# Justifying Born’s rule $P_{\alpha}=|\Psi_{\alpha}|^{2}$ using deterministic
chaos, decoherence, and the de Broglie-Bohm quantum theory

Aurélien Drezet1 1Univ. Grenoble Alpes, CNRS, Grenoble INP, Institut Neel,
F-38000 Grenoble, France<EMAIL_ADDRESS>
###### Abstract

In this work we derive Born’s rule from the pilot-wave theory of de Broglie
and Bohm. Based on a toy model involving a particle coupled to a environement
made of “qubits” (i.e., Bohmian pointers) we show that entanglement together
with deterministic chaos lead to a fast relaxation from any statistitical
distribution $\rho(x)$ (of finding a particle at point $x$) to the Born
probability law $|\Psi(x)|^{2}$. Our model is discussed in the context of
Boltzmann’s kinetic theory and we demonstrate a kind of H theorem for the
relaxation to the quantum equilibrium regime.

##  I Introduction and motivations

The work of Wojciech H. Zurek is universally recognized for its central
importance in the field of quantum foundations in particular concerning
decoherence and the understanding of the elusive border between the quantum
and classical realms Zurek1 . Zurek emphasized the role of pointer states and
environment induced superselection rules (einselections). In the recent years,
part of his work going beyond mere decoherence and averaging focussed on
quantum darwinism and envariance. Quantum darwinism main goal is to emphasize
the role of multiple copies of information records contained in the local
quantum environment. Envariance aims is to justify the existence and form of
quantum probabilities, i.e., deriving Born’s rule, from specific quantum
symmetries based on entanglement Zurek2 . In recent important reviews of his
work Zurek stressed the importance of some of theses concepts for discussing
the measurement problem in relation with various interpretations of quantum
mechanics Zurek3 ; Zurek4 . Recent works showed for instance the importance of
such envariance to establish Born’s rule in the many-worlds and many-minds
contexts Wallace ; Drezet2021 . While in his presentations Zurek generally
advocated a neutral position perhaps located between the Copenhagen and
Everett interpretations we believe his work on entanglement and decoherence
could have a positive impact on other interpretations such as the de Broglie
Bohm theory. We know that Zurek has always been prudent concerning Bohmian
mechanics (see for example his remarks in Zurek1996  p. 209) perhaps because
of the strong ontological price one has to pay in order to assume a nonlocal
quantum potential and surrealistic trajectories (present even if we include
decoherence Zurek3 ; Appleby ). Moreover, the aim of this work is to discuss
the pivotal role that quantum entanglement with an environment of “Bohmian
pointers” could play in order to justify Born’s rule in the context of such a
Bohmian interpretation. The goal is thus to suggest interesting and positive
implications that decoherence could have on ontologies different from
Everettian or Consistent histories approaches.
The de Broglie-Bohm quantum theory (BBQT) introduced by de Broglie in 1927
debroglie1927 ; Valentini ; debroglie1930  and rediscovered by Bohm in 1952
Bohm1952a ; Bohm1952b , is nowadays generally accepted as a satisfactory
interpretation of quantum mechanics, at least for problems dealing with non-
relativistic systems Hiley ; Holland . Within this regime the BBQT is a clean
deterministic formulation of quantum mechanics preserving the classical
concepts of point-like particles moving continuously in space-time. This
formulation is said to be empirically equivalent to the orthodox description
axiomatized by the Copenhagen school, meaning that the BBQT is able to justify
and reproduce the probabilistic predictions made by the standard quantum
measurement theory. More specifically, this implies recovering the famous Born
rule that connects the probability

$\displaystyle P_{\alpha}=|\Psi_{\alpha}|^{2}$ (1)

of observing an outcome $\alpha$ (associated with the quantum observable
$\hat{A}$) to the amplitude $\Psi_{\alpha}$ in the quantum state expansion
$|\Psi\rangle=\sum_{\alpha}\Psi_{\alpha}|\alpha\rangle$ (i.e.,
$|\alpha\rangle$ is an eigenstate of $\hat{A}$ for the observable eigenvalue
$\alpha$).
This issue has been a recurrent subject of controversies since the early
formulation of the BBQT (see for example Pauli’s objection in Pauli1953 ;
Keller1953 ). It mainly arizes because the BBQT is a deterministic mechanics
and therefore, like for classical statistical mechanics, probabilities in the
BBQT can only be introduced in relation with ignorance and uncertainty on the
initial conditions of the particle motions. Moreover, after more than one and
a half century of developments since Maxwell and Boltzmann times, it is well
recognized that the physical and rigorous mathematical foundation of
statistical mechanics is still debatable Uffink . The BBQT, which in some
sense generalizes and extends Newtonian mechanics, clearly inherits of these
difficulties constituting strong obstacles for defining a clean basis of its
statistical formulation. This fact strongly contrasts with standard quantum
mechanics for which randomness has been from the start axiomatized as genuine
and inevitable.
Over the years several responses have been proposed by different proponents of
the BBQT to justify Born’s rule (for recent reviews see Barrett1995 ;
Callender2007 ; Drezet2017 ). Here, we would like to focus on the oldest
approach which goes back to the work of David Bohm on deterministic and
molecular chaos. Indeed, already in 1951-1952 Bohm emphasized the fundamental
role of the disorder and chaotic motion of particles for justifying Born’s
rule Bohm1952a ; Bohm1952b . In his early work Bohm stressed that the
complexity of the de Broglie-Bohm dynamics during interaction processes, such
as quantum measurements, should drive the system to quantum equilibrium. In
other words, during interactions with an environment like a measurement
apparatus any initial probability distribution $\rho(X)\neq|\Psi(X)|^{2}$ for
particle in the configuration space (here
$X=[\mathbf{x}_{1},...,\mathbf{x}_{M}]\in\mathbb{R}^{3M}$ is a vector in the
$N$-particles configuration space) should evolve in time to reach the quantum
equilibrium limit $\rho(X)\rightarrow|\Psi(X)|^{2}$ corresponding to Born’s
rule. In this approach the relaxation process would be induced by both the
high sensitivity to changes in the initial conditions of the particle motions
(one typical signature of deterministic chaos) and by the molecular thermal
chaos resulting from the macroscopic nature of the interacting environment
(i.e., with $\sim 10^{23}$ degrees of freedom). Furthermore, in this strategy
Born’s rule $\rho(X)=|\Psi(X)|^{2}$ should appear as an attractor similar to
the microcanonical and canonical ensemble in thermodynamics. In 1953 Bohm
developed an example model Bohm1953  (see Potel2002  for a recent
investigation of this idea) where a quantum system randomly submitted to
several collisions with external particles constituting a bath was driven to
quantum equilibrium $\rho(X)=|\Psi(X)|^{2}$. In particular, during his
analysis Bohm sketched a quantum version of the famous Boltzmann $H$-theorem
for proving the irreversible tendency to reach Born’s rule (for other clues
that Bohm was strongly fascinated by deterministic chaos already in the 1950’s
see Bohm1955  and the original 1951 manuscript written by Bohm in 1951
Bohm1951  and rediscovered recently). However, in later works, specially in
the one made with Vigier BohmVigier1954  and then subsequently Hiley Hiley ,
Bohm modified the original de Broglie-Bohm dynamics by introducing stochastic
and fluctuating elements associated with a subquantum medium forcing the
relaxation towards quantum equilibrium $\rho(X)\rightarrow|\Psi(X)|^{2}$.
While this second semi-stochastic approach was motivated by general
philosophical considerations Bohm1957  proponents of the BBQT have felt
divided concerning the need for such a modification of the original framework.
In particular, starting in the 1990’s Valentini currently develops an approach
assuming the strict validity of the BBQT as an underlying deterministic
framework and introduces mixing and coarse-graining à la Tolman-Gibbs in the
configuration space in order to derive a Bohmian ‘subquantum’ version of the
$H$-theorem Valentini1991 ; ValentiniPhD . Importantly, in his work on the
‘subquantum heat-death’ (i.e., illustrated with many numerical calculations
Valentini2005 ; Valentini2012  often connected with cosmological studies
Valentini2007 ; Valentini2015 ) Valentini and coworkers stress the central
role of deterministic chaos in the mixing processes. Moreover, deterministic
chaos in the BBQT is a research topic in itself (for a recent review see
Efthymiopoulos2017 ; Contopoulos2020 ) and many authors (including Bohm Hiley
and Valentini Valentini2005 ; Valentini2012 ) have stressed the role of nodal-
lines associated with phase-singularities of the wavefunction for steering
deterministic chaos in the BBQT Frisk1997 ; Falsaperia2003 ; Wisniacki2005 .
Yet, it has also been pointed out Efthymiopoulos2017 ; Efthymiopoulos2012
that such a chaos is not generic enough to force the quantum relaxation
$\rho(X)=|\Psi(X)|^{2}$ for any arbitrary initial conditions
$\rho(X)\neq|\Psi(X)|^{2}$ (a reversibility objection à la Kelvin-Loschmidt is
already sufficient to see the impossibility of such an hypothetical deduction
Callender2007 ; Norsen2018 ).
In the present work we emphasize the role of an additional ingredient which
together with chaos and coarse graining can help and steer the quantum
dynamical relaxation $\rho(X)\rightarrow|\Psi(X)|^{2}$, namely, quantum
entanglement with the environment. The idea that quantum correlations must
play a central role in the BBQT for justifying Born’s rule is not new of
course. Bohm already emphasized the role of entanglement in his work Bohm1952b
; Hiley ; Bohm1953 . Moreover, in recent studies motivated by the Vigier-Bohm
analysis we developed a Fokker-Planck Drezet2017  and Langevin’s like
Drezet2018  description of relaxation to quantum equilibrium
$\rho(X)=|\Psi(X)|^{2}$ by coupling a small system $S$ to a thermal bath or
reservoir $T$ inducing a Brownian motion on $S$. We showed that under
reasonable assumptions we can justify a version of the $H$-theorem where
quantum equilibrium appears as a natural attractor. Furthermore, at the end of
Drezet2017  we sketched a even simpler strategy based on mixing together with
entanglement and involving deterministic chaotic iterative maps. After the
development of such an idea it came to our attention that a similar strategy
has been already developed in a elegant work by Philbin Philbin2015  and
therefore we didn’t include too much details concerning our model in
Drezet2017 . Here, we give the missing part and provide a more complete and
quantitative description of our scenario presented as an illustration of a
more general scheme. More precisely, we i) analyze the chaotic character of
the specific de Broglie Bohm dynamics associated with our toy model, ii) build
a Boltzmann diffusion equation for the probability distribution, and finally
iii) derive a simple $H$-theorem from which Born’s rule turns out to be an
attractor. We emphasize that our work, like the one of Philbin, suggests
interesting future developments for justifying Born’s rule and recovering
standard quantum mechanics within the BBQT.

##  II The status of Born’s rule in the de Broglie Bohm theory

We start with the wavefunction
$\psi(\mathbf{x},t)=R(\mathbf{x},t)e^{iS(\mathbf{x},t)/\hbar}$ obeying to
Schrödinger’s equation

$\displaystyle i\hbar\frac{\partial}{\partial
t}\psi(\mathbf{x},t)=\frac{-\hbar^{2}\boldsymbol{\nabla}^{2}}{2m}\psi(\mathbf{x},t)+V(\mathbf{x},t)\psi(\mathbf{x},t)$
(2)

for a single nonrelativistic particle with masses $m$ in the external
potentials $V(\mathbf{x},t)$ (we limit the analysis to a single particle but
the situation is actually generic). The BBQT leads to the first-order
‘guidance’ law of motion

$\displaystyle\frac{d}{dt}\mathbf{x}^{\psi}(t)=\mathbf{v}^{\psi}(\mathbf{x}^{\psi}(t),t)$
(3)

where
$\mathbf{v}^{\psi}(\mathbf{x},t)=\frac{1}{m}\boldsymbol{\nabla}S(\mathbf{x},t)$
defines an Eulerian velocity field and $\mathbf{x}^{\psi}(t)$ is a de Broglie
Bohm particle trajectory. Furthermore, from Eq. 2 we obtain the conservation
rule:

$\displaystyle-\frac{\partial}{\partial
t}R^{2}(\mathbf{x},t)=\boldsymbol{\nabla}\cdot[R^{2}(\mathbf{x},t)\mathbf{v}^{\psi}(\mathbf{x},t)]$
(4)

where we recognize $R^{2}(\mathbf{x},t)=|\psi(\mathbf{x},t)|^{2}$ the
distribution which is usually interpreted as Born’s probability density. Now,
in the abstract probability theory we assign to every points $\mathbf{x}$ a
density $\rho(\mathbf{x},t)$ corresponding to a fictitious conservative fluid
obeying the constraint

$\displaystyle-\frac{\partial}{\partial
t}\rho(\mathbf{x},t)=\boldsymbol{\nabla}\cdot[\rho(\mathbf{x},t)\mathbf{v}^{\psi}(\mathbf{x},t)].$
(5)

Comparing with Eq. 4 we deduce that the normalized distribution
$f(\mathbf{x},t)=\frac{\rho(\mathbf{x},t)}{R^{2}(\mathbf{x},t)}$ satisfies the
equation

$\displaystyle[\frac{\partial}{\partial
t}+\mathbf{v}^{\psi}(\mathbf{x},t)\cdot\boldsymbol{\nabla}]f(\mathbf{x},t):=\frac{d}{dt}f(\mathbf{x},t)=0.$
(6)

This actually means Bohm1953  that $f$ is an integral of motion along any
trajectory $\mathbf{x}^{\psi}(t)$. In particular, if $f(\mathbf{x},t_{in})=1$
at a given time $t_{in}$ and for any point $\mathbf{x}$ this holds true at any
time $t$. Therefore, Born’s rule being valid at a given time will be preserved
at any other time debroglie1930 ; Bohm1952a ; Bohm1953 . At that stage the
definition of the probability $\rho(\mathbf{x},t)d^{3}\mathbf{x}$ for finding
a particle in the infinitesimal volume $d^{3}\mathbf{x}$ is rather formal and
corresponds to a Bayesian-Laplacian interpretation where probabilities are
introduced as a kind of measure of chance.
Moreover, in BBQT the actual and measurable density of particles must be
defined using a collective or ensemble of $N$ independent systems prepared in
similar quantum states $\psi(\mathbf{x}_{i},t)$ with $i=1,...,N$. Yet, the
concept of independency in quantum mechanics imposes the whole statistical
ensemble with $N$ particles to be described by the total factorized
wavefunction

$\displaystyle\Psi_{N}(\mathbf{x}_{1},...,\mathbf{x}_{N},t)=\prod_{i=1}^{i=N}\psi(\mathbf{x}_{i},t)$
(7)

solution of the equation

$\displaystyle i\hbar\frac{\partial}{\partial
t}\Psi_{N}=[\sum_{i=1}^{i=N}\frac{-\hbar^{2}\boldsymbol{\nabla}_{i}^{2}}{2m}+V(\mathbf{x}_{i},t)]\Psi_{N}.$
(8)

For this quantum state $\Psi_{N}$ the BBQT allows us to build the velocity
fields
$\frac{d}{dt}\mathbf{x}^{\psi}_{i}(t)=\mathbf{v}^{\psi}(\mathbf{x}^{\psi}_{i}(t),t)$
where $\mathbf{x}^{\psi}_{i}(t):=\mathbf{x}^{\Psi_{N}}_{i}(t)$ define the de
Broglie Bohm paths for the uncorrelated particles (i.e., guided by the
individual and independent wave functions $\psi(\mathbf{x}_{i},t)$ and
Eulerian flows
$\mathbf{v}^{\Psi_{N}}_{i}(\mathbf{x}_{1},...,\mathbf{x}_{N},t)=\mathbf{v}^{\psi}(\mathbf{x}_{i},t)$).
Within this framework the actual density of particles $P(\mathbf{r},t)$ at
point $\mathbf{r}$ is given by

$\displaystyle
P(\mathbf{r},t)=\frac{1}{N}\sum_{k=1}^{k=N}\delta^{3}(\mathbf{r}-\mathbf{x}^{\psi}_{k}(t))$
(9)

which clearly obeys the conservation rule

$\displaystyle-\frac{\partial}{\partial
t}P(\mathbf{x},t)=\boldsymbol{\nabla}\cdot[P(\mathbf{x},t)\mathbf{v}^{\psi}(\mathbf{x},t)].$
(10)

Comparing with Eq. 6 we see that if
$\rho(\mathbf{x},t)=f(\mathbf{x},t)|\psi(\mathbf{x},t)|^{2}$ plays the role of
an abstract Laplacian probability, $P(\mathbf{r},t)$ instead represents the
frequentist statistical probability. Both concepts are connected by the weak
law of large numbers (WLLN) which is demonstrated in the limit
$N\rightarrow+\infty$ and leads to the equality
$\rho(\mathbf{x},t)=P(\mathbf{r},t)$, i.e.,

$\displaystyle
f(\mathbf{r},t)|\psi(\mathbf{r},t)|^{2}=\equiv\lim_{N\to+\infty}\frac{1}{N}\sum_{k=1}^{k=N}\delta^{3}(\mathbf{r}-\mathbf{x}^{\psi}_{k}(t))$
(11)

where the equality must be understood in the sense of a ‘limit in probability’
based on typicality and not as the more usual ‘point-wise limit’. We stress
that the application of the WLLN already relies on the Laplacian notion of
measure of chance since by definition in a multinomial Bernoulli process the
abstract probability density
$\rho_{N}(\mathbf{x}_{1},...,\mathbf{x}_{N},t)=\prod_{i=1}^{i=N}\rho(\mathbf{x}_{i},t)$
is used for weighting an infinitesimal volume of the $N$-particles
configuration space $d\tau_{N}:=\prod_{i=1}^{i=N}d^{3}\mathbf{x}_{i}$. It can
be shown that in the limit $N\rightarrow+\infty$ with the use of this measure
$\rho_{N}d\tau_{N}$ almost all possible configurations
$\mathbf{x}^{\psi}_{1}(t),...,\mathbf{x}^{\psi}_{N}(t)$ are obeying the
generalized Born’s rule
$P(\mathbf{r},t)=\rho(\mathbf{x},t)=f(\mathbf{x},t)|\psi(\mathbf{x},t)|^{2}$
(the fluctuation varying as $\frac{1}{\sqrt{N}}$). It is in that sense that
Eq. 11 is said to be typical where typical meaning valid for ‘overwhelmingly
many’, i.e., almost all configurations in the whole configuration space
weighted by $\rho_{N}d\tau_{N}$. The application of the law of large number to
the BBQT is well known and well established ValentiniPhD ; Durr1992a ;
Durr1992b  but has been the subject of intense controversies Norsen2018 ;
Durr1992a ; Valentini2020 ; Durr2019 . Issues concern 1) the interpretation of
$\rho_{N}$ as a probability density, i.e., in relation with the notion of
typicality, and 2) the choice of $f=1$ as natural and guided by the notion of
equivariance remark . To paraphrase David Wallace the only thing the law of
large numbers proves is that relative frequency tends to weight … with high
weight Wallacevideo . However, there is a certain circularity in the reasoning
which at least shows that the axiomatic of the probability calculus allows us
to identify an abstract probability such as $\rho d^{3}\mathbf{x}$ to a
frequency of occurrence like $Pd^{3}\mathbf{x}$. However, the law is unable to
guide us for selecting the good measure for weighting typical configurations
(the condition on equivariance remark  is only a mathematical recipe not a
physical consequence of a fundamental principle). Therefore, the value of the
$f$ function is unconstrained by the typicality reasoning without already
assuming the result Valentini2020 . In other words, it is impossible to deduce
Born’s rule without already assuming it.
Moreover, it is important to see that the relation
$\frac{d}{dt}f(\mathbf{x}^{\psi}(t),t)=0$ is playing the same role in the BBQT
for motions in the configuration space that Liouville’s theorem
$\frac{d}{dt}\eta(q(t),p(t),t)=0$ in classical statistical mechanics (where
$\eta(q,p,t)$ is the probability density in phase space $q,p$). Therefore,
with respect to the measure $d\Gamma=|\psi(\mathbf{x},t)|^{2}d^{3}\mathbf{x}$
(which is preserved in time along trajectories since
$\frac{d}{dt}d\Gamma_{t}=0$), the condition $f=1$ is equivalent to the
postulate of equiprobability used in standard statistical mechanics for the
microcanonical ensemble. Clearly, we see that the inherent difficulties
existing in classical statistical mechanics for justifying the microcanonical
ensemble are transposed in the BBQT for justifying Born’s rule, i.e., $f=1$.

##  III A deterministic and chaotic model for recovering Born’s rule within
the de Broglie Bohm quantum theory

###  III.1 The basic dynamics

As a consequence of the previous discussion we now propose a simple toy model
where the condition $f=1$ appears as an attractor, i.e, $f_{t}\rightarrow 1$
during a mixing process. We consider a single electron wave-packet impinging
on a beam-splitter. To simplify the discussion we consider an incident wave-
train with one spatial dimensions $x$ characterized by the wavefunction

$\displaystyle\psi_{0}(x,t)\simeq\Phi_{0}(x-v_{x}t)e^{i(k_{x}x-\omega_{k}t)}$
(12)

where we have the dispersion relation
$E_{k}:=\hbar\omega_{k}=\frac{\hbar^{2}k_{x}^{2}}{2m}$ and the (negative)
group velocity components $v_{x}=\frac{\hbar k_{x}}{m}<0$ with
$k_{x}=-|k_{x}|$. Furthermore, for mathematical consistency we impose
$\Phi_{0}\simeq const.=C$ in the spatial support region where the wave-packet
is not vanishing and the typical wavelength $\lambda=2\pi/|k_{x}|\ll L$ where
$L$ is a typical wave-packet spatial extension. If we assume Born’s rule
$|C|^{2}$ must be identified with a probability density and by normalization
this implies $C=1/\sqrt{L}$ (this point will be relevant only in Sec. III).
The beam-splitter is a rectangular potential barrier or well $V(x)=V_{0}$ with
$V_{0}$ a constant in the region $|x|<\epsilon/2\ll L$ and $V(x)=0$ otherwise.
During the interaction with the beam-splitter the whole wavefunction
approximately reads

$\displaystyle\psi(x,t)\simeq\psi_{0}(x,t)+R_{k}\psi_{1}(x,t)$ if
$x>\epsilon/2$
$\displaystyle\psi(x,t)\simeq\Phi_{0}(-v_{x}t)[A_{k}e^{iq_{x}x}+B_{k}e^{-iq_{x}x}]e^{-i\omega_{k}t}$
if $|x|<\epsilon/2$ $\displaystyle\psi(x,t)\simeq T_{k}\psi_{0}(x,t)$ if
$x<-\epsilon/2$ (13)

where
$\psi_{1}(x,t)=\Phi_{0}(x+v_{x}t)e^{-ik_{x}x}e^{-i\omega_{k}t}=\psi_{0}(-x,t)$,
and $R_{k}$ (reflection amplitude), $T_{k}$(transmission amplitude) and
$A_{k},B_{k}$ are Fabry-Perot coefficients computed in the limit where the
wave-packet is infinitely spatially extended. We have:

$\displaystyle
T_{k}=\frac{4q_{x}k_{x}}{(q_{x}+k_{x})^{2}}\frac{1}{e^{i(q_{x}-k_{x})\epsilon}-\frac{(q_{x}-k_{x})^{2}}{(q_{x}+k_{x})^{2}}e^{-i(k_{x}+q_{x})\epsilon}}$
$\displaystyle
R_{k}=iT_{k}\frac{k_{x}^{2}-q_{x}^{2}}{2q_{x}k_{x}}\sin{(q_{x}\epsilon)}$
$\displaystyle
A_{k}=R_{k}[\frac{q_{x}+k_{x}}{2q_{x}}e^{-i(q_{x}-k_{x})\epsilon/2}+\frac{q_{x}-k_{x}}{2q_{x}}e^{-i(q_{x}+k_{x})\epsilon/2}]$
$\displaystyle
B_{k}=R_{k}[\frac{q_{x}-k_{x}}{2q_{x}}e^{i(q_{x}+k_{x})\epsilon/2}+\frac{q_{x}+k_{x}}{2q_{x}}e^{i(q_{x}-k_{x})\epsilon/2}]$

where $q_{x}$ is given by the dispersion relation
$E_{k}:=\hbar\omega_{k}=\frac{\hbar^{2}q_{x}^{2}}{2m}+V_{0}$, i.e.,
$q_{x}^{2}-k_{x}^{2}=-2mV_{0}/\hbar$. As an illustration we choose
$\epsilon=\frac{1}{2}\frac{\lambda}{2\pi}$ and $q_{x}\simeq 2.5k_{x}$ (i.e.,
$V_{0}<0$) which leads to $T_{k}\simeq\frac{1}{\sqrt{2}}e^{i0.267\pi}$ and
$R_{k}=iT_{k}$ corresponding to a balanced 50/50 beam-splitter.
We need now to consider the problem from the point of view of the scattering
matrix theory. First, observe that for negative time $t_{in}<0$ (with
$|t_{in}|\gg L/|v_{x}|$) the incident wave-packet $\psi_{0}(x,t_{in})$ given
by Eq. 12 which is coming from the $x>0$ region with a negative group velocity
is transformed for large positive times $t_{f}>0$ (with $|t_{f}|\gg
L/|v_{x}|$) into the two non overlapping wave-packets

$\displaystyle\psi(x,t_{f})\simeq R_{k}\psi_{1}(x,t_{f})$ if $x>0$
$\displaystyle\psi(x,t_{f})\simeq T_{k}\psi_{0}(x,t_{f})$
$\displaystyle\textrm{if $x<0$ }.$ (15)

Since the wave packets are non-overlapping we write:

$\displaystyle\psi(x,t_{f})\simeq
R_{k}\psi_{1}(x,t_{f})+T_{k}\psi_{0}(x,t_{f}).$ (16)

Of course, the situation is symmetric: if an incident wave-packet
$\psi_{1}(x,t_{in})$ comes from the $x<0$ region with a positive group
velocity for $t_{in}<0$ we will obtain at the end, i.e., for $t_{f}>0$:

$\displaystyle\psi(x,t_{f})\simeq
T_{k}\psi_{1}(x,t_{f})+R_{k}\psi_{0}(x,t_{f}).$ (17)

The general case can thus be treated by superposition: an arbitrary initial
state $\psi(x,t_{in})=a_{+}\psi_{0}(x,t_{in})+a_{-}\psi_{1}(x,t_{in})$ for
negative times $t_{in}$ (with $|t_{in}|\gg L/|v_{x}|$) will evolve into

$\displaystyle\psi(x,t_{f})\simeq(a_{+}R_{k}+a_{-}T_{k})\psi_{1}(x,t_{f})$
$\displaystyle+(a_{+}T_{k}+a_{-}R_{k})\psi_{0}(x,t_{f})$ (18)

for positive times $t_{f}$ (with $|t_{f}|\gg L/|v_{x}|$). Writing
$a^{\prime}_{+}=a_{+}R_{k}+a_{-}T_{k}$ and
$a^{\prime}_{-}=a_{+}T_{k}+a_{-}R_{k}$ the different mode amplitudes we define
a 2$\times$2 unitary transformation

$\displaystyle\left(\begin{array}[]{c}a^{\prime}_{+}\\\
a^{\prime}_{-}\end{array}\right)=\left(\begin{array}[]{cc}R_{k}&T_{k}\\\
T_{k}&R_{k}\end{array}\right)\left(\begin{array}[]{c}a_{+}\\\
a_{-}\end{array}\right)$ (25)
$\displaystyle=\frac{e^{i0.267\pi}}{\sqrt{2}}\left(\begin{array}[]{cc}i&1\\\
1&i\end{array}\right)\left(\begin{array}[]{c}a_{+}\\\
a_{-}\end{array}\right).$ (30)

Moreover, consider now the point of view of the BBQT. Following this theory
the dynamics of the material point is obtained by integration of the guidance
equation

$\displaystyle\frac{d}{dt}x^{\psi}(t)=v^{\psi}(x^{\psi}(t),t)=\frac{\hbar}{m}\textrm{Im}[\frac{\partial}{\partial
x}\psi(x,t)|_{x=x^{\psi}(t)}]$ (31)

Figure 1: (a) Scattering of a 1D wavepacket impinging on a 50/50 beam-splitter
(BS). The colormap shows $Re[\Psi(x,t)]$ in the $t,x$ plane. The color (red
and yellow) lines are de Broglie-Bohm trajectories associated with this wave-
function (red and yellow trajectories are ending in two different wave-
packets. The dotted white lines are crosscuts as discussed in the main text.
(b) Shows a similar situation when a wave-packet impinges on the other input
gate.

which can easily be computed numerically. We illustrate in Fig. 1 the
interaction with the 50/50 beam-splitter characterized by Eq. 30 of a
rectangular wave-packet (i. e., $\Phi_{0}(x)=C$ if $|x|<L/2$ where $L$ is the
width of the wave-packet) incident from the $x>0$ region (i.e.,
$a_{+}=1,a_{-}=0$). As a remarkable feature we can see the so called Wiener
fringes debroglie1930  existing in the vicinity of the beam-splitter and which
strongly alter the de Broglie-Bohm trajectories. What is also immediately
visible is that the de Broglie Bohm trajectories $x^{\psi}(t)$ never cross
each other. This is a general property of this first order dynamics Holland ;
Hiley  which plays a central role in our analysis.
An interesting feature of this example concerns the density of ‘probability’
$|\psi(x,t)|^{2}$. Indeed, consider a time $t_{in}$ in the remote past before
the wave-packet coming from the positive region (i.e. like in Fig. 1)
interacts with the potential well. At that time the center of the wave-packet
is located at $x_{in}=v_{x}t_{in}>0$. Yet, since trajectories can not cross
each others we know that the ensemble $\gamma_{+}(t_{in})$ of all possible
particle positions at time $t_{in}$, i.e.,
$x^{\psi}(t_{in})\in[x_{in}-\frac{L}{2},x_{in}+\frac{L}{2}]$ is divided into
two parts. In the first part $\gamma_{+}^{(+)}(t_{in})$, i.e.,
$x^{\psi}(t_{in})\in[x_{in}+H,x_{in}+\frac{L}{2}]$ with $|H|<\frac{L}{2}$, all
particles evolve in the future (i.e., at time $t_{f}$) into the
$\psi_{1}(x,t_{f})$ reflected wave-packet (corresponding to the support
$\gamma_{+}(t_{f})$, i.e.,
$x^{\psi}(t_{f})\in[x_{f}-\frac{L}{2},x_{f}-\frac{L}{2}]$ with
$x_{f}=-v_{x}t_{f}>0$). In the second part $\gamma_{+}^{(-)}(t_{in})$, i.e.,
$x^{\psi}(t_{in})\in[x_{in}-\frac{L}{2},x_{in}+H]$, all the particles
necessarily end up their journey in the $\psi_{0}(x,t_{f})$ transmitted wave-
packet (corresponding to the support $\gamma_{-}(t_{f})$, i.e.,
$x^{\psi}(t_{f})\in[-x_{f}-\frac{L}{2},-x_{f}-\frac{L}{2}]$). Now, remember
that from the de Broglie-Bohm Liouville theorem the measure
$d\Gamma(x,t)=|\psi(x,t)|^{2}dx$ is preserved in time, i.e.,
$\frac{d}{dt}d\Gamma_{t}=0$. Therefore, the measure

$\displaystyle\Gamma_{+}(t_{f})=\int_{\gamma_{+}}|\psi(x,t)|^{2}dx=LC^{2}/2$
(32)

associated with the reflected wave necessarily equals the measure associated
with the segment $\gamma_{+}^{(+)}(t_{in})$, i.e.,

$\displaystyle\Gamma_{+}^{(+)}(t_{in})=(L/2-H)C^{2}=\Gamma_{+}(t_{f}).$ (33)

This leads to $H=0$ which in turn means that $\gamma_{+}^{(+)}(t_{in})$
corresponds to $x^{\psi}(t_{in})\in[x_{in},x_{in}+\frac{L}{2}]$ and
$\gamma_{+}^{(-)}(t_{in})$ to
$x^{\psi}(t_{in})\in[x_{in}-\frac{L}{2},x_{in}]$. This result is actually
general and holds for any symmetric wave-packet $\Phi_{0}(x)=\Phi_{0}(-x)$ if
we can neglect the overlap between $\Phi_{0}(x-v_{x}t_{f})$ and
$\Phi_{0}(x+v_{x}t_{f})$).
Moreover,

Figure 2: (a) Transformation from the initial $\gamma_{\pm}(t_{in})$ $x-$space
to the final $\gamma_{\pm}(t_{ou})$ $x-$space for the two situations shown
Fig. 1(a) and Fig. 1(b) respectively (depicted as blue lines and red lines
respectively). (b) Shows the same transformation using the $y$ coordinate
instead of the $x$ coordinate (as explained in the main text).

for the rectangular wave-packet we deduce from the de Broglie-Bohm Liouvillian
theorem $\frac{d}{dt}d\Gamma_{t}=0$ that any infinitesimal length element
$\delta x^{\psi}(t_{in})$ surrounding a point $x^{\psi}(t_{in})$ in
$\gamma_{+}(t_{in})$ evolves to the infinitesimal length $\delta
x^{\psi}(t_{f})=2\delta x^{\psi}(t_{in})$ surrounding the point
$x^{\psi}(t_{f})$ located in $\gamma_{\pm}(t_{f})$. This property can be used
to define a simple mapping between the initial coordinates
$x^{\psi}(t_{in})\in\gamma_{+}(t_{in})$ and the final outcome
$x^{\psi}(t_{f})\in\gamma_{+}(t_{f})\cup\gamma_{-}(t_{f})$. It is simpler to
introduce the normalized variables:

$\displaystyle
y(t_{in})=\frac{x^{\psi}(t_{in})-x_{in}}{2L}+\frac{3}{4}\in[\frac{1}{2},1]$ if
$x^{\psi}(t_{in})\in\gamma_{+}(t_{in})$ $\displaystyle
y(t_{f})=\frac{x^{\psi}(t_{f})-x_{f}}{2L}+\frac{3}{4}\in[\frac{1}{2},1]$ if
$x^{\psi}(t_{f})\in\gamma_{+}(t_{f})$ $\displaystyle
y(t_{f})=\frac{x^{\psi}(t_{f})+x_{f}}{2L}+\frac{1}{4}\in[0,\frac{1}{2}]$
$\displaystyle\textrm{ if $x^{\psi}(t_{f})\in\gamma_{-}(t_{f})$ }.$ (34)

The mapping between the two new ensembles (that we will continue to name
$\gamma_{+}(t_{in})$ and $\gamma_{+}(t_{f})\cup\gamma_{-}(t_{f})$) is thus
simply written:

$\displaystyle y(t_{f})=2y(t_{in})-1.$ (35)

The result of this mapping is illustrated using the $x$ coordinates in Fig.
2(a) or the $y$ coordinates in Fig. 2(b). In particular, it is visible that
the correspondence $y(t_{f})=F(y(t_{in}))$ is not always univocally defined.
This occurs at $x^{\psi}(t_{in})=x_{in}$ (i.e., $y(t_{in})=\frac{3}{4}$) which
evolves either as $x^{\psi}(t_{f})=x_{f}-L/2\in\gamma_{+}(t_{f})$ or
$x^{\psi}(t_{f})=-x_{f}+L/2\in\gamma_{-}(t_{f})$ corresponding to the single
value $y(t_{f})=\frac{1}{2}$. Physically, as shown in Fig 2(a), it means that
a point located at the center of the wave-packet $\psi_{0}(x,t_{in})$ is
unable to decide whether it should move into the reflected or transmitted
wave-packets: this is a point of instability. This apparently violates the
univocity of the de Broglie-Bohm dynamics Eq. 31 which imposes that at given
point, i.e., $x^{\psi}(t_{in})=x_{in}$, one and only one trajectory is
defined. However, we stress that this pathology is actually a consequence of
the oversimplification of our model consisting in assuming a idealized
rectangular wave packet $\Phi_{0}(x)=C$ if $|x|<L/2$ with abrupt boundaries at
$|x|=L/2$. In a real experiment with a Gaussian wave-packet the point
$x^{\psi}(t_{in})=x_{in}$ would be mapped at the internal periphery of the two
wave-packets constituting $\psi(x,t_{f})$ (this would correspond to the points
$x^{\psi}(t_{f})=\pm\epsilon/2\sim 0$ where the beam splitter is located). In
this regime our assumption of a finite support for $\Phi_{0}(x)$ is not
acceptable anymore.
The previous analysis was limited to the case of the wave-packet
$\psi_{0}(x,t_{in})$ coming from the $x>0$ region. However, in the symmetric
case of a wave-packet $\psi_{1}(x,t_{in})$ coming from the $x<0$ region (i.e.,
$a_{+}=0,a_{-}=1$) the situation is very similar (as shown in Fig. 2) with the
only differences that the $\gamma_{+}(t_{in})$ space is changed into
$\gamma_{-}(t_{in})$, i.e.,
$x^{\psi}(t_{in})\in[-x_{in}-\frac{L}{2},-x_{in}+\frac{L}{2}]$ and the roles
of $\gamma_{+}(t_{f})$ and $\gamma_{-}(t_{f})$ (which definitions are let
unchanged) are now permuted (i.e., $\gamma_{+}(t_{f})$ is now associated with
the transmitted wave-packet and $\gamma_{-}(t_{f})$ with the reflected one).
From the point of view of the BBQT the trajectories of Fig. 1(b) are obtained
by a mirror symmetry $x\rightarrow-x$ from Fig. 1(a). The new mapping
$x^{\psi}(t_{in})\rightarrow x^{\psi}(t_{f})$ is now well described by the
variable transformation:

$\displaystyle
y(t_{in})=\frac{x^{\psi}(t_{in})+x_{in}}{2L}+\frac{1}{4}\in[0,\frac{1}{2}]$ if
$x^{\psi}(t_{in})\in\gamma_{-}(t_{in})$ $\displaystyle
y(t_{f})=\frac{x^{\psi}(t_{f})-x_{f}}{2L}+\frac{3}{4}\in[\frac{1}{2},1]$ if
$x^{\psi}(t_{f})\in\gamma_{+}(t_{f})$ $\displaystyle
y(t_{f})=\frac{x^{\psi}(t_{f})+x_{f}}{2L}+\frac{1}{4}\in[0,\frac{1}{2}]$
$\displaystyle\textrm{ if $x^{\psi}(t_{f})\in\gamma_{-}(t_{f})$ }.$

which lets the definition of $y(t_{f})$ unchanged with respect to Eq. 34. The
mapping between the two ensembles $\gamma_{-}(t_{in})$ and
$\gamma_{+}(t_{f})\cup\gamma_{-}(t_{f})$ is now written:

$\displaystyle y(t_{f})=2y(t_{in})$ (37)

which is very similar to Eq. 35.

###  III.2 Entanglement and Bernoulli’s shift

If we regroup Eq. 34 and Eq. LABEL:transf2 together with Eq. 35 and Eq. 37 we
are tempted to recognize the well known Bernoulli map:

$\displaystyle y(t_{f})=2y(t_{in})$ $\displaystyle\textrm{mod}(1),$ (38)

which actually means

$\displaystyle y(t_{f})=2y(t_{in})-1$ if $y(t_{in})>\frac{1}{2}$
$\displaystyle y(t_{f})=2y(t_{in})$ if $y(t_{in})<\frac{1}{2}$ (39)

for $y(t_{f})$ and $y(t_{in})\in[0,1]$. This would physically corresponds to a
mapping
$\gamma_{+}(t_{in})\cup\gamma_{-}(t_{in})\rightarrow\gamma_{+}(t_{f})\cup\gamma_{-}(t_{f})$.
In classical physics such a mapping would be unproblematic since the two
dynamics given by Eq. 35 and Eq. 37 could be superposed without interference.
However, in quantum mechanics, and specially in the BBQT, the dynamics is
contextually guided by the whole wave function $\psi(x,t)$ and a general
superposition of states
$\psi(x,t_{in})=a_{+}\psi_{0}(x,t_{in})+a_{-}\psi_{1}(x,t_{in})$ evolves at
$t_{f}$ to the state $\psi(x,t_{in})$ given by Eq. 18. Consider for example
with Eq. 30 the unitary evolution

$\displaystyle\frac{\psi_{0}(x,t_{in})+i\psi_{1}(x,t_{in})}{\sqrt{2}}\rightarrow
ie^{i0.267\pi}\psi_{1}(x,t_{f}).$ (40)

Figure 3: Same as in Fig. 1 but for a symmetric superposition of the two wave-
functions impinging from the two input gates of BS. The superposition
principle forces the resulting wave-packet to end its journey in the exit gate
$\gamma_{+}(t_{f})$. The pilot-wave dynamics is strongly impacted by the
linearity of the superposition (compare with Fig. 1).

From the point of view of the BBQT (as illustrated in Fig. 3) we have a
mapping $\gamma_{+}(t_{in})\cup\gamma_{-}(t_{in})\rightarrow\gamma_{+}(t_{f})$
which has nothing to do with either Eq. 35 and Eq. 37 nor even Eq. 38. More
precisely, the mapping associated with Eq. 40 reads:

$\displaystyle y(t_{f})=\frac{y(t_{in})}{2}+\frac{1}{2}$ (41)

Therefore, the high contextuality of the BBQT leads (in agreement with wave-
particle duality) to new features induced by the coherence of the different
branches of the input wave function.
In order to make sense of the Bernoulli shift Eq. 38 in a simple way we modify
the properties of our beam-splitter by adding phase plates in the input and
output channels and while this is not mandatory for the generality of the
reasoning we will from now consider instead of Eq 30 the unitary relation

$\displaystyle\left(\begin{array}[]{c}a^{\prime}_{+}\\\
a^{\prime}_{-}\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
1&-1\end{array}\right)\left(\begin{array}[]{c}a_{+}\\\
a_{-}\end{array}\right).$ (48)

Figure 4: Skecth of the iterative procedure for entangling a initial wave-
packet with ‘Bohmian’ pointers providing an unambiguous which- path
information on the pilot wave particle motion (as explained in the main text).
The various pointers interacting at time $t_{0}$, $t_{1}$ … are sketched as
qubit states.

Furthermore, in order the break the coherence between the two input waves
$\psi_{0}(x,t_{in})$ and $\psi_{1}(x,t_{in})$ we introduce entanglement with
an external pointer qubit before entering the beam splitter. The pointer must
represent an unambiguous ‘which-path’ information concerning the moving
particle in the context of the BBQT. We represent the initial state of the
pointer by a wavefunction $\varphi^{1}_{in}(Z_{1})$ associated with the
coordinate $Z_{1}$ of the pointer (we assume $\int
dZ_{1}|\varphi^{1}_{in}(Z_{1})|^{2}=1$). The interaction leading to
entanglement works in the following way: Starting with an arbitrary state like
$A\psi_{0}(x,t_{0})+B\psi_{1}(x,t_{0})$ at time $t_{0}$ and a fixed initial
pointer state $\varphi^{1}_{in}(Z_{1})$ we obtain:

$\displaystyle(A\psi_{0}(x,t_{0})+B\psi_{1}(x,t_{0})\varphi^{1}_{in}(Z_{1})\rightarrow
A\psi_{0}(x,t_{0})\varphi^{1}_{\uparrow}(Z_{1})+B\psi_{1}(x,t_{0})\varphi^{1}_{\downarrow}(Z_{1}).$
(49)

Here we assume $\int dZ_{1}|\varphi^{1}_{\uparrow}(Z_{1})|^{2}=\int
dZ_{1}|\varphi^{1}_{\downarrow}(Z_{1})|^{2}=1$ and $\int
dZ_{1}\varphi^{1}_{\uparrow}(Z_{1})(\varphi^{1}_{\downarrow}(Z_{1}))^{\ast}=0$.
Additionally, in order to simplify the analysis we suppose the pointer-
particle interaction to be quasi-instantaneous and acting only at time
$t\simeq t_{0}$. Moreover, in BBQT the positions of the particle and pointer
play a fundamental ontic role. In order to have a genuine Bohmian which-path
information we thus require that the two pointer wavefunctions are well
localized and are not overlapping, i.e.,
$\varphi^{1}_{\downarrow}(Z_{1})\varphi^{1}_{\uparrow}(Z_{1})=0$ $\forall
Z_{1}$.
We now consider the following sequences of processes which are sketched on
Fig. 4. First, we prepare a non-entangled quantum system in the initial state
$\psi_{0}(x,t^{\prime}_{0})\varphi_{in}(Z)$ with $t^{\prime}_{0}\ll t_{0}$.
Before interacting with the qubit the particle wave-packet interacts with a
first beam-splitter BS0 like in the previous subsection. Using Eq. 48 and Eq.
49 this leads to:

$\displaystyle\psi_{0}(x,t^{\prime}_{0})\varphi^{1}_{in}(Z_{1})\rightarrow\frac{\psi_{1}(x,t_{0})+\psi_{0}(x,t_{0})}{\sqrt{2}}\varphi^{1}_{in}(Z_{1})\rightarrow\frac{\psi_{1}(x,t_{0})\varphi^{1}_{\uparrow}(Z_{1})+\psi_{0}(x,t_{0})\varphi^{1}_{\downarrow}(Z_{1})}{\sqrt{2}}.$
(50)

In order to use a probabilistic interpretation, i.e., Born’s rule we impose
the normalization $C=1/\sqrt{L}$ associated with the wave-packet $\Phi_{0}$
(see Eq. 12). Second, as shown on Fig. 4 the two wave-packets are moving in
free space and interact with two mirrors which reflect the beams into the
direction of a second beam-splitter BS1 where they cross (BS1 is the time
translation of the same beam-splitter but we will continue to use this
notation for simplicity). The main effect of the mirrors is to reverse the
direction of propagation of $\psi_{0}(x,t_{0})$ and $\psi_{1}(x,t_{0})$, i.e.,
$\psi_{0}(x,t_{0})\rightarrow-\psi_{1}(x,t^{\prime}_{1}+\frac{2D}{v_{x}})e^{i\chi}$
and
$\psi_{1}(x,t_{0})\rightarrow-\psi_{0}(x,t^{\prime}_{1}+\frac{2D}{v_{x}})e^{i\chi}$
with $t^{\prime}_{1}$ a time after the interaction and
$\chi=\frac{2D}{v_{x}}(\omega_{k}-k_{x}v_{x})$ a phase shift depending on the
distance $D$ between BS0 and any of the two mirrors ($-\frac{2D}{v_{x}}>0$ is
the travel-time taken by the center of the wave-packet for moving from BS0 to
BS1). At a time $t^{\prime}_{1}$ before crossing BS1 the quantum state reads
thus

$\displaystyle-e^{i\chi}\frac{\psi_{0}(x,t^{\prime}_{1}+\frac{2D}{v_{x}})\varphi^{1}_{\uparrow}(Z_{1})+\psi_{1}(x,t^{\prime}_{1}+\frac{2D}{v_{x}})\varphi^{1}_{\downarrow}(Z_{1})}{\sqrt{2}}.$
(51)

At a time $t_{1}\gg-\frac{2D+L}{v_{x}}$ after the interaction with BS1 the
quantum state reads (omitting the irrelevant phase factor)

$\displaystyle\frac{\psi_{0}(x,t^{\prime}_{1}+\frac{2D}{v_{x}})\varphi^{1}_{\uparrow}(Z_{1})+\psi_{1}(x,t^{\prime}_{1}+\frac{2D}{v_{x}})\varphi^{1}_{\downarrow}(Z_{1})}{\sqrt{2}}$
$\displaystyle\rightarrow\frac{\psi_{1}(x,t_{1}+\frac{2D}{v_{x}})\varphi^{1}_{\rightarrow}(Z_{1})+\psi_{0}(x,t_{1}+\frac{2D}{v_{x}})\varphi^{1}_{\leftarrow}(Z_{1})}{\sqrt{2}}$
(52)

where
$\varphi^{1}_{\rightarrow}=\frac{\varphi^{1}_{\uparrow}+\varphi^{1}_{\downarrow}}{\sqrt{2}}$
and
$\varphi^{1}_{\leftarrow}=\frac{\varphi^{1}_{\uparrow}-\varphi^{1}_{\downarrow}}{\sqrt{2}}$
are two orthogonal eigenstates. Now, if we write this quantum state during the
interaction with $BS_{1}$ as
$\Psi(x,Z,t)=\psi_{\uparrow}(x,t)\varphi^{1}_{\uparrow}(Z_{1})+\psi_{\downarrow}(x,t)\varphi^{1}_{\downarrow}(Z_{1})$
we can define the Bohmian particle velocity $\frac{d}{dt}x^{\Psi}(t)=v(x,Z,t)$
as:

$\displaystyle\frac{d}{dt}x^{\Psi}(t)=\frac{v_{\uparrow}(x,t)|\psi_{\uparrow}(x,t)\varphi^{1}_{\uparrow}(Z_{1})|^{2}+v_{\downarrow}(x,t)|\psi_{\downarrow}(x,t)\varphi^{1}_{\downarrow}(Z_{1})|^{2}}{|\psi_{\uparrow}(x,t)\varphi^{1}_{\uparrow}(Z_{1})|^{2}+|\psi_{\downarrow}(x,t)\varphi^{1}_{\downarrow}(Z_{1})|^{2}}$
(53)

where we introduced the two velocities
$v_{\uparrow/\downarrow}(x,t)=\frac{1}{m}\partial_{x}S_{\uparrow/\downarrow}(x,t)$
associated with the two wavefunctions $\psi_{\uparrow/\downarrow}(x,t)$. Eq.
53 relies on the ‘which-path’ constraint
$\varphi^{1}_{\downarrow}(Z_{1})\varphi^{1}_{\uparrow}(Z_{1})=0$ and therefore
we have here two different dynamics depending on the pointer position $Z_{1}$.
If $Z_{1}$ lies in the support of $\varphi^{1}_{\uparrow}(Z_{1})$ we have the
dynamics $\frac{d}{dt}x^{\Psi}(t)=v_{\uparrow}(x,t)$ corresponding to Fig.
1(a) whereas if $Z_{1}$ lies in the support of
$\varphi^{1}_{\downarrow}(Z_{1})$ we have the dynamics
$\frac{d}{dt}x^{\Psi}(t)=v_{\downarrow}(x,t)$ corresponding to Fig. 1(b).
The previous procedure for generating decohered Bohmian paths can be repeated
iteratively at the times $t_{2}$, $t_{3}$,… after interaction with the beam-
splitter BS2, BS3…(see Fig. 4). For this purpose we consider at time $t_{1}$
entanglement with a an additional pointer initially in the state
$\varphi^{2}_{in}(Z_{)}$ and we assume the transformation:

$\displaystyle\frac{\psi_{1}(x,t_{1}+\frac{2D}{v_{x}})\varphi^{1}_{\rightarrow}(Z_{1})+\psi_{0}(x,t_{1}+\frac{2D}{v_{x}})\varphi^{1}_{\leftarrow}(Z_{1})}{\sqrt{2}}\varphi^{2}_{in}(Z_{2})$
$\displaystyle\rightarrow\frac{\psi_{1}(x,t_{1}+\frac{2D}{v_{x}})\varphi^{1}_{\rightarrow}(Z_{1})\varphi^{2}_{\uparrow}(Z_{2})+\psi_{0}(x,t_{1}+\frac{2D}{v_{x}})\varphi^{1}_{\leftarrow}(Z_{1})\varphi^{2}_{\downarrow}(Z_{2})}{\sqrt{2}}.$
(54)

The wave-packets propagate into the interferometer and between the time
${t^{\prime}}_{2}$ and $t_{2}$ we obtain

$\displaystyle\frac{\psi_{0}(x,t^{\prime}_{2}+\frac{4D}{v_{x}})\varphi^{1}_{\rightarrow}(Z_{1})\varphi^{2}_{\uparrow}(Z_{2})+\psi_{1}(x,t^{\prime}_{2}+\frac{4D}{v_{x}})\varphi^{1}_{\leftarrow}(Z_{1})\varphi^{2}_{\downarrow}(Z_{2})}{\sqrt{2}}$
$\displaystyle\rightarrow\frac{\psi_{1}(x,t_{2}+\frac{4D}{v_{x}})\varphi^{12}_{\rightarrow}(Z_{1},Z_{2})+\psi_{0}(x,t_{2}+\frac{4D}{v_{x}})\varphi^{12}_{\leftarrow}(Z_{1},Z_{2})}{\sqrt{2}}$
(55)

with the orthonormal states
$\varphi^{12}_{\rightarrow}=\frac{1}{\sqrt{2}}(\varphi^{1}_{\rightarrow}\varphi^{2}_{\uparrow}+\varphi^{1}_{\leftarrow}\varphi^{2}_{\downarrow})$
and
$\varphi^{12}_{\leftarrow}(Z_{1},Z_{2})=\frac{1}{\sqrt{2}}(\varphi^{1}_{\rightarrow}\varphi^{2}_{\uparrow}-\varphi^{1}_{\leftarrow}\varphi^{2}_{\downarrow})$.
This can be generalized at any time $t_{n}$ after interaction with BSn:

$\displaystyle\frac{\psi_{0}(x,t^{\prime}_{n}+\frac{2nD}{v_{x}})\varphi^{1,...,n-1}_{\rightarrow}(Z_{1},...,Z_{n-1})\varphi^{n}_{\uparrow}(Z_{n})+\psi_{1}(x,t^{\prime}_{n}+\frac{2nD}{v_{x}})\varphi^{1,...,n-1}_{\leftarrow}(Z_{1},...,Z_{n-1})\varphi^{n}_{\downarrow}(Z_{n})}{\sqrt{2}}$
$\displaystyle\rightarrow\frac{\psi_{1}(x,t_{n}+\frac{2nD}{v_{x}})\varphi^{1,...,n}_{\rightarrow}(Z_{1},...,Z_{n})+\psi_{0}(x,t_{n}+\frac{2nD}{v_{x}})\varphi^{1,...,n}_{\leftarrow}(Z_{1},...,Z_{n})}{\sqrt{2}},$
(56)

with the orthonormal states
$\varphi^{1,...,n}_{\rightarrow/\leftarrow}=\frac{1}{\sqrt{2}}(\varphi^{1,...,n-1}_{\rightarrow}\varphi^{n}_{\uparrow}\pm\varphi^{1,...,n-1}_{\leftarrow}\varphi^{n}_{\downarrow})$.
Like for the interaction at BS1 (between $t^{\prime}_{1}$ and $t_{1}$) we can
define a Bohmian dynamical evolution similar to Eq. 53 but based on the wave
function

$\displaystyle\Psi(x,Z1,...Z_{n},t)=\psi_{\uparrow}(x,t)\varphi^{1,...,n-1}_{\rightarrow}(Z_{1},...,Z_{n-1})\varphi^{n}_{\uparrow}(Z_{n})$
$\displaystyle+\psi_{\downarrow}(x,t)\varphi^{1,...,n-1}_{\leftarrow}(Z_{1},...,Z_{n-1})\varphi^{2}_{\downarrow}(Z_{n}).$
(57)

We obtain the velocity

$\displaystyle\frac{d}{dt}x^{\Psi}(t)=\frac{v_{\uparrow}(x,t)|\psi_{\uparrow}(x,t)\varphi^{1,...,n-1}_{\rightarrow}(Z_{1},...,Z_{n-1})\varphi^{n}_{\uparrow}(Z_{n})|^{2}}{|\psi_{\uparrow}(x,t)\varphi^{1,...,n-1}_{\rightarrow}(Z_{1},...,Z_{n-1})\varphi^{n}_{\uparrow}(Z_{n})|^{2}+|\psi_{\downarrow}(x,t)\varphi^{1,...,n-1}_{\leftarrow}(Z_{1},...,Z_{n-1})\varphi^{2}_{\downarrow}(Z_{n})|^{2}}$
$\displaystyle+\frac{v_{\downarrow}(x,t)|\psi_{\downarrow}(x,t)\varphi^{1,...,n-1}_{\leftarrow}(Z_{1},...,Z_{n-1})\varphi^{2}_{\downarrow}(Z_{n})|^{2}}{|\psi_{\uparrow}(x,t)\varphi^{1,...,n-1}_{\rightarrow}(Z_{1},...,Z_{n-1})\varphi^{n}_{\uparrow}(Z_{n})|^{2}+|\psi_{\downarrow}(x,t)\varphi^{1,...,n-1}_{\leftarrow}(Z_{1},...,Z_{n-1})\varphi^{2}_{\downarrow}(Z_{n})|^{2}}$
(58)

which like Eq. 53 reduces to the one of the two dynamics i)
$\frac{d}{dt}x^{\Psi}(t)=v_{\uparrow}(x,t)$ if $Z_{n}$ lies in the support of
$\varphi^{n}_{\uparrow}(Z_{n})$ (i.e., corresponding to Fig. 1(a)), or ii)
$\frac{d}{dt}x^{\Psi}(t)=v_{\downarrow}(x,t)$ if $Z_{n}$ lies in the support
of $\varphi^{n}_{\downarrow}(Z_{n})$ (i.e., corresponding to Fig. 1(b)). The
full history of the particle in the interferometer depends on the positions
$Z_{1},...,Z_{n}$ taken by the various Bohmian pointers. In turn this
deterministic iterative process allows us to define a Bernoulli map for the
evolution.

###  III.3 Mixing, chaos, and relaxation to quantum equilibrium

The Bernoulli map is clearly defined from Eqs. 38,39 after introducing the
variable $y(t)$ replacing $x(t)$. Between $t^{\prime}_{n}$ and $t_{n}$ this
reads:

$\displaystyle y(t_{n})=2y(t^{\prime}_{n})$ $\displaystyle\textrm{mod}(1).$
(59)

Moreover, the $y(t^{\prime}_{n})$ coordinate at time $t^{\prime}_{n}$ is
obviously equal to $y(t_{n-1})$ at time $t_{n-1}$ (see Fig. 4) and therefore
we have the map

$\displaystyle y(t_{n})=2y(t_{n-1})$ $\displaystyle\textrm{mod}(1).$ (60)

This iterative Bernoulli map $y_{n}=F(y_{n-1})$ is one of the simplest and
most known chaotic map discussed in the literature Driebe ; Schuster . In
particular, its chaotic nature has been already studied in the context of the
BBQT Goldstein1992 ; Dewdney1996  (for different purposes as the one
considered here) and an attempt to use it for deriving Born’s rule has been
worked out Geiger  (without the entanglement used here and in Philbin2015 ;
Drezet2017 ).
The chaotic nature of the map is easy to obtain. Consider for example Fig. 5.

Figure 5: (a) Bernoulli map $y_{n}=F(y_{n-1})$ in the plane $y,y^{\prime}$
where the function $y^{\prime}=F(y)$ acts iteratively. The red and green line
are acting as mirrors during the process. The black and blue trajectories
correspond to different initial coordinates $y_{0}=0.22$ and $y_{0}=0.23$. (b)
The same Bernoulli map is shown a as function $y=y(n)$ of the iteration steps
$n=0,1,...$. The two chaotic trajectories shown in red and blue respectively
correspond to $y_{0}=0.22$ and $y_{0}=0.220001$ (see main text).

In Fig. 5(a) we show a standard representation of the iterative function
$y_{n}=F(y_{n-1})$ for two paths initially starting at $y_{0}=0.22$ and
$y_{0}=0.23$ and after few iterations the coordinates are apparently diverging
in a unpredictable way. This is even more clear in the representation of Fig.
5(b) where two trajectories $y(t_{n}):=y_{n}$ are shown with $y_{0}=0.22$ and
$y_{0}=0.220001$. Again, the motions becomes chaotic after few iterations and
the trajectories are strongly diverging. Mathematically, any number $y$ in the
interval $[0,1]$ is represented in binary notations as
$0.u_{1}u_{2}...u_{n}...$, i.e.,
$y=\frac{u_{1}}{2}+\frac{u_{2}}{4}+...+\frac{u_{n}}{2^{n}}+...$ where
$u_{n}=0$ or 1. The Bernoulli transformation $y^{\prime}=F(y)$ with
$y^{\prime}=\frac{u^{\prime}_{1}}{2}+\frac{u^{\prime}_{2}}{4}+...+\frac{u^{\prime}_{n}}{2^{n}}+...$
corresponds to the shift $u^{\prime}_{n}=u_{n-1}$ i.e., to the binary number
$0.u_{2}u_{3}...u_{n-1}...$. Iteratively this generates chaos since if the
$n^{th}$ term in
$y=\frac{u_{1}}{2}+\frac{u_{2}}{4}+...+\frac{u_{n}}{2^{n}}+...$ is known with
an uncertainty $\delta y_{i}=\frac{1}{2^{n}}$ after $n$ iterations this
uncertainty will grow up to $\delta y_{f}=1/2$. For example if $n=133$ and
$\delta y_{i}=2^{-133}\simeq 10^{-40}$ we have after only 40 iterations
completely lost any predictability in this dynamics (note that rational
numbers are periodical in the binary representation and therefore the sequence
will reappear periodically for rational numbers representing a null measure in
the segment $[0,1]$). It can be shown that this feature leads to randomness in
close analogy with ideal probabilistic coin tossing Ford . Therefore, any
uncertainty will ultimately lead to chaos. The Lyapunov divergence of this
Bernoulli map is readily obtained by considering as in Fig. 5 two trajectories
$y^{(A)}_{n}$ and $y^{(B)}_{n}=y^{(A)}_{n}+\delta y_{n}$ differing by a
infinitesimal number such that

$\displaystyle\delta y_{n}=2\delta y_{n-1}=2^{n}\delta y_{0}=e^{n\ln{2}}\delta
y_{0}$ (61)

where the positive Lyapunov exponent $\ln{2}$ characterizes this chaotic
dynamics. If we introduce the time delay $\delta t=-2D/v_{x}>0$ and define the
evolution time as $t_{n}=n\delta t$ we can rewrite the exponential divergence
in Eq. 61 as $e^{+t/\tau}$ where $\tau=\frac{\delta t}{\ln{2}}$ defines a
Lyapunov time.
Most importantly, the Bernoulli shift allows us to define a mixing property
for the probability distribution $\rho(y)$. More precisely we can consider at
any time $t_{n}$ the probability density
$\rho(x,t_{n})=\int...\int\rho(x,Z_{1},...,Z_{n},t_{n})dZ_{1}...dZ_{n}$ where
according to the BBQT we have
$\rho(x,Z_{1},...,Z_{n},t_{n})=f(x,Z_{1},...,Z_{n},t_{n})|\Psi(x,Z_{1},...Z_{n},t_{n})|^{2}$.
In this framework $\rho(x,t_{n})$ is a coarse-grained probability involving a
form of classical ignorance. In the following we will suppose that the
pointers are all in quantum equilibrium and therefore we have
$f(x,Z_{1},...,Z_{n},t_{n}):=f(x,t_{n})$ and
$\rho(x,t_{n})dx=f(x,t_{n})d\Gamma(x,t_{n})$ with
$d\Gamma(x,t_{n})=dx\int...\int|\Psi(x,Z_{1},...Z_{n},t_{n})|^{2}dZ_{1}...dZ_{n}$.
For the present purpose a key result of deterministic maps like
$y_{n}=F(y_{n-1})$ is the Perron-Frobenius theorem Driebe ; Schuster  which
allows us to introduce the operator $\hat{U}_{PF}$, i.e.,
$\mu(y,t_{n+1})=\hat{U}_{PF}\mu(y,t_{n})$ with be definition
$\rho(x,t)dx=\mu(y,t)dy$. For this we use the property for a trajectory

$\displaystyle\delta(y-y_{n+1})=\delta(y-F(y_{n}))=\int_{0}^{1}dY\delta(y-F(Y))\delta(Y-y_{n})$
(62)

and the fact that any density $\mu(w,t_{n})$ reads

$\displaystyle\int_{0}^{1}dy(t_{n})\mu(y(t_{n}),t_{n})\delta(w-y(t_{n}))=\int_{0}^{1}dy(t_{0})\mu(y(t_{0}),t_{0})\delta(w-y(t_{n}))$
(63)

(where we used Liouville’s theorem
$dy(t_{n})\mu(y(t_{n}),t_{n})=dy(t_{0})\mu(y(t_{0}),t_{0})$). Therefore, from
Eq. 62 we obtain

$\displaystyle\mu(y,t_{n+1})=\hat{U}_{PF}\mu(y,t_{n})=\int_{0}^{1}dY\delta(y-F(Y))\mu(Y,t_{n})$
(64)

which for the Bernoulli map means

$\displaystyle\mu(y,t_{n+1})=\hat{U}_{PF}\mu(y,t_{n})=\frac{1}{2}\left[\mu(\frac{y}{2},t_{n})+\mu(\frac{y+1}{2},t_{n})\right].$
(65)

Moreover, for the present wavefunction defined in term of the wave-packet
$\Phi_{0}(x)$ which is constant in amplitude in its support we can also write

$\displaystyle\tilde{f}(y,t_{n+1})=\hat{U}_{PF}\tilde{f}(y,t_{n})=\frac{1}{2}\left[\tilde{f}(\frac{y}{2},t_{n})+\tilde{f}(\frac{y+1}{2},t_{n})\right]$
(66)

with by definition $f(x,t)=\tilde{f}(y,t)$ using the transformation
$x\rightarrow y$ (see Eqs. 34,LABEL:transf2) and where
$\int_{\gamma_{+}(t_{n})\cup\gamma_{-}(t_{n})}dx\frac{|C|^{2}}{2}f(x,t_{n})=\int_{0}^{1}dy\tilde{f}(y,t_{n})=1$
involving the normalization $C=1/\sqrt{L}$. This iterative Perron-Frobenius
relation admits Bernoulli polynomial eigenstates defined by
$\frac{1}{2^{n}}B_{n}(y)=\hat{U}_{PF}B_{n}(y)$ with $B_{0}(y)=1$,
$B_{1}(y)=y-1/2$, $B_{2}(y)=y^{2}-y+1/6$,…Driebe .
It can be shown Driebe  that the $B_{m}(y)$ polynomials form a basis for the
probability function $\tilde{f}(y,t)$ and therefore it we we write
$\tilde{f}(y,t_{0})=\sum_{m=0}^{m=+\infty}A_{m}B_{m}(y)$ we have after $n$
iterations of the $\hat{U}_{PF}-$operator:

$\displaystyle\tilde{f}(y,t_{n})=\sum_{m=0}^{m=+\infty}A_{m}e^{-n\cdot
m\ln{2}}B_{m}(y).$ (67)

In this formula we have Driebe ; Prigogine2

$\displaystyle A_{m}=\int_{0}^{1}dy\tilde{f}(y,t_{0})\tilde{B}_{m}(y)$ (68)

where $\tilde{B}_{0}(y)=1$ and $\tilde{B}_{m}(y)=\lim_{\varepsilon\rightarrow
0^{+}}\frac{(-1)^{m-1}}{m!}\frac{d^{m-1}}{dy^{m-1}}[\delta(y-1+\varepsilon)-\delta(y-\varepsilon)]$
for $m\geq 1$. This leads to $A_{0}=\int_{0}^{1}\tilde{f}(y,t_{0})dy$ and
$A_{m}=\lim_{\varepsilon\rightarrow
0^{+}}\frac{1}{m!}\frac{d^{m-1}}{dy^{m-1}}[\tilde{f}(1-\varepsilon,t_{0})-\tilde{f}(0+\varepsilon,t_{0})]$.
Eq. 67 is important since it shows that in limit $n\rightarrow+\infty$ we have
necessarily $\tilde{f}(y,t_{n})\rightarrow A_{0}B_{0}(y)=A_{0}$. Moreover,
from the properties of the Bernoulli polynomials and the normalization of the
probability density we have necessarily $\int_{0}^{1}\tilde{f}(y,t)dy=A_{0}=1$
(with $\int_{0}^{1}dyB_{m}(y)=\delta_{0,m}$). Therefore we deduce

$\displaystyle\lim_{n\rightarrow+\infty}\tilde{f}(y,t_{n})=\lim_{n\rightarrow+\infty}f(x,t_{n})=1.$
(69)

Figure 6: Evolution of $\tilde{f}(y,t_{n}):=\tilde{f}_{n}(y)$ as a function of
$y$ for few $n$ values (using the Perron-Frobenius operator Eq. 66). The
initial distribution $\tilde{f}_{0}(y)$ (blue curve) was chosen to be
arbitrarily irregular. After few iterations $n\geq 2$ the function
$\tilde{f}_{n}(y)$ can no be distinguished from the line $\tilde{f}=f=1$
associated with quantum equilibrium (i.e., Born’s rule).

This result says that quantum equilibrium, and therefore Born’s rule is a
statistical attractor in the BBQT. Importantly, Eq. 67 shows that each term in
the sum is characterized by an exponential decay $e^{-mt_{n}/\tau}$ which is a
signature of stability (negative Lyapunov exponent) whereas the trajectories
(as we have shown in Eq. 61) have a positive Lyapunov exponent associated with
dynamical instability and chaos. These two pictures are thus clearly
complementary. This was already emphasized long ago by Prigogine in a
different context Prigogine ; Prigogine2 . As an illustration we show on Fig.
6 the transformation of an arbitrary (normalized) density
$\tilde{f}(y,t_{0})$: after only three applications of the Perron-Frobenius
operator the density is indistinguishable from the quantum equilibrium
$\tilde{f}=f=1$ which is therefore a very efficient attractor.
We emphasize that the iterative process sketched in Fig. 4 and associated with
states like Eqs. 56,57 involves in the end two branches $\psi_{0}(x,t_{n})$
and $\psi_{1}(x,t_{n})$ entangled with an environment of Bohmian pointers
characterized by
$\varphi^{1,...,n}_{\rightarrow/\leftarrow}=\frac{1}{\sqrt{2}}(\varphi^{1,...,n-1}_{\rightarrow}\varphi^{n}_{\uparrow}\pm\varphi^{1,...,n-1}_{\leftarrow}\varphi^{n}_{\downarrow})$.
Moreover, because of the orthogonality of theses pointer states the two
branches $\psi_{0}(x,t_{n})$ and $\psi_{1}(x,t_{n})$ can not interfere: they
are decohered. Still, in each of the two final wave-packets
$\psi_{0}(x,t_{N})$ and $\psi_{1}(x,t_{N})$ (after a large number of
iterations $N$) we have $f(x,t_{N})\simeq 1$ with a high accuracy. Therefore,
supposing that we now make a pinhole for selecting one of these two branches
we have prepared a quantum system satisfying Born’s rule
$\rho(x,t)\simeq|\psi(x,t)|^{2}$. Fundamentally, it means that if an entangled
system like the one we discussed is postselected by a filtering procedure we
can define subsystems for which Born rules is true and where quantum coherence
is maintained (this is the case with our two wave functions $\psi_{1}$ and
$\psi_{0}$ taken separately. For example the wave function $\psi_{0}(x,t)$ can
be collimated and sent into an interferometer in order to observe wave-
particle duality. All the systems following this guiding wave belong to a
statistical ensemble of particles obeying Born’s rule $f\simeq 1$. Therefore,
all the predictions of standard quantum mechanics will be reproduced with
these systems.
Although the present model is rudimentary it allows us to obtain precious
information on relaxation to quantum equilibrium. Indeed, observe that in the
continuous time approximation we have $\tilde{f}(y,t)\simeq
1+A_{1}e^{-t/\tau}B_{1}(y)$ which is a solution of the differential equation

$\displaystyle\frac{\partial\tilde{f}(y,t)}{\partial
t}=-\frac{\tilde{f}(y,t)-1}{\tau}$ (70)

This suggests a collision term in a Boltzmann-like equation and therefore an
extension of our model by writing

$\displaystyle\frac{df(x,t)}{dt}:=\partial_{t}f(x,t)+v_{\psi}(x,t)\partial_{x}f(x,t)=-\frac{f(x,t)-1}{\tau}$
(71)

or equivalently with $\rho(x,t)=f(x,t)|\psi(x,t)|^{2}$ and
$\partial_{t}|\psi(x,t)|^{2}+\partial_{x}(v_{\psi}(x,t)|\psi(x,t)|^{2})=0$:

$\displaystyle\partial_{t}\rho(x,t)+\partial_{x}(v_{\psi}(x,t)\rho(x,t))=-\frac{\rho(x,t)-|\psi(x,t)|^{2}}{\tau}.$
(72)

With such dynamics (with an effective broken time symmetry) it is useful to
introduce the Valentini entropy Valentini1991 :

$\displaystyle S_{t}:=-\int f(x,t)\ln{(f(x,t))}d\Gamma(x,t)$ (73)

with $d\Gamma(x,t)=|\psi(x,t)|^{2}$. From the previous equation we deduce

$\displaystyle\frac{d}{dt}S_{t}=-\int\frac{df_{t}}{dt}(1+\ln{f_{t}})d\Gamma_{t}=\int\frac{(f_{t}-1)}{\tau}(1+\ln{f_{t}})d\Gamma_{t}=\int\frac{(f_{t}-1)}{\tau}\ln{f_{t}}d\Gamma_{t}.$
(74)

Such a kinetic equation leads to a quantum version of the Boltzmann H-theorem
as it can be shown easily: First, we have by definition
$a\ln{b}+\frac{a}{b}-a\geq 0$ (with $a,b>0$) leading to
$(f-1)\ln{f}+\frac{f-1}{f}-f-1\geq 0$ if $f-1>0$, i.e., we obtain
$(f-1)\ln{f}\geq\frac{(f-1)^{2}}{f}$ if $f-1>0$. Moreover, we have also
$\ln{f}\leq f-1$ and thus if $f-1<0$ we have $(f-1)\ln{f}\geq(f-1)^{2}$. Now,
separating the full $\Gamma-$ space into two parts $\Gamma_{+}$ and
$\Gamma_{-}$ where $f-1\geq 0$ and $1-f\geq 0$ respectively we have:

$\displaystyle\frac{d}{dt}S_{t}=\int\frac{(f_{t}-1)}{\tau}\ln{f_{t}}d\Gamma_{t}\geq\int_{\Gamma_{+}}\frac{(f_{t}-1)^{2}}{f_{t}\tau}d\Gamma_{t}+\int_{\Gamma_{-}}\frac{(f_{t}-1)^{2}}{\tau}d\Gamma_{t}\geq
0.$ (75)

Therefore, Valentini’s entropy $S_{t}$ can not decrease and the equality
$\frac{d}{dt}S_{t}=0$ occurs iff $f=1$ corresponding to the quantum
equilibrium. This defines a H-theorem for the BBQT.

##  IV Conclusion and perspectives

The proposal discussed in this work is certainly schematic but it leads to
several interesting conclusions. First, since the dynamics maps used here is
deterministic and chaotic it shows that randomness is unavoidable in the BBQT.
As it was stressed by Prigogine Prigogine ; Prigogine2  we have here two
complementary descriptions: one with trajectories which can be associated with
the evolution map $y_{n+1}=F(y_{n})$ and the second with probability density
i.e., as given by the Perron-Frobenius transformation
$\tilde{f}(y,t_{n+1})=\hat{U}_{PF}\tilde{f}(y,t_{n})$. The two pictures are of
course not independent since for a single trajectory we have
$\delta(y-y_{n+1})=\hat{U}_{PF}\delta(y-y_{n})$ (i.e.,
$\tilde{f}(y,t_{n})=\delta(y-y_{n})=f(x,t_{n})=\frac{2}{|C|^{2}}\delta(x-x_{n})=2L\delta(x-x_{n})$).
Moreover, for a trajectory the probability distribution is singular and the
convergence to equilibrium is infinitely slow (this is connected to the fact
that the coefficients $A_{m}$ in Eq. 68 is given by an integral which is badly
defined for the singular Dirac distribution
$\tilde{f}(y,t_{0})=\delta(y-y_{0})$). Therefore, the infinite precision
required for computing such a chaotic path (due to the exponentially growing
deviation errors with time) leads for all practical computations to the strong
randomness above mentioned. To quote Ford Ford  “a chaotic orbit is random and
incalculable; its information content is both infinite and incompressible”.
Subsequently, because of the extreme sensitivity in the initial conditions
associated with the predictability horizon and the positive Lyapunov exponent,
the use of probability distributions in the BBQT seems (at least in our model)
unavoidable if we follow Prigogine reasoning. Indeed, for Prigogine dynamic
instability (and thus deterministic chaos) leads to probability. The
necessarily finite precision $\delta y_{0}$ used for determining the position
of a particle will grow exponentially with time to ultimately cover the whole
segment $[0,1]$. Therefore, if we assign a uniform ignorance probability
$\tilde{f}_{0}$ over the segment $\delta y_{0}$ (in which the particle is
located) then, i.e., subsequently after few iteration, we will have
$\tilde{f}_{t}=1$ over the whole segment. Yet, we stress that we don’t here
share all the conclusions obtained by Prigogine concerning determinism and
probability (for related and much more detailed criticisms see e.g., Bricmont
). Indeed, the BBQT (like classical mechanics that was considered by Prigogine
in Prigogine ; Prigogine2 ) is a fully deterministic theory with a clear
ontology in the 3D and configuration space. Therefore, while a trajectory
could be incalculable by any finite mean or algorithm, the path still
fundamentally exists for an idealized Laplacian daemon, i.e., having access to
an infinite computing power and precision for locating and defining the
particle motion. This metaphor is the core idea of Einstein’s realism:
Postulating the existence of a World independently of the presence or absence
of observers (even if the observers can be part of the World). From this ontic
perspective we need more than just ignorance in order to justify the use of
probability in statistical physics. Indeed, as emphasized long ago by Poincaré
the laws of the kinetic theory of gases still hold true even if we exactly
know the positions of all molecules - Poincare . There is something of
objective in the laws of statistical mechanics that goes beyond mere
ignorance: Otherwise how parameters such as diffusion constants would have
objective physical contents? This point was emphasized by Prigogine from the
very beginning and this constitutes the motivation for his program in order to
justify the objectivity of thermodynamics in general and the second law, i.e.,
irreversibility in particular. However, in our opinion the missing point in
Prigogine implication: “instability $\rightarrow$ probability $\rightarrow$
irreversibility” is the recognition that in a deterministic theory the laws
(chaotic or not) are not all but must be supplemented by specific initial
conditions ultimately having a cosmological origin. Indeed, if we suppose a
Universe made of only one electron described initially by the wavefunction
$\psi_{0}(x,t)$ and all the pointers involved in the iterative procedure
sketched in Fig. 4 then we must use the chaotic Bernoulli map
$y_{n+1}=F(y_{n})$ for this system or equivalently the Perron-Frobenius
evolution $\delta(y-y_{n+1})=\hat{U}_{PF}\delta(y-y_{n})$. As we explained
this system is unstable due to the presence of a positive Lyapunov exponent.
Moreover, if we want to make sense of the formulas 66 and 67 with the rapid
convergence to $\tilde{f}=f=1$ we must consider a sufficiently regular
distribution $\tilde{f}(y,t_{0})\neq\delta(y-y_{0})$. Now, as reminded in
Sec.II the application of the WLLN to a statistical ensemble requires a
‘metric’ of typicality associated with the Laplacian definition of
probability. In BBQT this metric reads
$\rho(\mathbf{r},t)=f(\mathbf{r},t)|\psi(\mathbf{r},t)|^{2}$ and the law of
large numbers leads to Eq. 11, i.e.,
$\rho(\mathbf{r},t)\equiv\lim_{N\to+\infty}\frac{1}{N}\sum_{k=1}^{k=N}\delta^{3}(\mathbf{r}-\mathbf{x}^{\psi}_{k}(t))$
defined probabilistically in the long run, i.e, for an infinitely long
sequence or infinite system. In our problem, this means that we consider an
infinite Gibbs ensemble of copies similar to our system described in Fig. 4.
Here, the presence of an infinite sum of Dirac distributions is expected to
lead to difficulties in connections with the chaotic map
$\delta(y-y_{n+1})=\hat{U}_{PF}\delta(y-y_{n})$. Therefore, in our problem if
the WLLN
$\rho(\mathbf{r},t)\equiv\lim_{N\to+\infty}\frac{1}{N}\sum_{k=1}^{k=N}\delta^{3}(\mathbf{r}-\mathbf{x}^{\psi}_{k}(t))$
is used to specify the initial distribution at time $t_{0}$ we must be aware
that this should preserve the chaotic description associated with the positive
Lyapunov exponent and Dirac distributions should be problematic. In order to
remove this unpleasant feature one must introduce coarse-graining as proposed
by Valentini Valentini1991 ; Valentini2020 . In our case this can be done by
using a regular weighting function $\Delta(u)$ such that
$\overline{\rho}(x,t)=\int du\delta(u)\rho(x-u,t)$ which in connection with
the WLLN leads to
$\overline{\rho}(x,t)\equiv\lim_{N\to+\infty}\frac{1}{N}\sum_{k=1}^{k=N}\Delta(x-x^{\psi}_{k}(t))$.
Coarse-graining of cells in the configuration space plays a central role in
the work of Valentini for defining a “subquantum H-theorem” Valentini1991 ;
ValentiniPhD . Here, we see that in connection with Prigogine work coarse-
graining must be supplemented with a dose of deterministic chaos and
entanglement in order to reach the quantum equilibrium regime. We believe that
these two picture complete each other very well.
To summarize, in this work, we have proposed a mechanism for relaxation to
quantum equilibrium in order to recover Born’s rule in the BBQT. The proposed
mechanism relies on entanglement with an environment of ‘Bohmian pointers’
allowing the system to be mixing. The scenario was developed for the case of a
single particle in 1D motion interacting with beam splitters and mirrors but
the model could be generalized to several situations involving collisions
between quanta and scattering with defects or other particles. The general
proposal is thus to consider the quantum relaxation to Born’s rule as a
genuine process in phases of matter where interactions between particles play
a fundamental role. This involves usual condensed matters or even plasma or
gases where collisions are mandatory. For example, if based on our toy model
we consider that interaction with the beam splitter and entanglement with
Bohmian pointers is a good qualitative model for discussing collisions between
molecules in the atmosphere, and if we remember that nitrogen molecules at a
temperature of 293 K and at a pressure of 1 bar involve typically a collision
frequency of 7 $10^{9}$ /s (which implies a fast dynamics for reaching quantum
equilibrium) we have thus a huge number $n$ of collisions per second
corresponding to a huge number of iterations in our Bernoulli-like process
based on the Perron-Frobenius operator $f(y,t_{n+1})=\hat{U}_{PF}f(y,t_{n})$.
Compared to Valentini’s framework Valentini1991 ; ValentiniPhD  where mixing
and relaxation to quantum equilibrium are associated with coarse-graining à la
Gibbs our approach emphasizes the role of information losses due to
entanglement with a local environment. In both cases, we obtain an increase of
entropy and a formulation of the $H-$theorem for the BBQT. Theses two views
are certainly complementary in the same way that Gibbs and Boltzmann
perspectives on entropy are related. This could have an impact on the
efficiency of quantum relaxation in the early stages of the Universe evolution
Valentini2007 ; Valentini2015 .

## References

  * (1) Zurek, W. H. Decoherence and the transition from quantum to classical. Physics Today 1991, 44, 36-44.
  * (2) Zurek, W. H. Quantum Darwinism, Classical reality and the Randomness of Quantum Jumps. Physics Today 2014, 67, 44-50.
  * (3) Zurek, W. H. Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Mod. Phys. 2004, 76, 1267-1305.
  * (4) Zurek, W. H. Emergence of the Classical from within the Quantum Universe. arXiv:2107.03378v1.
  * (5) Wallace, D. The ermerging Multiverse; Oxford University Press: Oxford, 2012.
  * (6) Drezet, A. Making sense of Born’s rule $p_{\alpha}=|\Psi_{\alpha}|^{2}$ with the many-minds interpretation Quantum Stud.: Math. Found. 2021, 8, 315–336.
  * (7) Zurek, W. Preferred Sets of States, Predictability, Classicality, and Environement Induced Decoherence. In Physical Origins of Time Asymmetry, eds. Halliwell, J.J.; Pérez-Mercader J.; and Zurek, W.H.; Cambridge University Press: Cambridge, 1996 pp. 175-212.
  * (8) Appleby, D. M. Bohmian Trajectories Post-Decoherence. Found. Phys., 1999 29, 1885-1916.
  * (9) De Broglie, L. La Mécanique Ondulatoire et la Structure Atomique de la Matière et du Rayonnement, J. Phys. Radium 1927, 8, 225-241; translated in: L. de Broglie and L. Brillouin, Selected papers on wave mechanics; Blackie and Son: Glasgow, 1930.
  * (10) Bacciagaluppi, G. and Valentini, A. Quantum theory at the crossroads: Reconsidering the 1927 Solvay Conference; Cambridge Univ. Press: Cambridge, 2009.
  * (11) De Broglie, L. Introduction à l’étude de la mécanique ondulatoire; Hermann, Paris, 1930. English translation as de Broglie, L. An introduction to the study of wave mechanics; Methuen: London, 1930.
  * (12) Bohm, D. A suggested Interpretation of the Quantum Theory in Terms of Hidden Variables I, Phys. Rev. 1952, 85, 166-179.
  * (13) Bohm, D. A suggested Interpretation of the Quantum Theory in Terms of Hidden Variables II, Phys. Rev. 1952, 85, 180-193.
  * (14) Bohm, D. and Hiley, B. J. The undivided Universe; Routledge: London, 1993.
  * (15) Holland, P. The quantum theory of motion; Cambridge University Press: London, 1993.
  * (16) Pauli, W. in Louis de Broglie Physicien et Penseur; Albin Michel, Paris, 1953, pp. 33-42.
  * (17) Keller, J. B. Bohm’s Interpretation of the Quantum Theory in Terms of “Hidden” Variables. Phys. Rev. 1953,89, 1040.
  * (18) Uffink, J. Compendium of the foundations of classical statistical physics. In Philosophy of physics (handbook of the philosophy of science), eds. Butterfield J. and Earman J.; Elsevier: Amsterdam, 2007 pp. 923-1074 .
  * (19) Barret, J. A. Introduction to Quantum Mechanics and the Measurement Problem. Topoi 1995, 14, 45-54 .
  * (20) Callender, C. The Emergence and Interpretation of Probability in Bohmian Mechanics.Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 2007, 38, 351-370.
  * (21) Drezet, A. How to Justify Born’s Rule using the Pilot Wave Theory of de Broglie? Ann. Fond. de Broglie 2017, 42, 103-131.
  * (22) Bohm, D. Proof that Probability Density Approaches $|\Psi|^{2}$ in Causal Interpretation of the Quantum Theory. Phys. Rev. 1953, 89, 458-466.
  * (23) Potel, G.; Muñoz-Aleñar, M.; Barranco, F.; Vigezzi, E. Stability properties of $|\Psi|^{2}$ in Bohmian dynamics. Phys. Lett. A 2002, 299, 125-130.
  * (24) Bohm, D. Schützer, W. The General Statistical Problem in Physics and the Theory of Probability. Nuovo Cim. 1995, 2, 1004-1047.
  * (25) Drezet, A. and Stock, B. A causal and continuous interpretation of the quantum theory: About an original manuscript by David Bohm sent to Louis de Broglie in 1951. Ann. Fond. de Broglie 2017, 42, 169-195.
  * (26) Bohm, D. Vigier, J. P. Model of the Causal Interpretation of Quantum Theory in Terms of a Fluid with Irregular Fluctuations. Phys. Rev. 1954, 96 208-217.
  * (27) Bohm, D. Causality and chance in modern physics; Routledge: London, 1957.
  * (28) Valentini, A. Signal-locality, uncertainty, and the subquantum H-theorem. I. Phys. Lett. A 1991, 156, 5-11.
  * (29) Valentini, A. On the pilot-wave theory of classical, quantum and subquantum physics, PhD Thesis; International school for advanced studies: Trieste, 1992.
  * (30) Valentini, A. and Westman, H. Dynamical Origin of Quantum Probabilities. Proc. R. Soc. A 2005, 461, 253-272.
  * (31) Towler, M. D.; Russell, N.J.; Valentini, A. Time scales for dynamical relaxation to the Born rule. Proc. R. Soc. A 2012 , 468, 990–1013.
  * (32) Valentini, A. Astrophysical and cosmological tests of quantum theory J. Phys. A: Math. Theor. 2007, 40, 3285-3303.
  * (33) Colin, S.; Valentini, A. Primordial quantum nonequilibrium and large-scale cosmic anomalies Phys. Rev. D 2015, 92, 043520.
  * (34) Efthymiopoulos, C.; Contopoulos, G. Tzemos, A. C. Chaos in de Broglie - Bohm quantum mechanics and the dynamics of quantum relaxation. Ann. Fond. de Broglie 2017, 45, 133-159.
  * (35) Contopoulos, G.; Tzemos, A. C. Chaos in Bohmian Quantum Mechanics: A Short Review. Regular and Chaotic Dynamics 2020, 25, 476-495.
  * (36) Frisk, H. Properties of the trajectories in Bohmian mechanics. Phys. Lett. A 1997, 227, 139-142.
  * (37) Falsaperla, P.; Fonte, G. On the motion of a single particle near a nodal line in the de Broglie–Bohm interpretation of quantum mechanics. Phys. Lett. A 2003, 316, 382-390.
  * (38) Wisniacki, D. A.; Pujals, E. R. Motion of vortices implies chaos in Bohmian mechanics. Europhys. Lett. 2005, 71, 159-165.
  * (39) Contopoulos, G.; Delis, N.; Efthymiopoulos, C. Order in de Broglie–Bohm quantum mechanics. J. Phys. A: Math. Theor. 2012, 45, 165301.
  * (40) Norsen, T. On the Explanation of Born-Rule Statistics in the de Broglie-Bohm Pilot-Wave Theory. Entropy 2018, 20, 422.
  * (41) Drezet, A. Brownian motion in the pilot wave interpretation of de Broglie and relaxation to quantum equilibrium. Ann. Fond. de Broglie 2018, 43, 23-50.
  * (42) Philbin, T. G. Derivation of quantum probabilities from deterministic evolution. Int. J. Quantum Found. 2015, 1, 175-184.
  * (43) Dürr, D.; Goldstein, S.; Zanghí, N. Quantum mechanics, randomness, and deterministic reality Phys. Lett. A 1992, 172, 6-12.
  * (44) Dürr, D.; Goldstein, S.; Zanghí, N. Quantum equilibrium and the origin of absolute uncertainty. J. Stat. Phys. 1992, 67, 843-907.
  * (45) Valentini A. Foundations of Statistical Mechanics and the Status of the Born Rule in de Broglie-Bohm Pilot-Wave Theory. In Statistical mechanics and scientific explanation: determinism, indeterminism and laws of nature; World Scientific: Singapore, 2020, pp. 423-477.
  * (46) Dürr, D. Struyve, W. Typicality in the foundations of statistical physics and Born’s rule. arXiv:1910.08049v1.
  * (47) Equivariance means here that the density $\rho$ must be an explicit function of $R$, i.e., $\rho=F(R)$ which admits the only solution $\rho=R^{2}$ Durr1992b ; Drezet2017 .
  * (48) Wallace, D. The Probability puzzle and many-worlds interpretation of quantum mechanics https://www.youtube.com/watch?v=8turL6Xnf9U (2015).
  * (49) D. J. Driebe, Fully chaotic maps and broken time symmetry; Springer: Dordrecht, 1999.
  * (50) Schuster, H. G. Deterministic chaos: An introduction; Physik-Verlag: Weinheim, 1984.
  * (51) Dürr, D.; Goldstein, S.; Zanghí, N. Quantum Chaos, Classical Randomness, and Bohmian Mechanics J. Stat. Phys. 1992, 68, 259-270.
  * (52) Dewdney, C.; Malik, Z. Measurement, decoherence and chaos in quantum pinball Phys. Lett. A 1996, 220, 183-188.
  * (53) Geiger, H.; Obermair, Helm, G. Ch. Quantum mechanics without statistical postulates. In Quantum communications, computing and measurement 3; Springer: Berlin, 2002, pp. 139-142.
  * (54) Ford, J. How random is a coin toss? Phys. Today 1983, 36, 40-47.
  * (55) Prigogine, I. The End of Certainty: Time, Chaos, and the New Laws of Nature; Free Press/Simon and Schuster: New York, 1997.
  * (56) Prigogine, I. Les Lois du Chaos; Flamarion: Paris, 1993.
  * (57) Bricmont, J. Science of Chaos or Chaos in Science. Physicalia Magazine 1995, 17, 159-208.
  * (58) Poincaré, H. Le hasard. Revue du mois, 1907, 3, 257-276.