# Least Information Principle for Quantum Mechanics

Jianhao M. Yang<EMAIL_ADDRESS>Qualcomm, San Diego, CA
92321, USA

###### Abstract

A least information principle is proposed here to derive the formulations of
non-relativistic quantum mechanics, including the uncertainty relation and the
Schrödinger equation in both position and momentum representations. The key
element for this principle is the introduction of new metrics to measure the
observable information due to vacuum fluctuations. In addition, the Planck
constant is understood as the basic discrete action unit to uncover observable
information of the dynamics behavior of a physical system. Besides recovering
the quantum mechanics formulation, the principle also brings in new results on
two fronts. Mathematically, defining the information metrics for vacuum
fluctuations using more general definitions of relative entropy results in a
generalized Schrödinger equation that depends on the order of relative
entropy. Conceptually, these information metrics are shown to be responsible
for manifesting entanglement effects without underlying physical interactions,
implying that entanglement effects are non-causal. The least information
principle further demonstrates the information essence of quantum mechanics,
and can be a mathematical tool to derive more advanced quantum theory.

##  I Introduction

Advancements of quantum information and quantum computing Nielsen ; Hayashi15
in recent decades have inspired the research efforts on the informational
foundation of quantum mechanics Rovelli:1995fv ; zeilinger1999foundational ;
Brukner:ys ; Brukner:1999qf ; Brukner:2002kx ; Fuchs2002 ;
brukner2009information ; Brukner:vn ; spekkens2007evidence ; Spekkens:2014fk ;
Paterek:2010fk ; gornitz2003introduction ; lyre1995quantum ; Hardy:2001jk ;
Dakic:2009bh ; masanes2011derivation ; Mueller:2012ai ; Masanes:2012uq ;
chiribella2011informational ; Mueller:2012pc ; Hardy:2013fk ;
kochen2013reconstruction ; 2008arXiv0805.2770G ; Hall2013 ; Hoehn:2014uua ;
Hoehn:2015 ; Stuckey ; Mehrafarin2005 ; Caticha2011 ; Caticha2019 ; Frieden ;
Reginatto . The search for informational foundation is motivated by the fact
that even though quantum mechanics has been extensively verified
experimentally, there are still many fundamental questions left unanswered. In
particular, it is still unclear whether quantum entanglement implies non-local
causal connection between two entangled objects, which has been the source of
contentions in understanding the EPR thought experiment EPR  and Bell
inequality Bell . Reformulating quantum mechanics based on information
principle can provides potential answers. For instance, Zeilinger
zeilinger1999foundational ; Brukner:ys  suggests that a foundational principle
for quantum mechanics is that _an elementary system carries 1 bit of
information_. Rovelli in his seminal paper on relational quantum mechanics
Rovelli:1995fv  postulates that _information extracted from a quantum system
is limited_. Inspired by these principles, Höhn reconstructs a concrete
quantum theory for a single qubit and N-qubit from elementary rules on an
observer’s information acquisition Hoehn:2014uua ; Hoehn:2015 . On the other
hand, various approaches based on entropy extremization have been proposed to
derive quantum mechanics. The entropic dynamics Caticha2011 ; Caticha2019
attempts to extract quantum mechanics as an application of the methods of
inference from maximizing Shannon entropy. Another variation approach based on
relative entropy is constructed to recover stochastic mechanics which in turn
can lead to the Schrödinger equation Yang2021 .

These research results are encouraging but with various limitations. An
interesting observation is that the role of Planck constant seems overlooked
in the search of information foundation for quantum mechanics. Planck constant
is the most fundamental constant in quantum mechanics since it gives the
connection from quantum mechanics to classical mechanics. That is, when
$\hbar\to 0$, quantum mechanics becomes classical mechanics. One would
naturally expect it retains such a role even if quantum mechanics is
reconstructed on information based principles. Thus it must be related to how
the amount of information is measured. In fact, Rovelli has pointed out that
his postulate on limited information implies the existence of Planck constant
Rovelli:1995fv , although it is unclear how $\hbar$ is used to measure the
amount of information in the subsequent reconstruction effort of quantum
theory in Rovelli:1995fv . The principle proposed by Zeilinger and Brukner
zeilinger1999foundational ; Brukner:ys  states that N elementary systems carry
N bits, and again does not show how the N bits are related to the Planck
constant. Such principle brings novel insight on entanglement for, say, a
bipartite system. Because if the 2 bits of information is exhausted in
specifying joint properties of the two subsystems, then nothing can be
specified for the individual subsystem. However, the question on whether
entanglement is due to non-local causal effort remains unanswered. The
limitation for the entropy extremization approaches in Caticha2011 ;
Caticha2019  and Yang2021  is their dependency on the stochastic mechanics as
underlying physical model Nelson , which suffers from the concerns of hidden
variables such as osmotic velocity, and its difficulty to explain non-local
behavior of multi-particle systems Nelsonbook .

The work presented here is a continuation of the searching for the information
foundations of quantum mechanics. We wish to construct a theory that clearly
shows how classical mechanics evolves to quantum mechanics from information
perspective, and brings new insights to understand quantum entanglement
without depending on certain non-local causal interaction. We also expect the
theory should produce results that are new to current quantum formulations. At
the center of our investigation effort is a least information principle.
Before proceeding further to describe the least information principle, it is
necessary to clarify the meaning of the term information. In essence,
information measures a degree of distinguishability. Here, the information we
are interested in is about the distinguishability of the system due to its
movement relative to a reference frame. The distinguishability should be
sufficient enough for future observation. We call such information as
_observable information_ FN01 . The least information principle is constructed
upon two key components. First, we assume that the Planck constant defines a
discrete action unit to measure whether the amount of information is
sufficiently large enough to be observable. That is, the Planck constant
determines the minimal amount of action effort that a system needs to
contribute in order to reveal information that is observable. With this
interpretation, the amount of observable information is connected to a
physical quantity such as action. As such, we now can relate the physical laws
to information theory. The second component is the introduction of new metrics
to measure the additional distinguishable information caused by the vacuum
fluctuations. It is assumed that there is a vacuum fluctuation that each
physical system will be experiencing constantly, and the vacuum fluctuation is
local, meaning that fluctuations of two subsystems are independent from each
other. The information metrics for vacuum fluctuations are defined in terms of
relative entropy that measures the information distances of different
probability distributions caused by vacuum fluctuations.

From the least information principle, a series of results are obtained. First,
by recursively applying least information principle in an infinitesimal time
interval and accumulated time interval, the uncertainty relation and the
Schrödinger equation are recovered; Although similar results have been
obtained in other research works Caticha2011 ; Caticha2019 ; Reginatto ;
Hall:2001 ; Hall:2002 , what is different here is the simplicity and
cleanness. There are no additional constants or Lagrangian multipliers
introduced, and no additional postulates. We will show that the same method
can be applied in the momentum representation to obtain the Schrödinger
equation in momentum representation. Imposing a no preferred representation
assumption results in the transformation theory between position and momentum
representations. Second, to demonstrate the least information principle can
produce new results, we define the information metrics for vacuum fluctuations
using the Rényi divergence. This leads to a generalized Schrödinger equation.
From a purely information origin perspective, the equation is legitimate;
Third, constructing quantum mechanics from information based principle brings
new insights on quantum entanglement. We will show that the information
metrics for vacuum fluctuations play a critical role in quantum entanglement.
Although the initial inseparable correlation in a bipartite system is created
by physical interaction, the preservation and manifestation of such
inseparable correlation are realized without further interaction. This is due
to the inseparability of information metrics for vacuum fluctuations, even
though the underlying vacuum fluctuation is local. This suggests that the
preservation and manifestation aspects of entanglement are not necessarily
causal.

The rest of the article is organized as follows. First we describe in detail
how the least information principle is constructed and what the underlying
assumptions are. Then we show how the basic quantum theory is recovered. This
follows by the derivation of a generalized Schrödinger equation not reported
in earlier research literature. Next, we give a detailed analysis on why
quantum entanglement is non-causal using the formulations developed here. We
then conclude the article after comprehensive discussions and comparisons to
previous relevant research works.

##  II The Least Information Principle

Consider a particle with mass $m$ moving from position $A$ to $B$. The motion
of the particle is a combination of two independent components, the classical
trajectory due to external potential and the random vacuum fluctuations around
any given position along the classical path. Due to the vacuum fluctuations,
there is no definite trajectory. What can we say in terms of observable
information for this physical scenario?

In classical mechanics, the classical trajectory follows the laws derived
through the least action principle. We wish to recast the least action
principle into a principle using information quantity. This requires us to
convert the classical action into an information metric. Action effort can be
understood, in the case of classical mechanics, the action $S_{c}$ for a
physical system to move from position $A$ to position $B$. We assume there is
a lower limit of action effort to reveal observable information, and the
smallest unit of action effort to reveal observable information is defined by
the Planck constant. Thus, given a finite amount of classical action $S_{c}$,
the amount of observable information is limited. It implies there is a limit
on the resolution of observable information. In other words, the Planck
constant determines the resolution of the observable information of the
dynamics behavior of a physical system. Formally, the assumption can be stated
as,

> _Assumption 1 – The action effort to reveal observable information is
> discrete. The basic discrete unit of action effort is determined by the
> Planck constant $\hbar$._

The word _reveal_ implies that the information is uncovered by the movement of
the physical system itself. As the system moves along $A$ to $B$, more and
more classical action occurs, and subsequently, more and more information is
revealed. For a movement with classical action $S_{c}=\int Ldt$ where $L$ is
the classical Lagrangian, the corresponding amount of information is defined
as $I_{p}:=2S_{c}/\hbar$. Such a definition appears similar to the definition
of the phase of probability amplitude for a trajectory of a particle in
Feynman’s path integral Feynman48 , $S/\hbar$. But the physical meaning is
quite different because we avoid postulating the concept of probability
amplitude here. This will be discussed in detail in Section VI.2. The reason
for factor 2 will become clear later in the derivation. We can intuitively
consider $I_{p}$ as the amount of observable information uncovered along a
classical trajectory _relative to a reference frame_. The laws of dynamics in
classical mechanics can be obtained through minimizing the action $S_{c}$, or,
equivalently, minimizing the information metrics $I_{p}$. This step of using
$I_{p}$ instead of $S_{c}$ appears trivial, but it is not. It recasts the
least action principle into a least information principle, and shifts the
working language to be information related. This paves the way to introduce
other information metrics due to vacuum fluctuations.

As mentioned in the introduction section, we assume there is a vacuum
fluctuation that each physical system will be experiencing constantly. Here we
are not intending to investigate the origin, or establish a physical model, of
such vacuum fluctuation. Instead, we make a minimal number of assumptions on
the underlying physical model, only enough so that we can apply the variation
principle based on information metrics. This is different from other theories
such as stochastic mechanics. The advantage of this approach is to avoid
keeping track of physical details that are irrelevant for predicting future
measurement results. The vacuum fluctuation is assumed to be local. Therefore,
for a composite system, the fluctuation of each subsystem is independent of
each other. We state the assumption as following:

> _Assumption 2 – A quantum system experiences vacuum fluctuations constantly.
> The fluctuations are completely random, and local._

A new information metric, $I_{f}$, is introduced into the least information
principle. $I_{f}$ is defined as a metric to measure the additional
distinguishable information revealed due to vacuum fluctuations _relative to
the classical trajectory_. Concrete form of $I_{f}$ will be defined later as a
function of Kullback-Leibler divergence $D_{KL}$, where $D_{KL}$ measures the
information distances of different probability distributions caused by vacuum
fluctuations. Thus, $I_{f}:=f(D_{KL})$ and the total amount of distinguishable
information is

$I=\frac{2}{\hbar}S_{c}+f(D_{KL}).$ (1)

Quantum theory can be derived through a variation approach to minimize such
information quantity, $\delta I=0$. When $\hbar\to 0$, the classical path
information $I_{p}\to\infty$, any finite amount of $I_{f}$ can be ignored.
Minimizing $I$ is then equivalent to minimizing $S_{c}$, resulting in the
dynamics laws of classical mechanics. However, according to the Assumption 1,
the resolution of observable information in terms of action effort is discrete
so that $\hbar\neq 0$, $I_{p}$ is finite. Therefore, the contribution from
$I_{f}$ must be included when minimizing the total amount of information. The
law of physical dynamics tends to reveal as little observable information as
possible.

Independent from the least information principle, we need another assumption
similar to the no preference of reference frame postulate in special
relativity. The observable information of the physical dynamics can be
expressed in different representations. Loosely speaking, a representation is
characterized by a set of variables with their values acting like coordinates
to describe the properties of the system Dirac . For instance, the position
representation uses position variable to describe the physical properties of
the system. Similarly, the momentum representation uses momentum variable to
describe the physical properties of the system. We assume that the total
observable information extracted in a representation is a complete description
of the dynamics of the system. The physical laws derived in other
representations do not offer additional power of predictions for future
measurement results. Consequently, the physical laws for the dynamics of the
system derived from different representations must be equivalent. As shown
later, from the same variation principle, we can derive Schrödinger equation
independently in both position and momentum representations. But we demand the
results must be equivalent. In summary, we have

> _Assumption 3 – There is no preferred representation for the law of physics
> derived in each representation._

Assumption 3 will lead the transformation formulation between position and
momentum representations.

With the least information principle and the underlying assumptions explained,
we now proceed to describe the results from applying this principle.

##  III Basic Quantum Formulation

###  III.1 Dynamics of Vacuum fluctuations and The Uncertainty Relation

First we consider the dynamics of a system only due to the random vacuum
fluctuations for an infinitesimal time internal $\Delta t$. This means we
neglect the external potential for the time being. Define the probability for
the system to transition from a 3-dimensional space position $x$ to another
position $x+w$, where $w=\Delta x$ is the displacement in 3-dimensional space
due to fluctuations, as $\wp(x+w|x)d^{3}w$. The classical action is
$S_{c}=\int\wp(x+w|x)Ld^{3}wdt$. Since we only consider the vacuum
fluctuations, the Lagrangian $L$ only contains the kinetic energy,
$L=\frac{1}{2}mv^{2}$. For an infinitesimal time internal $\Delta t$, one can
approximate the velocity $v=w/\Delta t$. This gives

$S_{c}=\frac{m}{2\Delta t}\int^{+\infty}_{-\infty}\wp(x+w|x)w^{2}d^{3}w.$ (2)

The information metrics $I_{f}$ is supposed to capture the additional
revelation of information due to vacuum fluctuations. Thus, it is naturally
defined as a relative entropy, or more specifically, the Kullback–Leibler
divergence, to measure the information distance between $\wp(x+w|x)$ and some
prior probability distribution. Since the vacuum fluctuations are completely
random, it is intuitive to assume the prior distribution with maximal
ignorance Caticha2019 ; Jaynes . That is, the prior probability distribution
is a uniform distribution $\mu$.

$\displaystyle I_{f}$ $\displaystyle=:D_{KL}(\wp(x+w|x)||\mu)$
$\displaystyle=\int\wp(x+w|x)ln[\wp(x+w|x)/\mu]d^{3}w.$

Combined with (2), the total amount of information defined in (1) is

$\displaystyle I=$ $\displaystyle\frac{m}{\hbar\Delta
t}\int\wp(x+w|x)w^{2}d^{3}w$
$\displaystyle+\int\wp(x+w|x)ln[\wp(x+w|x)/\mu]d^{3}w.$

Taking the variation $\delta I=0$ with respect to $\wp$ gives

$\delta I=\int(\frac{mw^{2}}{\hbar\Delta t}+ln\frac{\wp}{\mu}+1)\delta\wp
d^{3}w=0.$ (3)

Since $\delta\wp$ is arbitrary, one must have

$\frac{mw^{2}}{\hbar\Delta t}+ln\frac{\wp}{\mu}+1=0.$

The solution for $\wp$ is

$\wp(x+w|x)=\mu e^{-\frac{mw^{2}}{\hbar\Delta
t}-1}=\frac{1}{Z}e^{-\frac{mw^{2}}{\hbar\Delta t}},$ (4)

where $Z$ is a normalization factor that absorbs factor $\mu e^{-1}$. Equation
(4) shows that the transition probability density is a Gaussian distribution.
The variance $\langle w^{2}\rangle=\hbar\Delta t/2m$. Recalling that $w/\Delta
t=v$ is the approximation of velocity due to the vacuum fluctuations, we
denote $p_{f}=mv=mw/\Delta t$. Since $\langle p_{f}\rangle\propto\langle
w\rangle=0$, then $\langle(p+p_{f})^{2}-p^{2}\rangle=\langle
p_{f}^{2}\rangle$, and $p_{f}$ can be considered as the fluctuations of
momentum on top of the classical momentum. That is, $\Delta p=p_{f}=mw/\Delta
t$. Rearranging $\langle w^{2}\rangle=\hbar\Delta t/2m=\langle(\Delta
x)^{2}\rangle$ gives

$\langle\Delta x\Delta p\rangle=\frac{\hbar}{2}.$ (5)

This is the so-called _exact uncertainty relation_ Hall:2001 ; Hall:2002
where it is postulated with mathematical arguments. Here we derive it from the
variation principle of minimizing the amount of information due to vacuum
fluctuations. Now square both sides of (5) and applying the Cauchy–Schwarz
inequality gives

$\displaystyle\frac{\hbar^{2}}{4}$ $\displaystyle=\langle\Delta x\Delta
p\rangle^{2}=(\int\wp\Delta x\Delta pd^{3}w)^{2}$
$\displaystyle\leq\int\wp(\Delta x)^{2}d^{3}w\int\wp(\Delta p)^{2}d^{3}w$
$\displaystyle=\langle(\Delta x)^{2}\rangle\langle(\Delta p)^{2}\rangle.$

Taking square root of both sides results in

$\langle\Delta x\rangle\langle\Delta p\rangle\geq\hbar/2.$ (6)

###  III.2 Derivation of The Schrödinger Equation

We now turn to the dynamics for an cumulative period from $t_{A}\to t_{B}$
with external potential. In classical mechanics, the equation of motion is
described by the Hamilton-Jacobi equation,

$\frac{\partial S}{\partial t}+\frac{1}{2m}(\nabla S)^{2}+V=0.$ (7)

Suppose the initial condition is unknown, and define $\rho(x,t)$ as the
probability density for finding a particle in a given volume of the
configuration space. The probability density must satisfy the normalization
condition $\int\rho(x,t)d^{3}x=1$, and the continuity equation

$\frac{\partial\rho(x,t)}{\partial t}+\frac{1}{m}\nabla\cdot(\rho(x,t)\nabla
S)=0.$

The pair $(S,\rho)$ completely determines the motion of the classical
ensemble. As pointed out by Hall and Reginatto Hall:2001 ; Hall:2002 , the
Hamilton-Jacobi equation, and the continuity equation, can be derived from
classical action

$S_{c}=\int\rho\\{\frac{\partial S}{\partial t}+\frac{1}{2m}\nabla
S\cdot\nabla S+V\\}d^{3}xdt$ (8)

through fixed point variation with respect to $\rho$ and $S$, respectively.
Note that $S_{c}$ and $S$ are different physical variables, where $S_{c}$ can
be considered as the ensemble average of individual classical action $S$. The
amount of information for the motion of this ensemble between the two fixed
points is $I_{p}=2S_{c}/\hbar$ according to Assumption 1. To define the
information metrics for the vacuum fluctuations, $I_{f}$, we slice the time
duration $t_{A}\to t_{B}$ into $N$ short time steps
$t_{0}=t_{A},\ldots,t_{i},\ldots,t_{N-1}=t_{B}$, and each step is an
infinitesimal period $\Delta t$. In an infinitesimal time period at time
$t_{i}$, the particle not only moves according to the Hamilton-Jacobi equation
but also experiences random fluctuations. The probability density
$\rho(x,t_{i})$ alone is insufficient to encode all the information. Instead,
we need to consider $\rho(x+w,t_{i})$ for all possible $w$. Such additional
revelation of information is due to the vacuum fluctuations on top of the
classical trajectory. The proper measure of this distinction is the
information distance between $\rho(x,t_{i})$ and $\rho(x+w,t_{i})$. A natural
choice of such information measure is $D_{KL}(\rho(x,t_{i})||\rho(x+w,t_{i})$.
We then take the average of $D_{KL}$ over $w$. Denoting $E_{w}[\cdot]$ the
expectation value, and summing up such quantity for each infinitesimal time
interval, lead to the definition

$\displaystyle I_{f}$
$\displaystyle=:\sum_{i=0}^{N-1}E_{w}[D_{KL}(\rho(x,t_{i})||\rho(x+w,t_{i})]$
(9) $\displaystyle=\sum_{i=0}^{N-1}\int
d^{3}wd^{3}x\wp(x+w|x)\rho(x,t_{i})ln\frac{\rho(x,t_{i})}{\rho(x+w,t_{i})}.$
(10)

Notice that $\wp(x+w|x)$ is a Gaussian distribution given in (4). When $\Delta
t$ is small, only small $w$ will contribute to $I_{f}$. As shown in Appendix
A, when $\Delta t\to 0$, $I_{f}$ turns out to be

$I_{f}=\frac{\hbar}{4m}\int d^{3}xdt\rho(x,t)(\nabla ln\rho(x,t))^{2}.$ (11)

Eq. (11) contains the term related to Fisher information for the probability
density FriedenBook . Some literature directly adds Fisher information in the
variation method as a postulate to derive the Schrödinger equation Reginatto .
But (11) bears much more physical significance than Fisher information. It
shows that $I_{f}$ is proportional to $\hbar$. This is not trivial because it
avoids introducing additional arbitrary constants for the subsequent
derivation of Schrödinger equation. More importantly, defining $I_{f}$ using
the relative entropy opens up new results that cannot be obtained if $I_{f}$
is defined using Fisher information, because there are other generic forms of
relative entropy such as Rényi divergence. As will be seen later, by replacing
the Kullback–Leibler divergence with Rényi divergence, one will obtain a
generalized Schrödinger equation. Other authors also derive (11) using
mathematical arguments Hall:2001 ; Hall:2002 , while our approach is based on
intuitive information metrics. With (11), the total amount of information is

$I=\int\\{\frac{2}{h}\rho[\frac{\partial S}{\partial t}+\frac{1}{2m}\nabla
S\cdot\nabla S+V]+\frac{\hbar}{4m}\rho(\nabla ln\rho)^{2}\\}d^{3}xdt.$ (12)

Variation of $I$ with respect to $S$ gives the continuity equation, while
variation with respect to $\rho$ leads to

$\frac{\partial S}{\partial t}+\frac{1}{2m}\nabla S\cdot\nabla
S+V-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}\sqrt{\rho}}{\sqrt{\rho}}=0.$ (13)

The last term is the _Bohm’s quantum potential_ Bohm1952 . Bohm’s potential is
considered responsible for the non-locality phenomenon in quantum mechanics
Bohm2 . Historically, its origin is mysterious. Here we show that it
originates from the information metrics related to relative entropy, $I_{f}$.
The physical implications of this result will be discussed later. Defined a
complex function $\Psi=\sqrt{\rho}e^{iS/\hbar}$, the continuity equation and
the extended Hamilton-Jacobi equation (13) can be combined into a single
differential equation,

$i\hbar\frac{\partial\Psi}{\partial
t}=[-\frac{\hbar^{2}}{2m}\nabla^{2}+V]\Psi,$ (14)

which is the Schrödinger Equation.

In summary, by recursively applying the same least information principle on
the total information $\delta I=0$ in two steps, we recover the uncertainty
relation and the Schrödinger equation. The first step is for a short time
period to obtain the transitional probability density due to vacuum
fluctuations; Then the second step is for a cumulative time period to obtain
the dynamics law for $\rho$ and $S$. The applicability of the same variation
principle shows the consistency and simplicity of the theory, although the
form of Lagrangian is different in each step. In the first step, the
Lagrangian only contains the kinetic energy $L=mv^{2}/2$, which is in the form
of $L=\dot{x}p-H$ where $H$ is the classical Hamiltonian. In the second step,
we use a different form of classical Lagrangian $L^{\prime}=\partial
S/\partial t+H$. In classical mechanics, $L$ and $L^{\prime}$ can be related
through an extended canonical transformation. The choice of Lagrangian $L$ or
$L^{\prime}$ does not affect the form of Lagrange’s equations. Here we choose
$L^{\prime}=\partial S/\partial t+H$ as the classical Lagrangian in the second
step in order to use the pair of variables $(\rho,S)$ in the subsequent
variation procedure.

To demonstrate the simplicity of the least information principle, in Appendix
B, we apply the principle to derive the Schrödinger equation in an external
electromagnetic field. The interesting point is this example is that the
external electromagnetic field has no influence on the vacuum fluctuations.
This reconfirms that the information metrics $I_{f}$ is independent of the
external potential.

###  III.3 Transformation Between Position and Momentum Representations

The classical action $S_{c}$ and information metrics $I_{f}$ in (1) are so far
defined in the position representation, i.e., using position $x$ as variable.
However, there can be other observable quantities to serve as representation
variables. Momentum is one of such representation variables. We can find the
proper expressions for $S_{c}$ and $I_{f}$ in the momentum representation, and
follow the same variation principle to derive the quantum theory. By
Assumption 3, one would expect the law of dynamics in the momentum
representation is equivalent to that in the position representation derived
earlier. First let’s consider the effect of fluctuations in a short time step
$\Delta t$. The vacuum fluctuations occur not only in spatial space, but also
in momentum space. Denote the transition probability density for the vacuum
fluctuations as $\tilde{\wp}(p+\tilde{w}|p)$ where $\tilde{w}=\Delta p$ is due
to the momentum fluctuations. The classical Lagrangian without considering
external potential is $L=(p+\tilde{w})^{2}/2m$, and the average classical
action is

$S_{c}=\frac{\Delta
t}{2m}\int\tilde{\wp}(p+\tilde{w}|p)(p+\tilde{w})^{2}d^{3}\tilde{w}.$

Since $\langle\tilde{w}\rangle=0$, the only term contributed in the variation
with respect to $\tilde{\wp}$ is the one with $\langle\tilde{w}^{2}\rangle$.
Similar to the definition of $I_{f}$ in the position representation, here we
define $I_{f}=:D_{KL}(\tilde{\wp}(p+\tilde{w}|p)||\tilde{\mu})$ where
$\tilde{\mu}$ is a uniform probability density in the momentum space. Plugging
all these expressions into (1) and let $\delta I=0$ with respect to
$\tilde{\wp}$, one will obtain

$\tilde{\wp}(p+\tilde{w}|p)=\frac{1}{Z^{\prime}}e^{-\frac{\Delta
t}{m\hbar}\tilde{w}^{2}},$

and $Z^{\prime}$ is the normalization factor. The variance
$\langle\tilde{w}^{2}\rangle=\langle(\Delta p)^{2}\rangle=m\hbar/2\Delta t$.
This is also a Gaussian distribution but with a significant difference from
(4) in the position representation. That is, when $\Delta t\to 0$,
$\langle(\Delta p)^{2}\rangle\to\infty$ while $\langle(\Delta x)^{2}\rangle\to
0$. This implies that when $\Delta t\to 0$, the Gaussian distribution
$\tilde{\wp}$ becomes a uniform distribution. Note that $\Delta p\Delta
t=m\Delta x$, rearranging $\langle(\Delta p)^{2}\rangle=m\hbar/2\Delta t$
gives the same uncertainty relation in (5).

For illustration purposes, we will only derive the momentum representation of
Schrödinger equation for a free particle. Let $\varrho(p,t)$ be the
probability density in the momentum representation, the classical action is

$S_{c}=\int\varrho(p,t)\\{\frac{\partial S}{\partial
t}+\frac{p^{2}}{2m}\\}d^{3}pdt.$

$I_{f}$ is defined similarly to (9) as

$I_{f}=:\sum_{i=0}^{N-1}E_{\tilde{w}}[D_{KL}(\varrho(p,t_{i})||\varrho(p+\tilde{w},t_{i})].$
(15)

However, when $\Delta t\to 0$, $\tilde{\wp}(p+\tilde{w}|p)$ becomes an uniform
distribution, $I_{f}\to\infty$ independent of $\varrho$, as shown in the
Appendix D. This implies that $I_{f}$ does not contribute when taking
variation with respect to $\varrho$. Thus,

$\delta I=\delta\int\varrho(p,t)\\{\frac{2}{\hbar}\frac{\partial S}{\partial
t}+\frac{p^{2}}{m\hbar}\\}d^{3}pdt.$ (16)

Variation with respect to $\varrho$ gives

$\frac{\partial(S/\hbar)}{\partial t}+\frac{p^{2}}{2m\hbar}=0,$

and variation with respect to $S$ gives $\partial\varrho/\partial t=0$.
Defined $\psi=\sqrt{\varrho}e^{i(S/\hbar)}$, the two differential equations
are combined into a single differential equation,

$i\hbar\frac{\partial\psi}{\partial t}=\frac{p^{2}}{2m}\psi,$ (17)

which is the Schrödinger equation for a free particle in the momentum
representation. Recalled that in the position representation, the Schrödinger
equation for a free particle is $i\hbar\partial\Psi/\partial
t=[-(\hbar^{2}/2m)\nabla^{2}]\Psi$. The two equations are derived
independently from the variation of dynamics information defined in (1).
Assumption 3 demands that the two equations must be equivalent. To meet this
requirement, one sufficient condition is that the two wave functions are
transformed through

$\Psi(x,t)=(\frac{1}{\sqrt{2\pi\hbar}})^{3}\int e^{ip\cdot
x/\hbar}\psi(p,t)d^{3}p.$ (18)

This transformation justifies the introduction of operator
$\hat{p}=:-i\hbar\partial/\partial x$ to represent momentum in the position
representation, because using (18), one can verify that the expectation value
of momentum $\langle\psi(p,t)|p|\psi(p,t)\rangle$ can be computed as
$\langle\Psi(x,t)|\hat{p}|\Psi(x,t)\rangle$. Introduction of the momentum
operator $\hat{p}=:-i\hbar\partial/\partial x$ leads to the commutation
relation $[\hat{x},\hat{p}]=i\hbar$.

Now suppose in the momentum representation there is a different action unit
$\hbar_{p}\neq\hbar$. Repeating the same variation procedure gives a
Schrödinger equation for a free particle

$i\hbar_{p}\frac{\partial\psi}{\partial t}=\frac{p^{2}}{2m}\psi.$

To satisfy Assumption 3, the transformation function (18) needs to be modified
as

$\Psi(x,t)=(\frac{1}{\sqrt{2\pi\hbar}})^{3}\int e^{ip\cdot
x/\sqrt{\beta}\hbar}\psi(p,t)d^{3}p$

where $\beta=\hbar_{p}/\hbar$. Consequently,
$[\hat{x},\hat{p}]=i\hbar\sqrt{\beta}$. It is clear that the assumption of
having a different constant $\hbar_{p}\neq\hbar$ in momentum representation is
incompatible with the well established Dirac quantization condition
$[\hat{x},\hat{p}]=i\hbar$. By accepting $[\hat{x},\hat{p}]=i\hbar$, one must
reject $\hbar_{p}\neq\hbar$.

Deriving the Schrödinger equation, from the least information principle, in
the momentum representation with an external potential $V(x)\neq 0$ is a much
more complicated task. We will leave it for future research. However, the
theory for a free particle is sufficient to demonstrate why the Planck
constant must be the same in both position and momentum representations.

##  IV The Generalized Schrödinger Equation

The term $I_{f}$ is supposed to capture the additional information revealed by
the vacuum fluctuations, and is defined in (9) as the summation of the
expectation values of Kullback–Leibler divergence between $\rho(x,t)$ and
$\rho(x+w,t)$. However, there are more generic definitions of relative
entropy, such as the Rényi divergence Renyi ; Erven2014 . From an information
theoretic point of view, there is no reason to exclude alternative definitions
of relative entropy. Suppose we define $I_{f}$ based on Rényi divergence,

$\displaystyle I_{f}^{\alpha}$
$\displaystyle=:\sum_{i=0}^{N-1}E_{w}[D_{R}^{\alpha}(\rho(x,t_{i})||\rho(x+w,t_{i})]$
(19) $\displaystyle=\sum_{i=0}^{N-1}\int
d^{3}wd^{3}x\wp(w)\frac{1}{1-\alpha}ln\frac{\rho^{\alpha}(x,t_{i})}{\rho^{\alpha-1}(x+w,t_{i})}.$
(20)

Parameter $\alpha\in(0,1)\cup(1,\infty)$ is called the order of Rényi
divergence. When $\alpha\to 1$, $I_{f}^{\alpha}$ converges to $I_{f}$ as
defined in (9). In Appendix C, we show that using $I_{f}^{\alpha}$ and
following the same variation principle, we arrive at a similar extended
Hamilton-Jacobi equation as (13),

$\frac{\partial S}{\partial t}+\frac{1}{2m}\nabla S\cdot\nabla
S+V-\frac{\alpha\hbar^{2}}{2m}\frac{\nabla^{2}\sqrt{\rho}}{\sqrt{\rho}}=0,$
(21)

with an additional coefficient $\alpha$ appearing in the Bohm’s quantum
potential term. Defined $\Psi^{\prime}=\sqrt{\rho}e^{iS/\sqrt{\alpha}\hbar}$,
the continuity equation and the extended Hamilton-Jacobi equation (21) can be
combined into an equation similar to the Schrödinger equation (see Appendix
C),

$i\sqrt{\alpha}\hbar\frac{\partial\Psi^{\prime}}{\partial
t}=[-\frac{\alpha\hbar^{2}}{2m}\nabla^{2}+V]\Psi^{\prime}.$ (22)

When $\alpha=1$, the regular Schrödinger equation is recovered as expected.
Equation (22) gives a family of linear equations for each order of Rényi
divergence.

Interestingly, if we define $\hbar_{\alpha}=\sqrt{\alpha}\hbar$, then
$\Psi^{\prime}=\sqrt{\rho}e^{iS/\hbar_{\alpha}}$, and (22) becomes the same
form of regular Schrödinger equation with replacement of $\hbar$ with
$\hbar_{\alpha}$. It is as if there is an intrinsic relation between the order
of Rényi divergence and the Plank constant. This remains to be investigated
further. On the other hand, if the wave function is defined as usual without
the factor $\sqrt{\alpha}$, $\Psi^{\prime}=\sqrt{\rho}e^{iS/\hbar}$, it will
result in a nonlinear Schrödinger equation. This implies that the linearity of
Schrödinger equation depends on how the wave function is defined from the pair
of real variables $(\rho,S)$.

We also want to point out that $I_{f}^{\alpha}$ can be defined using Tsallis
divergence Tsallis ; Nielsen2011  as well, instead of using the Rényi
divergence,

$\displaystyle\begin{split}I_{f}^{\alpha}&=:\sum_{i=0}^{N-1}E_{w}[D_{T}^{\alpha}(\rho(x,t_{i})||\rho(x+w,t_{i})]\\\
&=\sum_{i=0}^{N-1}\int d^{3}w\wp(w)\frac{1}{1-\alpha}\int
d^{3}x\\{\frac{\rho(x,t_{i})^{\alpha}}{\rho(x+w,t_{i})^{\alpha-1}}-1\\}.\end{split}$
(23)

When $\Delta t\to 0$, it can be shown that the $I_{f}^{\alpha}$ defined above
converges into the same form as (47). Hence it results in the same generalized
Schrödinger equation (22).

Figure 1: Sketch of an entangled bipartite system. (1) Formation of
correlation. There is physical interaction between two subsystems for a period
of time at $t<0$. Suppose at $t=0$, the physical interaction is turned off.
Correlation between the two subsystems is established such that
$\rho\neq\rho_{1}\rho_{2}$. Here the up and down arrows symbolize the
correlated property, not related to spin. The composite system can be in a
superposition state, represented by two close shapes. (2) Preservation of the
correlation. The two subsystems move apart from each other without physical
interaction for $0<t<t_{m}$. The correlation encoded in
$\rho\neq\rho_{1}\rho_{2}$ is preserved since the law of dynamics, (28) in the
quantum mechanics case, admits solutions with $\rho\neq\rho_{1}\rho_{2}$. (3)
Manifestation of correlation. In the quantum case (3a), before the actual
measurement the wavefunction is inseparable. Measured results at time $t_{m}$
on the two subsystems are correlated. But in the classical case (3b)
measurements will find no correlation since $I_{f}=0$ and
$S_{C}=S_{C1}+S_{C2}$.

##  V New Insights on Entanglement

Entanglement is an essential feature of quantum mechanics. One of our goals
for searching the information foundation of quantum theory is to bring new
insight on the nature of entanglement. Is quantum entanglement a non-local
causal effect? Here we will show that the answer is encoded in the information
metrics $I_{f}$. Consider two subsystems with masses $m_{1}$ and $m_{2}$.
Suppose there is interaction between them for $t<0$ such that the joint
probability density at $t=0$ is inseparable,

$\rho(x_{1},x_{2},t=0)\neq\rho_{1}(x_{1},t=0)\rho_{2}(x_{2},t=0).$ (24)

At $t=0$, the interaction between them is turned off so that the external
potential $V(x_{1},x_{2})=V_{1}(x_{1})+V_{2}(x_{2})$ and the two subsystems
are moving away from each other. For $t\geq 0$, since there is no interaction,
the individual classical action is separable
$S(x_{1},x_{2},t)=S(x_{1},t)+S(x_{2},t)$. Consequently, ensemble average of
classical action

$\begin{split}S_{c}=&\int\rho(x_{1},x_{2},t)\\{(\frac{\partial S_{1}}{\partial
t}+\frac{1}{2m_{1}}\nabla_{1}S_{1}\cdot\nabla_{1}S_{1}+V_{1})\\\
&+(\frac{\partial S_{2}}{\partial
t}+\frac{1}{2m_{2}}\nabla_{2}S_{2}\cdot\nabla_{2}S_{2}+V_{2})\\}d^{3}x_{1}d^{3}x_{2}dt.\end{split}$
(25)

Even though the probability density $\rho(x_{1},x_{2},t)$ is inseparable, the
law of marginal probability still holds FN02 ,

$\displaystyle\int\rho(x_{1},x_{2},t)d^{3}x_{1}$
$\displaystyle=\rho_{2}(x_{2},t),$
$\displaystyle\int\rho(x_{1},x_{2},t)d^{3}x_{2}$
$\displaystyle=\rho_{1}(x_{1},t).$

Thus, $S_{c}=S_{c1}+S_{c2}$, with

$S_{c1}=\int\rho_{1}(x_{1},t)(\frac{\partial S_{1}}{\partial
t}+\frac{1}{2m_{1}}\nabla_{1}S_{1}\cdot\nabla_{1}S_{1}+V_{1})d^{3}x_{1}dt$
(26)

and a similar expression for $S_{c2}$. The Hamilton-Jacobi equation and the
continuity equation for each subsystem can be derived independently from
variations on $S_{c1}$ and $S_{c2}$, respectively. On the other hand, as shown
in the Appendix E, the information metric due to fluctuations is inseparable,
even though the vacuum fluctuations for the two subsystems are independent
each other,

$I_{f}=\int[\frac{\hbar}{4m_{1}}(\nabla_{1}ln\rho)^{2}+\frac{\hbar}{4m_{2}}(\nabla_{2}ln\rho)^{2}]\rho
d^{3}x_{1}d^{3}x_{2}dt.$ (27)

The inseparability is due to the fact that
$\rho(x_{1},x_{2},t)\neq\rho_{1}(x_{1},t)\rho_{2}(x_{2},t)$. Appendix E also
derives the Schrödinger equation of the bipartite system as

$i\hbar\frac{\partial\Psi}{\partial
t}=[-\frac{\hbar^{2}}{2m_{1}}\nabla_{1}^{2}-\frac{\hbar^{2}}{2m_{1}}\nabla_{2}^{2}+V_{1}+V_{2}]\Psi,$
(28)

where
$\Psi(x_{1},x_{2},t)=\sqrt{\rho(x_{1},x_{2},t)}e^{i(S_{1}+S_{2})/\hbar}$. If
and only if $\rho(x_{1},x_{2},t)=\rho_{1}(x_{1},t)\rho_{2}(x_{2},t)$ for _any
time $t\geq 0$_, this wave function can be separated into a product form,
$\Psi(x_{1},x_{2},t)=\Psi_{1}(x_{1},t)\Psi_{2}(x_{2},t)$ where

$\displaystyle\Psi_{1}(x_{1},t)=\sqrt{\rho_{1}(x_{1},t)}e^{iS_{1}/\hbar}$
$\displaystyle\Psi_{2}(x_{2},t)=\sqrt{\rho_{2}(x_{2},t)}e^{iS_{2}/\hbar}.$

But this condition is not satisfied given the initial condition at $t=0$ in
(24). Therefore, we must have
$\Psi(x_{1},x_{2},t)\neq\Psi_{1}(x_{1},t)\Psi_{2}(x_{2},t)$. This shows that
the bipartite wave function is inseparable, hence entangled, even though there
is no further interaction between the subsystems after $t\geq 0$, and the
underlying fluctuations are independent from each other.

The inseparability is enforced by the information metrics $I_{f}$, which is
the cumulative relative entropy produced by vacuum fluctuations. The dynamics
of the bipartite system can behave in such a way that $\rho(x_{1},x_{2},t)$
remains inseparable while minimizing the observable information. Although the
inseparability of $\rho(x_{1},x_{2},t)$ bears no consequence in classical
mechanics since $S_{c}$ itself is still separable, its effect is manifested in
quantum mechanics because of the existence of the term $I_{f}$. This is the
case even through the two subsystems are moving apart and without interaction,
and the vacuum fluctuations of the two subsystems are independent as shown in
Appendix E. Since $\rho(x_{1},x_{2},t)$ and consequently $I_{f}$ are
inseparable, they essentially global variables. One may insist that there must
be an hidden physical mechanism to keep $\rho(x_{1},x_{2},t)$ inseparable when
two subsystems move far apart without explicit interaction between them, and
such hidden physical mechanism must be non-local. It is debatable whether such
insistence is valid. The correlation between the two subsystems encoded in
$\rho(x_{1},x_{2},t)$ is purely informational. There is no fundamental reason
that such correlation information can be maintained only when there is
underlying physical interaction.

The analysis above shows that quantum entanglement can be understood with
three aspects. First aspect is the cause of entanglement. The initial
correlation between the two subsystems are initially created by physical
interaction between them at $t<0$ and encoded by the inseparability of
$\rho(x_{1},x_{2},t=0)$. Second aspect is the preservation of entanglement.
Schrödinger equation (28) allows such correlation to be preserved in
subsequent dynamics even though there is no further interaction. Actually the
inseparability of $\rho(x_{1},x_{2},t)$ is true in both classical and quantum
mechanics. However, the inseparability of $\rho(x_{1},x_{2},t)$ bears no
consequence in classical mechanics, since $S_{c}$ is still separable, as shown
in (26). Third aspect is the manifestation of entanglement effect. In quantum
mechanics, the effect of inseparability of $\rho(x_{1},x_{2},t)$ is manifested
through $I_{f}$ because without $I_{f}$ the dynamics of the composite system
just follows the laws of classical mechanics. Because of the inseparability of
$I_{f}$, the wavefunction of the bipartite system is also inseparable,even
though with no interaction, independent vacuum fluctuations. This implies
subsequent measurement will show the correlation of measurement results on the
two subsystems. Modern experiments have confirmed such correlation with
loophole free measurements Hensen . Fig.1 depicts these three aspects of
quantum entanglement.

To summarize, entanglement is a correlation encoded in initial physical
interaction between the two subsystems. It is causal in this sense at $t<0$.
Such correlation is preserved in later dynamics even if there is no further
interaction, and with independent vacuum fluctuations, of the two subsystems.
The effect of correlation is manifested by an informational metric $I_{f}$.
The preservation and manifestation aspects of entanglement are non-causal.

##  VI Discussion and conclusions

###  VI.1 Alternative Formulation of the Least Information Principle

Rewriting Eq. (1) as $J=S_{c}+(\hbar/2)I_{f}$, and performing the same
variation procedure will give the same laws of dynamics for the system.
However, the physical interpretation is different. One would need to consider
$(\hbar/2)I_{f}$ as additional action due to vacuum fluctuations. In other
words, one would compute the amount of observation information $I_{f}$ first,
then apply Assumption 1 to convert the information quantity to action
quantity. It is mathematically equivalent to Eq. (1). But the question is how
such action effort is realized physically. To answer this question, it
requires a physical model for the vacuum fluctuations at the sub-quantum
level. This is challenged and beyond the scope of this article. Alternatively,
we can interpret the least information principle based on Eq. (1) as
minimizing $I_{f}$ with the constraint of $S_{c}$ being a constant, and
$\hbar/2$ simply being a Lagrangian multiplier for such a constraint. Again,
mathematically, it is an equivalent formulation. But physically there are
challenges. Why should $S_{c}$ be a constant? Furthermore, regarding $\hbar/2$
as simply a Lagrangian multiplier misses the fundamental insight of Planck
constant as the discrete unit of action effort to reveal observable
information. We believe that the least information principle based on
Assumption 1, as we present in this article, gives the most intuitive physical
meaning of the mathematical formulation and without the need of a physical
model for the vacuum fluctuations.

###  VI.2 Implications of Assumption 1

The interpretation of Planck constant as the discrete action unit for
distinguisibility information reflects a fundamental physical limit. That is,
there is a lower bound of action needed to reveal observable information of
the dynamical behavior of a physical system. Smaller action will not reveal
observable information, information revealed by an action unit smaller than
$\hbar/2$ is indistinguishable and in-observable. In other words, the Planck
constant determines the resolution (in terms of action) of the observable
information of the dynamics behavior of a physical system. Historically the
Planck constant was first introduced to show that energy of radiation from a
black body is discrete. One can consider the discrete energy unit as the
smallest unit to be distinguished, or detected, in the black body radiation
phenomenon. Here, we just interpret Planck constant from an information
processing point of view. Interestingly, the postulate in special relativity
that the speed of light in vacuum is constant in all inertial reference frames
reflects another limit for information propagation. As pointed out by Landau
Landau , the constant speed of light actually is a consequence of a
fundamental physical limit that there is a limit of speed in the interaction
between two systems. The speed limit also implies that propagation of physical
information is not instant because information is propagated through physical
media such as light. Thus, we see the Planck constant and speed of light each
manifests a physical limit from information processing point of view, but from
a different angle.

The definition of $I_{p}=2S_{c}/\hbar$ in Assumption 1 appears similar to the
definition of the phase of probability amplitude for a trajectory of a
particle in Feynman’s path integral Feynman48 , $S/\hbar$, except the factor
of 2. However, there is no reason to associate $I_{p}$ to phase information
here, since we do not postulate a probability amplitude as a fundamental
quantity in the present work. In the path integral theory, $S/\hbar$ is not
assigned the meaning of observable information. Thus, it is not expected that
another information term like $I_{f}$ can be added into $S/\hbar$.
Fundamentally, the path integral theory does not interpret the Planck constant
as the quantum of action effort to reveal observable information. The
difference in not having a factor of 2 in path integral is purely due to
mathematical reason since $S/\hbar$ is associated with the probability
amplitude, whereas in our formulation, we deal with the variable of
probability density, which is the modular square of probability amplitude.
Nevertheless, both path integral and our formulation based on the least
information principle give the same Schrödinger equation. This is because both
theories start with the contribution of the classical path, then add the
additional contributions due to vacuum fluctuations, but in different ways. In
path integral, the summation of $e^{iS/\hbar}$ from all possible paths for the
probability amplitude effectively collects the contributions due to vacuum
fluctuations. On the other hand, in the least information principle, the
effort of vacuum fluctuations is manifested through the summation of the
Kullback-Leibler divergence as defined in (9).

###  VI.3 Comparisons with Relevant Research Works

In the original paper for Relational Quantum Mechanics (RQM) Rovelli:1995fv ,
Rovelli proposes two postulates from information perspective. The first
postulate, _there is a maximum amount of relevant information that can be
extracted from a system_ , is in the same spirit with Assumption 1. Rovelli
has pointed out that his first postulate implies the existence of Planck
constant. But the reconstruction effort of quantum theory in Rovelli:1995fv
does not define an information metric and how $\hbar$ is used to compute the
amount of information. Here we make explicit mathematical connections between
$\hbar$ and information metrics in (1), leading to the least information
principle to reconstruct quantum mechanics. Conceptually, we make it more
clear the connection between the Planck constant and the discreteness of
action effort to reveal observable information, which we believe simplifies
the subsequent reconstruction. The second postulate in Rovelli:1995fv , _it is
always possible to acquire new information about a system_ , is motivated to
explain the complementarity in quantum theory Hoehn:2014uua ; Hoehn:2015 .
This postulate appears not needed in our theory in terms of explaining
complementarity. Instead, we assume there is no preferred representation for
physical laws, which is more intuitive. The no preferred representation
assumption allows us to derive the transformation formulation between position
and momentum representations, and consequently the commutative relation
$[\hat{x},\hat{p}]=i\hbar$. The uncertainty relation $\Delta x\Delta
p\geq\hbar/2$ is a consequence of applying the least information principle in
the infinitesimal time step, as shown in Section III.1. Other authors proposed
postulates similar to the no preferred representation assumption, such as no
preferred measurement Mehrafarin2005 , no preferred reference frame Stuckey .
However, these postulates are proposed in very different contexts.

Inspired by the relational quantum mechanics, and the Zeilinger-Brukner
informational interpretation of quantum theory zeilinger1999foundational ;
Brukner:ys , Höhn gives a successful reconstruction of quantum theory for a
single qubit and N-qubit from elementary rules on an observer’s information
acquisition Hoehn:2014uua ; Hoehn:2015 . One of the basic postulates in this
reconstruction is the limited information principle advocated in
Rovelli:1995fv ; zeilinger1999foundational ; Brukner:ys . It states that _an
observer can acquire maximally $N$ independent bits of information about a
physical system_ where $N$ is a natural number. Even though we share the same
idea that the amount of information that can be extracted from a system is
limited, the meaning of information in Hoehn:2014uua ; Hoehn:2015  is quite
different from the observable information in our works. The mathematical
expressions of information are also different. We assume the Planck constant
is related to the discreteness of action effort to reveal observable
information, which is not the case in Hoehn:2014uua ; Hoehn:2015 .
Consequently, the Schrödinger equation derived in Hoehn:2015  does not give
the concrete form of Hamiltonian, while our derivation here shows the concrete
form of the Hamiltonian is as shown in (14).

The entropic dynamics approach to quantum mechanics Caticha2011 ; Caticha2019
bears some similarity with the theory presented in this work. For instance,
the formulations are carried out with two steps, an infinitesimal time step
and a cumulative time period. It also aims to derive the physical dynamics by
extremizing entropy. However, the entropic dynamics approach relies on another
postulate on energy conservation to complete the derivation of Schrödinger
equation. The theory presented in this paper has the advantage of simplicity
since it recursively applies the same least information principle in both an
infinitesimal time step and a cumulative time period. The entropic dynamics
approach also requires several seemingly arbitrary constants in the
formulation, while we only need the Planck constant $\hbar$ and its meaning is
clearly given in Assumption 1.

The derivation of Schrödinger equation in Section III.2 starts from (42) which
is proposed by Hall and Reginatto Hall:2001 ; Hall:2002 . Mathematically, we
arrive at the same extended Hamilton-Jacobi equation (13) as that in Hall:2001
; Hall:2002 . However, the derivation approaches are very different. Hall and
Reginatto assume an exact uncertainty relation (5), while in our theory (5) is
derived from the least information principle in a infinitesimal time step. We
clearly show the information origin of the Bohm’s penitential term, while Hall
and Reginatto derive it by assuming the random fluctuations in momentum space
and the exact uncertainty relation.

###  VI.4 Conclusions

We propose a least information principle to demonstrate how classical
mechanics becomes quantum mechanics from the information perspective. The
least information principle makes clear use of a property of the Planck
constant that it defines the discrete unit of action effort associated with
the observable information. Classical mechanics corresponds to a physical
theory when such a lower limit of discrete action effort is approximated as
zero. However, to describe a quantum system, the theory must factor in the
discreteness of action effort to reveal observable information. New
information metrics for the observable information revealed from vacuum
fluctuations are introduced. These metrics are defined in terms of relative
entropy to measure the information distances of different probability
distributions caused by local vacuum fluctuations. To derive quantum theory,
the least information principle seeks to minimize the amount of information
revealed from both classical trajectory and vacuum fluctuations. Nature
appears to reveal as least information as possible for future observation.
This principle allows us to elegantly derive the uncertainty relation between
position and momentum, and the Schrödinger equations in both position and
momentum representations. Adding the no preferred representation assumption,
we obtain the transformation formulation between position and momentum
representations. The Planck constant must be the same in different
presentations in order to be compatible with the Dirac commutative relation
between position and momentum.

Furthermore, the theory provides new insight on the nature of entanglement.
The correlation in entanglement is manifested due to the inseparability of the
information metrics $I_{f}$, even though the vacuum fluctuations of two
subsystems are independent. It suggests that although the initial formation of
entanglement needs physical interaction, the manifestation of entanglement is,
on the other hand, non-causal. Similarly, $I_{f}$ is responsible for the
origin of Bohm’s quantum potential, which is widely considered as non-local.
However, just as entanglement, such non-locality originates from an
informational requirement and does not need to be causal.

Utilizing Rényi divergence in the least information principle leads to a
generalized Schrödinger equation (22) that depends on the order of Rényi
divergence. Given the extensive experimental confirmations of the normal
Schrödinger equation, it is inconceivable that one will find physical
scenarios for which the generalized Schrödinger equation with $\alpha\neq 1$
is applicable. However, the generalized Schrödinger equation is legitimate
from an information perspective. It confirms that the least information
principle can produce new results.

The least information principle can also be applied to field theory to obtain
the Schrödinger functional equation for a massive scalar field Newpaper . We
believe the principle can be further extended to derive the Dirac equation for
an electron with spin. The information perception on the speed of light and
Planck constant is also intriguing. That is, the speed of light is the upper
limit of propagating information, while the Planck constant is considered as
the lower limit of action effort to reveal observable information of a
physical object. This suggests that the information perception is a natural
avenue to unify relativity and quantum theories. We hope that the results and
discussions presented in this work help to further the understanding of the
information essence of quantum theory.

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##  Appendix A The Derivation of Schrödinger Equation

The key step in deriving Schrödinger equation is to prove (11) from (9). To do
this, one first takes the Taylor expansion of $\rho(x+w,t)$ around $x$

$\rho(x+w,t_{i})=\rho(x,t_{i})+\nabla\rho(x,t_{i})w+\frac{1}{2}\nabla^{2}\rho(x,t_{i})w^{2}+O(w^{2}).$
(29)

The expansion is legitimate because (4) shows that the variance of fluctuation
displacement $w$ is proportional to $\Delta t$. As $\Delta t\to 0$, only very
small $w$ is possible. Then

$\displaystyle ln\frac{\rho(x+w,t_{i})}{\rho(x,t_{i})}$
$\displaystyle=ln(1+\frac{\nabla\rho}{\rho}w+\frac{\nabla^{2}\rho}{2\rho}w^{2})=\frac{\nabla\rho}{\rho}w+\frac{\nabla^{2}\rho}{2\rho}w^{2}-\frac{1}{2}(\frac{\nabla\rho}{\rho}w+\frac{\nabla^{2}\rho}{2\rho}w^{2})^{2}$
(30)
$\displaystyle=\frac{\nabla\rho}{\rho}w+\frac{\nabla^{2}\rho}{2\rho}w^{2}-\frac{1}{2}(\frac{\nabla\rho}{\rho})^{2}w^{2}+O(w^{2}).$
(31)

Substitute the above expansion into (9),

$\displaystyle E_{w}[D_{KL}(\rho(x,t_{i}))]$ $\displaystyle=-\int\wp
d^{3}wd^{3}x[(\nabla\rho)w+\frac{1}{2}(\nabla^{2}\rho)w^{2}-\frac{1}{2}\rho(\nabla
ln\rho)^{2}w^{2}]$ (32) $\displaystyle=-\int d^{3}x\\{(\nabla\rho)\langle
w\rangle+\frac{1}{2}[(\nabla^{2}\rho)-\rho(\nabla ln\rho)^{2})\langle
w^{2}\rangle]\\}=\frac{1}{2}\langle w^{2}\rangle\int dx[\rho(\nabla
ln\rho)^{2}-(\nabla^{2}\rho)].$ (33)

The last step uses the fact that $\langle w\rangle=0$. Integrating the last
term and assuming $\rho$ is a smooth function such that its spatial gradient
approaches zero when $x\to\pm\infty$, we have

$\int d^{3}x\nabla^{2}\rho=\nabla\rho(x,t)|^{+\infty}_{-\infty}=0.$ (34)

Substitute $\langle w^{2}\rangle=\hbar\Delta t/2m$ into (33) and then into
(9),

$I_{f}=\sum_{i=0}^{N-1}E_{w}[D_{KL}(\rho(x,t_{i})||\rho(x+w,t_{i})]=\sum_{i=0}^{N-1}\frac{\hbar\Delta
t}{4m}\int d^{3}x\rho(\nabla ln\rho)^{2}=\frac{\hbar}{4m}\int
d^{3}xdt\rho(\nabla ln\rho)^{2},$ (35)

which is Eq. (11). The next step is to derive (13). Variation of $I$ given in
(12) with respect to $\rho$ gives

$\delta I=\int\\{\frac{2}{h}[\frac{\partial S}{\partial t}+\frac{1}{2m}\nabla
S\cdot\nabla
S+V]\delta\rho+\frac{\hbar}{4m}[2\frac{\nabla\rho}{\rho}\delta\nabla\rho-\frac{(\nabla\rho)^{2}}{\rho^{2}}\delta\rho]\\}d^{3}xdt.$
(36)

Integration by part for the term with $\delta\nabla\rho$, we have

$\int\frac{\nabla\rho}{\rho}\delta\nabla\rho
d^{3}x=-\int\nabla(\frac{\nabla\rho}{\rho})\delta\rho
d^{3}x=\int(\frac{(\nabla\rho)^{2}}{\rho^{2}}-\frac{\nabla^{2}\rho}{\rho})\delta\rho
d^{3}x$ (37)

Insert (37) back to (36),

$\delta I=\int\\{\frac{2}{h}[\frac{\partial S}{\partial t}+\frac{1}{2m}\nabla
S\cdot\nabla
S+V]+\frac{\hbar}{4m}[\frac{(\nabla\rho)^{2}}{\rho^{2}}-2\frac{\nabla^{2}\rho}{\rho}]\\}\delta\rho
d^{3}xdt.$ (38)

Taking $\delta I=0$ for arbitrary $\delta\rho$, we must have

$\frac{\partial S}{\partial t}+\frac{1}{2m}\nabla S\cdot\nabla
S+V+\frac{\hbar^{2}}{8m}[\frac{(\nabla\rho)^{2}}{\rho^{2}}-2\frac{\nabla^{2}\rho}{\rho}]=0.$
(39)

One can verify that
$[\frac{(\nabla\rho)^{2}}{\rho^{2}}-2\frac{\nabla^{2}\rho}{\rho}]=-4\frac{\nabla^{2}\sqrt{\rho}}{\sqrt{\rho}}$.
Substituting it back to (39) gives the desired result in (13).

##  Appendix B Charge Particle in An External Electromagnetic Field

Suppose a particle of charge $q$ and mass $m$ is placed in an electromagnetic
field with vector potential A and scalar potential $\phi$. Without random
fluctuations, the particle moves along a classical trajectory determined by
the classical Hamilton-Jacobi equation:

$\frac{\partial S}{\partial t}+\frac{1}{2m}(\nabla
S-q\textbf{A})^{2}+q\phi=0.$ (40)

Compared to (7), a generalized momentum term $(\nabla S-q\textbf{A})$ replaces
the original momentum $\nabla S$ Nelson ; FeynmanNotes . Similarly, the
continuity equation becomes

$\frac{\partial\rho}{\partial t}+\frac{1}{m}\nabla\cdot(\rho(\nabla
S-q\textbf{A}))=0.$ (41)

These two equations can be derived through fixed point variation on the
average classical action

$S_{c}=\int\rho\\{\frac{\partial S}{\partial t}+\frac{1}{2m}(\nabla
S-q\textbf{A})^{2}+q\phi\\}d^{3}xdt.$ (42)

Thus, observable information from the classical trajectory can be defined as
$I_{p}=2S_{c}/\hbar$. In addition, the particle also experiences constant
fluctuations around the classical trajectory. We assume the external
electromagnetic field has no influence on the vacuum fluctuations. This means
$I_{f}$ defined in (9) is applicable here. Variation of the total observable
information $I_{p}+I_{f}$ with respect to $\rho$ gives the extended Hamilton-
Jacobi equation

$\frac{\partial S}{\partial t}+\frac{1}{2m}(\nabla
S-q\textbf{A})^{2}+q\phi-\frac{\hbar^{2}}{2m}\frac{\nabla^{2}\sqrt{\rho}}{\sqrt{\rho}}=0.$
(43)

Defined $\Psi=\sqrt{\rho}e^{iS/\hbar}$, the continuity equation and the
extended Hamilton-Jacobi equation (43) are combined into a single differential
equation,

$i\hbar\frac{\partial\Psi}{\partial
t}=[\frac{1}{2m}(i\hbar\nabla+q\textbf{A})^{2}+q\phi]\Psi,$ (44)

which is the Schrödinger equation in an external electromagnetic field on the
condition $\nabla\cdot\textbf{A}=0$.

##  Appendix C Rényi Divergence and the Generalized Schrödinger Equation

Based on the definition of $I_{f}^{\alpha}$ in (19), and starting from (29),
we have

$\displaystyle\frac{\rho^{\alpha}(x,t_{i})}{\rho^{\alpha-1}(x+w,t_{i})}$
$\displaystyle=\frac{\rho^{\alpha}}{\rho^{\alpha-1}(1+\frac{\nabla\rho}{\rho}w+\frac{\nabla^{2}\rho}{2\rho}w^{2})^{\alpha-1}}$
$\displaystyle=\rho\\{1+(1-\alpha)(\frac{\nabla\rho}{\rho}w+\frac{\nabla^{2}\rho}{2\rho}w^{2})+\frac{1}{2}\alpha(\alpha-1)(\frac{\nabla\rho}{\rho}w+\frac{\nabla^{2}\rho}{2\rho}w^{2})^{2}\\}$
$\displaystyle=\rho+(1-\alpha)[(\nabla\rho)w+\frac{1}{2}(\nabla^{2}\rho)w^{2}]+\frac{1}{2}\alpha(\alpha-1)\frac{(\nabla\rho)^{2}}{\rho}w^{2}+O(w^{2})$

Given the normalization condition $\int\rho d^{3}x=1$, and the regularity
assumption of $\rho$, $\int\nabla\rho d^{3}x=0$, we have

$ln\\{\int\frac{\rho^{\alpha}(x,t_{i})}{\rho^{\alpha-1}(x+w,t_{i})}d^{3}x\\}=ln\\{1+\frac{1}{2}\alpha(\alpha-1)w^{2}\int\frac{(\nabla\rho)^{2}}{\rho}d^{3}x\\}=\frac{1}{2}\alpha(1-\alpha)w^{2}\int\frac{(\nabla\rho)^{2}}{\rho}d^{3}x.$

Thus, $I_{f}^{\alpha}$ is simplified as

$\displaystyle I_{f}^{\alpha}$ $\displaystyle=\sum_{i=0}^{N-1}\int
d^{3}wd^{3}x\wp(w)\frac{1}{1-\alpha}ln\frac{\rho^{\alpha}(x,t_{i})}{\rho^{\alpha-1}(x+w,t_{i})}$
(45) $\displaystyle=\sum_{i=0}^{N-1}\int d^{3}w\wp(w)\frac{1}{2}\alpha
w^{2}\int
dx\frac{(\nabla\rho)^{2}}{\rho}=\sum_{i=0}^{N-1}\frac{1}{2}\alpha\langle
w^{2}\rangle\int d^{3}x\frac{(\nabla\rho)^{2}}{\rho}$ (46)
$\displaystyle=\sum_{i=0}^{N-1}\frac{\alpha\hbar}{4m}\Delta t\int
d^{3}x\frac{(\nabla\rho)^{2}}{\rho}=\frac{\alpha\hbar}{4m}\int
d^{3}xdt\frac{(\nabla\rho)^{2}}{\rho}.$ (47)

Compared to (35), the only difference from $I_{f}$ is an additional
coefficient $\alpha$, i.e., $I_{f}^{\alpha}=\alpha I_{f}$. Equation (21) can
be derived by repeating the calculation in Section A. To obtain the
generalized Schrödinger equation, we define
$\Psi^{\prime}=\sqrt{\rho}e^{iS/\sqrt{\alpha}\hbar}$, then take the partial
derivative over time,

$\displaystyle\frac{\partial\Psi^{\prime}}{\partial t}$
$\displaystyle=\frac{1}{2\rho}\frac{\partial\rho}{\partial
t}\Psi^{\prime}+\frac{i}{\sqrt{\alpha}\hbar}\frac{\partial S}{\partial
t}\Psi^{\prime}.$

Multiplying $i\sqrt{\alpha}\hbar/\Psi^{\prime}$ both sides, and applying the
continuity equation and extended Hamilton-Jaccobi function (21), we get

$\displaystyle\frac{i\sqrt{\alpha}\hbar}{\Psi^{\prime}}\frac{\partial\Psi^{\prime}}{\partial
t}=\frac{i\sqrt{\alpha}\hbar}{2\rho}\frac{\partial\rho}{\partial
t}-\frac{\partial S}{\partial
t}=-\frac{i\sqrt{\alpha}\hbar}{2m\rho}\nabla(\rho\nabla S)+\frac{1}{2m}(\nabla
S)^{2}+V-\frac{\alpha\hbar^{2}}{2m}\frac{\nabla^{2}\sqrt{\rho}}{\sqrt{\rho}}.$
(48)

Taking the gradient of $\Psi^{\prime}=\sqrt{\rho}e^{iS/\sqrt{\alpha}\hbar}$,
and using $\rho=\Psi^{\prime}\Psi^{\prime*}$, one can obtain the following
identities

$\displaystyle\nabla S$
$\displaystyle=\frac{i\sqrt{\alpha}\hbar}{2}(\frac{\nabla\Psi^{\prime*}}{\Psi^{\prime*}}-\frac{\nabla\Psi^{\prime}}{\Psi^{\prime}})$
$\displaystyle\frac{\nabla(\rho\nabla S)}{\rho}$
$\displaystyle=\frac{i\sqrt{\alpha}\hbar}{2}(\frac{\nabla^{2}\Psi^{\prime*}}{\Psi^{\prime*}}-\frac{\nabla^{2}\Psi^{\prime}}{\Psi^{\prime}})$
$\displaystyle\frac{\nabla^{2}\sqrt{\rho}}{\sqrt{\rho}}$
$\displaystyle=\frac{1}{2}(\frac{\nabla^{2}\Psi^{\prime*}}{\Psi^{\prime*}}+\frac{\nabla^{2}\Psi^{\prime}}{\Psi^{\prime}})-\frac{1}{4}(\frac{\nabla\Psi^{\prime*}}{\Psi^{\prime*}}-\frac{\nabla\Psi^{\prime}}{\Psi^{\prime}})^{2}.$

Substitute these identities into (48),

$\displaystyle\frac{i\sqrt{\alpha}\hbar}{\Psi^{\prime}}\frac{\partial\Psi^{\prime}}{\partial
t}$
$\displaystyle=\frac{\alpha\hbar^{2}}{4m}(\frac{\nabla^{2}\Psi^{\prime*}}{\Psi^{\prime*}}-\frac{\nabla^{2}\Psi^{\prime}}{\Psi^{\prime}})-\frac{\alpha\hbar^{2}}{8m}(\frac{\nabla\Psi^{\prime*}}{\Psi^{\prime*}}-\frac{\nabla\Psi^{\prime}}{\Psi^{\prime}})^{2}+V-\frac{\alpha\hbar^{2}}{4m}(\frac{\nabla^{2}\Psi^{\prime*}}{\Psi^{\prime*}}+\frac{\nabla^{2}\Psi^{\prime}}{\Psi^{\prime}})+\frac{\alpha\hbar^{2}}{8m}(\frac{\nabla\Psi^{\prime*}}{\Psi^{\prime*}}-\frac{\nabla\Psi^{\prime}}{\Psi^{\prime}})^{2}$
$\displaystyle=-\frac{\alpha\hbar^{2}}{2m}\frac{\nabla^{2}\Psi^{\prime}}{\Psi^{\prime}}+V.$

Multiplying $\Psi^{\prime}$ both sides, we arrive at the generalized
Schrödinger equation (22).

##  Appendix D  Schrödinger equation for a Free Particle in Momentum
Representation

In deriving the Schrödinger equation for a free particle in momentum
representation, we need to prove that $I_{f}$, defined in (15), does not
contribute in the variation procedure with respect to $\varrho(p,t)$, as long
as $\varrho(p,t)$ is a regular smooth function. We provide an intuitive proof
here that is sufficiently convincing. A more mathematically rigorous proof is
desirable in future research. First, we note that the Kullback–Leibler
divergence is a special case of Rényi divergence $D^{\alpha}_{R}$ when the
order $\alpha=1$. Second, we make use of the fact that the Rényi divergence is
non-decreasing as a function of its order $\alpha$ Erven2014 . Thus,

$D_{KL}(\varrho(p,t_{i})||\varrho(p+\tilde{w},t_{i})\geq
D^{\frac{1}{2}}_{R}(\varrho(p,t_{i})||\varrho(p+\tilde{w},t_{i}).$ (49)

Given the non-negativity of divergence Nielsen , the expectation value of
$D_{KL}$ and $D^{\frac{1}{2}}_{R}$ with respect to transition probability
density $\tilde{\wp}(p+\tilde{w}|p)$ also satisfies the inequality,

$\displaystyle
E_{\tilde{w}}[D_{KL}(\varrho(p,t_{i})||\varrho(p+\tilde{w},t_{i})]$
$\displaystyle\geq
E_{\tilde{w}}[D^{\frac{1}{2}}_{R}(\varrho(p,t_{i})||\varrho(p+\tilde{w},t_{i})]$
(50) $\displaystyle=-2\int d^{3}\tilde{w}\tilde{\wp}(p+\tilde{w}|p)ln[\int
dp\sqrt{\varrho(p,t_{i})\varrho(p+\tilde{w},t_{i})}].$ (51)

As shown in the main text, as $\Delta t\to 0$, the variance
$\langle\tilde{w}^{2}\rangle\to\infty$, and $\wp(p+\tilde{w}|p)$ becomes a
uniform function with respect to $\tilde{w}$. This means that any value of
$\tilde{w}$ contributes equally in calculating the divergence
$D^{\frac{1}{2}}_{R}$. However, the integral inside the logarithm function
basically depends on the overlap between functions $\varrho(p,t_{i})$ and
$\varrho(p+\tilde{w},t_{i})$. We will ignore the case when $\varrho(p,t_{i})$
is a constant because in that case
$D_{KL}(\varrho(p,t_{i})||\varrho(p+\tilde{w},t_{i})=0$. Assuming
$\varrho(p,t_{i})$ is a smooth function. For a free particle with finite
energy, the momentum is also finite. Combining this fact with the
normalization condition $\int\varrho(p,t_{i})d^{3}p=1$, we must have
$\lim_{|p|\to\infty}\varrho(p,t_{i})=0$. Thus, the overlap between functions
$\varrho(p)$ and $\varrho(p+\tilde{w})$ will be sufficiently small when
$\tilde{w}$ is sufficiently large,

$\lim_{|\tilde{w}|\to\infty}\int
d^{3}p\sqrt{\varrho(p,t_{i})\varrho(p+\tilde{w},t_{i})}\to 0.$ (52)

The implies

$-2\lim_{|\tilde{w}|\to\infty}ln[\int
d^{3}p\sqrt{\varrho(p,t_{i})\varrho(p+\tilde{w},t_{i})}]\to+\infty.$ (53)

Given the non-negativity of $D^{\frac{1}{2}}_{R}$ for any $\tilde{w}$, and the
probability distribution for each $\tilde{w}$ becomes uniform, the value of
right hand side of (51) will be dominated by large $\tilde{w}$, and the result
is approaching positive infinity. Hence, the left hand side of (51) is also
approaching positive infinity. This result is independent of the specific
functional form of $\varrho(p,t_{i})$ assuming that $\varrho(p,t_{i})$ is a
smooth continuous function. Consequently, variation of $E_{\tilde{w}}[D_{KL}]$
with respect to $\varrho(p,t_{i})$ does not give any constraint to
$\varrho(p,t_{i})$,

$\frac{\delta
E_{\tilde{w}}[D_{KL}(\varrho(p,t_{i})||\varrho(p+\tilde{w},t_{i})]}{\delta\varrho(p,t_{i})}=0.$
(54)

Since this is true for every time moment $t_{i}$, from the definition of
$I_{f}$ in (15), we conclude that $\delta I_{f}/\delta\varrho=0$. Note that if
defining $I_{f}$ using Fisher information, instead of Kullback–Leibler
divergence $D_{KL}$, as the information metrics, one will not reach the
conclusion that $I_{f}$ is a infinite number independent of $\varrho$.

##  Appendix E The Separability of Information Metrics $I_{f}$ for a Bipartite
System

We assume that from $t\geq 0$, there is no interaction between the two
subsystems of the bipartite system. It is also natural to assume that the
vacuum fluctuations for each subsystem are independent from each other. This
implies that transition probability is separable,

$\wp(x_{1}+w_{1},x_{2}+w_{2},t_{i}|x_{1},x_{2},t_{i})=\wp_{1}(x_{1}+w_{1},t_{i}|x_{1},t_{i})\wp_{2}(x_{2}+w_{2},t_{i}|x_{2},t_{i}).$
(55)

Extend the definition of $I_{f}$ in (9) to the bipartite system:

$\displaystyle I_{f}$
$\displaystyle=:\sum_{i=0}^{N-1}E_{w}[D_{KL}(\rho(x_{1},x_{2},t_{i})||\rho(x_{1}+w_{1},x_{2}+w_{2},t_{i})]$
$\displaystyle=\sum_{i=0}^{N-1}\int
d^{3}w_{1}d^{3}w_{2}d^{3}x_{1}d^{3}x_{2}\wp(x_{1}+w_{1},x_{2}+w_{2},t_{i}|x_{1},x_{2},t_{i})\rho(x_{1},x_{2},t_{i})ln\frac{\rho(x_{1},x_{2},t_{i})}{\rho(x_{1}+w_{1},x_{2}+w_{2},t_{i})}$
$\displaystyle=\sum_{i=0}^{N-1}\int
d^{3}w_{1}d^{3}w_{2}d^{3}x_{1}d^{3}x_{2}\wp_{1}(x_{1}+w_{1},t_{i}|x_{1},t_{i})\wp_{2}(x_{2}+w_{2},t|x_{2},t_{i})\rho(x_{1},x_{2},t_{i})ln\frac{\rho(x_{1},x_{2},t_{i})}{\rho(x_{1}+w_{1},x_{2}+w_{2},t_{i})}.$

Expanding the logarithm function similarly to (30),

$\displaystyle
ln\frac{\rho(x_{1},x_{2},t_{i})}{\rho(x_{1}+w_{1},x_{2}+w_{2},t_{i})}=$
$\displaystyle\frac{\nabla_{1}\rho}{\rho}w_{1}+\frac{\nabla_{2}\rho}{\rho}w_{2}+\frac{\nabla_{1}^{2}\rho}{2\rho}w_{1}^{2}+\frac{\nabla_{2}^{2}\rho}{2\rho}w_{2}^{2}$
(56)
$\displaystyle-\frac{1}{2}(\frac{\nabla_{1}\rho}{\rho})^{2}w_{1}^{2}-\frac{1}{2}(\frac{\nabla_{2}\rho}{\rho})^{2}w_{2}^{2}+O(w_{1}^{2})+O(w_{2}^{2}).$
(57)

we have

$\displaystyle I_{f}=$ $\displaystyle-\sum_{i=0}^{N-1}\int
d^{3}w_{1}d^{3}w_{2}d^{3}x_{1}d^{3}x_{2}\wp_{1}\wp_{2}[(\nabla_{1}\rho)w_{1}+\frac{1}{2}(\nabla_{1}^{2}\rho)w_{1}^{2}-\frac{1}{2}\rho(\nabla_{1}ln\rho)^{2}w_{1}^{2}$
$\displaystyle+(\nabla_{2}\rho)w_{2}+\frac{1}{2}(\nabla_{2}^{2}\rho)w_{2}^{2}-\frac{1}{2}\rho(\nabla_{2}ln\rho)^{2}w_{2}^{2}]$
$\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{i=0}^{N-1}\int
d^{2}x_{1}d^{3}x_{2}\\{\langle
w_{1}^{2}\rangle[\rho(\nabla_{1}ln\rho)^{2}-(\nabla_{1}^{2}\rho)]+\langle
w_{2}^{2}\rangle[\rho(\nabla_{2}ln\rho)^{2}-(\nabla_{2}^{2}\rho)]\\}$

The last step uses the fact that $\langle w_{1}\rangle=\langle
w_{2}\rangle=0$. Taking the same assumption as (34) that $\rho$ is a regular
function and its gradient with respect to $x_{1}$ or $x_{2}$ approaches zero
when $x\to\pm\infty$, we have

$I_{f}=\frac{1}{2}\sum_{i=0}^{N-1}\int d^{2}x_{1}d^{3}x_{2}\\{\langle
w_{1}^{2}\rangle\rho(\nabla_{1}ln\rho)^{2}+\langle
w_{2}^{2}\rangle\rho(\nabla_{2}ln\rho)^{2}\\}.$

Substituting $\langle w_{1}^{2}\rangle=\hbar\Delta t/2m_{1}$ and $\langle
w_{2}^{2}\rangle=\hbar\Delta t/2m_{2}$, and taking $\Delta t\to 0$, we get

$I_{f}=\int
d^{2}x_{1}d^{3}x_{2}dt\\{\frac{\hbar}{4m_{1}}\rho(\nabla_{1}ln\rho)^{2}+\frac{\hbar}{4m_{2}}\rho(\nabla_{2}ln\rho)^{2}\\}.$
(58)

Combined with (25), the total amount of information for the bipartite system
is

$I=\frac{2}{\hbar}\int
d^{2}x_{1}d^{3}x_{2}dt\rho(x_{1},x_{2},t)\\{\frac{\partial S}{\partial
t}+\frac{1}{2m_{1}}(\nabla_{1}S)^{2}+\frac{1}{2m_{2}}(\nabla_{2}S)^{2}+V_{1}+V_{2}+\frac{\hbar^{2}}{8m_{1}}(\nabla_{1}ln\rho)^{2}+\frac{\hbar^{2}}{8m_{2}}(\nabla_{2}ln\rho)^{2}\\}$
(59)

where $S=S_{1}+S_{2}$. Variations with respect to $S$ and $\rho$,
respectively, give two equations,

$\displaystyle\frac{\partial\rho}{\partial
t}+\frac{1}{m_{1}}\nabla_{1}(\rho\nabla_{1}S)+\frac{1}{m_{2}}\nabla_{2}(\rho\nabla_{2}S)=0;$
(60) $\displaystyle\frac{\partial S}{\partial
t}+\frac{1}{2m_{1}}(\nabla_{1}S)^{2}+\frac{1}{2m_{2}}(\nabla_{2}S)^{2}+V_{1}+V_{2}-\frac{\hbar^{2}}{2m_{1}}\frac{\nabla_{1}^{2}\sqrt{\rho}}{\sqrt{\rho}}-\frac{\hbar^{2}}{2m_{2}}\frac{\nabla_{2}^{2}\sqrt{\rho}}{\sqrt{\rho}}=0.$
(61)

Defined $\Psi(x_{1},x_{2},t)=\sqrt{\rho(x_{1},x_{2},t)}e^{iS/\hbar}$ where
$S=S_{1}+S_{2}$, it can be verified that that the two equations above are
equivalent to the Schrödinger equation in (28).

Equation (58) shows that $I_{f}$ is inseparable since
$\rho(x_{1},x_{2},t)\neq\rho_{1}(x_{1},t)\rho_{2}(x_{2},t)$. On the other
hand, suppose $\rho(x_{1},x_{2},t)=\rho_{1}(x_{1},t)\rho_{2}(x_{2},t)$, then
$\nabla_{1}ln\rho=\nabla_{1}(ln\rho_{1}+ln\rho_{2})=\nabla_{1}ln\rho_{1}$.
Similarly, $\nabla_{2}ln\rho=\nabla_{2}ln\rho_{2}$, then

$\displaystyle I_{f}$ $\displaystyle=\int
d^{2}x_{1}d^{3}x_{2}dt\\{\frac{\hbar}{4m_{1}}\rho_{1}\rho_{2}(\nabla_{1}ln\rho_{1})^{2}+\frac{\hbar}{4m_{2}}\rho_{1}\rho_{2}(\nabla_{2}ln\rho_{2})^{2}\\}$
(62) $\displaystyle=\frac{\hbar}{4m_{1}}\int
d^{3}x_{1}dt\rho_{1}(\nabla_{1}ln\rho_{1})^{2}+\frac{\hbar}{4m_{2}}\int
d^{3}x_{2}dt\rho_{2}(\nabla_{2}ln\rho_{2})^{2}=I_{f1}+I_{f2},$ (63)

and is clearly separable into two independent terms, where

$I_{f1}=\frac{\hbar}{4m_{1}}\int\rho_{1}(\nabla_{1}ln\rho_{1})^{2}d^{3}x_{1}dt;\text{
}I_{f2}=\frac{\hbar}{4m_{2}}\int\rho_{2}(\nabla_{2}ln\rho_{2})^{2}d^{3}x_{2}dt.$
(64)