_0$ is equal to the size of the symmetric difference between $\supp v$ and $\supp v_0$. As $v$ is a vertex of $\mathcal{V}$, this distance must be finite and then $\supp v_0$, which is equal to $\Gamma_+$, must be finite.

In the same way, if $v(0^\infty) = 1$, then $v(x)= 1$ for every vertex of $\Gamma$ and, therefore, $\Gamma_+^c$ is finite.
\end{proof}
\begin{prop}
Let $\omega$ be a sequence in $\{0,1,2\}^\infty$. If the action $G_\omega \actsGroup \mathcal{X}$ has a bounded orbit, then $G_\omega$ fixes a vertex of $\mathcal{X}$.
\end{prop}
\begin{proof}
We want to show that every bounded orbit has a center which will be a fixed point. The cube complex $\mathcal{X}$ is not locally finite and is therefore not a complete metric space. Then, the existence of such a center can not be proved using the usual general results and finding this point requires some work. To do this, we will embed $\mathcal{X}$ in a Hilbert space, where every bounded orbit has a center, and prove that this point comes from a vertex of $\mathcal{X}$.

We consider the Hilbert space of square summable functions on the vertices of $\Gamma$:
\begin{equation*}
\ell^2(V(\Gamma)) = \{ \phi : \Gamma \rightarrow \field{R} : \sum_{x \in V(\Gamma)} \phi(x) ^2 < \infty \}.
\end{equation*}
The set $\mathcal{V}$ of vertices of $\mathcal{X}$ is also a set of functions on the vertices of $\Gamma$ and can be embeded isometrically in $\ell^2(V(\Gamma))$ via
\begin{align*}
p: \mathcal{V} &\rightarrow \ell^2(V(\Gamma)) \\
	v &\mapsto p_v
\end{align*}
where $v_0$ is the vertex defined in Definition~\ref{D:v0} and
\begin{equation*}
p_v(x)=
\begin{cases}
1 & \text{if }v_0(x) \neq v(x) \\
0 & \text{if }v_0(x) = v(x)
\end{cases}.
\end{equation*}
We define the action $G_\omega \actsGroup \ell^2(V(\Gamma))$ as
\begin{equation*}
(g\phi)(x) =
\begin{cases}
\phi(g^{-1} x) &\text{if } v_0(g^{-1}x) = v_0(x) \\
1- \phi(g^{-1} x) &\text{if } v_0(g^{-1}x) \neq v_0(x)
\end{cases}
\end{equation*}
for every $g \in G_\omega$, $\phi \in \ell^2(V(\Gamma))$ and $x \in \Gamma$. It can be checked that this action is compatible with the action $G_\omega \actsGroup \mathcal{V}$, in the sense that $g p_v = p_{gv}$.

A bounded set of a Hilbert space has a unique center, see \cite[Lemma 2.2.7]{Bekka2008}.
Suppose that there exists a bounded orbit in $\mathcal{X}$. The embedding $p$ sends this orbit to a bounded orbit in $\ell^2(V(\Gamma))$ of which the center $c$ is a fixed point. We claim that, as the action of $G_\omega$ on the vertices $\Gamma$ is transitive, this function $c$ can only take values in $\{0,1\}$. Indeed, for every pair $x$ and $y$ of vertices of $\Gamma$, there exists an element $g$ of $G_\omega$ such that $g^{-1}x = y$. If $v_0(x) = v_0(y)$, the equality $c(x) = gc(x)$ implies that $c(x) = c(y)$. If $v_0(x) \neq v_0(y)$, then
\begin{equation*}
c(x) = gc(x) = 1-c(g^{-1}x) = 1-c(y).
\end{equation*}
In the same way, we can show that $c(y) = 1-c(x)$ and then $c(x)$ and $c(y)$ take values only in $\{0,1\}$.

By the definition of $\ell^2(V(\Gamma))$, all but finitely many values of $c$ must be equal to $0$. Then, using the definition of $p$, the point $c$ is the image of the vertex of $\mathcal{X}$ which is equal to $v_0$ everywhere except on the vertices of $\Gamma$ where c is equal to 1.
\end{proof}
The two previous propositions imply the following theorem.
\begin{thm}\label{T:bounded_orbit}
Let $\omega$ be a sequence in $\{0,1,2\}^\infty$. Then, the action $G_\omega \actsGroup \mathcal{X}$ does not have a bounded orbit for every $\omega$.
\end{thm}
\subsection{Properness and faithfulness}

In this section, we will consider the subfamily of Grigorchuk groups $G_\omega$ which are indexed by sequences $\omega$ in $\{0,1,2\}^\infty$ without repetition, i.e. $\omega_i \neq \omega_{i+1}$ for every $i$, and prove that their action on $\mathcal{X}$ is proper and faithful. This subfamily contains uncountably many groups.

The idea of the proof is to show that stabilizing a subset of vertices of $\Gamma$ is a strong condition and only few elements of $G_\omega$ succeed. We will decompose this proof into different steps. Firstly, we will compute the stabilizers of two particular subsets of $V(\Gamma)$. Secondly, we will show that the stabilizer of a subset of vertices of $\Gamma$ which has a finite symmetric difference with the two particular subsets, is also finite. Finally, we will explain how these stabilizers of subsets can be related with the stabilizers of the cubes of $\mathcal{X}$.

Let us begin by defining precisely the notion of stabilizer of a subset of $V(\Gamma)$ and the 2 particular subsets that we will study.
\begin{defn}
Let $\Omega \subset V(\Gamma)$ be a subset of vertices of $\Gamma$ and $G_\omega$ a Grigorchuk group. The stabilizer of $\Omega$ for the action of $G_\omega \actsGroup \mathcal{X}$ is
\begin{equation*}
\stab_\omega(\Omega) = \{g \in G_\omega : g\Omega = \Omega \}.
\end{equation*}
\end{defn}
\begin{defn}
Two particular subsets of $V(\Gamma)$ that we will study are $\Gamma_+$, which we have already described above, and $\widetilde{\Gamma}_+ = \Gamma_+ \setminus \{0^\infty \}$.
\end{defn}
The following lemma explains the behavior of the vertices of $\Gamma$ between these two subsets when we add a digit at the beginning of their binary writing.
\begin{lem} \label{L:prefix_gamma}
Let $x$ be a vertex of $\Gamma$. Then
\begin{itemize}
\item $x \in \Gamma_+ \Leftrightarrow 0 x \in \widetilde{\Gamma}_+^c$.
\item $x \in \Gamma_+^c \Leftrightarrow 0x \in \widetilde{\Gamma}_+$.
\item $x \in \Gamma_+^c \Rightarrow 1x \in \widetilde{\Gamma}_+$.
\item $x \in \widetilde{\Gamma}_+ \Leftrightarrow 0x \in \Gamma_+^c$.
\item $x \in \widetilde{\Gamma}_+ \Leftrightarrow 1x \in \widetilde{\Gamma}_+^c$.
\item $x \in \widetilde{\Gamma}_+^c \Leftrightarrow 0x \in \Gamma_+$.
\item $x \in \widetilde{\Gamma}_+^c \Leftrightarrow 1x \in \widetilde{\Gamma}_+$.
\end{itemize}
\end{lem}
\begin{proof}
We proved in Proposition~\ref{P:Gamma_description} that $\Gamma_+$ can be fully described by looking at the parity of the position of the last digit 1. This allows us to give the following descriptions:
\begin{align*}
\Gamma_+ &= \{0^\infty\} \cup \{x_1x_2 \ldots x_{2n+1}0^\infty\} \\
\Gamma_+^c &= \{x_1x_2 \ldots x_{2n}0^\infty\} \\
\widetilde{\Gamma}_+ &=  \{x_1x_2 \ldots x_{2n+1}0^\infty \} \\
\widetilde{\Gamma}_+^c &=  \{0^\infty\} \cup\{x_1x_2 \ldots x_{2n}0^\infty \}
\end{align*}
where every last $x_i$ is equal to $1$. Every point of the lemma can be proved by direct computations using these descriptions.
\end{proof}
We will study the restrictions of elements of $\stab_\omega(\Gamma_+)$ and $\stab_\omega(\widetilde{\Gamma}_+)$ on the first level of the tree.
\begin{lem}\label{L:stab_projections}
Let $\omega$ be a sequence in $\{0,1,2\}^\infty$ and $g$ an element of $G_\omega$.
\begin{enumerate}
\item If $g$ stabilizes the first level of the tree and $g \in \stab_\omega(\Gamma_+)$, then $g_0,g_1 \in \stab_{\sigma\omega}(\widetilde{\Gamma}_+)$.
\item If $g$ stabilizes the first level of the tree and $g \in \stab_\omega(\widetilde{\Gamma}_+)$, then $g_0 \in \stab_{\sigma\omega}(\Gamma_+)$ and $g_1 \in \stab_{\sigma\omega}(\Gamma_+) \cap \stab_{\sigma\omega}(\widetilde{\Gamma}_+)$.
\item If $g$ does not stabilize the first level of the tree and $g \in \stab_\omega(\widetilde{\Gamma}_+)$, then $g_0,g_1 \in \stab_{\sigma\omega}(\Gamma_+) \cap \stab_{\sigma\omega}(\widetilde{\Gamma}_+)$.
\end{enumerate}

\end{lem}
\begin{proof}
Let's start by noting that an element $g$ is in $\stab_\omega(\Gamma_+)$ (resp. $\stab_\omega(\widetilde{\Gamma}_+)$) if and only if it is in $\stab_\omega(\Gamma_+^c)$ (resp. $\stab_\omega(\widetilde{\Gamma}_+^c)$). The proof is an application of Lemma~\ref{L:prefix_gamma}.
\begin{enumerate}
\item
Let $g$ be an element of $\stab_\omega(\Gamma_+)$ which stabilizes the first level of the tree. Then,
\begin{align*}
x \in \widetilde{\Gamma}_+ 	& \Rightarrow  0 x \in \Gamma_+^c \\
						& \Rightarrow  g(0 x) \in \Gamma_+^c \\
						& \Rightarrow  0g_0(x) \in \Gamma_+^c \\
						& \Rightarrow  g_0(x) \in \widetilde{\Gamma}_+ \\
						& \Rightarrow  g_0 \in \stab_{\sigma\omega}(\widetilde{\Gamma}_+),
\end{align*}
\begin{align*}
x \in \widetilde{\Gamma}_+^c 	& \Rightarrow  1 x \in \Gamma_+ \\
							& \Rightarrow  g(1 x) \in \Gamma_+ \\
							& \Rightarrow  1g_1(x) \in \Gamma_+ \\
							& \Rightarrow  g_1(x) \in \widetilde{\Gamma}_+^c  \\
							& \Rightarrow  g_1 \in \stab_{\sigma\omega}(\widetilde{\Gamma}_+),
\end{align*}
\item
Let $g$ be an element of $\stab_\omega(\widetilde{\Gamma}_+)$ which stabilizes the first level of the tree. Then,
\begin{align*}
x \in \Gamma_+^c 	&\Rightarrow  0 x \in \widetilde{\Gamma}_+ \\
					&\Rightarrow  g(0 x) \in \widetilde{\Gamma}_+ \\
					&\Rightarrow  0g_0(x) \in \widetilde{\Gamma}_+ \\
					&\Rightarrow  g_0(x) \in \Gamma_+^c \\
					&\Rightarrow  g_0 \in \stab_{\sigma\omega}(\Gamma_+),
\end{align*}
\begin{align*}
x \in \widetilde{\Gamma}_+ & \Rightarrow  1 x \in \widetilde{\Gamma}_+^c \\
						&\Rightarrow  g(1 x) \in \widetilde{\Gamma}_+^c  \\
						& \Rightarrow  1g_1(x) \in \widetilde{\Gamma}_+^c \\
						& \Rightarrow  g_1(x) \in \widetilde{\Gamma}_+ \\
						& \Rightarrow  g_1 \in \stab_{\sigma\omega}(\widetilde{\Gamma}_+),
\end{align*}
\begin{align*}
x \in \Gamma_+^c	&\Rightarrow 1x \in \widetilde{\Gamma}_+ \\
					&\Rightarrow g(1x) \in \widetilde{\Gamma}_+ \\
					&\Rightarrow 1g_1(x) \in \widetilde{\Gamma}_+ \\
					&\Rightarrow g_1(x) \in \Gamma_+^c \\
					&\Rightarrow g_1 \in \stab_{\sigma\omega}(\Gamma_+).
\end{align*}
\item
Let $g$ be an element of $\stab_\omega(\widetilde{\Gamma}_+)$ which does not stabilize the first level of the tree. Then,
\begin{align*}
x \in \Gamma_+			&\Rightarrow	0x \in \widetilde{\Gamma}_+^c \\
							&\Rightarrow	g(0x) \in \widetilde{\Gamma}_+^c \\
							&\Rightarrow	1g_0(x) \in \widetilde{\Gamma}_+^c \\
							&\Rightarrow 	g_0(x) \in \widetilde{\Gamma}_+ \subset \Gamma_+ \\
							&\Rightarrow  	g_0 \in \stab_{\sigma\omega}(\Gamma_+),
\end{align*}
\begin{align*}
x \in \widetilde{\Gamma}_+\subset \Gamma_+		&\Rightarrow	0x \in \widetilde{\Gamma}_+^c \\
							&\Rightarrow	g(0x) \in \widetilde{\Gamma}_+^c \\
							&\Rightarrow	1g_0(x) \in \widetilde{\Gamma}_+^c \\
							&\Rightarrow	g_0(x) \in \widetilde{\Gamma}_+ \\
							&\Rightarrow  	g_0 \in \stab_{\sigma\omega}(\widetilde{\Gamma}_+),
\end{align*}
\begin{align*}
x \in \Gamma_+^c			&\Rightarrow	1x \in \widetilde{\Gamma}_+ \\
							&\Rightarrow	g(1x) \in \widetilde{\Gamma}_+ \\
							&\Rightarrow	0g_1(x) \in \widetilde{\Gamma}_+ \\
							&\Rightarrow 	g_1(x) \in \Gamma_+^c \\
							&\Rightarrow  	g_1 \in \stab_{\sigma\omega}(\Gamma_+),
\end{align*}
\begin{align*}
x \in \widetilde{\Gamma}_+^c 	&\Rightarrow	1x \in \widetilde{\Gamma}_+ \\
							&\Rightarrow	g(1x) \in \widetilde{\Gamma}_+ \\
							&\Rightarrow	0g_1(x) \in \widetilde{\Gamma}_+ \\
							&\Rightarrow	g_1(x) \in \widetilde{\Gamma}_+^c \\
							&\Rightarrow  	g_1 \in \stab_{\sigma\omega}(\widetilde{\Gamma}_+). \\
\end{align*}
\end{enumerate}
\end{proof}

We recall the following classical lemma which gives an estimate of the size of restrictions of an element which fixes the first level of the tree.
\begin{lem} \label{L:reduction_lemma}
Let $\omega$ be a sequence in $\{0,1,2\}^\infty$ and $g$ an element of $G_\omega$ which stabilizes the first level of the tree. Then,
\begin{equation*}
l(g_i) \leq \frac{l(g) +1}{2} \quad i =1,2
\end{equation*}
where $l(g)$ is the length of $g$ with respect to the corresponding generating set.
\end{lem}
\begin{proof}
It is a straightforward generalization of \cite[Lemma VIII.46]{DelaHarpe2000}.
\end{proof}

We can now compute explicitly the stabilizers of $\Gamma_+$ and $\widetilde{\Gamma}_+$.
\begin{prop} \label{P:stab_gamma}
Let $\omega$ be a sequence without repetition. Then,
\begin{align*}
\stab_\omega(\Gamma_+) &= \gen{a,u_{\omega,1}} \iso D_8\\
\stab_\omega(\widetilde{\Gamma}_+) &= \gen{b_\omega,c_\omega,d_\omega} \iso \field{Z}/2\field{Z} \times \field{Z}/2\field{Z}
\end{align*}
where $u_{\omega,1}$ is the generator in $\{b_\omega,c_\omega,d_\omega\}$ such that $u_{\omega,1} 0^\infty = 0^\infty$, or explicitly,
\begin{equation*}
u_{\omega,1} =
\begin{cases}
b_\omega & \omega_1 = 2 \\
c_\omega & \omega_1 = 1 \\
d_\omega & \omega_1 = 0
\end{cases}.
\end{equation*}
\end{prop}
\begin{proof}
We will proceed by induction on the length of the elements of the group  in parallel for the 2 parts of the proposition and all the sequences $\omega$ without repetition.

Let us begin by some notations. We denote by $u_{\omega,n}$ the generator in $\{b_\omega,c_\omega,d_\omega\}$ such that $u_{\omega,n}1^n0^\infty = 1^n0^\infty $, or explicitly,
\begin{equation*}
u_{\omega,n} =
\begin{cases}
b_\omega & \omega_n = 2 \\
c_\omega & \omega_n = 1 \\
d_\omega & \omega_n = 0
\end{cases}.
\end{equation*}
For the initial step, we verify the statements by hand for the elements of length smaller or equal to 2. It is clear that $\gen{a,u_{\omega,1}} \subset \stab_\omega(\Gamma_+)$ and $\gen{b_\omega,c_\omega,d_\omega} \subset \stab_\omega(\widetilde{\Gamma}_+)$. For all the other elements, an example of a vertex that comes out of the sets is given in Figure~\ref{F:induction_stab_Gamma}.
\begin{figure}
\centering
\begin{subfigure}{1\textwidth}
\centering
\include{pictures/graph_gamma_induction}
\caption{The neighborhood of $0^\infty$ on the graph $\Gamma_\omega$. The third generators which is not $u_{\omega,1} or u_{\omega,2}$ is denoted by $u$.}
\end{subfigure}
\par\bigskip
\begin{subfigure}[b]{0.4\textwidth}
\centering
\begin{tabular}{c | c}
$u_{\omega,2}$ 	& $0^\infty$ \\ \hline
$w$ 	& $0^\infty$ \\ \hline
$au_{\omega,2}$ 	& $0^\infty$ \\ \hline
$aw$ 	& $0^\infty$ \\ \hline
$u_{\omega,2}a$ 	& $10^\infty$ \\ \hline
$wa$ 	& $10^\infty$ \\
\end{tabular}
\caption{For every element of $G_\omega$ of length smaller to $2$ which is not in $\gen{a,u_{\omega,1}}$, we exhibit a vertex of $\Gamma_+$ whose image is in $\Gamma_+^c$.}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.4\textwidth}
\centering
\begin{tabular}{c | c}
$a$ 	& $10^\infty$ \\ \hline
$au_{\omega,1}$ 	& $1010^\infty$ \\ \hline
$au_{\omega,2}$ 	& $10^\infty$ \\ \hline
$aw$ 	& $1010^\infty$ \\ \hline
$u_{\omega,1}a$ 	& $10^\infty$ \\ \hline
$u_{\omega,2}a$ 	& $10^\infty$ \\ \hline
$wa$ 	& $10^\infty$ \\
\end{tabular}
\caption{For every element of $G_\omega$ of length smaller to $2$ of $G_\omega$ which is not in $\gen{u_{\omega,1},u_{\omega,2},u}$, we exhibit a vertex of $\widetilde{\Gamma}_+$ whose image is in $\widetilde{\Gamma}_+^c$.}
\end{subfigure}
\caption{Initial step for the elements of small length.} \label{F:induction_stab_Gamma}
\end{figure}

We now suppose that, for a fixed integer $n\geq3$ and all the admissible sequences $\omega$, all the elements of length at most $n$ are in the desired subgroups. Recall that the stabilizer of the first level of the tree has an explicit generating set:
\begin{equation*}
 \stab_\omega(1) = \gen{b_\omega,c_\omega,d_\omega,ab_\omega a,ac_\omega a,a d_\omega a}.
\end{equation*}
The restrictions of these generators are
\begin{align*}
b_\omega &= (I_\omega^2,b_{\sigma\omega}) \\
c_\omega &= (I_\omega^1,c_{\sigma\omega}) \\
d_\omega &= (I_\omega^0,d_{\sigma\omega}) \\
ab_\omega a &= (b_{\sigma\omega},I_\omega^2) \\
a c_\omega a &= (c_{\sigma\omega},I_\omega^1) \\
a d_\omega a &= (d_{\sigma\omega},I_\omega^0).
\end{align*}

Let $g$ be an element of $G_\omega$ of length $n+1$. We note that if $\sigma$ is a sequence without repetition, $\sigma\omega$ is also. There are 3 cases to prove.
\begin{enumerate}
\item{\textbf{If $g$ is in $\stab_\omega(\Gamma_+)$.}}
We notice that $g$ is in $\stab_\omega(\Gamma_+)$ if and only if $ag$ also is. By supposing that $g$ can be of length $n+2$, we can assume that $g$ stabilizes the first level of the tree. By Lemmas~\ref{L:stab_projections} and \ref{L:reduction_lemma}, $g_0$ and $g_1$ are of length at most $n$ and are in $\stab_{\sigma \omega}(\widetilde{\Gamma}_+)$. Using induction's hypothesis, $g_0$ and $g_1$ are in $\gen{b_{\sigma \omega},c_{\sigma \omega},d_{\sigma \omega}}$. As the restriction $g_0$ does not contain the generator $a$ in its writing, the element $g$ can only contain the generator $u_{\omega,1}$ of the set $\{b_\omega, c_\omega, d_\omega\}$. In the same way, the restriction $g_1$ does not contain the generator $a$ and then the element $g$ can only contain the generator $au_{\omega,1}a$ of the set $\{a b_\omega a, ac_\omega a, ad_\omega a\}$. Therefor, $g$ is in $\gen{u_{\omega,1},au_{\omega,1}a}$, but we may have multiply $g$ by $a$, then $g$ is in $\gen{a, u_{\omega,1}}$.

\item{\textbf{If $g$ is in $\stab_\omega(\widetilde{\Gamma}_+)$ and stabilizes the first level.}}
Combining Lemmas~\ref{L:stab_projections} and \ref{L:reduction_lemma}, the restriction $g_0$ is in $\gen{a,u_{\sigma \omega,1}}$ and the restriction $g_1$ is in $\gen{u_{\sigma \omega,1}}$. The form of $g_0$ allows the element $g$ to contain only the generator $au_{\omega,2} a$ of the set $\{a b_{\sigma \omega} a,a c_{\sigma \omega}a,a d_{\sigma \omega}a \}$ because this is the only one which projects on $u_{\sigma \omega,1}$. The restriction $g_1$ does not contain the generator $a$, then the element $g$ can only contain the element $a u_{\omega,1} a$ of $\{a b_{\sigma \omega} a,a c_{\sigma \omega}a,a d_{\sigma \omega}a \}$. As $\omega_1 \neq \omega_2$, the element $a u_{\omega,1} a$ is not equal to the element $au_{\omega,2} a$ and then $g$ is in $\gen{b_\omega, c_\omega , d_\omega }$.

\item{\textbf{If $g$ is in $\stab_\omega(\widetilde{\Gamma}_+)$ and does not stabilize the first level.}}
Once again by combining Lemmas~\ref{L:stab_projections} and \ref{L:reduction_lemma}, the restrictions $g_0$ and $g_1$ are in $\gen{u_{\sigma \omega,1}}$. The form of $g_0$ allows only the generators $\{u_{\omega,1}, a u_{\omega,2} \}$ and $g_1$ only the generators $\{u_{\omega,2}, a u_{\omega,1} \}$. As $\omega_1 \neq \omega_2$, these generators are different and then $g$ is equal to $1$.
\end{enumerate}

For the explicit isomorphisms, it is left to the reader to generalize the case of $G_{(012)^\infty}$ proved in \cite[VIII.B.10]{DelaHarpe2000} and \cite[VIII.B.16]{DelaHarpe2000}.
\end{proof}
The second step of the proof is to show that it is also difficult to stabilize subsets which differ only on a finite number of vertices with $\Gamma_+$.
\begin{prop}\label{P:Finite_symm_finite_stab}
Let $\omega$ be a sequence without repetition and $\Omega$ a subset of vertices of $\Gamma$ such that $\Omega \Delta \Gamma_+$ is finite. Then $\stab_\omega(\Omega)$ is finite.
\end{prop}
\begin{proof}
We denote by $\Lambda$ the symmetric difference $\Omega \Delta \Gamma_+$. This is a finite set, therefore there exists an even integer $n$ such that, for every $x$ in $\Lambda$, there exists $x' \in \{0,1\}^n$ such that $x=x'0^\infty$.

Let $x$ be an arbitrary prefix of length $n$ such that $x0^\infty$ is an element of $\Omega$ and let $y$ be an element of $\{0,1\}^\infty$. We claim that
\begin{equation*}
xy \in \Omega \text{ if and only if } y \in \Gamma_+.
\end{equation*}

The claim is trivial if $y = 0^\infty$. Now, let us assume that $y$ is not equal to $0^\infty$. As the length of $x$ is even, the parities of the positions of the last 1 of $y$ and of $xy$ are equal. Then $y$ is in $\Gamma_+$ if and only if $xy$ is also in $\Gamma_+$. Moreover, the element $xy$ is in $\Omega$ if and only if $xy$ is in $\Gamma_+$. Indeed, all the elements of the symmetric difference $\Lambda$ have a prefix of length shorter or equal than $n$ which is not the case of $xy$ as $y$ is not equal to $0^\infty$. The claim is therefore true.

In the same way, let $x$ be a prefix of length $n$ such that $x0^\infty$ is not an element of $\Omega$ and let $y$ be an element of $\{0,1\}^\infty$, we claim that
\begin{equation*}
xy \in \Omega \text{ if and only if } y \in \widetilde{\Gamma}_+.
\end{equation*}

Let $K= \stab_\omega(\Omega) \cap \stab_\omega(n)$ be the subgroup of elements which stabilizes $\Omega$ as well as the $n$\th{} level of the tree. For every vertex $x$ of the $n$\th{} level of the tree and every element $g$ of $K$, the restriction $g_x$ is an element of $\stab_{\sigma^n\omega}(\Gamma_+) \cup \stab_{\sigma^n\omega}(\widetilde{\Gamma}_+)$.

Indeed, if $x0^\infty$ is in $\Omega$, the following equivalences are true for all $y \in \Gamma_+$:
\begin{align*}
y \in \Gamma_+ 	&\Leftrightarrow	 xy \in \Omega \\
				&\Leftrightarrow	 g(xy) \in \Omega \\
				&\Leftrightarrow	 xg_x(y) \in \Omega \\
				&\Leftrightarrow	 g_x(y) \in \Gamma_+
\end{align*}
and then $g_x$ is in $\stab_{\sigma^n\omega}(\Gamma_+)$. In the same way, if $x0^\infty$ is not in $\Omega$:
\begin{align*}
y \in \widetilde{\Gamma}_+ 	&\Leftrightarrow	 xy \in \Omega \\
										&\Leftrightarrow	 g(xy) \in \Omega \\
										&\Leftrightarrow	 xg_x(y) \in \Omega \\
										&\Leftrightarrow	 g_x(y) \in \widetilde{\Gamma}_+
\end{align*}
and then $g_x$ is in $\stab_{\sigma^n\omega}(\widetilde{\Gamma}_+)$.
Therefore, we can define the following embedding:
\begin{equation*}
K \hookrightarrow \prod_{v \in \{0,1\}^n} \left( \stab_{\sigma^n\omega}(\Gamma_+) \cup \stab_{\sigma^n\omega}(\widetilde{\Gamma}_+) \right).
\end{equation*}
We proved in Proposition~\ref{P:stab_gamma} that $\stab_{\sigma^n\omega}(\Gamma_+)$ and $\stab_{\sigma^n\omega}(\widetilde{\Gamma}_+)$ are finite and then $K$ is also finite. As $K$ is a subgroup of $\stab_\omega(\Omega)$ of index at most $2^n$, the stabilizer is finite.
\end{proof}
\begin{cor}
Let $\omega$ be a sequence without repetition, $v$ a vertex of $\mathcal{X}$ with a support $\Lambda$ and $n$ an even integer such that the prefix of the elements of $\Lambda \Delta \Gamma_+$ are at most of length $n$. Then,
\begin{equation*}
|\stab_\omega(v) | \leq 8 \cdot 4 \cdot 4^n
\end{equation*}
\end{cor}
\begin{proof}
It is a direct computation using the embedding above, the cardinals of $D_8$ and $\field{Z}/2\field{Z} \times \field{Z} / 2\field{Z}$ and $[G_\omega,\stab_\omega(n)]=2^n$.
\end{proof}
We can now relate these stabilizers of subsets and the stabilizers of the cubes of $\mathcal{X}$.
\begin{prop}\label{P:proper}
Let $\omega$ be a sequence without repetition, then $\stab_\omega(C) = \{g \in G_\omega : gC = C\}$ is finite for every cube $C$ of $\mathcal{X}$.
\end{prop}
\begin{proof}
We begin by the case of the vertices.

Let $v$ be a vertex of $\mathcal{X}$. Let us denote by $\Omega \subset \Gamma$ the support of $v$. It is straightforward that
\begin{equation*}
\stab_\omega(v) =  \stab_\omega(\Omega).
\end{equation*}
The vertex $v$ is in the connected cube $\mathcal{X}$, then the symmetric $\Omega \Delta \Gamma_+$ is finite and then $\stab_\omega(\Omega)$ is finite by Proposition~\ref{P:Finite_symm_finite_stab}.
Now, we will show that every $n$-cube have also finite stabilizer.

Let us suppose by contradiction that there exists a $n$-cube $C$ with an infinite stabilizer. The elements of this stabilizer send vertices of $C$ to vertices of $C$. We pick a vertex $v$ on the boundary of the cube and we consider its orbit by the action of $\stab_\omega(C)$. As this subgroup is infinite, there is at least one vertex $w$ of $C$, which is reached infinitely many times. We define the set $K$ as
\begin{equation*}
K \coloneqq \{g \in \stab_\omega(C) : g.v = w\}.
\end{equation*}
It is infinite by the choose of $w$. For an element $g$ of $K$, the subset $ Kg^{-1} $ is in $\stab_\omega(w)$ and is therefor infinite, which is a contradiction with Proposition~\ref{P:Finite_symm_finite_stab}.
\end{proof}
We can now prove the main theorem of this section.
\begin{thm}\label{T:Properness_action}
Let $\omega$ be an element of $\{0,1,2\}^\infty$ without repetition. Then the action of $G_\omega$ on $\mathcal{X}$ is proper and faithful.
\end{thm}
\begin{proof}
The properness is a consequence of Proposition~\ref{P:proper}.

To prove that the action is faithful, we need to show that
\begin{equation*}
\smashoperator[r]{\bigcap_{\substack{\Omega \subset V(\Gamma) \\ |\Omega \Delta V(\Gamma)|<\infty}}} \stab_\omega(\Omega) = \bigcap_{v \in \mathcal{V}} \stab_\omega(v)  = \{1\}.
\end{equation*}
By Proposition~\ref{P:stab_gamma}, the intersection $\stab_\omega(\Gamma) \cap \stab_\omega(\widetilde{\Gamma}_+) = \{1, d_\omega\}$. Moreover, $d_\omega$ does not stabilizes $\Gamma_+ \setminus\{0^\infty, 1010^\infty\}$ because $d_\omega 101^\infty = 10^\infty$. Then, the intersection above is trivial.
\end{proof}
We can further refine the description of the stabilizers.
\begin{prop}\label{P:subgroup_stab}
Let $\omega$ be a sequence without repetition and $H$ be a finite subgroup of $G_\omega$. Then, there exists a vertex $v$ of $\mathcal{X}$ such that $H < \stab_\omega(v)$.
\end{prop}
\begin{proof}
We consider the subset $\Lambda \subset \Gamma$ defined as
\begin{equation*}
\Lambda = \bigcup_{h \in H} h \Gamma_+.
\end{equation*}
In the proof of Proposition~\ref{P:Stab_connected_comp}, we showed that the symmetric difference $\Gamma_+ \Delta g.\Gamma_+$ is always finite and so is $\Lambda \Delta \Gamma_+$. Therefor, the vertex $v = \chi_\Lambda$ is a vertex of $\mathcal{X}$ and , as $h\Lambda = \Lambda$, it is stabilized by $H$.
\end{proof}
\begin{cor}
Let $\omega$ be a sequence without repetition. The stabilizers of the vertices of $\mathcal{X}$ are arbitrary large.
\end{cor}
\begin{proof}
The order of the elements of $G_\omega$ is not bounded, then there exist finite arbitrary large finite subgroups in $G_\omega$. By the previous proposition, we can also find aribtrary large stabilizers of vertices.
\end{proof}
\section{Classifying space of proper actions}
\begin{defn}
A classifying space of proper actions of a group $G$, denoted by $\EG$, is a topological space which admits a proper $G$-action and with the following property: if $X$ is any space with a proper $G$-action, then there exists a $G$-equivariant map $f: X \rightarrow \EG$ and any two $G$-equivariant maps  $X \rightarrow \EG$ admit a $G$-equivariant homotopie between them.
\end{defn}
As proved in Theorem~\ref{T:Properness_action}, the cube complex $\mathcal{X}$ is a topological space which admits a proper $G_\omega$-action if $\omega$ does not contain repetition. We will show that it satisfies the property above.
\begin{thm}\label{T:model}
Let $\omega \in \{0,1,2\}^\infty$ be a sequence without repetition. Then the CAT(0) cube complex $\mathcal{X}$ is a model for $\EG$, the classifying space of proper actions of $G_\omega$.
\end{thm}
\begin{proof}
We use the reformulation of the universality definition proved in \cite[Prop 1.8]{Baum1994}. Applied to our situation, it means that in order to prove that CAT(0) cube complex is a model for $\EGom$, it is enough to verify the following two points.
\begin{enumerate}
\item If $H$ is a finite subgroup of $G_\omega$, then there exists a point $x$ in $\mathcal{X}$ which is fixed by $H$.
\item View $\mathcal{X} \times \mathcal{X}$ as a space endowed with the usual diagonal action $g(x_0, x_1) = (gx_0, gx_1)$. Denote by $p_0,p_1: \mathcal{X} \times \mathcal{X} \rightarrow \mathcal{X}$ the two projections on each component, then there exists a $G$-equivariant homotopie between $p_0$ and $p_1$.
\end{enumerate}
The first point is a consequence of Proposition~\ref{P:subgroup_stab}. For the second one, we define the following homotopy
\begin{equation*}
h((x_0,x_1),t) = t x_0 + (1-t) x_1.
\end{equation*}
This is well-defined as $\mathcal{X}$ is convex.
\end{proof}
\section{Questions}
A group action $G \actsGroup X$ of a group $G$ on a metric space $X$ is called \emph{metrically proper} if $d(x,g_nx) \rightarrow \infty$ for every infinite family $\{g_n\}$ of elements of $G$ and every point $x$ in $X$. This property is strictly stronger than the definition of proper action used above if the space $X$ is not locally compact. For example, consider the Cayley graph of a free group generated by a countable infinite family of generators and the canonical associated action of the group on it. All the stabilizers are trivial hence the action is proper, but all the generators send the identity at distance $1$ and so it is not metrically proper. We don't know if the actions $G_\omega \actsGroup \mathcal{X}$ are metrically proper.
\begin{question}
Are there any sequences $\omega$ for which the action of $G_\omega$ on $\mathcal{X}$ is metrically proper?
\end{question}
Our intuition tells us that this is the case for the groups $G_\omega$ where the action is proper. A positive answer will give an elegant proof of the Haagerup property for these groups by \cite{Cherix2004}. This property is known for all Grigorchuk groups using the fact that they have subexponential growth, whereas here we would have a more elementary proof.

Grigorchuk groups are examples of branch groups \cite{Bartholdi2003}. There are other classes of branch groups which share properties used in the construction of $\mathcal{X}$ and tools used in the proof (Schreier graph with more than one end, restriction of the action on the subtrees, reduction lemma, etc).
\begin{question}
Is it possible to do the same construction for other finitely generated branch groups with a Schreier graph with at least 2 ends to obtain an action without bounded orbit, faithful, proper on a CAT(0) cube complex?
\end{question}


text:
\section*{Introduction}

Biological systems are constantly exposed to changing environments but
manage to adapt to them by gathering information about the conditions of
their surroundings and processing this information to arrive at best strategies.
This ability to gather and process information about
the surrounding environment does not require sophisticated organs like the
brain of an animal. For instance, slime mould -- a large brainless amoeba-like
cell -- can solve computationally difficult problems in a relatively short time
\cite{slime_1,slime_2}. When the root of a plant encounters another root, it
can detect whether it is the root of the same plant or it belongs to another
plant \cite{GN}.
The parasitic plant Cuscuta (dodder) are capable of processing olfactory
signals to identify host locations and distinguish plants of differing nutrient
levels \cite{Moraes}.
Or, recent experiments suggest that common bean plants can
detect the existence of objects in their vicinities without physical contacts
\cite{Calvo}. These, and
countless other examples, beg the questions how basic biological systems
such as a single cell are capable of processing and storing information,
and how can one model the dynamics of biological systems resulting from
their information processing capabilities.

The phenomenon of responding to the changing environment, however,
is not restricted to biological systems. Take, for instance, a quantum
system like an atom or even a single particle. When a
quantum system is immersed in an
open environment, its state responds to the conditions of the environment.
The dynamics of the system can then be interpreted as
the result of signal processing, for, as demonstrated below,
the dynamical equation governing the wave function of the system (the
stochastic unravelling of the Lindblad equation) is precisely the equation for
the optimal signal detection. Hence a quantum system in an
open environment behaves as if it is a micro information processor, just like an
aggregate of quantum particles forming biological systems. From this point
of view it is natural to enquire whether the dynamical equation for the
evolution of the wave function in an open environment can be used to also
describe behaviours of biological systems.

The purpose of this paper is to explore the hypothesis that signal
processing capability is fundamental to laws of nature, applicable
to a wide range of systems including both quantum and biological,
and examine its
consequences. Evidently there are innumerable -- if not all -- biological
phenomena that may be addressed from the viewpoint of signal processing.
For definiteness we shall focus on the motions of plants, in
particular, the phenomena known as
heliotropism or phototropism -- the solar or light
tracking of plants \cite{Harmer1}, and gravitropism --
the gravitational-field tracking of plant roots \cite{Sato}.
While scientific
study of heliotropism and gravitropism
goes a long way \cite{Darwin,Bose,Yin}, the understanding
of the biokinematics
as well as the purposes for heliotropism and gravitropism
are only emerging in recent years
\cite{Harmer2,Goyal,Baluska,Kiss}. Our purpose here is
to characterise qualitatively the dynamical behaviours of plants that result
from information processing by employing familiar models used in open
quantum systems. This not only demonstrates how models of signal
detection are universally applicable to capture behaviours of quantum as
well as biological systems, but also provides a new tool to model, and
hence to make predictions about the statistics, of plant motions. Specifically,
for a qualitative modelling of
light-tracking motions of plants we shall employ a
stochastic Schr\"odinger equation for which the Lindblad operator is
given by the position operator \cite{Diosi1}, but for the
Hamiltonian we take it to be simply the momentum operator. Borrowing
techniques of signal processing, we derive an exact solution to the
dynamical equation which describes the one-dimensional tracking of a
deterministic motion. Similarly, for gravity-tracking motions
of plant-root orientations we employ a stochastic Schr\"odinger equation
where both the Lindblad operator and the Hamiltonian are given by the
Pauli matrix, for which the exact solutions are known \cite{BH1}.

The significance of the introduction of dynamical models for plant motions
under uncertain environments -- a concept hitherto missing in the literature
-- is that such models allow for the quantification of the information processed,
as well as information erased. From Landauer's principle
\cite{LR}, then, erasure of information is accompanied by energy consumption
and heat production.
We believe that this information erasure process, resulting in the increase of
entropy, must lie at the heart of ageing, or arrow of time, in biological systems.
This point of view has been advocated in the past \cite{BCP,Trewavas},
though the
introduction of concrete dynamical models for describing the behaviours of
biological systems in response to the changes of their environmental conditions
under the influence of noise has hitherto been missing.
In particular, our model provides a rough estimate for the amount of
energy consumption in plants due to erasure of processed information.
For instance,
if a daisy flower were to process information at the Landauer limit,
then our model predicts
that the flowers will consume of order $10^{-2}$ eV of energy overnight
per each information-bearing cell by erasing the processed
information to track the sun during the day.
In the case of circumnutation of bean plants, our estimates
show that the erasure
of information about the location of a climbing support requires about
$1$ eV of energy per each information-bearing cell at the Landauer limit.
This indicates that inefficient information processing can be detrimental
to their survival.
The fact that most plants produce little heat, in particular, supports our
hypothesis that they are able to process
information at a level significantly closer to the Landauer limit than mechanical
devices at our disposal.

\section*{Signal processing quantum dynamics}

We begin by establishing the relation between the stochastic Schr\"odinger
equation
associated with Lindbladean dynamics and signal processing. This
connection has long been envisaged by Belavkin and others \cite{Belavkin},
but was made explicit by Brody \& Hughston
\cite{BH1,BH2,BH3} as an effective tool for solving, and
thus arriving at weak solutions to certain types of stochastic
Schr\"odinger equations. To start, consider the following
problem in signal processing. We have an unknown quantity of interest,
e.g., a signal or a message, represented by the random variable $L$. Let
us assume that $L$ takes discrete values $\{l_i\}$
with the probabilities $\{p_i\}$.
The true value of $L$ is unknown to the observer,
who merely receives information about the value of the signal $L$
that is obscured by noise. Assuming that noise is additive and is
modelled by a Brownian motion $\{B_t\}$, and that the signal is
revealed in time at a constant rate $\sigma$, the noisy observation
of the signal is characterised by the information process
\begin{eqnarray}
\xi_t = \sigma L t + B_t.
\end{eqnarray}
In the literature of signal detection, the signal-plus-noise
time series $\{\xi_t\}$ such as the one defined here is called the observation
process.
However, because $\{\xi_t\}$ models the flow of information, while in the
quantum context the term ``observation'' can have multiple meanings, we
shall refer to $\{\xi_t\}$ as the information process.
Note that if there is a large number of random additive contributions to noise,
then the law of large numbers implies that it is reasonable to assume that
additive noise is normally distributed, making Brownian motion a reasonable
candidate to model noise.
The best estimate of $L$ that minimises the quadratic
error, given the observed time series $\{\xi_s\}_{s\leq t}$ up to time $t$, is
given by the conditional expectation
\begin{eqnarray}
\langle L\rangle_t = \sum_{i} l_i \, {\mathbb P}\left(L=l_i|\{\xi_s\}_{s\leq t}
\right) .
\end{eqnarray}
The problem thus reduces to working out the conditional probability
$\pi_{it} = {\mathbb P}(L=l_i|\{\xi_s\}_{s\leq t})$ that $L=l_i$ given the
observation. From the Markov property of the information process
and the fact that $L=\lim_{t\to\infty}\xi_t/(\sigma t)$, the conditional
probability reduces to a simpler expression $\pi_{it} = {\mathbb P}(L=l_i|\xi_t)$,
which can easily be determined by use of the Bayes formula:
\begin{eqnarray}
{\mathbb P}(L=l_i|\xi_t) = \frac{{\mathbb P}(L=l_i)\,\rho(\xi_t|L=l_i)}
{\sum_j {\mathbb P}(L=l_j)\,\rho(\xi_t|L=l_j)} \, .
\end{eqnarray}
Here $\rho(\xi_t|L=l_j)$ denotes the density function of $\xi_t$ conditional
on the event that $L=l_j$.
From ${\mathbb P}(L=l_i)=p_i$, and the fact that conditional on $L=l_i$ the
random variable $\xi_t$ is normally distributed with mean $\sigma l_i t$
and variance $t$, we deduce that
\begin{eqnarray}
\rho(\xi|L=l_i)=\frac{1}{\sqrt{2\pi t}} \exp\left(-
\frac{(\xi-\sigma l_i t)^2}{2t}\right),
\end{eqnarray}
and hence, upon substitution, that
\begin{eqnarray}
\pi_{it} = \frac{p_i\,\exp\left(\sigma l_i\, \xi_t - \frac{1}{2} \sigma^2 l_i^2 t\right) }
{\sum_j p_j\,\exp\left(\sigma l_j\, \xi_t - \frac{1}{2} \sigma^2 l_j^2 t\right) } \, .
\label{eq:3}
\end{eqnarray}

We now consider taking the stochastic differential of the conditional
probability process $\{\pi_{it}\}$. Because $\{\pi_{it}\}$ is a smooth function
of the two variables $t,\xi_t$, by use of Ito's formula
\begin{eqnarray}
\mbox{$\rm d$} f(t,\xi_t) = {\dot f}(t,\xi_t) \, \mbox{$\rm d$} t + f'(t,\xi_t) \, \mbox{$\rm d$}\xi_t + \mbox{$\textstyle \frac{1}{2}$} f''(t,\xi_t) \, \mbox{$\rm d$} t
\end{eqnarray}
for any smooth function $f(t,x)$ of two variables, where dot denotes differentiation
with respect to $t$ and dash denotes differentiation with respect to $x$, a
calculation shows that
\begin{eqnarray}
\mbox{$\rm d$}\pi_{it} = \sigma \pi_{it} \big( l_i - \langle L\rangle_t \big) \left( \mbox{$\rm d$}\xi_t - \sigma
\langle L\rangle_t \, \mbox{$\rm d$} t \right).
\end{eqnarray}
Introducing the process $\{W_t\}$ according to
\begin{eqnarray}
W_t = \xi_t - \sigma \int_0^t \langle L\rangle_s \, \mbox{$\rm d$} s   ,
\end{eqnarray}
we can show that $\{W_t\}$ thus defined is a standard Brownian motion
under the physical probability measure ${\mathbb P}$
\cite{BH2}. In signal detection, the process $\{W_t\}$ arising in this manner
is called the innovations process \cite{Kailath}, and it reveals the arrival of
new information.
That is, the increment $\mbox{$\rm d$}\xi_t$ in the observed time series contains both
new information as well as redundant known information about the signal
$L$. Removing $\sigma\langle L\rangle_t \,\mbox{$\rm d$} t$ from this increment, we
are left with only the new information that was not already known at time
$t$. The fact that $\{W_t\}$ represents new information, however, follows
only if $\langle L\rangle_t$ is the ``best'' estimate of $L$
that minimises the uncertainty (conditional variance) of
$L$, exhausting all the
relevant information gathered up to time $t$. We shall return to the
significance of this statement in open quantum systems shortly, but for
now we conclude that the increments $\mbox{$\rm d$}\pi_{it} = \sigma\pi_{it}
( l_i - \langle L\rangle_t) \,\mbox{$\rm d$} W_t$ of the conditional probability process
are proportional to the increments of the innovations process.

If we define $\phi_{it}=\sqrt{\pi_{it}}$ to be the square root probability, then
another application of Ito's formula shows that
\begin{eqnarray}
\mbox{$\rm d$}\sqrt{\pi_{it}} = \frac{1}{2\sqrt{\pi_{it}}}\,\mbox{$\rm d$}\pi_{it} -
\frac{1}{8\pi_{it}\sqrt{\pi_{it}}}\,(\mbox{$\rm d$}\pi_{it})^2 ,
\end{eqnarray}
and hence from $(\mbox{$\rm d$}\pi_{it})^2=\sigma^2 \pi_{it}^2
(l_i-\langle L\rangle_t)^2\mbox{$\rm d$} t$ that
\begin{eqnarray}
\mbox{$\rm d$}\phi_{it} = \textstyle{\frac{1}{2}} \sigma \big( l_i - \langle L\rangle_t \big) \,
\phi_{it} \, \mbox{$\rm d$} W_t - \textstyle{\frac{1}{8}} \sigma^2
\big( l_i - \langle L\rangle_t \big)^2 \phi_{it} \, \mbox{$\rm d$} t.
\label{eq:x9}
 \end{eqnarray}
With this expression in mind, let us consider a Hilbert space ${\mathcal H}$
associated
with a physical system, on which an operator ${\hat L}$ is defined, whose
eigenvalues are $\{l_i\}$ and eigenstates are $\{|l_i\rangle\}$. Assume that
the Hamiltonian ${\hat H}$ of the system, whose eigenvalues are $\{E_i\}$,
commutes with ${\hat L}$, and that the eigenvalues of the two operators are
not degenerate (the degenerate case can equally be treated \cite{BH2}).
For simplicity let
the initial state of the system be given by a pure state $|\psi_0\rangle$, and
let $p_i=|\langle l_i|\psi_0\rangle|^2$. Let the process $|\psi_t\rangle$ be
given by
\begin{eqnarray}
|\psi_t\rangle = \sum_i \phi_{it} \, \mbox{$\rm e$}^{{\rm i}(\theta_i-E_i t)}\, | l_i\rangle,
\label{eq:9}
\end{eqnarray}
where $\{\theta_i\}$ are arbitrary constant and $\phi_{it}$ satisfies
(\ref{eq:x9}).
Then from the discussion above it should be evident, by taking the stochastic
differential of (\ref{eq:9}), that $|\psi_t\rangle$ \textit{is}
the solution to the stochastic Schr\"odinger equation
\begin{eqnarray}
\mbox{$\rm d$}|\psi_t\rangle =  - \mbox{$\rm i$} {\hat H}\, |\psi_t\rangle \, \mbox{$\rm d$} t +
\textstyle{\frac{1}{2}} \sigma \big( {\hat L} -
\langle {\hat L}\rangle_t \big) \, |\psi_t\rangle \, \mbox{$\rm d$} W_t - \textstyle{\frac{1}{8}}
\sigma^2 \big( {\hat L} - \langle {\hat L}\rangle_t \big)^2 |\psi_t\rangle \, \mbox{$\rm d$} t ,
\label{eq:8}
\end{eqnarray}
where
\begin{eqnarray}
\langle {\hat L}\rangle_t = \langle\psi_t|{\hat L}|\psi_t\rangle =
 \sum_{i} l_i \, {\mathbb P}\left(L=l_i|\xi_t\right) ,
\end{eqnarray}
and the constants $\{\theta_i\}$ are fixed by the relative phases of the
initial condition $|\psi_0\rangle$. This is
the method introduced to find solutions to the so-called
energy-based stochastic Schr\"odinger equation for which ${\hat L}={\hat H}$,
that is, when the Lindblad operator is given by the
Hamiltonian itself \cite{BH1}. But (\ref{eq:8}) is the dynamical equation for
the wave function of
the open quantum system that gives the stochastic unravelling of the
corresponding Lindblad equation
\begin{eqnarray}
\partial_t {\hat\rho} = -\mbox{$\rm i$}[{\hat H},{\hat \rho}] + \frac{1}{4}\sigma^2
\left[ {\hat L} {\hat\rho} {\hat L} - \frac{1}{2} \left(
{\hat L}^2 \rho + {\hat\rho} {\hat L}^2 \right) \right]
\label{eq:z14}
\end{eqnarray}
for the reduced density matrix ${\hat\rho}$ of the system, so we are
left with the signal-processing solution (\ref{eq:9})
to the stochastic unravelling dynamics, where $\phi_{it}=\sqrt{\pi_{it}}$ and
where $\pi_{it}$ is given by (\ref{eq:3}). Putting it differently, if we define a
matrix ${\hat\rho}$ by setting its $(i,j)$ element to be
\begin{eqnarray}
{\hat\rho}_{ij} = \mbox{$\rm e$}^{-{\rm i}(E_i-E_j)t} \, {\mathbb E}\left[ \sqrt{\pi_{it}\,
\pi_{jt}} \right] ,
\end{eqnarray}
where ${\mathbb E}[-]$ denotes
expectation over all random paths $\{\xi_t\}$, and if $\{\pi_{it}\}$ is the
solution (\ref{eq:3}) to the signal detection problem, then the matrix
${\hat\rho}$ satisfies the Lindblad equation (\ref{eq:z14}) for the
dynamics of the open quantum system. In other words,
the density matrix defined by ${\hat\rho}={\mathbb E}[
|\psi_t\rangle\langle\psi_t|]$ solves the Lindblad equation
(\ref{eq:z14}) when the state vector $|\psi_t\rangle$ is the solution to
(\ref{eq:8}). We remark that here we have restricted our discussion to
Lindblad operators that are Hermitian, but an analogous conclusion
can be deduced when they are not Hermitian, leading
to the so-called quantum filtering equations \cite{Gough}.

More generally, if $[{\hat H},{\hat L}]\neq0$, then we introduce a
time-reversed state $|\varphi_t\rangle = \mbox{$\rm e$}^{{\rm i}{\hat H}t} |\psi_t\rangle$,
and let ${\hat L}_t = \mbox{$\rm e$}^{{\rm i}{\hat H}t} \, {\hat L} \, \mbox{$\rm e$}^{-{\rm i}{\hat H}t}$.
Then a short calculation shows that the stochastic Schr\"odinger equation
for $|{\varphi_t}\rangle$ is given by
\begin{eqnarray}
\mbox{$\rm d$} |\varphi_t\rangle =
\mbox{$\textstyle \frac{1}{2}$} \sigma ({\hat L}_t-\langle {\hat L}_t\rangle)
|\varphi_t\rangle \, {\mbox{$\rm d$}}W_t
- \textstyle{\frac{1}{8}}
\sigma^2({\hat L}_t-\langle {\hat L}_t\rangle)^2 |\varphi_t\rangle
\, {\mbox{$\rm d$}}t  ,
\label{eq:10}
\end{eqnarray}
where $\langle {\hat L}_t\rangle = \langle{\varphi}_t| \,{\hat L}_t \,
|\varphi_t\rangle$. Notice that in the time-reversed representation the
unitary term associated with the Hamiltonian is removed from the
dynamical equation. Thus, the solution for $|\varphi_t\rangle$ is obtained
by finding the signal detection problem for which the time-dependent
signal process $L(t)$ corresponds to the spectrum of the quantum operator
$\mbox{$\rm e$}^{{\rm i}{\hat H}t} \, {\hat L} \, \mbox{$\rm e$}^{-{\rm i}{\hat H}t}$, and for the
(non-Markovian) information process we have
\begin{eqnarray}
\xi_t = \sigma \int_0^t L(s)\,\mbox{$\rm d$} s + B_t .
\end{eqnarray}
Specifically, letting $\pi_{it}={\mathbb P}(L(t)=l_i|\{\xi_s\}_{s\leq t})$, the
solution to (\ref{eq:10}) can be constructed by setting $|\varphi_t\rangle
= \sum_i \sqrt{\pi_{it}} \, \mbox{$\rm e$}^{{\rm i}\theta_i}\, | l_i\rangle$.

It is worth remarking that a stochastic unravelling of the Lindblad equation
for the reduced density matrix of the system in terms of a random pure-state
dynamics is by no means unique, and thus is not restricted to the Ito
stochastic differential equations of the form (\ref{eq:8}). Indeed, the
underlying noise type need not take the form of a Brownian motion:
depending on the context of the experiment the noise type may well be
modelled more appropriately from a considerably broader family of
processes known as L\'evy processes, of which Brownian motion is just
one example. However, for each noise type (e.g., a Poisson process, a
gamma process, a variance gamma process, and so on) there is a
canonical way of formulating the signal detection problem \cite{BHY},
and deduce the corresponding
conditional density process $\{\pi_{it}\}$. Then by defining $\phi_{it}=
\sqrt{\pi_{it}}$ and substituting the result in (\ref{eq:9}) we obtain the
corresponding stochastic unravelling of the same Lindblad equation.
Hence in each case we are left with the problem of signal processing
in one form or another. It is also worth noting that the
open quantum dynamics for the density matrix may take a form more
general than the familiar Lindblad equation. For example, if the Markovian
approximation used to deduce the dynamics of the reduced density matrix
is not applicable, then one is lead to a corresponding stochastic unravelling
for the wave function evolution that is manifestly non-Markovian \cite{DGS}.
Whether these more general equations admit an
explicit signal detection interpretation remains an open question, though
we would speculate from the structure of the unravelling equation \cite{DGS}
that the answer will be affirmative.

At any rate, the foregoing analysis demonstrates that the stochastic
unravelling of the
Lindblad equation \textit{necessarily takes the form of the optimal signal
detection}. In the Brownian case
the dynamical equation (\ref{eq:10}) that gives rise to a seemingly random
and yet purposeful evolution is a Hilbert space formulation
of the Kushner equation \cite{Kushner,Bucy}
in signal processing. Importantly, the Wiener process
$\{W_t\}$ appearing in (\ref{eq:10}) does not represent noise, but is the
innovations process. Hence the evolution of the wave function
follows the random path that is determined by the ``best'' estimate of
the underlying signalling problem: It is not the case that the state
of an open quantum system is randomly perturbed by means of a noisy
Brownian motion (as in the classical Brownian particle). For sure the state
is influenced by the noise $\{B_t\}$, but its evolution follows the path
determined only by the arrival of new information resulting from the
interaction with the environment regarding the stable state of the system,
corresponding to the eigenstates of the Lindblad operator.
In other words, the wave function of the system is the representation of
the conditions of their uncertain environments.

\section*{Quantum dynamics for motion tracking}

This feature of quantum dynamics resembles, at least at an intuitive level,
the behaviour of biological systems. With this in mind we attempt to model
motions of biological systems -- plants in this case -- in
response to changing environments using stochastic Schr\"odinger equations.
To characterise the tracking motion we consider a model of the type
discussed, for instance, by Diosi \cite{Diosi1}, in which the Lindblad
operator is the
position operator: ${\hat L}=\frac{1}{2}\sigma{\hat Q}$, where $\sigma$ is a
constant. The effect of the Lindbladean part of the dynamics is to
localise the wave function at the hidden ``true'' value of
the position, which in the present context may represent the location of the
light source. Plants do not possess sophisticated visual sensors, so
\textit{a priori} the position of the light source is hidden to them. However,
they possess a range of photoreceptors
to detect and analyse incident light so as to regulate their responses to the
environments \cite{Huq}. From the viewpoint of signal detection, the target
-- e.g., the position of the sun -- is moving, and this will be modelled by
choosing the Hamiltonian to be just the momentum operator: ${\hat H}=
\mu{\hat P}$, where $\mu$ is the rate parameter. Thus we have the
dynamical equation
\begin{eqnarray}
\mbox{$\rm d$} \psi_t(x) = -\mbox{$\rm i$} \mu {\hat P} \, \psi_t(x)\, \mbox{$\rm d$} t  +
\textstyle{\frac{1}{2}} \sigma \big( {\hat Q} -
\langle {\hat Q}\rangle_t \big) \psi_t(x) \, \mbox{$\rm d$} W_t - \textstyle{\frac{1}{8}}
\sigma^2 \big( {\hat Q} - \langle {\hat Q}\rangle_t \big)^2 \psi_t(x) \, \mbox{$\rm d$} t .
\label{eq:11}
\end{eqnarray}
The idea that we propose here therefore is to regard the squared wave
function $\pi_t(x)=|\psi_t(x)|^2$, where $\psi_t(x)$ is the solution to
(\ref{eq:11}), as representing the probability distribution of the location of
the light source, as ``perceived'' by the plant (along the east-to-west straight
line, which we take to approximate the actual motion of the sun on the
upper hemisphere via vertical projection;
the model here is understood to capture the qualitative behaviour of motion
tracking for illustrative purposes). In other words, the integral
$\int x\,\pi_t(x)\,\mbox{$\rm d$} x$
represents the best estimate of the location of the light source arrived at
by the plant from the data gathered through a range of receptors.

For simplicity let us assume that the initial state is a standard Gaussian
state: $\psi_0(x) = (2\pi)^{-1/4} \exp(-\frac{1}{4}x^2)$. The signal detection
solution to (\ref{eq:11}) is then as follows. We let $X$ be a standard normal
random variable with mean zero and variance one, and consider the ``signal''
process $X_t = X + \mu t$. The observation of the signal, however, is
obscured by a Brownian noise, giving rise to the information process
\begin{eqnarray}
\xi_t = \sigma \int_0^t X_s \, \mbox{$\rm d$} s + B_t.
\end{eqnarray}
Then by considering the conditional density
${\mathbb P}(X_t=x|\{\xi_s\}_{s\leq t})$ that determines the best estimate
for the location of the target
we deduce, after a short calculation, that the full solution to (\ref{eq:11}) --
not merely the asymptotic steady-state solution \cite{GRW,Diosi2} -- with the
Gaussian initial state is given by
\begin{eqnarray}
\psi_t(x) = \left( \frac{1+\sigma^2t}{2\pi}\right)^{\frac{1}{4}}
\exp\left( -\frac{\left[ (1+\sigma^2t)x - \sigma\xi_t - \mu t (1+\frac{1}{2}
\sigma^2 t ) \right]^2 }{4(1+\sigma^2t)} \right) ,
\end{eqnarray}
where the innovations Brownian motion $\{W_t\}$ in (\ref{eq:11})
is related to the information process $\{\xi_t\}$ according to
\begin{eqnarray}
\mbox{$\rm d$} W_t = \mbox{$\rm d$} \xi_t - \sigma \left( \frac{\sigma\xi_t + \mu t (1+\frac{1}{2}
\sigma^2 t)}{1+\sigma^2t} \right) \mbox{$\rm d$} t .
\end{eqnarray}

In the present case where the initial state is Gaussian, the best estimate
for the position of the light source:
\begin{eqnarray}
\int_{-\infty}^\infty x\,\pi_t(x)\,\mbox{$\rm d$} x = \frac{\sigma\xi_t + \mu t (1+\frac{1}{2}
\sigma^2 t)}{1+\sigma^2t}
\label{eq:x22}
\end{eqnarray}
is a linear function of the information process $\{\xi_t\}$, which seems
reasonable. More generally, in the case of an arbitrary initial state
$\psi_0(x)$, writing $p(x)=|\psi_0(x)|^2$, letting $X$ be a random variable
with the density $p(x)$, and setting $X_t=X+\mu t$, the solution to
(\ref{eq:11}) is determined by the conditional probability process:
\begin{eqnarray}
\pi_t(x) = \frac{p(x) \, \exp\left( \sigma(x-\mu t)\left(\xi_t-\frac{1}{2}\sigma
xt\right)\right)}{\int p(x) \, \exp\left( \sigma(x-\mu t)\left(\xi_t-\frac{1}{2}\sigma
xt\right)\right) \, {\rm d}x} .
\end{eqnarray}
That is, $\psi_t(x)=\mbox{$\rm e$}^{{\rm i}\theta(x)}\sqrt{\pi_t(x)}$, where $\theta(x)$
in the initial phase.
The relatively simple model constructed here appears
effective in characterising the simple solar-tracking motion whereby the
plant orients towards the direction of the sun but with small errors. In
particular, if the distribution of the error in the orientation of the plant
leaves and flowers were normally distributed, then we find that the
Gaussian model (\ref{eq:x22}) would be appropriate.

\section*{Quantifying the information extracted}

We are interested in the question on how much information extraction is
needed so as to deduce the location of the unknown moving target $\{X_t\}$.
In a spontaneous localisation model like (\ref{eq:11}) the large-time behaviour
of the system is not physically relevant, for, in the limit $t\to\infty$ the wave
function converges to (the square root of) a delta function at the point corresponding to
the true value of $X$, but this requires an
infinite amount of information,
or, equivalently, infinite reduction in entropy.

In the case of solar tracking of plants, on the other, the sun is not a point
particle. Likewise, the leaves or flowers are not positioned perfectly
perpendicular to the incident light. Hence the localisation
model (\ref{eq:11}) is only relevant up to the time when a good progress is
made in terms of tracking the moving sun. This intuitive idea can
be made precise by studying the uncertainty (variance) measure of $X$: A
good progress is made if the conditional variance of $X$ is reduced to a
fraction of the initial variance \cite{BH2,LH,Adler}.
In the present context,
this timescale $\tau$ is given by $\tau\sim1/\sigma^2 \Delta X^2$,
where $\Delta X^2$ is the initial variance of $X$.

To quantify the information extraction, therefore, we consider the reduction
$S_0-S_\tau$ of the Shannon-Wiener entropy
\begin{eqnarray}
S_t = -\int \pi_t(x) \log \pi_t(x) \mbox{$\rm d$} x,
\end{eqnarray}
where $\pi_t(x)=|\psi_t(x)|^2$.
Note that we are not concerned with the von Neumann entropy
here, which represents the observer's knowledge, or lack of it, of the system;
whereas the Shannon-Wiener entropy represents the system's lack of
knowledge of the environment. To see this, we recall that density matrix of
the system is given by
\begin{eqnarray}
\rho_t(x,y) = {\mathbb E} \left[ {\overline{\psi_t(x)}} \psi_t(y)\right],
\end{eqnarray}
where ${\mathbb E}$ denotes expectation over all random paths for the
information process $\{\xi_t\}$, and it is $\rho_t(x,y)$ that fulfils the
deterministic Lindbladean dynamical equation describing the time
evolution of the open system. Because an (intelligent) observer
is unaware which paths the information process has chosen, but knows
only of its statistical distribution, the ``state'' of the system as perceived
by the external observer is given by the density matrix $\rho_t(x,y)$.
The von Neumann entropy
\begin{eqnarray}
-{\rm tr}(\rho\ln\rho) = -\iint \rho_t(x,y)\ln\rho_t(y,x)\, \mbox{$\rm d$} y\, \mbox{$\rm d$} x
\end{eqnarray}
thus represents the observer's
lack of knowledge of the exact state of the system. In contrast, since
$\pi_t(x)=|\psi_t(x)|^2$ represents the (unintelligent) system's knowledge
of the environment, gathered by optimally processing the noisy
information about the environment, it is the Shannon-Wiener entropy
that represents the system's lack of knowledge of the environment.

Because the entropy process
$\{S_t\}$ is stochastic, and depends on the information process $\{\xi_t\}$, it
can increase or decrease, but on average it decreases \cite{BH2}.
Hence we are interested in the averaged entropy reduction
$\Delta S =S_0-{\mathbb E}[S_\tau]$ for the information gain.
In the special case
where the initial state is Gaussian, however, the associated entropy is
deterministic and is given by
\begin{eqnarray}
S_t = \frac{1}{2}\left[1+\log(2\pi) - \log(1+\sigma^2t)\right],
\end{eqnarray}
from which it follows that $\Delta S = \frac{1}{2}\log(1+\sigma^2\tau)$.
On the other hand the initial uncertainty of $X$ in this example is unity
so that $\tau=\sigma^{-2}$. Hence $\Delta S =
\frac{1}{2}\log 2$ and we find that the
amount of information processed to track the motion is half of
one bit. Allowing for a variability in the initial wave function, we thus conclude
that the amount of information processed to track the motion is at most of
order few bits.

\section*{Information erasure and heat production}

It seems reasonable to assume that much of the information processed for
plant movements are not stored in the plant indefinitely. (Note that this need
not apply universally. For instance, there are suggestions that Mimosa plants
can be trained to learn certain behaviours, which it can remember for more
than a month \cite{GRDM}.) But the loss of processed information has to be
distinguished from information that are encoded in their genes. Take,
for instance, the seed of a plant buried in dark soil. Information
encoded in the seed cells sends roots towards the direction of the gravitational
pull, and stems in the opposite direction -- the so-called gravitropism.
However, plants \textit{a priori} do
not possess information about their environments, so
measurements are performed to detect the direction of the
gravitational field. In particular, if the direction of the gravitational force changes,
then roots will change the orientation of their growths to
accommodate this change of the environment \cite{Knight}.

We are not concerned here with the actual mechanism by which plans
detect gravitational field, which has only been unravelled relatively recently
\cite{Baluska,Kiss}. What concerns us is the information-processing aspect
of the growth orientation selection, and this can be modelled heuristically
using a stochastic Lindbladean dynamical equation for a two-level system
with ${\hat H},\,{\hat L}\propto{\hat\sigma}_z$; the solution to which in the
quantum context are known \cite{BH1,BH2}. Specifically, we can think of
a uniform magnetic field in the vertical $z$-direction surrounding a
spin-$\frac{1}{2}$ particle in quantum mechanics as the analogue of the
gravitational field surrounding the seed in biology.
The direction of the spin then represents the orientation
of the root growth.
Under the dynamics governed by such stochastic Schr\"odinger
equation the spin of the particle, which initially could be pointing in any
orientation, will follow a random path such that eventually it will line up
parallel to
the direction of the field, and this reorientation of spin can be interpreted as
modelling the root reorientation in gravitropism.
Thus in this model the information process is
$\xi_t=\sigma L t + B_t$, where the random variable $L$ takes the value
$+1$ with probability $p=|\langle\uparrow\!|\psi_0\rangle|^2$, and takes the
value $-1$ with probability $1-p=|\langle\downarrow\!|\psi_0\rangle|^2$.
Here, $|\psi_0\rangle$ is the initial wave function of the system, and
$(|\!\uparrow\rangle,|\!\downarrow\rangle)$
are the two eigenstates of the Pauli matrix ${\hat\sigma}_z$, corresponding
to the spin-up and spin-down states. The effect
of the Hamiltonian ${\hat H}=g{\hat\sigma}_z$ is then to generate the
nutation of the root with angular frequency determined by the parameter
$g$, while the effect of the Lindblad operator ${\hat L}=\gamma
{\hat\sigma}_z$ is to generate gravitropism at a rate that is governed by
the parameter $\gamma$.

Because the problem here is of binary nature, of order one bit of
information is processed to decide the growth orientation. This
information is transient;
measurements are performed repeatedly (by continuous monitoring) to
reaffirm the orientation of the gravitational field \cite{Sato}.
Thus, one bit of information must be regularly erased from each
information-bearing unit inside the cells; most likely in the elongation
zone cells, for, it seems implausible that such information
is transmitted and stored elsewhere in the plant, because this will require
additional resource for error corrections.

Returning to heliotropism, take, for instance, the case of
sunflowers. While mature sunflowers are permanently facing east, young
sunflowers trace the motion of the sun. After sunset, the flowers then turn
around from west to east and await the sunrise. This indicates a memory
effect. Indeed, if one were to rotate a pot of a young sunflower plant by, say,
$\pi/2$, then the sunflower now turns back and forth between north
and south for a few days before stopping \cite{Harmer2}. In other words,
the memory is lost only after a while. According to our rough estimate above,
this results in the erasure of at most few bits of information. On the other
hand, in the case of a daisy flower, overnight the flower tend to orient in a
random direction, indicating a faster loss of information.

It is known that information erasure is necessarily
accompanied by energy consumption and hence heat production, resulting
in an increase of environmental entropy \cite{LR}. In particular, the minimum
amount of energy consumption required for the erasure of one bit of
information at temperature $T$ is $k_{\rm B}T\log 2$. Based on
our estimate, a daisy flower at summer-night temperature
(say, 290 K) will thus consume at least of order $10^{-2}$~eV of energy per
each information-encoding unit. This
in turn results in heat production. A detection of this effect is likely to be
difficult, for, the magnitude of energy involved for information erasure
based on our estimate is likely to be rather small compared to that for
plant movements.
Further, the environment of plants are far removed from the highly
controlled environment in which such an effect has been detected in a
quantum system \cite{Berut}.

The precise mechanism of how plants store information is not fully
understood. Hence it is not known how many information-storing
units are contained in a given plant, making it difficult to estimate the total
energy cost of information erasure. To get a better
intuition for the scale involved, therefore, let us consider,
as an example, the circumnutation of
a common bean plant \cite{Calvo} by assuming that each cell contains at
most a single such unit. For the bean plant to identify the location of an object
like a pole to climb up within a $\pm10^\circ$ angular window with $95\%$
confidence, say in a recent experimental setup \cite{Calvo},
it must process between 15 and 20 bits of information. This estimate
follows from the assumption that the \textit{a priori} probability of finding
an object is uniform over the circle around the plant. Assuming that the
\textit{a posteriori} distribution can be described by any one of the standard
circular distributions such as the von Mises or the wrapped normal distribution,
one arrives at this estimate. Now if the
object is removed, this information must be erased. If the plant can process
information relatively close to the Landauer limit, say, $10^4$ times the
limit, then the cost of erasure is of order $10^{4}$~eV, which would be
approximately just
below $1\%$ of the total energy consumed by the cell \cite{Milo}. Intuitively this
is a plausible figure, given the limited energy resource available to biological
systems. Or, putting the matter differently, if information is processed
significantly less efficiently, say at the level of our everyday computers, then
the erasure cost becomes too high for survival. Our hypothesis
that biological systems can process information close to the Landauer limit is
consistent with the empirical observation that plants produce very little heat
(except for thermogenic plants that purposefully produce heat
\cite{Robinson}).

\section*{Second law in biology}

In accordance with the Landauer principle the erasure of information will
increase the entropy of the environment. It is tempting therefore to conjecture
that biological systems operate by extracting information from their
environments, processing them, and arriving at the best estimate of the
state of the environment for the purpose of adaptation.
This will result in the gain of information, and thus
reduction in entropy. However, some, or much of the processed information
is lost, resulting in increasing entropy. The process of information erasure
then must lie at the heart of ageing -- or arrow of time -- in biological systems.

It should be added that while the present paper concerns the dynamical
mechanism of the information-processing of the environmental conditions of
biological systems, our estimates show that the erasure cost, and hence
entropy production, of genetic information encoded
\textit{a priori} in biological systems would be substantially more significant
than that of the \textit{a posteriori} processed information. This is consistent
with the empirical fact that the life span of a biological system on average
reduces when genetic information encoded in the system is lost. Hence in
terms of biological arrow of time, we argue that the erasure of processed
information will not constitute the dominant contributing factor.

\section*{Model calibration}

How can model parameters be estimated against data? In our simple tracking
model we have the rate parameter $\sigma$, which determines the timescale
that plants can orient towards the light source. For instance, leaves of Cornish
mallow can reorient as rapidly as $2\pi/9$ per hour \cite{Koller}, indicating
that $\sigma$ takes a large value. (Recall that the response
timescale is inverse proportional to $\sigma^2$ so that a rapid response
would imply a large $\sigma$.) If plants exhibit a memory effect, like
circadian rhythm of sunflowers \cite{Harmer2}, then either the
variance of the signal is small or else the value of $\sigma$ is
small. In the case of gravitropism, using the model described above, if we let
${\hat L}=\gamma{\hat\sigma}_z$,
then the parameter $\gamma$ can be calibrated from studying the timescale
of root reorientation by turning the growing plant still in the soil upside-down.
Figure~\ref{fig1} shows how the response time of the plant can be slow or
fast, depending on the value of $\gamma$.

\begin{figure}[h]
      \centering
       \subfloat[$\gamma=0.25$: slow reorientation]
       {\includegraphics[width=0.45\textwidth]{fig1-1.eps}}\hfill
       \subfloat[$\gamma=0.95$: fast reorientation]
       {\includegraphics[width=0.45\textwidth]{fig1-2.eps}}\hfill
\caption{\textit{Model simulation of the binary root reorientation}. In a model with
${\hat L}=\gamma{\hat\sigma}_z$, assume that the plant is initially 95\%
confident that the gravitational force is pointing down (the signal $L=-1$); but
at time zero it is flipped upside-down (the signal $L=+1$). Depending on the
value of the parameter $\gamma$ the
reorientation (and hence the loss of the initial memory) can be slow or fast.
Four sample paths for the order parameter $\langle {\hat\sigma}_z\rangle_t$
are shown here each for the two chosen values of $\gamma$.
The order parameter represents the vertical component
of the unit directional vector of the root. Because the reorientation timescale
is proportional to $\gamma^{-2}$, although in all cases the
root will eventually point down, this process can take a long time if $\gamma$
is small (the left panel); whereas for a larger value of $\gamma$ the
reorientation takes place rapidly (the right panel).}
\label{fig1}
\end{figure}

\section*{Discussion}

In summary, we have shown how stochastic unravelling of the Lindblad
equation for open system dynamics in quantum mechanics necessarily
takes the form of the equation for the best information processing in signal
detection. This observation suggests that the notion of an
optimal information-processing
capability is fundamental to the laws of nature at the quantum level. Such a
statement may at first seem counterintuitive, for, one tends to associate
some form of intelligence to the concept of information processing. In physics
one is more accustomed to the idea of a variational principle to arrive at
laws of nature. However, our conclusion can in fact be reached by means of
a variational argument, albeit in noisy environments. Namely, the dynamical
equation for the state of the system can be derived by demanding that the
average entropy reduction of the system is maximised.
Indeed, the entropy and variance -- the two uncertainty
measures -- are closely related in the context of signal processing.
This follows from
the fact \cite{BH2} that the Shannon-Wiener entropy associated
with the stochastic Schr\"odinger equation (\ref{eq:8}) satisfies the relation
\begin{eqnarray}
{\mathbb E}\left[ S_\tau \right] = \mbox{$\textstyle \frac{1}{2}$} \sigma^2 \int_\tau^\infty
{\mathbb E}\big[\Delta L_t^2\big]\, \mbox{$\rm d$} t ,
\end{eqnarray}
where $\Delta L_t^2=\langle\psi_t| ({\hat L}-\langle{\hat L}\rangle_t)^2
|\psi_t\rangle$
is the conditional variance of the Lindblad operator. Hence minimisation of
the quadratic error is linked to maximisation of the entropy reduction.

Given that biological systems like plants use sophisticated mechanisms
to process external signals concerning light, water, temperature, gravity,
organic compounds, and
so on, and adapt their behaviours accordingly, we postulated that the stochastic
unravelling of the Lindblad equation in open quantum systems can be
applied to model the dynamical behaviours of plants in open environments.
We considered a localisation model to capture
qualitatively the tracking motion, to arrive at a rough
estimate of the quantity of information processed. The idea that information
processing in biological systems must be viewed as fundamental has been
advocated before \cite{Quadtler,Binder,Mescher},
but here we have been able to
quantify this in a dynamical context. Empirical observations suggest that
some of the processed information is erased, from which we postulated
(a) that biological systems must operate relatively close to the Landauer
limit of computation; and (b) that information erasure must underlie the ageing
of biological systems in a fundamental way.

It is worth remarking that the specific models of stochastic Schr\"odinger
equations considered above for the characterisation of plant motions can
be presented by use of purely classical signal detection techniques. While
this is consistent with our thesis that signal processing capability is
fundamental to laws of nature including quantum theory, it also implies
that the resulting estimates in the context of biology,
at least in the examples considered here, in principle
could have been obtained
purely in terms of classical signal detection, without referencing
quantum theory.
That said, it would be technically more challenging to pose
a model for the spherical nutation of roots using a purely classical language,
whereas our quantum spin-$\frac{1}{2}$ model offers a highly effective and
simple treatment of the matter. More importantly,
we believe that in many realistic setups in
biology, more adequate descriptions can be obtained by means of more
quantum-mechanical models for which the commutator of the Hamiltonian
${\hat H}$ and the Lindblad operator ${\hat L}$ is nontrivial (that is,
$[{\hat H},{\hat L}]$ neither vanishes nor is proportional to the identity). In
the biological context, the Lindblad operator represents adaptation, while
the Hamiltonian represents changes of the environment. (For example, in
the simple motion-tracking model considered here, the Lindblad operator
has the effect of orienting towards the location of the sun,
whereas the Hamiltonian changes the location of the sun.) It is inadvertently
the case in biology that adaptation is possible so long as environmental
changes are sufficiently slow, whereas a fast change can be catastrophic to
the survival of the
biological system, leading to a very different dynamical behaviour. This is
already seen even in the simple phenomenon of heliotropism \cite{Harmer1}.
Such a transition can be described by a stochastic
Schr\"odinger equation for which $[{\hat H},{\hat L}]\neq0$ in a nontrivial
way, because in such a model one typically encounters a phase transition in
the dynamical behaviour of the system \cite{Bassi,BL}, depending on the
relative strengths of ${\hat H}$ and ${\hat L}$.
Conversely, without the lack of commutativity it is not
possible to describe such critical phenomena seen in biology and ecology;
yet, the notion of incompatible observables is one of the signatures of
quantum theory. We argue that this observation
justifies our unified approach to model dynamical behaviours of quantum
and biological systems in open environments, and hope to develop more
general theories elsewhere.

We conclude by remarking that there is an ongoing debate in plant science
concerning the notion of consciousness and sentience
in plants \cite{Reber,Taiz,Calvo2,Calvo3}. This follows from advances in
detecting
plants' remarkable abilities in observing, analysing, and adapting to the
changing environments that surround them. Our postulate that
information-processing capability is part of the laws of nature, thus not
requiring any intelligence, might shed a new light on this debate.

\section*{Acknowledgements}

The author thanks L. P. Hughston,
B. K. Meister, E. M. Graefe, G. Piskov, P. Calvo, F. Balu\v{s}ka,
J. Al-Khalili, A. Rocco, S. Saunders, and A. Trewavas for stimulating
discussions; and acknowledges support from the Russian Science
Foundation (grant 20-11-20226), the London Mathematical
Society (grant URB-2021-43), and the John Templeton Foundation
(grant 62210). The opinions expressed in this publication are those of
the author and do not necessarily reflect the views of the
John Templeton Foundation.

redpajama_set_name: RedPajamaArXiv