# The Stochastic-Quantum Theorem

Jacob A. Barandes<EMAIL_ADDRESS>Jefferson Physical Laboratory,
Harvard University, Cambridge, MA 02138

(August 28, 2024)

###### Abstract

This paper introduces several new classes of mathematical structures that have
close connections with physics and with the theory of dynamical systems. The
most general of these structures, called generalized stochastic systems,
collectively encompass many important kinds of stochastic processes, including
Markov chains and random dynamical systems. This paper then states and proves
a new theorem that establishes a precise correspondence between any
generalized stochastic system and a unitarily evolving quantum system. This
theorem therefore leads to a new formulation of quantum theory, alongside the
Hilbert-space, path-integral, and quasiprobability formulations. The theorem
also provides a first-principles explanation for why quantum systems are based
on the complex numbers, Hilbert spaces, linear-unitary time evolution, and the
Born rule. In addition, the theorem suggests that by selecting a suitable
Hilbert space, together with an appropriate choice of unitary evolution, one
can simulate any generalized stochastic system on a quantum computer, thereby
potentially opening up an extensive set of novel applications for quantum
computing.

##  I Introduction

In the development of physical theories, it sometimes turns out that existing
definitions are too conceptually limiting, and that more flexible definitions
are needed. Working with more flexible definitions at a higher level of
abstraction or generality may make it easier to discover new connections or
prove new theorems, which would then also apply down at the lower level of the
original definitions.

This paper will argue that by appropriately generalizing standard definitions
of dynamical systems, one can obtain novel classes of mathematical structures
that encompass an extensive array of physically important models. As with a
traditionally defined dynamical system, each such mathematical structure
describes a physical system moving deterministically or stochastically along
some trajectory in a configuration space, albeit with a more general set of
laws than according to standard definitions. (For pedagogical treatments of
the standard theory of dynamical systems, see, for instance, [1, 2, 3].)

This paper also states and proves a new theorem showing that despite being
based on trajectories in configuration spaces, the newly introduced class of
_generalized stochastic systems_ actually includes all quantum systems. As a
consequence, this _stochastic-quantum theorem_ offers a more conceptually
transparent way to understand quantum systems, with superpositions no longer
regarded as literal blends of physical states. The theorem also provides a
first-principles explanation for features of quantum theory that are usually
taken to be axiomatic, including Hilbert spaces over the complex numbers,
linear-unitary evolution, and the Born rule.

Seen from another point of view, this _stochastic-quantum correspondence_
yields an alternative way to formulate quantum theory, in the language of
trajectories unfolding stochastically in configuration spaces. This
alternative formulation is distinct from the traditional Hilbert-space
formulation [4, 5], the path-integral formulation [6, 7, 8], and the quasi-
probability formulation [9, 10].

From a more practical perspective, turning this stochastic-quantum
correspondence around suggests that unitarily evolving quantum systems can be
put to work simulating a very broad class of non-Markovian stochastic
processes, thereby potentially opening up an extensive suite of new
applications for quantum computers.

Section II begins by defining deterministic generalizations of dynamical
systems, followed by the introduction of important distinctions between
_indivisible_ , _divisible_ , and _Markovian_ dynamics. Section III provides a
generalized definition of a system with stochastic laws, shows how to
represent such a generalized stochastic system in the formalism of linear
algebra, describes connections between this work and the existing research
literature, defines the relationship between a composite system and its
subsystems, and introduces the crucial notion of a _unistochastic system_.
Section IV states the stochastic-quantum theorem, whose proof is this paper’s
primary goal, and then discusses some important corollaries and provides a
simple example of the theorem in practice. Section V lays out the theorem’s
proof, which entails explicitly constructing the claimed correspondence
between stochastic systems and quantum systems along the way. Section VI
concludes the paper with a brief discussion of future work.

##  II Deterministic Systems

###  II.1 Generalized Dynamical Systems

Dynamical systems are abstract mathematical structures that usefully model
many deterministic physical processes. According to the standard definition
[1, 2, 3], a dynamical system consists of a map representing some kind of
evolution law that can be repeatedly applied to the elements of some set of
states. A dynamical system is usually assumed to be _divisible_ , in the sense
that one can ‘divide up’ its evolution law over any time duration into well-
defined evolution laws that describe intermediate time durations. The more
general case would be an _indivisible_ dynamical system that might lack this
feature.

The terms ‘divisible’ and ‘indivisible’ for dynamical laws are remarkably new.
The terminology appears to be due to Wolf and Cirac, who introduced it in a
2008 paper [11] on quantum channels.111Note that this terminology is unrelated
to the much older concept of _infinite divisibility_ , which refers to a
probability distribution that can be expressed as the probability distribution
of a sum of any integer number of independent and identically distributed
random variables.

To accommodate the eventual possibility of indivisible evolution, this paper
will define a _generalized dynamical system_ to mean a tuple of the form

$\mathopen{}\mathclose{{}\left(\mathcal{X},\mathcal{T},f}\right)$ (1)

that consists of the following data.

  * •

The symbol $\mathcal{X}$ denotes a set that will be called the generalized
dynamical system’s _state space_ (or _phase space_), and whose individual
elements $i\in\mathcal{X}$ denote the system’s _(allowed) states_.

  * •

Note that $\mathcal{X}$ may or may not be a finite set, and it may or may not
involve additional structure in its definition, such as measure-theoretic
structure or a vector-space structure. For the purposes of this paper, no such
additional structure will be specified or assumed. More broadly, for reasons
of brevity and simplicity, this paper will entirely set aside measure-
theoretic considerations that arise for the case of uncountable sets.

  * •

The symbol $\mathcal{T}$ denotes a _set of times_ $t\in\mathcal{T}$, where
$\mathcal{T}$ may or may not be isomorphic to a subset of the real line
$\mathbb{R}$ under addition.

  * •

The symbol $f$ denotes a map

$f:\mathcal{X}\times\mathcal{T}\to\mathcal{X}$ (2)

that will be called the system’s _dynamical map_. This dynamical map $f$ takes
as inputs any state $i$ and any time $t$, and outputs a state
$f\mathopen{}\mathclose{{}\left(i,t}\right)\in\mathcal{X}$:

$i,t\mapsto
f\mathopen{}\mathclose{{}\left(i,t}\right)\in\mathcal{X}\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }i\in\mathcal{X},\,t\in\mathcal{T}}\right].$ (3)

  * •

Fixing the time $t$ turns $f$ into a _time-dependent dynamical map_

$f_{t}:\mathcal{X}\to\mathcal{X}$ (4)

defined by

$i\mapsto f_{t}\mathopen{}\mathclose{{}\left(i}\right)\equiv
f\mathopen{}\mathclose{{}\left(i,t}\right)\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }i\in\mathcal{X}}\right].$ (5)

  * •

Without any important loss of generality, the set of times $\mathcal{T}$ will
be assumed to include an element denoted by $0$ and called the _initial time_.
It will be further assumed that at the initial time $0$, the time-dependent
dynamical map $f_{t}$ trivializes to the identity map
$\mathrm{id}_{\mathcal{X}}$ on $\mathcal{X}$:

$f_{0}=\mathrm{id}_{\mathcal{X}},\quad\textrm{or}\quad
f_{0}\mathopen{}\mathclose{{}\left(i}\right)=i\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }i\in\mathcal{X}}\right].$ (6)

  * •

One can regard the argument $i$ appearing in the expression
$f_{t}\mathopen{}\mathclose{{}\left(i}\right)$ as an initial state of the
system at the initial time $0$, with the time-dependent dynamical map $f_{t}$
then describing the evolution of that state $i$ from the initial time $0$ to
the time $t$.

  * •

Given a fixed state $i$, the set of states

$\\{f_{t}\mathopen{}\mathclose{{}\left(i}\right)\,|\,t\in\mathcal{T}\\}\subset\mathcal{X}$
(7)

describes the _orbit_ , or _trajectory_ , of the initial state $i$ through the
system’s state space $\mathcal{X}$ according to the dynamical map $f$.

In many applications, one takes the set of times $\mathcal{T}$ to be a
_semigroup_ , meaning that the definition of $\mathcal{T}$ includes an
associative binary operation $\star$ (which is often denoted instead by $+$ in
the commutative case):

$t,t^{\prime}\mapsto t\star
t^{\prime}\in\mathcal{T}\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}t,t^{\prime}\in\mathcal{T}}\right],$ (8) $(t\star t^{\prime})\star
t^{\prime\prime}=t\star(t^{\prime}\star
t^{\prime\prime})\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}t,t^{\prime},t^{\prime\prime}\in\mathcal{T}}\right].$ (9)

One usually also takes the initial time $0$ to be the identity element under
this binary operation,

$0\star t=t\star 0=t\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}t\in\mathcal{T}}\right],$ (10)

in which case $\mathcal{T}$ becomes a _monoid_ , meaning a semigroup with an
identity element. If, furthermore, every time $t$ has an inverse $t^{\prime}$
such that $t\star t^{\prime}=t^{\prime}\star t=0$, then $\mathcal{T}$ becomes
a _group_.

###  II.2 Divisible Dynamical Systems

In the most general case, a generalized dynamical system will not provide a
way to evolve a system from a _non-initial_ time $t^{\prime}\neq 0$ to another
time $t$. A generalized dynamical system will also generically lack any means
of ‘dividing up’ the evolution from $0$ to $t\neq 0$ into well-defined forms
of evolution over intermediate time durations between $0$ and $t$ (even
assuming that the set of times $\mathcal{T}$ has a notion of ordering). To
make contact with the kinds of dynamical systems considered more widely in the
research literature, it will therefore be necessary to introduce a somewhat
less general class of mathematical structures.

This paper will define a _divisible dynamical system_ to be a tuple of the
form

$\mathopen{}\mathclose{{}\left(\mathcal{X},\mathcal{T},g}\right).$ (11)

  * •

Here $\mathcal{X}$ is a state space and $\mathcal{T}$ is a set of times
forming a monoid, whose identity element $0$ plays the role of an initial
time, as usual.

  * •

The symbol $g$ denotes a map

$g:\mathcal{X}\times\mathcal{T}^{2}\to\mathcal{X}$ (12)

that will be called the divisible dynamical system’s _transition map_. This
transition map $g$ takes as inputs any state $i$ and any pair of times
$t,t^{\prime}$, and outputs a state
$g\mathopen{}\mathclose{{}\left(i,\mathopen{}\mathclose{{}\left(t,t^{\prime}}\right)}\right)\in\mathcal{X}$:

$\displaystyle i,t,t^{\prime}$ $\displaystyle\mapsto
g\mathopen{}\mathclose{{}\left(i,\mathopen{}\mathclose{{}\left(t,t^{\prime}}\right)}\right)\in\mathcal{X}$
(13) $\displaystyle\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}i\in\mathcal{X},\,t,t^{\prime}\in\mathcal{T}}\right].$

The transition map $g$ here should be understood as describing the evolution
or transition of the state $i$ at the time $t^{\prime}$ to the state
$g\mathopen{}\mathclose{{}\left(i,\mathopen{}\mathclose{{}\left(t,t^{\prime}}\right)}\right)$
at the time $t$.

  * •

Fixing two times $t,t^{\prime}$ turns $g$ into a _time-dependent transition
map_

$g_{t\leftarrow t^{\prime}}:\mathcal{X}\to\mathcal{X}$ (14)

defined by

$\displaystyle i$ $\displaystyle\mapsto g_{t\leftarrow
t^{\prime}}\mathopen{}\mathclose{{}\left(i}\right)\equiv
g\mathopen{}\mathclose{{}\left(i,\mathopen{}\mathclose{{}\left(t,t^{\prime}}\right)}\right)$
(15) $\displaystyle\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}i\in\mathcal{X}}\right].$

This time-dependent transition map will be required to trivialize to the
identity map $\mathrm{id}_{\mathcal{X}}$ on $\mathcal{X}$ when $t^{\prime}=t$,

$\displaystyle g_{t\leftarrow t}$
$\displaystyle=\mathrm{id}_{\mathcal{X}}\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }t\in\mathcal{T}}\right],$ (16)
$\displaystyle\negthickspace\negthickspace\negthickspace\negthickspace\textrm{or}\quad
g_{t\leftarrow
t}\mathopen{}\mathclose{{}\left(i}\right)=i\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }i\in\mathcal{X},\,t\in\mathcal{T}}\right],$

as well as satisfy the _divisibility condition_

$\displaystyle g_{t\leftarrow t^{\prime\prime}}$ $\displaystyle=g_{t\leftarrow
t^{\prime}}\circ g_{t^{\prime}\leftarrow t^{\prime\prime}}$ (17)
$\displaystyle\negthickspace\negthickspace\negthickspace\mathopen{}\mathclose{{}\left[\textrm{for
all }t,t^{\prime},t^{\prime\prime}\in\mathcal{T}}\right],$

where $\circ$ denotes function composition. The divisibility condition means
that the time-dependent transition map $g_{t\leftarrow t^{\prime\prime}}$
factorizes or ‘divides up’ into a part $g_{t^{\prime}\leftarrow
t^{\prime\prime}}$ that carries out the evolution from $t^{\prime\prime}$ to
$t^{\prime}$, followed by a part $g_{t\leftarrow t^{\prime}}$ that carries out
the evolution from $t^{\prime}$ to $t$.

  * •

Choosing the time $t^{\prime}$ in the time-dependent transition map
$g_{t\leftarrow t^{\prime}}$ to be the initial time $0$ naturally defines a
time-dependent dynamical map (4),

$f_{t}\equiv g_{t\leftarrow 0}\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }t\in\mathcal{T}}\right],$ (18)

which then also defines an overall dynamical map
$f:\mathcal{X}\times\mathcal{T}\to\mathcal{X}$ according to
$f\mathopen{}\mathclose{{}\left(i,t}\right)\equiv
f_{t}\mathopen{}\mathclose{{}\left(i}\right)$. The trivialization condition
$g_{0\leftarrow 0}=\mathrm{id}_{\mathcal{X}}$ from (16) ensures that $f_{t}$
satisfies the corresponding trivialization condition
$f_{0}=\mathrm{id}_{\mathcal{X}}$ in (6). It follows that every divisible
dynamical system is, in particular, a generalized dynamical system, one that
includes the additional structure that corresponds to having a transition map
$g$.

  * •

Meanwhile, setting $t^{\prime\prime}=0$ in the divisibility condition (17) on
the transition map $g$ yields the subsidiary divisibility condition

$f_{t}=g_{t\leftarrow t^{\prime}}\circ
f_{t^{\prime}}\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}t,t^{\prime}\in\mathcal{T}}\right],$ (19)

which means that the time-dependent dynamical map $f_{t}$ ‘divides up’ into a
part $f_{t^{\prime}}$ that carries out the evolution from the initial time $0$
to $t^{\prime}$, followed by a part $g_{t\leftarrow t^{\prime}}$ that carries
out the evolution from $t^{\prime}$ to $t$.

  * •

It also follows from the subsidiary divisibility condition (19), together with
the trivialization condition $f_{0}=\mathrm{id}_{\mathcal{X}}$ in (6), that
the time-dependent dynamical map $f_{t}$ has a well-defined inverse
$f_{t}^{-1}$ given by

$f_{t}^{-1}=g_{0\leftarrow t}\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }t\in\mathcal{T}}\right].$ (20)

That is, a divisible dynamical system’s time-dependent dynamical maps $f_{t}$
are invertible.222If the set of times $\mathcal{T}$ has a well-defined
ordering relation, and if one modifies the definition of a divisible dynamical
system $\mathopen{}\mathclose{{}\left(\mathcal{X},\mathcal{T},g}\right)$ by
restricting the transition map (12) so that $g_{t\leftarrow t^{\prime}}$ is
only defined when $t^{\prime}$ comes before $t$ according to that ordering
relation, then the arguments leading to (20) will break down. In that case,
the time-dependent dynamical maps $f_{t}$ will not necessarily have inverses.

###  II.3 Markovian Dynamical Systems

A divisible dynamical system will be called _memoryless_ , or _Markovian_ , if
it has the special property

$g_{t\star t^{\prime}\leftarrow t^{\prime}}=g_{t\leftarrow
0}\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}t,t^{\prime}\in\mathcal{T}}\right].$ (21)

Equivalently, using $f_{t}\equiv g_{t\leftarrow 0}$ from (18), one can write
this special property as

$g_{t\star t^{\prime}\leftarrow
t^{\prime}}=f_{t}\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}t,t^{\prime}\in\mathcal{T}}\right].$ (22)

It then follows immediately from the subsidiary divisibility condition (19)
that the dynamical map $f:\mathcal{X}\times\mathcal{T}\to\mathcal{X}$ gives a
semigroup action of the set of times $\mathcal{T}$ on the state space
$\mathcal{X}$, in the sense that the semigroup operation $\star$ on
$\mathcal{T}$ is mapped to function composition:

$f_{t\star t^{\prime}}=f_{t}\circ
f_{t^{\prime}}\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}t,t^{\prime}\in\mathcal{T}}\right].$ (23)

This equation is an example of _Markovianity_ , or a _Markov property_.
Roughly speaking, a Markov property implies that time evolution depends only
on duration, and not on history.

Notice that the Markov property (23) is phrased directly in terms of a
dynamical map $f$, so it can be imposed on a generalized dynamical system
$\mathopen{}\mathclose{{}\left(\mathcal{X},\mathcal{T},f}\right)$, without any
need to invoke a transition map $g$. In that case, it also follows that there
is no longer a meaningful distinction between a generalized dynamical system
$\mathopen{}\mathclose{{}\left(\mathcal{X},\mathcal{T},f}\right)$ and a
divisible dynamical system
$\mathopen{}\mathclose{{}\left(\mathcal{X},\mathcal{T},g}\right)$. As such, if
the Markov property (23) is imposed, then the resulting mathematical structure
will simply be called a _Markovian dynamical system_.

What most references call a ‘dynamical system’ corresponds to what this paper
would call a Markovian dynamical system. That is, a dynamical system, without
any further qualifiers, is a tuple
$\mathopen{}\mathclose{{}\left(\mathcal{X},\mathcal{T},f}\right)$ for which
$\mathcal{X}$ is a state space, $\mathcal{T}$ is a set of times forming a
monoid, and $f:\mathcal{X}\times\mathcal{T}\to\mathcal{X}$ is a dynamical map
satisfying the Markov property (23). It follows that a generalized dynamical
system (1), as befits its name, really is a generalization of the kinds of
dynamical systems that are usually considered in textbooks, in the research
literature, and in many applications.

In the most general cases, however, generalized dynamical systems will not
satisfy the Markov property (23). Nor will it necessarily be possible to
obtain a given generalized dynamical system
$\mathopen{}\mathclose{{}\left(\mathcal{X},\mathcal{T},f}\right)$ by starting
with a divisible dynamical system
$\mathopen{}\mathclose{{}\left(\mathcal{X},\mathcal{T},g}\right)$ and then
defining a dynamical map $f$ according to $f_{t}\equiv g_{t\leftarrow 0}$ for
all times $t$, as in (18). Indeed, in light of the inversion formula (20), the
absence of any inverse time-dependent dynamical maps $f_{t}^{-1}$ would
present an immediate obstruction to deriving a dynamical map $f$ from a
transition map $g$.

A generalized dynamical system
$\mathopen{}\mathclose{{}\left(\mathcal{X},\mathcal{T},f}\right)$ that cannot
be derived from a divisible dynamical system
$\mathopen{}\mathclose{{}\left(\mathcal{X},\mathcal{T},g}\right)$ will lack
the structure needed to divide up its time evolution into intermediate
durations, and will therefore be called an _indivisible dynamical system_.

##  III Stochastic Systems

###  III.1 Generalized Stochastic Systems

This paper will be primarily concerned with a mathematical structure that
replaces the deterministic behavior of a generalized dynamical system (1) with
probabilistic, or stochastic, behavior. A _generalized stochastic system_ will
be defined to mean a tuple of the form

$\mathopen{}\mathclose{{}\left(\mathcal{C},\mathcal{T},\Gamma,p,\mathcal{A}}\right)$
(24)

that consists of the following data.

  * •

The symbol $\mathcal{C}$ will denote a set called the system’s _configuration
space_ , and the elements of $\mathcal{C}$ will be called the system’s
_(allowed) configurations_. Configurations and configuration spaces will play
an analogous role for generalized stochastic systems that states and state
spaces play for generalized dynamical systems.

  * •

The reason for switching the terminology from ‘states’ to ‘configurations’ is
conceptual. In applications of the theory of dynamical systems to physical
situations, such as in classical Hamiltonian mechanics, the notion of a
‘state’ is often taken to include rates of change or momenta in addition to
configurations, because defining a ‘state’ in that way can make it possible to
obtain deterministic laws in the form of first-order differential equations.
For a generalized stochastic system, by contrast, the system’s probabilistic
laws may mean that rates of change and momenta are not well-defined in the
absence of specifying a trajectory. In that case, the only available notion of
a ‘state’ is limited to the more rudimentary notion of a ‘configuration.’

  * •

For the purposes of this paper, $\mathcal{C}$ will be assumed to be a finite
set, with some (possibly very large) number $N$ of elements denoted by
$1,2,3,\dots,N$, and labeled abstractly by Latin letters $i,j,\dots$. By an
abuse of notation, the symbol $\mathcal{C}$ will sometimes be used to refer to
the generalized stochastic system as a whole.

  * •

The symbol $\mathcal{T}$ will again denote a _set of times_ , including a time
$0$ that will be called the system’s _initial time_.

  * •

The symbol $\Gamma$ will denote a map

$\Gamma:\mathcal{C}^{2}\times\mathcal{T}\to\mathopen{}\mathclose{{}\left[0,1}\right]\subset\mathbb{R}$
(25)

that will be called the system’s _stochastic map_ , where
$\mathopen{}\mathclose{{}\left[0,1}\right]$ denotes the closed unit interval
$0\leq x\leq 1$. Each value of this stochastic map will be labeled as

$\displaystyle\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle\equiv\Gamma\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(i,j}\right),t}\right)$
(26)
$\displaystyle\negthickspace\negthickspace\negthickspace\mathopen{}\mathclose{{}\left[\textrm{for
all }i,j\in\mathcal{C},\,t\in\mathcal{T}}\right],$

and will be called the _conditional probability_ or _transition probability_
$p\mathopen{}\mathclose{{}\left(i,t|j,0}\right)$ for the system to be in its
$i$th configuration at the time $t$, given that the system is in its $j$th
configuration at the initial time $0$:

$\displaystyle\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle\equiv
p\mathopen{}\mathclose{{}\left(i,t|j,0}\right)\in\mathopen{}\mathclose{{}\left[0,1}\right]$
(27)
$\displaystyle\negthickspace\negthickspace\negthickspace\mathopen{}\mathclose{{}\left[\textrm{for
all }i,j\in\mathcal{C},\,t\in\mathcal{T}}\right].$

  * •

The stochastic map $\Gamma$ will be required to satisfy the standard
normalization condition

$\displaystyle\sum_{i=1}^{N}\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle=\sum_{i=1}^{N}p\mathopen{}\mathclose{{}\left(i,t|j,0}\right)=1$
(28)
$\displaystyle\negthickspace\negthickspace\negthickspace\mathopen{}\mathclose{{}\left[\textrm{for
all }j\in\mathcal{C},\,t\in\mathcal{T}}\right],$

as well as the initial condition

$\Gamma_{ij}\mathopen{}\mathclose{{}\left(0}\right)\equiv\delta_{ij}\equiv\begin{cases}1&\textrm{for
}i=j,\\\ 0&\textrm{for }i\neq j,\end{cases}$ (29)

where $\delta_{ij}$ is the usual Kronecker delta.

  * •

The symbol $p$ will denote a map

$p:\mathcal{C}\times\mathcal{T}\to\mathopen{}\mathclose{{}\left[0,1}\right]\subset\mathbb{R}$
(30)

that will be called the system’s _probability distribution_. Each value of
this map will be labeled as

$\displaystyle p_{i}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle\equiv
p\mathopen{}\mathclose{{}\left(i,t}\right)\in\mathopen{}\mathclose{{}\left[0,1}\right]$
(31)
$\displaystyle\negthickspace\negthickspace\negthickspace\mathopen{}\mathclose{{}\left[\textrm{for
all }i\in\mathcal{C},\,t\in\mathcal{T}}\right],$

and will be called the _standalone probability_ for the system to be in its
$i$th configuration at the time $t$.

  * •

Only the standalone probabilities
$p_{1}\mathopen{}\mathclose{{}\left(0}\right),\dots,p_{N}\mathopen{}\mathclose{{}\left(0}\right)$
at the initial time $0$ will be taken to be freely adjustable, subject to the
standard normalization condition

$\sum_{j=1}^{N}p_{j}\mathopen{}\mathclose{{}\left(0}\right)=1.$ (32)

The standalone probabilities at every other time $t$ will be assumed to be
defined by the following _Bayesian marginalization condition_ :

$\displaystyle p_{i}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle\equiv\sum_{j=1}^{N}\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)p_{j}\mathopen{}\mathclose{{}\left(0}\right)$
(33)
$\displaystyle=\sum_{j=1}^{N}p\mathopen{}\mathclose{{}\left(i,t|j,0}\right)p_{j}\mathopen{}\mathclose{{}\left(0}\right)$
$\displaystyle\negthickspace\negthickspace\negthickspace\mathopen{}\mathclose{{}\left[\textrm{for
all }i\in\mathcal{C},\,t\in\mathcal{T}}\right].$

The normalization condition (28) on the stochastic map $\Gamma$ and the
normalization condition (32) on the standalone probabilities at the initial
time $0$ then together ensure that the probability distribution $p$ satisfies
the standard normalization condition more generally:

$\sum_{i=1}^{N}p_{i}\mathopen{}\mathclose{{}\left(t}\right)=1\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }t\in\mathcal{T}}\right].$ (34)

  * •

Note that the definition of the stochastic map $\Gamma$ is independent of the
choice of standalone probabilities
$p_{1}\mathopen{}\mathclose{{}\left(0}\right),\dots,p_{N}\mathopen{}\mathclose{{}\left(0}\right)$
at the initial time $0$. That is, $\Gamma$ can be freely adjusted
independently of those initial standalone probabilities. Importantly, the
Bayesian marginalization condition (33) therefore defines a _linear
relationship_ between the standalone probabilities
$p_{1}\mathopen{}\mathclose{{}\left(0}\right),\dots,p_{N}\mathopen{}\mathclose{{}\left(0}\right)$
at the initial time $0$ and the standalone probabilities
$p_{1}\mathopen{}\mathclose{{}\left(t}\right),\dots,p_{N}\mathopen{}\mathclose{{}\left(t}\right)$
at any other time $t$. In the work ahead, it will be argued that this linear
relationship is ultimately responsible for the linearity of time evolution in
quantum theory.

  * •

The stochastic map $\Gamma$ will _not_ be assumed to have anything like a
Markov or divisibility property. More precisely, no assumption will be made
that given any pair of times $t,t^{\prime}$, there will exist a time
$t^{\prime\prime}$ such that the following Markov property holds:

$\displaystyle\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle=\sum_{k=1}^{N}\Gamma_{ik}\mathopen{}\mathclose{{}\left(t^{\prime\prime}}\right)\Gamma_{kj}\mathopen{}\mathclose{{}\left(t^{\prime}}\right)$
(35)
$\displaystyle\negthickspace\negthickspace\negthickspace\mathopen{}\mathclose{{}\left[\textrm{for
all }i,j\in\mathcal{C}}\right].$

Nor will it be assumed that $\Gamma$ satisfies even the _weaker_ divisibility
condition that for any pair of times $t,t^{\prime}$, there exists a set of
real-valued, non-negative quantities $X_{ik}\geq 0$ for which333Note that if
the quantities $\Gamma_{kj}\mathopen{}\mathclose{{}\left(t^{\prime}}\right)$
are regarded as forming an $N\times N$ stochastic matrix, a term to be defined
shortly, then the inverse matrix, if it exists, will always have negative
entries unless both matrices are permutation matrices. It follows that one
cannot safely define the non-negative quantities $X_{ij}$ by invoking inverse
matrices.

$\displaystyle\sum_{i=1}^{N}X_{ij}$ $\displaystyle=1\textrm{\quad
and}\quad\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)=\sum_{k=1}^{N}X_{ik}\Gamma_{kj}\mathopen{}\mathclose{{}\left(t^{\prime}}\right)$
(36) $\displaystyle\qquad\mathopen{}\mathclose{{}\left[\textrm{for all
}i,j\in\mathcal{C}}\right].$

That is, a generalized stochastic system will generically be _non-Markovian_
and _indivisible_ , in a sense that parallels the notion of indivisibility for
a generalized dynamical system, as introduced earlier in this paper.

  * •

This notion of non-Markovianity represents a distinct way to generalize the
Markovian case, as compared with most forms of non-Markovianity described in
textbooks and in the research literature. According to those more traditional
forms of non-Markovianity, one assumes that the set of times $\mathcal{T}$ has
an ordering relation, and one further assumes the existence of higher-order
conditional probabilities
$p\mathopen{}\mathclose{{}\left(i,t|j_{1},t_{1};j_{2},t_{2};\dots}\right)$
that are conditioned on arbitrarily many times $t_{1},t_{2},\dots$. From that
more traditional standpoint, the system is Markovian if and only if the latest
conditioning time $t_{1}$ always screens off all the earlier conditioning
times $t_{2},\dots$, so that
$p\mathopen{}\mathclose{{}\left(i,t|j_{1},t_{1};j_{2},t_{2};\dots}\right)=p\mathopen{}\mathclose{{}\left(i,t|j_{1},t_{1}}\right)$.
The definition of a generalized stochastic system presented in this paper does
not assume the existence of such higher-order conditional probabilities in the
first place.

  * •

The symbol $\mathcal{A}$ will denote a commutative algebra of maps of the form

$A:\mathcal{C}\times\mathcal{T}\to\mathbb{R}$ (37)

under the usual rules of function arithmetic, and will be called the system’s
_algebra of random variables_. The individual values of each _random variable_
$A\in\mathcal{A}$ will be labeled as

$a_{i}\mathopen{}\mathclose{{}\left(t}\right)\equiv
A\mathopen{}\mathclose{{}\left(i,t}\right)\in\mathbb{R}\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }i\in\mathcal{C},\,t\in\mathcal{T}}\right].$ (38)

Each such value $a_{i}\mathopen{}\mathclose{{}\left(t}\right)$ will be called
the _magnitude_ of the random variable $A$ when the system is in its $i$th
configuration at the time $t$. No assumption will be made here that these
magnitudes are all distinct, even at any fixed time $t$.

  * •

For the purposes of this paper, the algebra of random variables $\mathcal{A}$
will always be taken to be maximal, in the sense of containing every well-
defined map of the form (37).

  * •

Fixing the time $t$, the _expectation value_
$\mathopen{}\mathclose{{}\left\langle
A\mathopen{}\mathclose{{}\left(t}\right)}\right\rangle$ of a random variable
$A$ will denote its statistical average according to the probability
distribution $p$ at the time $t$:

$\mathopen{}\mathclose{{}\left\langle
A\mathopen{}\mathclose{{}\left(t}\right)}\right\rangle\equiv\sum_{i=1}^{N}a_{i}\mathopen{}\mathclose{{}\left(t}\right)p_{i}\mathopen{}\mathclose{{}\left(t}\right).$
(39)

###  III.2 Ingredients from Linear Algebra

Given a generalized stochastic system
$\mathopen{}\mathclose{{}\left(\mathcal{C},\mathcal{T},\Gamma,p,\mathcal{A}}\right)$,
with $\mathcal{C}$ a configuration space of finite integer size $N$, it will
be convenient to introduce some formalism from linear algebra.

  * •

Fixing the time $t$, let $p\mathopen{}\mathclose{{}\left(t}\right)$, called
the system’s _(time-dependent) probability vector_ , denote the $N\times 1$
column vector whose $i$th entry is the $i$th standalone probability
$p_{i}\mathopen{}\mathclose{{}\left(t}\right)$:

$p\mathopen{}\mathclose{{}\left(t}\right)\equiv\begin{pmatrix}p_{1}\mathopen{}\mathclose{{}\left(t}\right)\\\
\vdots\\\ p_{N}\mathopen{}\mathclose{{}\left(t}\right)\end{pmatrix}.$ (40)

  * •

Again fixing the time $t$, let
$\Gamma\mathopen{}\mathclose{{}\left(t}\right)$, called the system’s _(time-
dependent) transition matrix_ , denote the $N\times N$ matrix for which the
entry in the $i$th row, $j$th column is the transition probability
$\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)\equiv
p\mathopen{}\mathclose{{}\left(i,t|j,0}\right)$:

$\displaystyle\Gamma\mathopen{}\mathclose{{}\left(t}\right)\equiv\begin{pmatrix}\Gamma_{11}\mathopen{}\mathclose{{}\left(t}\right)&\Gamma_{12}\mathopen{}\mathclose{{}\left(t}\right)\\\
\Gamma_{21}\mathopen{}\mathclose{{}\left(t}\right)&\ddots\\\
&&\Gamma_{NN}\mathopen{}\mathclose{{}\left(t}\right)\end{pmatrix}$
$\displaystyle=\begin{pmatrix}p\mathopen{}\mathclose{{}\left(1,t|1,0}\right)&p\mathopen{}\mathclose{{}\left(1,t|2,0}\right)\\\
p\mathopen{}\mathclose{{}\left(2,t|1,0}\right)&\ddots\\\
&&p\mathopen{}\mathclose{{}\left(N,t|N,0}\right)\end{pmatrix}.$ (41)

It follows that for any time $t$, the transition matrix
$\Gamma\mathopen{}\mathclose{{}\left(t}\right)$ is a _(column) stochastic
matrix_ in the mathematical sense, meaning that its entries are all non-
negative real numbers,

$\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)\geq
0\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}i,j\in\mathcal{C},\,t\in\mathcal{T}}\right],$ (42)

and that its columns each sum to $1$, as required in (28). The initial
condition (29) on the stochastic map $\Gamma$ then becomes the statement that
the transition matrix $\Gamma\mathopen{}\mathclose{{}\left(0}\right)$ at the
initial time $0$ is just the $N\times N$ identity matrix $\mymathbb{1}$:

$\Gamma\mathopen{}\mathclose{{}\left(0}\right)=\mymathbb{1}\equiv\begin{pmatrix}1&0\\\
0&\ddots\\\ &&1\end{pmatrix}.$ (43)

Observe that the Bayesian marginalization condition (33) naturally takes the
form of matrix multiplication:

$p\mathopen{}\mathclose{{}\left(t}\right)=\Gamma\mathopen{}\mathclose{{}\left(t}\right)p\mathopen{}\mathclose{{}\left(0}\right)\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }t\in\mathcal{T}}\right].$ (44)

###  III.3 Connections with Other Constructions

There is a definite sense in which a generalized stochastic system
$\mathopen{}\mathclose{{}\left(\mathcal{C},\mathcal{T},\Gamma,p,\mathcal{A}}\right)$,
as introduced in (24), provides a probabilistic extension of a generalized
dynamical system
$\mathopen{}\mathclose{{}\left(\mathcal{X},\mathcal{T},f}\right)$, as
introduced in (1). Indeed, if the stochastic map
$\Gamma:\mathcal{C}^{2}\times\mathcal{T}\to\mathopen{}\mathclose{{}\left[0,1}\right]$
from (25) outputs only the trivial probabilities $1$ and $0$, then $\Gamma$ is
effectively deterministic. In that case, one can naturally set
$\mathcal{X}\equiv\mathcal{C}$ and define a dynamical map
$f:\mathcal{X}\times\mathcal{T}\to\mathcal{X}$ according to

$\displaystyle f\mathopen{}\mathclose{{}\left(j,t}\right)$
$\displaystyle=i\quad\textrm{if and only
if}\quad\Gamma\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(i,j}\right),t}\right)=1$
(45) $\displaystyle\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}i,j\in\mathcal{X},\,t\in\mathcal{T}}\right].$

A generalized stochastic system also naturally generalizes a mathematical
structure called a _stochastic dynamical system_ , or _random dynamical
system_ [12, 13].

  * •

The definition of a random dynamical system starts with what this paper would
call a generalized dynamical system
$\mathopen{}\mathclose{{}\left(\mathcal{X},\mathcal{T},f}\right)$, with
$\mathcal{T}$ a monoid, and replaces the single dynamical map $f$ with a
probabilistic family or ensemble of dynamical maps
$\\{f_{\omega}\,|\,\omega\in\Omega\\}$ that are indexed by a point $\omega$ in
some sample space $\Omega$. That is, a randomly sampled point $\omega$ in
$\Omega$ picks out an entire dynamical map $f_{\omega}$, as a whole.

  * •

For any randomly sampled point $\omega$ and any fixed time $t$, one can define
a map

$f_{\omega,t}:\mathcal{X}\to\mathcal{X}$ (46)

by

$i\mapsto f_{\omega,t}\mathopen{}\mathclose{{}\left(i}\right)\equiv
f_{\omega}\mathopen{}\mathclose{{}\left(i,t}\right)\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }i\in\mathcal{X}}\right].$ (47)

A random dynamical system is then assumed to satisfy an initial condition that
generalizes (6),

$f_{\omega,0}=\mathrm{id}_{\mathcal{X}}\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }\omega\in\Omega}\right],$ (48)

as well as a Markov property that generalizes (23),

$\displaystyle f_{\omega^{\prime},t\star t^{\prime}}$
$\displaystyle=f_{\theta\mathopen{}\mathclose{{}\left(\omega^{\prime},t^{\prime}}\right),t}\circ
f_{\omega^{\prime},t^{\prime}}$ (49)
$\displaystyle\negthickspace\negthickspace\negthickspace\mathopen{}\mathclose{{}\left[\textrm{for
all }\omega^{\prime}\in\Omega,\,t,t^{\prime}\in\mathcal{T}}\right].$

Here $\theta:\Omega\times\mathcal{T}\to\Omega$ is a map that is part of the
definition of the random dynamical system, and specifies a deterministic rule
for updating the sample point $\omega^{\prime}$ and the dynamical map
$f_{\omega^{\prime}}$ to accommodate replacing the original initial time $0$
with the effectively new initial time $t^{\prime}$. That is, $\theta$ is
necessary to account for any (purportedly deterministic) evolution in the
underlying source of randomness itself.

  * •

It follows from the foregoing definitions that the conditional probability
$p\mathopen{}\mathclose{{}\left(i,t|j,0}\right)$ for the system to be in its
$i$th state at the time $t$, given that the system is in its $j$th state at
the initial time $0$, is obtained by adding up the probabilities for all the
points $\omega$ in the sample space $\Omega$ whose corresponding dynamical
maps $f_{\omega}$ take the state $j$ at the initial time $0$ and yield the
state $i$ at the final time $t$:

$\displaystyle p\mathopen{}\mathclose{{}\left(i,t|j,0}\right)$
$\displaystyle=\textrm{probability}\mathopen{}\mathclose{{}\left(\\{\omega\in\Omega\,|\,f_{\omega,t}\mathopen{}\mathclose{{}\left(j}\right)=i\\}}\right)$
(50)
$\displaystyle\negthickspace\negthickspace\negthickspace\mathopen{}\mathclose{{}\left[\textrm{for
all }i,j\in\mathcal{X},\,t\in\mathcal{T}}\right].$

  * •

Notice that a random dynamical system assigns a well-defined probability to
any set of dynamical maps $\\{f_{\omega}\,|\,\omega\in\Omega^{\prime}\\}$ that
corresponds to a measurable subset $\Omega^{\prime}$ of the sample space
$\Omega$. Thus, after fixing a state $j$ at the initial time $0$, a random
dynamical system assigns a well-defined probability to any set of trajectories
of the form
$\\{\\{f_{\omega,t}\mathopen{}\mathclose{{}\left(j}\right)\,|\,t\in\mathcal{T}\\}\,|\,\omega\in\Omega^{\prime}\\}$.
(The nested subset notation here is just intended to make clear that this
expression refers to an indexed set of indexed sets.) In particular, by
generalizing the right-hand side of (50) to impose more conditions on the
system’s dynamical maps, one can define conditional probabilities
$p\mathopen{}\mathclose{{}\left(i_{1},t_{1};i_{2},t_{2};\dots|j,0}\right)$
that involve arbitrarily many times. Random dynamical systems therefore
contain far more information than the generalized stochastic systems defined
in this paper, meaning that a generalized stochastic system indeed represents
a less constrained, more general kind of mathematical structure.

In a sense, the absolutely _most_ _general_ kind of stochastic mathematical
structure is a _stochastic process_ , which essentially requires only the
specification of a state space $\mathcal{X}$, a set of times $\mathcal{T}$, an
initial probability distribution $p$, and a set of one or more time-dependent
random variables $\mathcal{A}$. (See [14, 15, 16, 17] for textbook
treatments.) Importantly, the definition of stochastic process lacks the
specification of a _dynamical law_. By requiring a dynamical law in the form
of a stochastic map $\Gamma$, a generalized stochastic system
$\mathopen{}\mathclose{{}\left(\mathcal{C},\mathcal{T},\Gamma,p,\mathcal{A}}\right)$
is not quite as general as a stochastic process, but will still be general
enough to encompass a large class of physical and mathematical models.

In particular, one can regard any _Markov chain_ as a special case of a
generalized stochastic system. The starting place is to assume that the set of
times $\mathcal{T}$ is isomorphic to the integers,
$\mathcal{T}\cong\mathbb{Z}$, with each time $t=n\,\delta t$ an integer number
$n\in\mathbb{Z}$ of steps of some fixed, elementary time scale $\delta t$. One
then further assumes that for each integer $n$, the time-dependent transition
matrix $\Gamma\mathopen{}\mathclose{{}\left(n\,\delta t}\right)$ originally
defined in (41) can be expressed as the $n$th power of the transition matrix
$\Gamma\mathopen{}\mathclose{{}\left(\delta t}\right)$ that implements the
evolution for just the first time step $\delta t$:

$\Gamma\mathopen{}\mathclose{{}\left(n\,\delta
t}\right)=\mathopen{}\mathclose{{}\left[\Gamma\mathopen{}\mathclose{{}\left(\delta
t}\right)}\right]^{n}\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}n\in\mathbb{Z}}\right].$ (51)

More broadly, a generalized stochastic system can therefore be understood as a
kind of non-Markovian generalization of a Markov chain.

As this paper will also show, the class of generalized stochastic systems
essentially includes all quantum systems as well.

###  III.4 Composite Systems and Subsystems

Introducing a notation of primes and tildes now to distinguish between
specific generalized stochastic systems, a generalized stochastic system
$\begin{pmatrix}\tilde{\mathcal{C}},\tilde{\mathcal{T}},\tilde{\Gamma},\tilde{p},\tilde{\mathcal{A}}\end{pmatrix}$
will be called a _composite system_ if its configuration space
$\tilde{\mathcal{C}}$ is naturally expressible as a nontrivial Cartesian
product of two sets $\mathcal{C}$ and $\mathcal{C}^{\prime}$:

$\tilde{\mathcal{C}}=\mathcal{C}\times\mathcal{C}^{\prime}.$ (52)

A composite system has the following salient features.

  * •

Letting $N$ denote the size of $\mathcal{C}$, and letting $N^{\prime}$ denote
the size of $\mathcal{C}^{\prime}$, the composite system’s configuration space
$\tilde{\mathcal{C}}$ has size $\tilde{N}\equiv NN^{\prime}$.

  * •

Labeling the elements of $\mathcal{C}$ by unprimed Latin letters $i,j,\dots$,
and labeling the elements of $\mathcal{C}^{\prime}$ by primed Latin letters
$i^{\prime},j^{\prime},\dots$, the composite system’s stochastic map
$\tilde{\Gamma}:\tilde{\mathcal{C}}^{2}\times\tilde{\mathcal{T}}\to\mathopen{}\mathclose{{}\left[0,1}\right]$
has individual values that will be denoted by

$\displaystyle\tilde{\Gamma}_{ii^{\prime},jj^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle\equiv\tilde{\Gamma}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(i,i^{\prime}}\right),\mathopen{}\mathclose{{}\left(j,j^{\prime}}\right)}\right),t}\right)$
$\displaystyle\equiv\tilde{p}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(i,i^{\prime}}\right),t|\mathopen{}\mathclose{{}\left(j,j^{\prime}}\right),0}\right)$
(53) $\displaystyle\negthickspace\negthickspace\negthickspace[\textrm{for all
}i,j\in\mathcal{C},\,i^{\prime},j^{\prime}\in\mathcal{C}^{\prime},\,t\in\tilde{\mathcal{T}}].$

The normalization condition (28) on the stochastic map $\tilde{\Gamma}$ takes
the form

$\displaystyle\sum_{i=1}^{N}\sum_{i^{\prime}=1}^{N^{\prime}}\tilde{\Gamma}_{ii^{\prime},jj^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)=1$
(54) $\displaystyle\quad[\textrm{for all
}j\in\mathcal{C},\,j^{\prime}\in\mathcal{C}^{\prime},\,t\in\tilde{\mathcal{T}}].$

  * •

The composite system’s probability distribution
$\tilde{p}:\tilde{\mathcal{C}}\times\tilde{\mathcal{T}}\to\mathopen{}\mathclose{{}\left[0,1}\right]$
has individual values at any fixed time $t$ that will be denoted by

$\displaystyle\tilde{p}_{ii^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle\equiv
p\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(i,i^{\prime}}\right),t}\right)$
(55) $\displaystyle\negthickspace\negthickspace\negthickspace[\textrm{for all
}i\in\mathcal{C},\,i^{\prime}\in\mathcal{C}^{\prime},\,t\in\tilde{\mathcal{T}}].$

The normalization condition (34) on the probability distribution $\tilde{p}$
takes the form

$\sum_{i=1}^{N}\sum_{i^{\prime}=1}^{N^{\prime}}\tilde{p}_{ii^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)=1\quad[\textrm{for
all }t\in\tilde{\mathcal{T}}],$ (56)

and the Bayesian marginalization condition (33) becomes the definition

$\displaystyle\tilde{p}_{ii^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle\equiv\sum_{j=1}^{N}\sum_{j^{\prime}=1}^{N^{\prime}}\tilde{\Gamma}_{ii^{\prime},jj^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)\tilde{p}_{jj^{\prime}}\mathopen{}\mathclose{{}\left(0}\right)$
(57) $\displaystyle\negmedspace\negmedspace\negmedspace[\textrm{for all
}i\in\mathcal{C},\,i^{\prime}\in\mathcal{C}^{\prime},\,t\in\tilde{\mathcal{T}}].$

Given a composite system
$(\tilde{\mathcal{C}},\tilde{\mathcal{T}},\tilde{\Gamma},\tilde{p},\tilde{\mathcal{A}})$
with configuration space
$\tilde{\mathcal{C}}=\mathcal{C}\times\mathcal{C}^{\prime}$, as in (52), the
composite system’s set of times $\tilde{\mathcal{T}}$ trivially defines two
sets of times $\mathcal{T}$ and $\mathcal{T}^{\prime}$ according to

$\mathcal{T}\equiv\mathcal{T}^{\prime}\equiv\tilde{\mathcal{T}}.$ (58)

Meanwhile, the composite system’s probability distribution $\tilde{p}$ defines
two probability distributions
$p:\mathcal{C}\times\mathcal{T}\to\mathopen{}\mathclose{{}\left[0,1}\right]$
and
$p^{\prime}:\mathcal{C}^{\prime}\times\mathcal{T}^{\prime}\to\mathopen{}\mathclose{{}\left[0,1}\right]$
according to the respective marginalization formulas

$\displaystyle p_{i}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle\equiv\sum_{i^{\prime}=1}^{N^{\prime}}\tilde{p}_{ii^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)\quad[\textrm{for
all }i\in\mathcal{C},\,t\in\mathcal{T}],$ (59) $\displaystyle
p_{i^{\prime}}^{\prime}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle\equiv\sum_{i=1}^{N}\tilde{p}_{ii^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)\quad[\textrm{for
all }i^{\prime}\in\mathcal{C}^{\prime},\,t\in\mathcal{T}^{\prime}].$ (60)

These two probability distributions $p$ and $p^{\prime}$ do not necessarily
turn $\mathcal{C}$ and $\mathcal{C}^{\prime}$ into generalized stochastic
systems of their own, in the sense of (24), due to the generic lack of well-
defined stochastic maps
$\Gamma:\mathcal{C}^{2}\times\mathcal{T}\to\mathopen{}\mathclose{{}\left[0,1}\right]$
and $\Gamma^{\prime}:\mathcal{C}^{\prime
2}\times\mathcal{T}^{\prime}\to\mathopen{}\mathclose{{}\left[0,1}\right]$.
Indeed, in place of the Bayesian marginalization condition (33), one instead
has the relations

$\displaystyle p_{i}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle\equiv\sum_{i^{\prime}=1}^{N^{\prime}}\sum_{j=1}^{N}\sum_{j^{\prime}=1}^{N^{\prime}}\tilde{\Gamma}_{ii^{\prime},jj^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)\tilde{p}_{jj^{\prime}}\mathopen{}\mathclose{{}\left(0}\right)$
(61) $\displaystyle\quad\qquad[\textrm{for all
}i\in\mathcal{C},\,t\in\mathcal{T}],$ $\displaystyle
p_{i^{\prime}}^{\prime}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle\equiv\sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{j^{\prime}=1}^{N^{\prime}}\tilde{\Gamma}_{ii^{\prime},jj^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)\tilde{p}_{jj^{\prime}}\mathopen{}\mathclose{{}\left(0}\right)$
(62) $\displaystyle\quad\qquad[\textrm{for all
}i^{\prime}\in\mathcal{C}^{\prime},\,t\in\mathcal{T}^{\prime}].$

The two sets $\mathcal{C}$ and $\mathcal{C}^{\prime}$ will be called the
configuration spaces of _subsystems_ of the composite system
$(\tilde{\mathcal{C}},\tilde{\mathcal{T}},\tilde{\Gamma},\tilde{p},\tilde{\mathcal{A}})$.

###  III.5 Unistochastic Systems

Recall that an $N\times N$ matrix $U$ with complex-valued entries is called
_unitary_ if it satisfies

$U^{\dagger}U=UU^{\dagger}=\mymathbb{1}.$ (63)

Here $\dagger$ denotes the adjoint operation, meaning complex conjugation
combined with the transpose operation.

A stochastic matrix $X$ is called _unistochastic_ if each of its entries
$X_{ij}$ is the modulus-squared $|U_{ij}|^{2}$ of the corresponding entry
$U_{ij}$ of a unitary matrix $U$:

$X=\begin{pmatrix}X_{11}&X_{12}\\\ X_{21}&\ddots\\\
&&X_{NN}\end{pmatrix}=\begin{pmatrix}|U_{11}|^{2}&|U_{12}|^{2}\\\
|U_{21}|^{2}&\ddots\\\ &&|U_{NN}|^{2}\end{pmatrix}.$ (64)

The study of unistochastic matrices was initiated in a 1954 paper [18] by
Horn, who originally called them ‘ortho-stochastic matrices.’ Unistochastic
matrices were given their modern name in 1989 by Thompson [19, 20], and today
_orthostochastic matrices_ refer to the special case in which $U$ can be taken
to be a real orthogonal matrix, meaning a unitary matrix with entries that are
all real-valued.

Every unistochastic matrix is _doubly stochastic_ , or _bistochastic_ ,
meaning that its columns _and_ rows each sum to $1$.444Proof:
$\sum_{i}|V_{ij}|^{2}=\sum_{i}\overline{V_{ij}}V_{ij}=\mathopen{}\mathclose{{}\left[V^{\dagger}V}\right]_{jj}=1$
and
$\sum_{j}|V_{ij}|^{2}=\sum_{j}V_{ij}\overline{V_{ij}}=\mathopen{}\mathclose{{}\left[VV^{\dagger}}\right]_{ii}=1$,
where bars denote complex conjugation. QED All $2\times 2$ doubly stochastic
matrices are unistochastic,555Proof: Every $2\times 2$ doubly stochastic
matrix is of the form
$\mathopen{}\mathclose{{}\left(\begin{smallmatrix}x&1-x\\\
1-x&x\end{smallmatrix}}\right)$, where $0\leq x\leq 1$, and the matrix
$\mathopen{}\mathclose{{}\left(\begin{smallmatrix}\sqrt{x}&-\sqrt{1-x}\\\
\sqrt{1-x}&\sqrt{x}\end{smallmatrix}}\right)$ is easily seen to be unitary.
QED but this equivalence does not extend beyond the $2\times 2$ case.666For
example, the $3\times 3$ doubly stochastic matrix
$\frac{1}{2}\mathopen{}\mathclose{{}\left(\begin{smallmatrix}0&1&1\\\ 1&0&1\\\
1&1&0\end{smallmatrix}}\right)$ is provably not unistochastic. Again beyond
the $2\times 2$ case, not every unistochastic matrix is orthostochastic.777For
example, the $3\times 3$ matrix
$\frac{1}{\sqrt{3}}\mathopen{}\mathclose{{}\left(\begin{smallmatrix}1&1&1\\\
1&z&z^{2}\\\ 1&z^{2}&z\end{smallmatrix}}\right)$ is unitary for
$z=\exp\mathopen{}\mathclose{{}\left(2\pi i/3}\right)$, so the $3\times 3$
matrix $\frac{1}{3}\mathopen{}\mathclose{{}\left(\begin{smallmatrix}1&1&1\\\
1&1&1\\\ 1&1&1\end{smallmatrix}}\right)$ is unistochastic. But this
unistochastic matrix is not orthostochastic, because there does not exist a
set of three mutually orthogonal $3\times 1$ column vectors that feature only
$1$s and $-1$s as entries.

A generalized stochastic system
$\mathopen{}\mathclose{{}\left(\mathcal{C},\mathcal{T},\Gamma,p,\mathcal{A}}\right)$
whose stochastic map $\Gamma$ defines a unistochastic matrix
$\Gamma\mathopen{}\mathclose{{}\left(t}\right)$ at every time $t$ will be
called a _unistochastic system_. One of the main goals of this paper will be
to establish that the study of generalized stochastic systems essentially
reduces to the study of unistochastic systems, which will also turn out to
correspond to unitarily evolving quantum systems.

##  IV The Stochastic-Quantum Theorem

###  IV.1 Statement of the Theorem

With all these preliminaries now in place, this paper’s next major goal will
be to prove the following theorem, which is a new result.

$\boxed{\begin{subarray}{c}{\displaystyle\textbf{The Stochastic-Quantum
Theorem}}\\\ \ \\\ {\displaystyle\textrm{Every generalized stochastic
system}}\\\ {\displaystyle\textrm{can be regarded as a subsystem of a}}\\\
{\displaystyle\textrm{unistochastic system.}}\end{subarray}}$ (65)

Remarkably, to study the class of generalized stochastic systems, this theorem
implies that it suffices to restrict one’s attention to the subclass of
unistochastic systems.

###  IV.2 Corollaries of the Theorem

The proof of the stochastic-quantum theorem (65) will involve the construction
of a representation of the given generalized stochastic system in the
formalism of Hilbert spaces, and will show that every generalized stochastic
system corresponds to a unitarily evolving quantum system on a Hilbert space.
This paper will therefore establish an important new correspondence between
generalized stochastic systems and quantum systems, and thereby turn some of
the puzzling axiomatic ingredients of quantum theory—the complex numbers,888
_Time-reversal transformations_ for quantum systems are carried out by _anti-
unitary operators_ , which take the general form $VK$, where $V$ is unitary
and $K$ is an abstract operator that carries out complex-conjugation. By
definition, $K^{2}=1$ and $Ki=-iK$, where $i\equiv\sqrt{-1}$ is the imaginary
unit. One can therefore show that $i$, $K$, and $iK$ generate a Clifford
algebra called the _pseudo-quaternions_ [21]. As a consequence, there is a
sense in which quantum systems are actually defined over the pseudo-
quaternions, rather than merely over the complex numbers. Hilbert spaces,
linear-unitary time evolution, and the Born rule in particular—into the output
of a theorem.

One can also read this _stochastic-quantum correspondence_ in the other
direction, as the statement that all generalized stochastic systems can be
modeled in terms of unitarily evolving quantum systems. From this perspective,
unitarily evolving quantum systems actually represent the most general way to
model a system with stochastic dynamical laws.

###  IV.3 A Simple Example

The notion of embedding a generalized stochastic system into a unistochastic
system does not always require an elaborate construction. As a simple example,
consider a _discrete_ Markovian dynamical system
$\mathopen{}\mathclose{{}\left(\mathcal{X},\mathcal{T},f}\right)$ whose state
space $\mathcal{X}$ has some finite size $N$, whose set of times $\mathcal{T}$
is isomorphic to the integers $\mathbb{Z}$ under addition, and whose dynamical
map $f:\mathcal{X}\times\mathcal{T}\to\mathcal{X}$ satisfies the appropriate
version of the Markov property (23):

$f_{t+t^{\prime}}=f_{t}\circ
f_{t^{\prime}}\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}t,t^{\prime}\in\mathcal{T}\cong\mathbb{Z}}\right].$ (66)

Based on these definitions and assumptions, it follows from the Markov
property and the trivialization condition $f_{0}=\mathrm{id}_{\mathcal{X}}$
from (6) that the time-dependent dynamical map $f_{t}$ is invertible for any
time $t\in\mathcal{T}$, with inverse given by

$f_{t}^{-1}=f_{-t}\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}t\in\mathcal{T}\cong\mathbb{Z}}\right].$ (67)

Markovian dynamical systems of this kind provide a discretization of many of
the kinds of deterministic, time-reversible physical systems that show up in
classical physics.

Letting $\delta t$ be the elementary duration of a single time step, and
letting $n\in\mathbb{Z}$ denote an integer number of time steps, the system’s
specific state $i$ at the time $t=n\,\delta t$ can be identified as an
$N\times 1$ column vector with a $1$ in its $i$th entry and $0$s in all its
other entries. Moreover, the time-dependent dynamical map $f_{n\,\delta t}$
can be expressed as the $n$th power of a fixed _permutation matrix_ $\Sigma$,
meaning a matrix consisting of only $1$s and $0$s, with a single $1$ in each
row and in each column.

Every permutation matrix is unitary, and is also unistochastic, because
computing the modulus-squares of $1$s and $0$s gives back $1$s and $0$s.
Hence, a discrete Markovian dynamical system defined in this way is already a
unistochastic system, and therefore trivially satisfies the stochastic-quantum
theorem (65).

Interestingly, there exists an analytic interpolation of this _discrete-time_
unistochastic system that yields a corresponding _continuous-time_
unistochastic system, at the cost of introducing nontrivial stochasticity into
the intervals between the discrete time steps. This new unistochastic system
is based on a time-dependent unistochastic matrix
$\Gamma\mathopen{}\mathclose{{}\left(t}\right)$ whose entries are the modulus-
squares of the corresponding entries of the $N\times N$ unitary time-evolution
operator defined by
$U\mathopen{}\mathclose{{}\left(t}\right)\equiv\Sigma^{t/\delta t}$, and
therefore satisfies $\Gamma\mathopen{}\mathclose{{}\left(n\,\delta
t}\right)=\Sigma^{n}$ for any integer $n$. One can even go on to define an
$N\times N$ self-adjoint _Hamiltonian_ $H$ as the infinitesimal generator of
$U\mathopen{}\mathclose{{}\left(t}\right)$, so there is ultimately a
_Schrödinger equation_ underlying this system.999To define the unitary matrix
$\Sigma^{t/\delta t}$, one starts by using the fact that every permutation
matrix is also unitary to write the permutation matrix $\Sigma$ as
$V^{\dagger}DV$ for some unitary matrix $V$ and some diagonal matrix $D$,
where the entries of $D$ are the eigenvalues of $\Sigma$. Because $\Sigma$ is
unitary, its eigenvalues are all phase factors
$\exp\mathopen{}\mathclose{{}\left(i\theta_{m}}\right)$, where $m=1,\dots,N$,
and where each phase $\theta_{m}$ is a real number. Setting all the
eigenvalues to the power $t/\delta t$, so that they each take the new form
$\exp\mathopen{}\mathclose{{}\left(i\theta_{m}t/\delta t}\right)$, one ends up
with the matrix $\Sigma^{t/\delta t}\equiv V^{\dagger}D^{t/\delta t}V$, which
is still unitary and now depends analytically on the time parameter $t$. The
quantum-theoretic Hamiltonian matrix $H$ for this system then has energy
eigenvalues defined by $E_{m}\equiv-\hbar\theta_{m}/\delta t$ for
$m=1,\dots,N$, and is diagonalized by the same unitary matrix $V$ that
diagonalizes $\Sigma$.

##  V The Hilbert-Space Representation

###  V.1 The Time-Evolution Operator

To commence the proof of the stochastic-quantum theorem (65), one starts with
a given generalized stochastic system
$\mathopen{}\mathclose{{}\left(\mathcal{C},\mathcal{T},\Gamma,p,\mathcal{A}}\right)$.
The non-negativity (42) of the system’s transition probabilities,
$\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)\geq 0,$ means that each
transition probability can be written as the modulus-squared of a (non-unique)
complex number $\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)$:

$\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)=\mathopen{}\mathclose{{}\left|\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)}\right|^{2}\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }i,j\in\mathcal{C},\,t\in\mathcal{T}}\right].$ (68)

For each fixed time $t$, the complex numbers
$\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)$ collectively form their
own $N\times N$ matrix, which will be called the system’s _time-evolution
operator_ $\Theta\mathopen{}\mathclose{{}\left(t}\right)$:

$\Theta\mathopen{}\mathclose{{}\left(t}\right)\equiv\begin{pmatrix}\Theta_{11}\mathopen{}\mathclose{{}\left(t}\right)&\Theta_{12}\mathopen{}\mathclose{{}\left(t}\right)\\\
\Theta_{21}\mathopen{}\mathclose{{}\left(t}\right)&\ddots\\\
&&\Theta_{NN}\mathopen{}\mathclose{{}\left(t}\right)\end{pmatrix}.$ (69)

(As a concession to terminological conventions, the terms ‘matrix’ and
‘operator’ will be used more-or-less interchangeably in what follows.)

The normalization condition (28) on the system’s transition matrix
$\Gamma\mathopen{}\mathclose{{}\left(t}\right)$ then becomes the summation
condition

$\sum_{i=1}^{N}\mathopen{}\mathclose{{}\left|\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)}\right|^{2}=1\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }j\in\mathcal{C},\,t\in\mathcal{T}}\right],$ (70)

which can roughly be regarded as a generalization of a unitarity constraint.
In keeping with the initial condition
$\Gamma\mathopen{}\mathclose{{}\left(0}\right)=\mymathbb{1}$ from (43), the
time-evolution operator $\Theta\mathopen{}\mathclose{{}\left(0}\right)$ at the
initial time $0$ will be taken to be the $N\times N$ identity matrix
$\mymathbb{1}$:

$\Theta\mathopen{}\mathclose{{}\left(0}\right)\equiv\mymathbb{1}\equiv\begin{pmatrix}1&0\\\
0&\ddots\\\ &&1\end{pmatrix}.$ (71)

###  V.2 The Hilbert Space

The time-evolution operator $\Theta\mathopen{}\mathclose{{}\left(t}\right)$
acts on an $N$-dimensional _Hilbert space_ $\mathcal{H}$ defined as the space
$\mathbb{C}^{N}$ of $N\times 1$ column vectors with complex-valued entries,
together with the standard inner product $v^{\dagger}w$ for all
$v,w\in\mathbb{C}^{N}$:

$\mathcal{H}\equiv\mathbb{C}^{N}.$ (72)

The standard orthonormal basis $e_{1},\dots,e_{N}$ is defined by

$e_{1}\equiv\mathopen{}\mathclose{{}\left(\begin{smallmatrix}1\\\ 0\\\
\vdots\\\ 0\\\ 0\end{smallmatrix}}\right),\quad\dots,\quad
e_{N}\equiv\mathopen{}\mathclose{{}\left(\begin{smallmatrix}0\\\ 0\\\
\vdots\\\ 0\\\ 1\end{smallmatrix}}\right).$ (73)

These vectors are labeled as $e_{i}$, where each value of $i=1,\dots,N$
denotes a configuration in the system’s configuration space $\mathcal{C}$, so
this basis will be called the system’s _configuration basis_.

There exists an associated set of rank-one projectors $P_{1},\dots,P_{N}$
defined by

$P_{i}\equiv
e_{i}e_{i}^{\dagger}=\mathrm{diag}(0,\dots,0,\underset{\mathclap{\begin{subarray}{c}\uparrow\\\
i\textrm{th
entry}\end{subarray}}}{1},0,\dots,0)\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }i\in\mathcal{C}}\right].$ (74)

These _configuration projectors_ $P_{1},\dots,P_{N}$ form a _projection-valued
measure (PVM)_ [22, 23], meaning that they satisfy the conditions of _mutual
exclusivity_ ,

$P_{i}P_{j}=\delta_{ij}P_{i}\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}i,j\in\mathcal{C}}\right],$ (75)

with $\delta_{ij}$ again the usual Kronecker delta, and _completeness_ ,

$\sum_{i=1}^{N}P_{i}=\mymathbb{1},$ (76)

with $\mymathbb{1}$ again the $N\times N$ identity matrix.

###  V.3 The Dictionary

It follows from a short calculation that the relationship
$\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)=|\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)|^{2}$
in (68) can be expressed in terms of a matrix trace as

$\displaystyle\boxed{\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)=\mathrm{tr}\big{(}\Theta^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)P_{i}\Theta\mathopen{}\mathclose{{}\left(t}\right)P_{j}\big{)}}$
(77) $\displaystyle\quad\qquad\qquad\mathopen{}\mathclose{{}\left[\textrm{for
all }i,j\in\mathcal{C},\,t\in\mathcal{T}}\right].$

This ‘dictionary’ essentially translates between the theory of generalized
stochastic systems, as expressed by the left-hand side, and a corresponding
Hilbert-space representation, as expressed by the right-hand side. This
equation will turn out to form the core of a _stochastic-quantum
correspondence_ that will play a crucial role in the proof of the stochastic-
quantum theorem.

###  V.4 The Density Matrix

Inserting the dictionary (77) into the Bayesian marginalization condition (33)
yields the following equation:

$p_{i}\mathopen{}\mathclose{{}\left(t}\right)=\mathrm{tr}\mathopen{}\mathclose{{}\left(P_{i}\rho\mathopen{}\mathclose{{}\left(t}\right)}\right)\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }i\in\mathcal{C},\,t\in\mathcal{T}}\right].$ (78)

Here $\rho\mathopen{}\mathclose{{}\left(t}\right)$, which will be called the
system’s _density matrix_ , is an $N\times N$ time-dependent, self-adjoint,
unit-trace, generically non-diagonal matrix defined for any fixed time $t$ by

$\rho\mathopen{}\mathclose{{}\left(t}\right)\equiv\Theta\mathopen{}\mathclose{{}\left(t}\right)\rho\mathopen{}\mathclose{{}\left(0}\right)\Theta^{\dagger}\mathopen{}\mathclose{{}\left(t}\right),$
(79)

where its value at the initial time $0$ is the following $N\times N$ diagonal
matrix:

$\rho\mathopen{}\mathclose{{}\left(0}\right)\equiv\sum_{j=1}^{N}p_{j}\mathopen{}\mathclose{{}\left(0}\right)P_{j}\equiv\begin{pmatrix}p_{1}\mathopen{}\mathclose{{}\left(0}\right)&0\\\
0&\ddots\\\ &&p_{N}\mathopen{}\mathclose{{}\left(0}\right)\end{pmatrix}.$ (80)

Importantly, one sees from this analysis that the linearity of the Bayesian
marginalization condition (33) underlies the linearity of the relationship
(79) between the system’s density matrix
$\rho\mathopen{}\mathclose{{}\left(0}\right)$ at the initial time $0$ and its
density matrix $\rho\mathopen{}\mathclose{{}\left(t}\right)$ at other times
$t$.

For any fixed time $t$, one can similarly express the expectation value (39)
of a random variable $A$ as

$\mathopen{}\mathclose{{}\left\langle
A\mathopen{}\mathclose{{}\left(t}\right)}\right\rangle=\mathrm{tr}\mathopen{}\mathclose{{}\left(A\mathopen{}\mathclose{{}\left(t}\right)\rho\mathopen{}\mathclose{{}\left(t}\right)}\right).$
(81)

Here $A\mathopen{}\mathclose{{}\left(t}\right)$ denotes the $N\times N$
diagonal matrix

$A\mathopen{}\mathclose{{}\left(t}\right)\equiv\sum_{i=1}^{N}a_{i}\mathopen{}\mathclose{{}\left(t}\right)P_{i}\equiv\begin{pmatrix}a_{1}\mathopen{}\mathclose{{}\left(t}\right)&0\\\
0&\ddots\\\ &&a_{N}\mathopen{}\mathclose{{}\left(t}\right)\end{pmatrix}.$ (82)

Observe, in particular, that the magnitudes
$a_{1}\mathopen{}\mathclose{{}\left(t}\right),\dots,a_{N}\mathopen{}\mathclose{{}\left(t}\right)$
of the random variable $A$ become the _eigenvalues_ of the $N\times N$ matrix
$A\mathopen{}\mathclose{{}\left(t}\right)$.

Notice also that the formula for the standalone probability
$p_{i}\mathopen{}\mathclose{{}\left(t}\right)$ in (78) and the formula for the
expectation value $\mathopen{}\mathclose{{}\left\langle
A\mathopen{}\mathclose{{}\left(t}\right)}\right\rangle$ in (81) are both
special cases of the _Born rule_.

###  V.5 An Aside on ‘Classical Wave Functions’

Pausing for a moment, recall the smooth unistochastic interpolation of a
discrete Markovian dynamical system described in Subsection IV.3. For that
system, the unitary time-evolution operator
$U\mathopen{}\mathclose{{}\left(t}\right)\equiv\Sigma^{t/\delta t}$
trivializes to a permutation matrix $\Sigma^{n}$ at every integer time step
$n\,\delta t$. It follows that the system’s density matrix
$\rho\mathopen{}\mathclose{{}\left(t}\right)$, as defined in terms of its
initial value $\rho\mathopen{}\mathclose{{}\left(0}\right)$ from (79), reduces
to a diagonal matrix $\rho\mathopen{}\mathclose{{}\left(n\,\delta t}\right)$
at each integer time step. Taking the square root of each of its diagonal
entries $p_{1}\mathopen{}\mathclose{{}\left(n\,\delta
t}\right),\dots,p_{N}\mathopen{}\mathclose{{}\left(n\,\delta t}\right)$, and
allowing for arbitrary phase factors, one obtains a ‘classical wave function’
with components $\Psi_{1}\mathopen{}\mathclose{{}\left(n\,\delta
t}\right),\dots,\Psi_{N}\mathopen{}\mathclose{{}\left(n\,\delta t}\right)$
satisfying $|\Psi_{i}\mathopen{}\mathclose{{}\left(n\,\delta
t}\right)|^{2}=p_{i}\mathopen{}\mathclose{{}\left(n\,\delta t}\right)$ and
capturing precisely the same information as the diagonal density matrix
$\rho\mathopen{}\mathclose{{}\left(n\,\delta t}\right)$.

This classical wave function is the starting place for a representation of
classical deterministic physics known popularly as the _‘Koopman-von Neumann’
formulation_ , due to its superficial resemblance to work by Koopman [24] and
von Neumann [25, 26] in the 1930s. However, as pointed out explicitly in [27],
Koopman and von Neumann were actually using Hilbert spaces to represent
observables, rather than to represent probability distributions. The
formulation of classical physics in terms of ‘classical wave functions’ is
more properly due to Schönberg [28], Loinger [29], Della Riccia and Wiener
[30], and Sudarshan [31].

###  V.6 The Kraus Decomposition

Returning to the proof, and fixing the time $t$, and letting $\beta$ denote an
integer from $1$ to $N$ that will play a conceptually different role from a
configuration index, let $K_{\beta}\mathopen{}\mathclose{{}\left(t}\right)$
denote the $N\times N$ matrix whose $\beta$th column agrees with the $\beta$th
column of the time-evolution operator
$\Theta\mathopen{}\mathclose{{}\left(t}\right)$, with $0$s in all its other
entries. That is, the entry in the $i$th row, $j$th column of
$K_{\beta}\mathopen{}\mathclose{{}\left(t}\right)$ is given by

$K_{\beta,ij}\mathopen{}\mathclose{{}\left(t}\right)\equiv\delta_{\beta
j}\Theta_{ij}\mathopen{}\mathclose{{}\left(t}\right)\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }\beta,i,j\in\mathcal{C},\,t\in\mathcal{T}}\right].$ (83)

It follows from the summation condition (70) on
$\Theta\mathopen{}\mathclose{{}\left(t}\right)$ that these new matrices
satisfy the _Kraus condition_ :

$\sum_{\beta=1}^{N}K_{\beta}^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)K_{\beta}\mathopen{}\mathclose{{}\left(t}\right)=\mymathbb{1}\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }t\in\mathcal{T}}\right].$ (84)

Moreover, one can write the dictionary (77) as

$\displaystyle\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle=\sum_{\beta=1}^{N}\mathrm{tr}\big{(}K_{\beta}^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)P_{i}K_{\beta}\mathopen{}\mathclose{{}\left(t}\right)P_{j}\big{)}$
(85)
$\displaystyle\negthickspace\negthickspace\negthickspace\mathopen{}\mathclose{{}\left[\textrm{for
all }i,j\in\mathcal{C},\,t\in\mathcal{T}}\right],$

and one can express the time-evolution rule (79) for the system’s density
matrix as

$\rho\mathopen{}\mathclose{{}\left(t}\right)=\sum_{\beta=1}^{N}K_{\beta}\mathopen{}\mathclose{{}\left(t}\right)\rho\mathopen{}\mathclose{{}\left(0}\right)K_{\beta}^{\dagger}\mathopen{}\mathclose{{}\left(t}\right).$
(86)

The matrices
$K_{1}\mathopen{}\mathclose{{}\left(t}\right),\dots,K_{N}\mathopen{}\mathclose{{}\left(t}\right)$
are therefore _Kraus operators_ [32], and (86) gives a _Kraus decomposition_
of $\rho\mathopen{}\mathclose{{}\left(t}\right)$.

Abstracting these results, one obtains a _completely positive trace-preserving
map_ , or _quantum channel_ ,

$\mathcal{E}_{t}:\mathbb{C}^{N\times N}\to\mathbb{C}^{N\times N},$ (87)

defined on the algebra $\mathbb{C}^{N\times N}$ of $N\times N$ matrices over
the complex numbers according to

$X\mapsto\mathcal{E}_{t}\mathopen{}\mathclose{{}\left(X}\right)\equiv\sum_{\beta=1}^{N}K_{\beta}\mathopen{}\mathclose{{}\left(t}\right)XK_{\beta}^{\dagger}\mathopen{}\mathclose{{}\left(t}\right).$
(88)

###  V.7 Dilation

By the _Stinespring dilation theorem_ [33, 34], the quantum channel (87) can
be _purified_ , meaning made into a form of _unitary_ time-evolution.
Specifically, for some integer $\tilde{N}$ in the _bounded_ interval

$N\leq\tilde{N}\leq N^{3},$ (89)

there exists an $\tilde{N}\times\tilde{N}$ unitary matrix
$\tilde{U}\mathopen{}\mathclose{{}\left(t}\right)$ on a potentially enlarged
or _dilated_ Hilbert space

$\tilde{\mathcal{H}}\equiv\mathbb{C}^{\tilde{N}}$ (90)

such that the dictionary (77) can be written as101010From the starting
assumptions presented here, one can sketch the following proof: Given $N\times
N$ Kraus matrices $K_{\beta}\mathopen{}\mathclose{{}\left(t}\right)$, with
$\beta=1,\dots,N$, define an $N^{3}\times N^{2}$ matrix
$\tilde{V}\mathopen{}\mathclose{{}\left(t}\right)$ according to
$\tilde{V}_{\mathopen{}\mathclose{{}\left(i\beta
m}\right)\mathopen{}\mathclose{{}\left(jl}\right)}\mathopen{}\mathclose{{}\left(t}\right)\equiv
K_{\beta,ij}\mathopen{}\mathclose{{}\left(t}\right)\delta_{lm}$, treating
$\mathopen{}\mathclose{{}\left(i\beta m}\right)$ as the first index of
$\tilde{V}\mathopen{}\mathclose{{}\left(t}\right)$ and treating
$\mathopen{}\mathclose{{}\left(jl}\right)$ as its second index. One can show
that this matrix satisfies
$\tilde{V}^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)\tilde{V}\mathopen{}\mathclose{{}\left(t}\right)=\mymathbb{1}_{N^{2}\times
N^{2}}$, so it defines a partial isometry, which can always be extended to a
unitary $N^{3}\times N^{3}$ matrix
$\tilde{U}_{\mathopen{}\mathclose{{}\left(i\beta
m}\right)\mathopen{}\mathclose{{}\left(ja}\right)}\mathopen{}\mathclose{{}\left(t}\right)$
by adding $N^{3}-N^{2}$ additional columns that are mutually orthogonal with
each other and with the previous $N^{2}$ columns already in
$\tilde{V}\mathopen{}\mathclose{{}\left(t}\right)$, where the new index $a$
runs through $N^{2}$ possible values. These additional columns can always be
chosen so that at the initial time $0$, where
$V\mathopen{}\mathclose{{}\left(0}\right)=\mymathbb{1}$ is the $N\times N$
identity matrix, they make $\tilde{U}\mathopen{}\mathclose{{}\left(0}\right)$
coincide with the $N^{3}\times N^{3}$ identity matrix. The last step is to
show that $\tilde{U}\mathopen{}\mathclose{{}\left(t}\right)$ satisfies (91),
whose right-hand side reduces to
$\sum_{\beta,m}|\tilde{U}_{\mathopen{}\mathclose{{}\left(i\beta
m}\right)\mathopen{}\mathclose{{}\left(jj^{\prime}}\right)}\mathopen{}\mathclose{{}\left(t}\right)|^{2}=\sum_{\beta}|K_{\beta,ij}\mathopen{}\mathclose{{}\left(t}\right)|^{2}$.
QED

$\displaystyle\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle=\mathrm{tr}\mathopen{}\mathclose{{}\left(\mathrm{tr}^{\prime}\mathopen{}\mathclose{{}\left(\tilde{U}^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)[P_{i}\otimes\mymathbb{1}^{\prime}]\tilde{U}\mathopen{}\mathclose{{}\left(t}\right)[P_{j}\otimes
P_{j^{\prime}}^{\prime}]}\right)}\right)$ (91)
$\displaystyle\quad\qquad[\textrm{for all
}i,j\in\mathcal{C},\,t\in\mathcal{T}].$

The formula (91) involves a number of ingredients.

  * •

The dilated Hilbert space $\tilde{\mathcal{H}}$ is defined as the tensor
product

$\tilde{\mathcal{H}}\equiv\mathcal{H}\otimes\mathcal{H}^{\prime},$ (92)

where $\mathcal{H}\equiv\mathbb{C}^{N}$ is the system’s original Hilbert space
(72), and where

$\mathcal{H}^{\prime}\equiv\mathbb{C}^{N^{\prime}}$ (93)

is an _ancillary_ Hilbert space whose dimension $N^{\prime}$ satisfies
$\tilde{N}=NN^{\prime}\leq N^{3}$. That is, $N^{\prime}$ is an integer lying
in the bounded interval

$1\leq N^{\prime}\leq N^{2}.$ (94)

  * •

The first trace $\mathrm{tr}\mathopen{}\mathclose{{}\left(\cdots}\right)$ in
(91) denotes the _partial trace_ over just the original Hilbert space
$\mathcal{H}$, and the second trace
$\mathrm{tr}^{\prime}\mathopen{}\mathclose{{}\left(\cdots}\right)$ similarly
denotes the partial trace over the ancillary Hilbert space
$\mathcal{H}^{\prime}$.

  * •

Importantly, for each fixed time $t$,
$\tilde{U}\mathopen{}\mathclose{{}\left(t}\right)$ is an
$\tilde{N}\times\tilde{N}$ _unitary_ matrix that reduces to the
$\tilde{N}\times\tilde{N}$ identity matrix
$\tilde{U}\mathopen{}\mathclose{{}\left(0}\right)=\tilde{\mymathbb{1}}$ at the
initial time $0$.

  * •

The symbol $\mymathbb{1}^{\prime}$ denotes the $N^{\prime}\times N^{\prime}$
identity matrix on the ancillary Hilbert space $\mathcal{H}^{\prime}$.

  * •

Letting $e_{1}^{\prime},\dots,e_{N^{\prime}}^{\prime}$ denote the standard
orthonormal basis for the ancillary Hilbert space $\mathcal{H}^{\prime}$, in
analogy with the configuration basis (73) for the system’s original Hilbert
space $\mathcal{H}$, and letting the primed Latin letters
$i^{\prime},j^{\prime},\dots$ each denote an element of an ancillary
configuration space $\mathcal{C}^{\prime}$ consisting of the integers from $1$
to $N^{\prime}$, the symbol $P_{i^{\prime}}^{\prime}$ denotes the rank-one
projector

$\displaystyle P_{i^{\prime}}^{\prime}$ $\displaystyle\equiv
e_{i^{\prime}}^{\prime}e_{i^{\prime}}^{\prime\dagger}=\mathrm{diag}(0,\dots,0,\underset{\mathclap{\begin{subarray}{c}\uparrow\\\
i^{\prime}\textrm{th entry}\end{subarray}}}{1},0,\dots,0)$ (95)
$\displaystyle\quad\mathopen{}\mathclose{{}\left[\textrm{for all
}i^{\prime}\in\mathcal{C}^{\prime}}\right],$

which is an $N^{\prime}\times N^{\prime}$ diagonal matrix with a $1$ in its
$i^{\prime}$th diagonal entry and $0$s in all its other entries. These
projectors form a projection-valued measure (PVM) on $\mathcal{H}^{\prime}$
satisfying the conditions of mutual exclusivity,

$P_{i^{\prime}}^{\prime}P_{j^{\prime}}^{\prime}=\delta_{i^{\prime}j^{\prime}}P_{i^{\prime}}^{\prime}\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }i^{\prime},j^{\prime}\in\mathcal{C}^{\prime}}\right],$ (96)

and completeness,

$\sum_{i^{\prime}=1}^{N^{\prime}}P_{i^{\prime}}^{\prime}=\mymathbb{1}^{\prime}.$
(97)

  * •

Note that the left-hand side of (91) is insensitive to the specific choice of
$j^{\prime}\in\mathcal{C}^{\prime}$ on the right-hand side.

Extending the foregoing construction, and fixing the time $t$, one obtains an
$\tilde{N}\times\tilde{N}$ transition matrix given by the new dictionary

$\displaystyle\tilde{\Gamma}_{ii^{\prime},jj^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle\equiv\tilde{\mathrm{tr}}\big{(}\tilde{U}^{\dagger}\mathopen{}\mathclose{{}\left(t}\right)\tilde{P}_{ii^{\prime}}\tilde{U}\mathopen{}\mathclose{{}\left(t}\right)\tilde{P}_{jj^{\prime}}\big{)}$
(98)
$\displaystyle\negthickspace\negthickspace\negthickspace\mathopen{}\mathclose{{}\left[\textrm{for
all
}i,j\in\mathcal{C},\,i^{\prime},j^{\prime}\in\mathcal{C}^{\prime},\,t\in\mathcal{T}}\right].$

Here the trace is now over the dilated Hilbert space $\tilde{\mathcal{H}}$,
and

$\tilde{P}_{ii^{\prime}}\equiv
P_{i}\otimes\tilde{P}_{i^{\prime}}\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }i\in\mathcal{C},\,i^{\prime}\in\mathcal{C}^{\prime}}\right]$ (99)

defines a rank-one projector on $\tilde{\mathcal{H}}$.

Unfolding the notation, the dilated dictionary (98) reduces to the statement
that

$\displaystyle\tilde{\Gamma}_{ii^{\prime},jj^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle=|\tilde{U}_{ii^{\prime},jj^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)|^{2}$
(100)
$\displaystyle\negthickspace\negthickspace\negthickspace\mathopen{}\mathclose{{}\left[\textrm{for
all
}i,j\in\mathcal{C},\,i^{\prime},j^{\prime}\in\mathcal{C}^{\prime},\,t\in\mathcal{T}}\right].$

That is, each entry
$\tilde{\Gamma}_{ii^{\prime},jj^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)$
of the $\tilde{N}\times\tilde{N}$ transition matrix
$\tilde{\Gamma}\mathopen{}\mathclose{{}\left(t}\right)$ is the modulus-squared
of the corresponding entry
$\tilde{U}_{ii^{\prime},jj^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)$
of an $\tilde{N}\times\tilde{N}$ _unitary_ matrix
$\tilde{U}\mathopen{}\mathclose{{}\left(t}\right)$. Again, a stochastic matrix
with this special feature is called a _unistochastic matrix_.

###  V.8 The Dilated Generalized Stochastic System

One can now define a _dilated_ generalized stochastic system
$(\tilde{\mathcal{C}},\tilde{\mathcal{T}},\tilde{\Gamma},\tilde{p},\tilde{\mathcal{A}})$
in the following way.

  * •

Let the dilated system’s configuration space $\tilde{\mathcal{C}}$ be defined
as the Cartesian product

$\tilde{\mathcal{C}}\equiv\mathcal{C}\times\mathcal{C}^{\prime},$ (101)

where $\mathcal{C}$ is the original system’s configuration space, with $N$
elements labeled by unprimed Latin letters $i,j,\dots$, and where
$\mathcal{C}^{\prime}$ is the configuration space of an ancillary subsystem,
with $N^{\prime}$ elements labeled by primed Latin letters
$i^{\prime},j^{\prime},\dots$.

  * •

Let the dilated system’s set of times $\tilde{\mathcal{T}}$ be the original
system’s set of times $\mathcal{T}$:

$\tilde{\mathcal{T}}\equiv\mathcal{T}.$ (102)

  * •

Let
$\tilde{\Gamma}:\tilde{\mathcal{C}}^{2}\times\tilde{\mathcal{T}}\to\mathopen{}\mathclose{{}\left[0,1}\right]$
be the stochastic map defined according to the dilated dictionary (100). Then
for each fixed time $t$, the $\tilde{N}\times\tilde{N}$ matrix
$\tilde{\Gamma}\mathopen{}\mathclose{{}\left(t}\right)$ is unistochastic.
Moreover, by construction,
$\tilde{\Gamma}\mathopen{}\mathclose{{}\left(t}\right)$ satisfies the
marginalization condition

$\displaystyle\sum_{i^{\prime}=1}^{N^{\prime}}\tilde{\Gamma}_{ii^{\prime},jj^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle=\Gamma_{ij}\mathopen{}\mathclose{{}\left(t}\right)$ (103)
$\displaystyle\negthickspace\negthickspace\negthickspace\mathopen{}\mathclose{{}\left[\textrm{for
all
}i,j\in\mathcal{C},\,j^{\prime}\in\mathcal{C}^{\prime},\,t\in\mathcal{T}}\right],$

where, as in (91), the value of $j^{\prime}\in\mathcal{C}^{\prime}$ is
irrelevant.

  * •

Let the map
$\tilde{p}:\tilde{\mathcal{C}}\to\mathopen{}\mathclose{{}\left[0,1}\right]$ be
the probability distribution defined by

$\displaystyle\tilde{p}_{ii^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)$
$\displaystyle\equiv\sum_{j=1}^{N}\tilde{\Gamma}_{ii^{\prime},jj^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)p_{j}\mathopen{}\mathclose{{}\left(0}\right)$
(104)
$\displaystyle\negthickspace\negthickspace\negthickspace\mathopen{}\mathclose{{}\left[\textrm{for
all
}i\in\mathcal{C},\,i^{\prime},j^{\prime}\in\mathcal{C}^{\prime},\,t\in\mathcal{T}}\right].$

It follows from the marginalization condition (103), together with the
Bayesian marginalization condition (33), that

$\sum_{i^{\prime}=1}^{N^{\prime}}\tilde{p}_{ii^{\prime}}\mathopen{}\mathclose{{}\left(t}\right)=p_{i}\mathopen{}\mathclose{{}\left(t}\right)\quad\mathopen{}\mathclose{{}\left[\textrm{for
all }i\in\mathcal{C},\,t\in\mathcal{T}}\right],$ (105)

as in (59).

  * •

Let the algebra $\tilde{\mathcal{A}}$ of random variables be the set of all
maps of the form
$\tilde{A}:\tilde{\mathcal{C}}\times\tilde{\mathcal{T}}\to\mathbb{R}$.

These results establish that
$(\tilde{\mathcal{C}},\tilde{\mathcal{T}},\tilde{\Gamma},\tilde{p},\tilde{\mathcal{A}})$
is a composite unistochastic system, and that the original generalized
stochastic system
$\mathopen{}\mathclose{{}\left(\mathcal{C},\mathcal{T},\Gamma,p,\mathcal{A}}\right)$
can be regarded as one of its subsystems. This conclusion completes the proof
of the stochastic-quantum theorem (65). QED

###  V.9 The Corresponding Quantum System

Corresponding to the dilated unistochastic system
$(\tilde{\mathcal{C}},\tilde{\mathcal{T}},\tilde{\Gamma},\tilde{p},\tilde{\mathcal{A}})$
is a quantum system based on a Hilbert space $\tilde{\mathcal{H}}$ of
dimension $\tilde{N}\leq N^{3}$, with linear-unitary time evolution encoded in
the unitary time-evolution operator
$\tilde{U}\mathopen{}\mathclose{{}\left(t}\right)$.

Unistochastic matrices are not generally orthostochastic, meaning that they
are not guaranteed to be expressible in terms of real orthogonal matrices. As
a consequence, if one is given a generalized stochastic system whose $N\times
N$ transition matrix $\Gamma\mathopen{}\mathclose{{}\left(t}\right)$ is
_already_ unistochastic, then there is no guarantee that the corresponding
$N\times N$ unitary time-evolution operator
$U\mathopen{}\mathclose{{}\left(t}\right)$ can be assumed to be a real
orthogonal matrix. The stochastic-quantum correspondence therefore implies
that in order to provide Hilbert-space representations for the most general
kinds of generalized stochastic systems, the complex numbers $\mathbb{C}$ will
be an important feature of quantum theory.

Whether in that case or more generally, of course, one is always free to start
with an $N\times N$ time-evolution operator
$\Theta\mathopen{}\mathclose{{}\left(t}\right)$ in (68) whose individual
entries are all real. With that choice, the $\tilde{N}\times\tilde{N}$ unitary
time-evolution operator $\tilde{U}\mathopen{}\mathclose{{}\left(t}\right)$
obtained from the Stinespring dilation theorem will likewise be real, and will
therefore be an orthogonal matrix.

However, it is important to keep in mind that from the point of view of the
stochastic-quantum correspondence, a generalized stochastic system’s Hilbert-
space representations are convenient fictions, and so one is entirely free to
assume that they involve the complex numbers anyway. Assuming that a given
choice of Hilbert-space representation is defined over the complex numbers,
rather than merely over the real numbers, allows one to take advantage of the
spectral theorem, eigenvectors of the time-evolution operator
$\tilde{U}\mathopen{}\mathclose{{}\left(t}\right)$, and anti-unitary
operators. Assuming appropriate smoothness conditions in time, one can further
make use of a self-adjoint Hamiltonian
$\tilde{H}\mathopen{}\mathclose{{}\left(t}\right)$ with real-valued energy
eigenvalues, as well as the Schrödinger equation. To the extent that these
mathematical constructs are often taken by textbooks to be indisputable
features of quantum systems, the complex numbers become an avoidable part of
quantum theory.111111Even if one assumes that a given choice of Hilbert-space
representation is defined over the complex numbers, one can always double the
dimension of the Hilbert space from $N$ to $2N$ and represent the imaginary
unit $i\equiv\sqrt{-1}$ by the $2\times 2$ real matrix
$\mathopen{}\mathclose{{}\left(\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}}\right)$, in which case all $N\times N$ unitary matrices
become $2N\times 2N$ real orthogonal matrices. Interestingly, one can then
also represent the complex-conjugation operation $K$ needed for anti-unitary
operators as a $2\times 2$ real matrix
$\mathopen{}\mathclose{{}\left(\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}}\right)$. However, this approach still ultimately
preserves the algebraic structure of the complex numbers in the Hilbert space,
in the sense that they will correspond to a subalgebra of matrices on the
overall Hilbert space that commute with the Hamiltonian and all the system’s
observables. Hence, this approach does not truly eliminate the complex numbers
from the Hilbert-space formalism of quantum theory.

##  VI Discussion and Future Work

The unitarily evolving quantum system that lies on the other side of the
stochastic-quantum correspondence is not limited to a _commutative_ algebra of
observables represented by operators that are diagonal in the configuration
basis.

Indeed, as explained in other work [35], one can model the quantum measurement
process of an observable represented by an _arbitrary_ self-adjoint operator
in terms of a unistochastic system that contains a subject system, a measuring
device, and an environment, all as explicitly defined subsystems. By taking
the measuring device’s allowed configurations to correspond to definite
readings of outcomes, and by taking the overall unistochastic system’s
transition matrix to be based on precisely the type of unitary time-evolution
operator employed in standard textbook treatments of the measurement process,
one inevitably finds that the measuring device ends up in one of its
measurement-reading configurations with a stochastic probability given by the
general form of the _Born rule_. Hence, in principle, one has access to a
quantum system’s entire _noncommutative_ algebra of observables.

With the stochastic-quantum correspondence in hand, one can refer other exotic
features of quantum systems back to their associated generalized stochastic
systems to give those features a more physically transparent interpretation.
For example, as shown in [35], _interference_ and _entanglement_ can be
understood as artifacts of the generic indivisibility of a generalized
stochastic system’s dynamics. (It is also shown in [35] how this
interpretative framework navigates the various _no-go theorems_ that have
appeared in the research literature over the past century.)

This overall approach to quantum foundations therefore sheds new light on some
of the strangest features of quantum theory, in addition to suggesting novel
applications of quantum computers. This approach might even provide a helpful
stepping stone for the development of self-consistent generalizations of
quantum theory itself.

## Acknowledgments

The author would like to thank Scott Aaronson, David Baker, Craig Callender,
Iris Cong, Jordan Cotler, Jenann Ismael, David Kagan, Logan McCarty, Arjun
Mirani, and Jill North for helpful conversations.

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