example $\triangle$ remark $\triangle$

# Double or nothing: a Kolmogorov extension theorem for multitime
(bi)probabilities in quantum mechanics

Davide Lonigro  Department Physik, Friedrich-Alexander-Universität Erlangen-
Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany  Dipartimento di Matematica,
Università degli Studi di Bari Aldo Moro, via E. Orabona 4, 70125 Bari, Italy
Istituto Nazionale di Fisica Nucleare, Sezione di Bari, via G. Amendola 173,
70126 Bari, Italy  Fattah Sakuldee  The International Centre for Theory of
Quantum Technologies, University of Gdańsk, Jana Bażyńskiego 1A, 80-309
Gdańsk, Poland  Łukasz Cywiński  Institute of Physics, Polish Academy of
Sciences, al. Lotników 32/46, PL 02-668 Warsaw, Poland  Dariusz Chruściński
Institute of Physics, Faculty of Physics, Astronomy and Informatics, Nicolaus
Copernicus University, Grudziadzka 5/7, 87-100 Toruń, Poland  Piotr Szańkowski
Institute of Physics, Polish Academy of Sciences, al. Lotników 32/46, PL
02-668 Warsaw, Poland<EMAIL_ADDRESS>
###### Abstract

The multitime probability distributions obtained by repeatedly probing a
quantum system via the measurement of an observable generally violate
Kolmogorov’s consistency property. Therefore, one cannot interpret such
distributions as the result of the sampling of a single trajectory. We show
that, nonetheless, they do result from the sampling of one _pair_ of
trajectories. In this sense, rather than give up on trajectories, quantum
mechanics requires to double down on them. To this purpose, we prove a
generalization of the Kolmogorov extension theorem that applies to families of
complex-valued bi-probability distributions (that is, defined on pairs of
elements of the original sample spaces), and we employ this result in the
quantum mechanical scenario. We also discuss the relation of our results with
the quantum comb formalism.

##  1 Introduction

The theory of stochastic processes finds one of its cornerstones in
Kolmogorov’s extension theorem [1]. Under suitable assumptions, the theorem
ensures that any consistent family of multitime joint probabilities—typically
used to describe the distribution of measurement outcomes of a physical system
repeatedly probed by the experimenter—arises from an underlying “master
probability” on a space of trajectories. Here, focusing on the case of
discrete-valued outcomes in a finite set $\Omega$, “consistent” means the
following: given any $n$-tuple of times $t_{1}<t_{2}<\ldots<t_{n}$, one has

$\sum_{f_{j}\in\Omega}P_{t_{n},\ldots,t_{j},\dots,t_{1}}(f_{n},\ldots,f_{j},\ldots,f_{1})=P_{t_{n},\ldots,\cancel{t_{j}},\ldots,t_{1}}(f_{n},\ldots,\cancel{f_{j}},\ldots,f_{1}),$
(1.1)

the two functions in the above equation respectively representing the $n$-time
and $(n-1)$-time probability distributions. The extension theorem then asserts
the existence of a probability measure $\mathcal{P}[f]\,[\mathcal{D}f]$ on the
space of trajectories $t\mapsto f(t)\in\Omega$, from which all multitime
probability distributions can be obtained as restrictions to discrete time
grids. In terms of functional integrals, this corresponds to the formal
relation

$P_{t_{n},\ldots,t_{1}}(f_{n},\ldots,f_{1})=\int\Big{(}\prod_{j=1}^{n}\delta_{f(t_{j}),f_{j}}\Big{)}\mathcal{P}[f][\mathcal{D}f],$
(1.2)

that is, the $n$-time probability $P_{t_{n},\ldots,t_{1}}(f_{n},\ldots,f_{1})$
is the outcome of summing over all trajectories $f(t)$ that pass through the
specified values at the corresponding times,
$f(t_{1})=f_{1},\ldots,f(t_{n})=f_{n}$. Equivalently, all multitime
distributions can be obtained by an underlying stochastic process. The
phenomenological importance of the extension theorem cannot be understated: it
guarantees that, when consistent, the measured chronological sequences are
nothing but discrete-time samples of an underlying trajectory that is traced
over time in accordance with the dynamical laws of the system.

Such a simple paradigm is a defining feature of classical theories, but it is
nowhere to be found in quantum mechanics111 Generalizations of the Kolmogorov
consistency condition for a quantum counterpart of stochastic processes, known
as “quantum combs” or “process matrices”, were already investigated in the
literature on multitime quantum processes [2, 3]. In principle, such a
construction is based on a different generalization of the trajectory picture
than in our work, and hence the resulting analyses are given in different
setups. We will revisit the connection of our results to the quantum comb
formalism in Section 4.3. . The multitime probability distributions obtained
by repeatedly probing a quantum system do not, in general, satisfy the
consistency property (1.1). The standard Hilbert space–based formalism of
quantum mechanics reflects this basic observation. A quantum system initially
prepared in a state $\hat{\rho}$, evolving via a unitary propagator
$\hat{U}_{t,t_{0}}$ generated by a (possibly time-dependent) Hamiltonian
$\hat{H}(t)$, and periodically probed with a measuring device represented by a
family of orthogonal projectors $\\{\hat{P}(f)\\}_{f\in\Omega}$ (physically,
the spectral resolution of the probed observable) yields a family of multitime
probabilities given by

$P_{t_{n},\ldots,t_{1}}(f_{n},\dots,f_{1})=\operatorname{tr}\bigg{[}\Big{(}\prod_{j=n}^{1}\hat{P}_{t_{j}}(f_{j})\Big{)}\hat{\rho}\Big{(}\prod_{j=1}^{n}\hat{P}_{t_{j}}(f_{j})\Big{)}\bigg{]},$
(1.3)

where $\hat{P}_{t}(f)=\hat{U}_{0,t}\hat{P}(f)\hat{U}_{t,0}$ and
$\prod_{j=n}^{1}\hat{A}_{j}$ ($\prod_{j=1}^{n}\hat{A}_{j})$ indicates an
ordered composition, $\hat{A}_{n}\cdots\hat{A}_{1}$
($\hat{A}_{1}\cdots\hat{A}_{n}$). From Eq. (1.3) one easily shows that the
consistency (1.1) is generally violated, albeit it can be satisfied for
specific systems [4, 5, 6, 7, 8, 9]. Therefore, in general there is no
underlying stochastic process generating the family of probability
distributions representing the phenomenology of the system—in other words, we
cannot interpret quantum mechanical probabilities as the result of the probing
of one underlying, objective trajectory.

###  1.1 Main result

The violation of consistency by the distributions (1.3), and the consequent
incompatibility with the Kolmogorov extension theorem, forces the quantum
theory to abandon the trajectory picture interpretation, that can be instead
regarded as the foundational tenet of classical theories. Indeed, there is no
way to reconstruct the outcomes of a quantum mechanical experiment from one
trajectory. In this paper we will show that, nevertheless, two trajectories
are enough for this purpose.

The basic idea is remarkably simple. Instead of the probabilities (1.3), we
shall consider the following family of complex-valued functions defined on the
space of pairs of $n$-tuples,

$Q_{t_{n},\ldots,t_{1}}(f_{n}^{+},f_{n}^{-};\ldots,f_{1}^{+},f_{1}^{-})=\operatorname{tr}\bigg{[}\Big{(}\prod_{j=n}^{1}\hat{P}_{t_{j}}(f^{+}_{j})\Big{)}\hat{\rho}\Big{(}\prod_{j=1}^{n}\hat{P}_{t_{j}}(f^{-}_{j})\Big{)}\bigg{]}.$
(1.4)

They are normalized,
$\sum_{f^{\pm}_{n},\ldots,f_{1}^{\pm}}Q_{t_{n},\ldots,t_{1}}(f_{n}^{+},f_{n}^{-};\ldots;f_{1}^{+},f_{1}^{-})=1$,
and their “diagonal” values coincide with the probability distributions (1.3),
$Q_{t_{n},\ldots,t_{1}}(f_{n},f_{n};\ldots;f_{1},f_{1})=P_{t_{n},\ldots,t_{1}}(f_{n},\ldots,f_{1})$.
As such, we will refer to these functions as bi-probability distributions,
even though $Q_{t_{n},\ldots,t_{1}}$, being complex-valued, are not proper
probability distributions themselves. A simple calculation shows that the
family of bi-probabilities do satisfy a consistency–like condition,

$\sum_{f_{j}^{+},f_{j}^{-}}\\!\\!\\!Q_{t_{n},\ldots,t_{j},\ldots,t_{1}}(f_{n}^{+},f_{n}^{-};...;f_{j}^{+},f_{j}^{-};...;f_{1}^{+},f_{1}^{-})=Q_{t_{n},\ldots,\cancel{t_{j}},\ldots,t_{1}}(f_{n}^{+},f_{n}^{-};\ldots;\cancel{f_{j}^{+},f_{j}^{-}};\ldots;f_{1}^{+},f_{1}^{-}).$
(1.5)

We will denote this property as bi-consistency to stress the fact that the sum
has to be performed separately on the two $n$-tuples
$(f_{n}^{+},\ldots,f_{1}^{+})$ and $(f_{n}^{-},\ldots,f_{1}^{-})$.

Motivated by this observation, we will prove that a generalization of
Kolmogorov’s theorem applies to quantum mechanics: there exists a master,
complex-valued bi-probability measure
$\mathcal{Q}[f^{+},f^{-}][\mathcal{D}f^{+}][\mathcal{D}f^{-}]$ on the space of
trajectory pairs $t\mapsto(f^{+}(t),f^{-}(t))$, such that a relation analogous
to Eq. (1.2) holds:

$Q_{t_{n},\dots,t_{1}}(f_{n}^{+},f_{n}^{-};\ldots;f_{1}^{+},f_{1}^{-})=\iint\Big{(}\prod_{j=1}^{n}\delta_{f^{+}(t_{j}),f^{+}_{j}}\,\delta_{f^{-}(t_{j}),f^{-}_{j}}\Big{)}\mathcal{Q}[f^{+},f^{-}][\mathcal{D}f^{+}][\mathcal{D}f^{-}],$
(1.6)

whence, in particular,

$P_{t_{n},\dots,t_{1}}(f_{n};\dots;f_{j})=\iint\Big{(}\prod_{j=1}^{n}\delta_{f^{+}(t_{j}),f_{j}}\,\delta_{f^{-}(t_{j}),f_{j}}\Big{)}\mathcal{Q}[f^{+},f^{-}][\mathcal{D}f^{+}][\mathcal{D}f^{-}],$
(1.7)

and the $n$-time probability $P_{t_{n},\ldots,t_{1}}(f_{n},\ldots,f_{j})$ is
an outcome of the superposition of all pairs of trajectories
$(f^{+}(t),f^{-}(t))$ such that both pass through the same sequence of values
$f_{1},\dots,f_{n}$ at the corresponding times $t_{1},\dots,t_{n}$. We refer
to Theorem 3.1 for the precise statement. As a “bonus”, the link between
quantum mechanics and the classical theory of stochastic processes becomes
manifest: Eq. (1.7) reduces to its classical counterpart (1.2) in the limit in
which the only pairs of trajectories $(f^{+},f^{-})$ that contribute to the
integral are those with $f^{+}(t)=f^{-}(t)$ at all times.

The significance of this result reaches beyond the context of sequential
measurements. The bi-probability formalism is a general purpose
parameterization that find its use in virtually any context of quantum theory
[10, 11, 12, 9]. One illustrative example of such an application is the
description of open system dynamics. Let
$\hat{\rho}_{O}\mapsto\Lambda_{t}\hat{\rho}_{O}$ be the dynamical map for an
open system $O$ that is coupled to an observable
$\sum_{f\in\Omega}f\hat{P}(f)=\hat{F}$ via the Hamiltonian
$\hat{H}_{OE}(t)=\hat{H}_{O}\otimes\hat{1}+\hat{1}\otimes\hat{H}(t)+\lambda\hat{V}_{O}\otimes\hat{F}$,
where the original system plays the role of the environment. Then
$\Lambda_{t}$ can be parameterized as a “bi-average” over trajectory pairs
[10, 12, 9],

$\displaystyle\Lambda_{t}\hat{\rho}_{O}$
$\displaystyle=\operatorname{tr}_{E}\Big{[}\big{(}\mathcal{T}\mathrm{e}^{-\mathrm{i}\int_{0}^{t}\hat{H}_{OE}(s)\mathrm{d}s}\big{)}\hat{\rho}_{O}\otimes\hat{\rho}\,\big{(}\mathcal{T}\mathrm{e}^{-\mathrm{i}\int_{0}^{t}\hat{H}_{OE}(s)\mathrm{d}s}\big{)}^{\dagger}\Big{]}$
$\displaystyle=\iint\mathrm{e}^{-\mathrm{i}\int_{0}^{t}\big{(}\hat{H}_{O}+\lambda
f^{+}(s)\hat{V}_{O}\big{)}\mathrm{d}s}\hat{\rho}_{O}\,\mathrm{e}^{+\mathrm{i}\int_{0}^{t}\big{(}\hat{H}_{O}+\lambda
f^{-}(s)\hat{V}_{O}\big{)}\mathrm{d}s}\,\mathcal{Q}[f^{+},f^{-}][\mathcal{D}f^{+}][\mathcal{D}f^{-}].$
(1.8)

As such, the extension theorem proven here provides the rigorous mathematical
justification for the use of the bi-trajectory picture in the bi-probability
formulation, a justification that was previously lacking.

###  1.2 Strategy and outline

The goal of the paper is achieved in two steps:

  * •

the generalization of the extension theorem to complex-valued
(bi)probabilities is stated and subsequently proven in Section 2;

  * •

the generalized extension theorem is shown in Section 3 to be applicable to
quantum mechanical bi-probabilities given by Eq. (1.4).

The first step is made necessary by the fact that Kolmogorov’s extension
theorem, in its standard formulation (see e.g. [13, 14]), only handles
positive and normalized distributions. Fortunately, it is possible to adapt
the proof of its standard version to the case of complex-valued functions.
Notably, an additional requirement with respect to the standard one is needed:
for a family of general bi-probabilities
$\\{B_{t_{n},\dots,t_{1}}:n\in\mathbb{N},t_{1}<\dots<t_{n}\\}$ to be
extendable it must satisfy, on top of bi-consistency, a uniform bound
condition:

$\sup\Big{\\{}\sum_{f_{n}^{+},f_{n}^{-}}\cdots\sum_{f_{1}^{+},f_{1}^{-}}\left|B_{t_{n},\dots,t_{1}}(f_{n}^{+},f_{n}^{-};\dots,f_{1}^{+},f_{1}^{-})\right|\,:\,n\in\mathbb{N},\,t_{1}<\cdots<t_{n}\Big{\\}}<\infty,$
(1.9)

which, while obvious for normalized positive distributions, is generally
nontrivial when signed or complex-valued functions are involved, like in this
case. A thorough explanation of the mathematics behind this construction (even
beyond the discrete scenario discussed here) is reported in Appendix A.

In the second step we go back to quantum mechanics and prove that the
functions defined in Eq. (1.4) do satisfy the assumptions of the generalized
extension theorem. This will essentially boil down to showing that the crucial
property (1.9) does hold as long as the family of quantum bi-probabilities is
confined to a finite time window. In fact, demonstrating uniform boundedness
for quantum mechanical bi-probabilities turns out to be the main technical
challenge in the way towards the desired result.

The concluding discussion of our results, their connections with related
concepts found in the literature, and possible future developments, are
gathered in Section 4. The summary and final remarks are collected in Section
5.

##  2 Kolmogorov extension theorem for multitime bi-probabilities

Before we investigate the extension theorem for the specific family of bi-
probabilities encountered in the quantum setting (cf. Eq. (1.4)), we shall
discuss how it operates in the general context. Let $\Omega$ be a set of real
numbers, and $I$ an arbitrary (continuous or discrete) index set. Physically,
$\Omega$ represents the set of possible outcomes of a single experiment, and
$I$ could be regarded as the time interval over which the experiment or the
dynamics of the system takes place. In the abstract context considered here,
the purpose of $I$ is to keep track of the order of the supplied arguments.
Now, consider a family of functions

$\displaystyle\mathbb{B}_{I}$
$\displaystyle:=\\{B_{t_{n},\ldots,t_{1}}(f_{n}^{+},f_{n}^{-};\ldots;f_{1}^{+},f_{1}^{-})\
:\ n\in\mathbb{N},\,t_{1},\ldots,t_{n}\in I,\,t_{1}<\ldots<t_{n}\\},$ (2.1)

where each member is an abstract complex-valued function of a pair of
$n$-tuples,

$\displaystyle\Omega^{n}\times\Omega^{n}\ni(\bm{f}^{+}_{n},\bm{f}^{-}_{n})$
$\displaystyle\mapsto
B_{\bm{t}_{n}}(\bm{f}^{+}_{n},\bm{f}^{-}_{n})\in\mathbb{C}.$ (2.2)

Here and in the following, we will be using the following notation: $n$-tuples
of elements either of $\Omega$ or of $I$ are written as quasi-vectors
$\bm{f}_{n}=(f_{n},\ldots,f_{1})$ and $\bm{t}_{n}=(t_{n},\ldots,t_{1})$, with
the subscript in $\bm{f}_{n}$ and $\bm{t}_{n}$ denoting the number of
elements. Even when not explicitly indicated, we shall always assume that the
times $t_{1},\ldots,t_{n}\in I$ are distinct and ordered,
$t_{1}<t_{2}<\ldots<t_{n}$. Besides, with a slight abuse of notation, we will
use either of the following symbols to denote the values of $B_{\bm{t}_{n}}$:

$B_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-}),\qquad
B_{\bm{t}_{n}}(f_{n}^{+},f_{n}^{-};\ldots;f_{1}^{+},f_{1}^{-}),\qquad
B_{t_{n},\ldots,t_{1}}(f_{n}^{+},f_{n}^{-};\ldots;f_{1}^{+},f_{1}^{-}),$ (2.3)

the second or third one being used whenever we will need to single out one
particular pair of entries.

Presently the goal is to examine the specific mechanisms behind the extension
theorem for the family of complex-valued functions. Hence, we shall leave the
family $\mathbb{B}_{I}$ largely unconstrained except for two properties that
are sufficient for the extension to work.

######  Assumption 2.1.

The family of functions $\mathbb{B}_{I}$ satisfy the following properties:

  1. (B1)

Bi-consistency: for every $n\in\mathbb{N}$, $j=1,\dots,n$, and every
$\bm{t}_{n}=(t_{n},\dots,t_{1})$, we have

$\sum_{f^{+}_{j},f^{-}_{j}\in\Omega}B_{\bm{t}_{n}}(\bm{f}^{+}_{n},\bm{f}^{-}_{n})=B_{t_{1},\dots,\cancel{t_{j}},\dots,t_{n}}(f^{+}_{n},f^{-}_{n};\dots;\cancel{f^{+}_{j},f^{-}_{j}};\dots;f^{+}_{1},f^{-}_{1}).$
(2.4)

  2. (B2)

Uniform boundedness:

$\sup\Big{\\{}\|B_{\bm{t}_{n}}\|_{1}\,:\,B_{\bm{t}_{n}}\in\mathbb{B}_{I}\Big{\\}}<\infty,\quad\text{where}\quad\|B_{\bm{t}_{n}}\|_{1}:=\sum_{\bm{f}^{+}_{n},\bm{f}^{-}_{n}}\left|B_{\bm{t}_{n}}(\bm{f}^{+}_{n},\bm{f}^{-}_{n})\right|.$
(2.5)

Note that the uniform boundedness is trivially satisfied in the case of
standard (that is, positive-valued) probability distributions. Indeed, if the
functions satisfy $0\leq B_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})\leq 1$,
and are normalized,
$\sum_{\bm{f}^{\pm}_{n}}B_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})=1$, then
by definition $\|B_{\bm{t}_{n}}\|_{1}=1$ for the whole family. For this
reason, in the traditional statement of the extension theorem, the second
assumption would be superfluous. However, for our purposes (cf. Section 3), we
shall be dealing with the complex-valued case, whence the need for (B2).

The idea at the root of the extension theorem is that each distribution in
$\mathbb{B}_{I}$, each defining a discrete complex-valued measure on
$\Omega^{n}\times\Omega^{n}$, can be unambiguously obtained from a unique,
complex-valued “master measure” on the space $\Omega^{I}\times\Omega^{I}$ of
pairs of trajectories, i.e., the space of functions

$\displaystyle I\ni t\mapsto(f^{+}(t),f^{-}(t))\in\Omega\times\Omega$ (2.6)

Technically, complex-valued measures are set functions defined on a properly
chosen class of sets—a $\sigma$-algebra. There is indeed a standard procedure
for constructing a $\sigma$-algebra on spaces of trajectories, cf. Definition
A.2 in the Appendix A collecting all technical details; this construction
shall be tacitly assumed in the following.

######  Theorem 2.2 (Extension theorem for bi-probabilities).

Let $\mathbb{B}_{I}$ satisfy Assumptions (B1) and (B2). Then there exists a
unique complex-valued measure
$\mathcal{B}_{I}[f^{+},f^{-}][\mathcal{D}f^{+}][\mathcal{D}f^{-}]$ on the
space of trajectory pairs that uniquely extends every member of
$\mathbb{B}_{I}$ in the following sense:
$B_{\bm{t}_{n}}(\bm{f}^{+}_{n},\bm{f}^{-}_{n})$ equals the integral of
$\mathcal{B}_{I}[f^{+},f^{-}][\mathcal{D}f^{+}][\mathcal{D}f^{-}]$ on the set
of all pairs of trajectories satisfying $f^{\pm}(t_{j})=f^{\pm}_{j}$,
$j=1,\dots,n$, i.e.,

$\displaystyle
B_{\bm{t}_{n}}(\bm{f}^{+}_{n},\bm{f}_{n}^{-})=\iint\Big{(}\prod_{j=1}^{n}\delta_{f^{+}(t_{j}),f_{j}^{+}}\,\delta_{f^{-}(t_{j}),f_{j}^{-}}\Big{)}\mathcal{B}_{I}[f^{+},f^{-}][\mathcal{D}f^{+}][\mathcal{D}f^{-}].$
(2.7)

This theorem is a special case (see Remark A.9) of a more general statement,
Theorem A.8, that essentially generalizes the traditional Kolmogorov extension
theorem to include complex-valued measures. The full proof of this
generalization, together with an overview of all the needed mathematical
theory, is presented in Appendix A. Referring to the appendix for all due
details, below we sketch a simplified formulation of the proof, adapted to the
case examined here. This should suffice in illustrating the ideas behind it
and the necessity of both assumptions (B1) and (B2).

###### Sketch of the proof.

In plain words, the theorem asserts that, for a bi-consistent and uniformly
bounded family of multi-variable functions, it is possible to take the limit
of infinitely many arguments. However, instead of performing the limit on the
functions $B_{\bm{t}_{n}}$ themselves, the idea is to carry it out at the
level of the corresponding averages—that is, the linear functionals defined by

$\displaystyle\operatorname{E}_{\bm{t}_{n}}[X_{\bm{t}_{n}}]:=\sum_{\bm{f}_{n}^{\pm}}B_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})X_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})$
(2.8)

with $X_{\bm{t}_{n}}:\,\Omega^{n}\times\Omega^{n}\to\mathbb{C}$. The proof
then boils down to demonstrating the existence of a unique “master average”,
that is, a functional $\mathcal{E}_{I}$ satisfying the following property:
given any continuous
$\mathcal{X}_{I}:\Omega^{I}\times\Omega^{I}\to\mathbb{C}$, we have

$\displaystyle\mathcal{E}_{I}[\mathcal{X}_{I}]=\lim_{\bm{t}_{n}\to
I}\operatorname{E}_{\bm{t}_{n}}[X_{\bm{t}_{n}}],$ (2.9)

for any sequence of multi-variable functions $(X_{\bm{t}_{n}})_{\bm{t}_{n}}$
such that $\|X_{\bm{t}_{n}}-\mathcal{X}_{I}\|_{\infty}\to 0$ as $\bm{t}_{n}\to
I$, where $\|\cdot\|_{\infty}$ denotes the supremum norm. The
Stone–Weierstrass theorem ensures that such sequence always exists for any
continuous $\mathcal{X}_{I}$. Here, $\bm{t}_{n}\to I$ is used as a shorthand
for the corresponding set limit, $\lim_{n\to\infty}I_{n}\cap I=I$, with

$\displaystyle I_{n}:=\\{t\in I\,:\,\bm{t}_{n}=(t_{n},\ldots,t_{1}),\
t=t_{n},\ldots,t_{1}\\}.$ (2.10)

where the sequence $(\bm{t}_{n})_{n=1}^{\infty}$ is chosen in such a way that
$I_{1}\subset I_{2}\subset\cdots\subset I$.

We must therefore show that the convergence of $(X_{\bm{t}_{n}})_{\bm{t}_{n}}$
to a given $\mathcal{X}_{I}$ implies the convergence of
$(\operatorname{E}_{\bm{t}_{n}}[X_{\bm{t}_{n}}])_{\bm{t}_{n}}$, independently
of the choice of the sequence of functions. To this end it is sufficient and
necessary to verify that
$(\operatorname{E}_{\bm{t}_{n}}[X_{\bm{t}_{n}}])_{\bm{t}_{n}}$ is a Cauchy
sequence, which in this case follows from (B1) and (B2),

$\displaystyle\big{|}\operatorname{E}_{\bm{t}_{n}}[X_{\bm{t}_{n}}]-\operatorname{E}_{\bm{t}_{m}}[X_{\bm{t}_{m}}]\big{|}$
$\displaystyle=\big{|}\operatorname{E}_{\bm{t}_{n}\cup\bm{t}_{m}}[X_{\bm{t}_{n}}]-\operatorname{E}_{\bm{t}_{n}\cup\bm{t}_{m}}[X_{\bm{t}_{m}}]\big{|}=\big{|}\operatorname{E}_{\bm{t}_{n}\cup\bm{t}_{m}}[X_{\bm{t}_{n}}-X_{\bm{t}_{m}}]\big{|}$
$\displaystyle\leq\|X_{\bm{t}_{n}}-X_{\bm{t}_{m}}\|_{\infty}\|B_{\bm{t}_{n}\cup\bm{t}_{m}}\|_{1}$
$\displaystyle\leq\Big{(}\sup_{B_{\bm{t}_{k}}\in\mathbb{B}_{I}}\|B_{\bm{t}_{k}}\|_{1}\Big{)}\|X_{\bm{t}_{n}}-X_{\bm{t}_{m}}\|_{\infty}\xrightarrow{\bm{t}_{n},\bm{t}_{m}\to
I}0,$ (2.11)

with $\bm{t}_{n}\cup\bm{t}_{m}$ indicating $N$-tuple $\bm{t}_{N}$ consisting
of elements from both $\bm{t}_{n}$ and $\bm{t}_{m}$ (and thus, corresponding
to set $I_{N}=I_{n}\cup I_{m}$). The first two equalities in (2), where the
linearity of $\operatorname{E}$ is used on two functions with unequal tuples
$\bm{t}_{n}\neq\bm{t}_{m}$, are enabled by bi-consistency (B1): indeed, if we
take e.g. $\bm{t}_{n}=(t_{n},\ldots,t_{1})$ and
$\bm{t}_{m}=(t_{n},\ldots,\cancel{t_{j}},\ldots,t_{1})$, then

$\displaystyle\operatorname{E}_{\bm{t}_{m}}[X_{\bm{t}_{m}}]$
$\displaystyle=\sum_{\bm{f}^{\pm}_{m}}B_{\bm{t}_{m}}(\bm{f}_{m}^{+},\bm{f}_{m}^{-})X_{\bm{t}_{m}}(\bm{f}_{m}^{+},\bm{f}_{m}^{-})=\sum_{\bm{f}_{m}^{\pm}}\Big{(}\sum_{f_{j}^{\pm}}B_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})\Big{)}X_{\bm{t}_{m}}(\bm{f}_{m}^{+},\bm{f}_{m}^{-})$
$\displaystyle=\sum_{\bm{f}_{n}^{\pm}}B_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})X_{\bm{t}_{m}}(\bm{f}_{m}^{+},\bm{f}_{m}^{-})=\operatorname{E}_{\bm{t}_{n}}[X_{\bm{t}_{m}}]=\operatorname{E}_{\bm{t}_{n}\cup\bm{t}_{m}}[X_{\bm{t}_{m}}].$
(2.12)

Of course, the last inequality in (2) is due to the uniform boundedness (B2)
of $\mathbb{B}_{I}$.

Thus, the existence of a functional $\mathcal{E}_{I}$ satisfying Eq. (2.9) has
been proven. Since measures and the averages are in a one-to-one
correspondence (Theorem A.10), this ultimately guarantees the existence of a
measure on trajectories satisfying

$\displaystyle\iint\mathcal{X}_{I}[f^{+},f^{-}]\,\mathcal{B}_{I}[f^{+},f^{-}][\mathcal{D}f^{+}][\mathcal{D}f^{-}]$
$\displaystyle=\mathcal{E}_{I}[\mathcal{X}_{I}]$ (2.13)

for every continuous
$\mathcal{X}_{I}:\,\Omega^{I}\times\Omega^{I}\to\mathbb{C}$. Finally, to prove
Eq. (2.7), take $\bm{t}_{n}\in I^{n}$ and $\bm{f}^{\pm}_{n}\in\Omega^{n}$, let
$\chi_{\bm{t}_{n}}$ be the corresponding characteristic function:

$\chi_{\bm{t}_{n}}(\bm{\phi}_{n}^{+},\bm{\phi}_{n}^{-})=\delta_{\bm{\phi}^{+}_{n},\bm{f}_{n}^{+}}\delta_{\bm{\phi}_{n}^{-},\bm{f}_{n}^{-}}=\prod_{j=1}^{n}\delta_{\phi^{+}_{j},f^{+}_{j}}\delta_{\phi^{-}_{j},f^{-}_{j}}$
(2.14)

and define $\chi_{I}$ by

$\chi_{I}[f^{+},f^{-}]:=\chi_{\bm{t}_{n}}\left(f^{+}(t_{n}),f^{-}(t_{n});\ldots;f^{+}(t_{1}),f^{-}(t_{1})\right)=\prod_{j=1}^{n}\delta_{f^{+}(t_{j}),f^{+}_{j}}\delta_{f^{-}(t_{j}),f^{-}_{j}}.$
(2.15)

Then by construction
$\mathrm{E}_{\bm{t}_{n}}[\chi_{\bm{t}_{n}}]=\mathcal{E}_{I}[\chi_{I}]$, and a
direct check shows that the former and the latter respectively correspond to
the left- and right-hand sides of Eq. (2.7).

∎

##  3 The extension theorem in quantum mechanics

Let us now consider a quantum system represented by a finite-dimensional
Hilbert space $\mathcal{H}$, the dynamics of which is described by the unitary
evolution operator ($\hbar=1$):

$\displaystyle\hat{U}_{t,t^{\prime}}$
$\displaystyle=\mathcal{T}\mathrm{e}^{-\mathrm{i}\int_{t^{\prime}}^{t}\hat{H}(s)ds}=\sum_{k=0}^{\infty}(-\mathrm{i})^{k}\int_{t^{\prime}}^{t}\mathrm{d}s_{k}\int_{t^{\prime}}^{s_{k}}\mathrm{d}s_{k-1}\cdots\int_{t^{\prime}}^{s_{2}}\mathrm{d}s_{1}\prod_{j=k}^{1}\hat{H}(s_{j}),$
(3.1)

with $\hat{H}(t)$ being the (possibly time-dependent) Hamiltonian of the
system. The initial state of the system is represented by a density operator
$\hat{\rho}$ on $\mathcal{H}$, and it is considered here to be fixed.

In this setting, one constructs the family of quantum bi-probabilities
associated with the time interval $I\subseteq\mathbb{R}_{+}$ and an observable
$F$ represented by a discrete-valued projection-valued measure (PVM)
$\\{\hat{P}(f)\\}_{f\in\Omega}$, with $\Omega$ being a set of finite
cardinality,

$\displaystyle\mathbb{Q}_{I}^{F}:=\\{Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}^{-}_{n})\
:\ n\in\mathbb{N},\,0=t_{0}<t_{1}<\ldots<t_{n},\,t_{1},\ldots,t_{n}\in I\\}$
(3.2)

consisting of functions defined according to

$Q_{\bm{t}_{n}}(\bm{f}^{+}_{n},\bm{f}^{-}_{n}):=\operatorname{tr}\Big{[}\Big{(}\prod_{j=n}^{1}\hat{P}_{t_{j}}(f^{+}_{j})\Big{)}\hat{\rho}\,\Big{(}\prod_{j=1}^{n}\hat{P}_{t_{j}}(f^{-}_{j})\Big{)}\Big{]},$
(3.3)

with $\hat{P}_{t}(f)=\hat{U}_{0,t}\hat{P}(f)\hat{U}_{t,0}$.

Bi-probabilities find their use in multiple contexts of standard quantum
theory as a flexible parameterization of the dynamics of the observable they
are associated with. Examples include the descriptions of the
quantum–classical transition [9] and, as already mentioned in Section 1.1, of
open system dynamics [10, 11, 12]. Of course, the family of bi-probabilities
also contains the description of sequential measurements of the observable $F$
[9, 11]. Accordingly, together with $\mathbb{Q}_{I}^{F}$, we shall also
consider the family of probabilities describing sequences of projective
measurements of $F$ performed on the system,

$\displaystyle\mathbb{P}_{I}^{F}:=\\{P_{\bm{t}_{n}}(\bm{f}_{n})\ :\
n\in\mathbb{N},\ 0=t_{0}<t_{1}<\ldots<t_{n},\ t_{1},\ldots,t_{n}\in I\\},$
(3.4)

where each member function is defined as the diagonal part of the
corresponding quantum bi-probability:

$P_{\bm{t}_{n}}(\bm{f}_{n}):=Q_{\bm{t}_{n}}(\bm{f}_{n},\bm{f}_{n})=\operatorname{tr}\Big{[}\Big{(}\prod_{j=n}^{1}\hat{P}_{t_{j}}(f_{j})\Big{)}\hat{\rho}\,\Big{(}\prod_{j=1}^{n}\hat{P}_{t_{j}}(f_{j})\Big{)}\Big{]}.$
(3.5)

######  Theorem 3.1 (Extension theorem for quantum bi-probabilities).

Let $\mathcal{H}$ be a Hilbert space with dimension
$\dim\mathcal{H}=d<\infty$, $\hat{\rho}$ a density operator on $\mathcal{H}$,
$\\{\hat{P}(f)\\}_{f\in\Omega}$ a projection-valued measure, and $\hat{H}(t)$
a continuously time-dependent Hamiltonian over a compact interval
$I\subset\mathbb{R}$. Let $\mathbb{Q}^{F}_{I}$ the corresponding family of
complex bi-probability distributions as defined in Eq. (3.3). Then there
exists a complex-valued measure
$\mathcal{Q}_{I}[f^{+},f^{-}]\,[\mathcal{D}f^{+}][\mathcal{D}f^{-}]$ on the
space of trajectory pairs that uniquely extends $\mathbb{Q}^{F}_{I}$ in the
following sense:

$Q_{\bm{t}_{n}}(\bm{f}^{+}_{n},\bm{f}^{-}_{n})=\iint\Big{(}\prod_{j=1}^{n}\delta_{f^{+}(t_{j}),f^{+}_{j}}\,\delta_{f^{-}(t_{j}),f^{-}_{j}}\Big{)}\mathcal{Q}_{I}[f^{+},f^{-}][\mathcal{D}f^{+}][\mathcal{D}f^{-}],$
(3.6)

and, in particular,

$P_{\bm{t}_{n}}(\bm{f}_{n})=\iint\Big{(}\prod_{j=1}^{n}\delta_{f^{+}(t_{j}),f_{j}}\,\delta_{f^{-}(t_{j}),f_{j}}\Big{)}\mathcal{Q}_{I}[f^{+},f^{-}][\mathcal{D}f^{+}][\mathcal{D}f^{-}].$
(3.7)

This result follows from Theorem 2.2, provided that the family
$\mathbb{Q}_{I}^{F}$ is bi-consistent (B1) and uniformly bounded (B2).
Accordingly, the remainder of this section is devoted to showing that
$\mathbb{Q}^{F}_{I}$ does indeed satisfy both of these properties; while bi-
consistency will simply follow by an immediate computation, uniform
boundedness will be proven as a non-trivial consequence of the physical
properties of the dynamics in quantum systems. We postpone the discussion on
this result to Section 4.

###  3.1 Properties of quantum bi-probabilities

We shall start by listing some basic properties of the families
$\mathbb{Q}^{F}_{I}$ and $\mathbb{P}^{F}_{I}$ that follow directly from the
definition (3.3).

######  Proposition 3.2.

The family of quantum bi-probabilities $\mathbb{Q}_{I}^{F}$ as in Eq. (3.3)
has the following properties:

  1. (Q1)

normalization:

$\sum_{\bm{f}_{n}^{+},\bm{f}_{n}^{-}\in\Omega^{n}}Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})=1.$

  2. (Q2)

causality:

$Q_{t_{n},\ldots,t_{1}}(f^{+}_{n},f^{-}_{n};\dots;f^{+}_{1},f^{-}_{1})=\delta_{f^{+}_{n},f^{-}_{n}}Q_{t_{n},\dots,t_{1}}(f^{+}_{n},f^{-}_{n};\dots;f^{+}_{1},f^{-}_{1}).$

  3. (Q3)

positive semidefiniteness: given any function $\Omega^{n}\ni\bm{f}_{n}\mapsto
Z(\bm{f}_{n})\in\mathbb{C}$,

$\sum_{\bm{f}_{n}^{+},\bm{f}_{n}^{-}}Z(\bm{f}_{n}^{+})Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})Z(\bm{f}_{n}^{-})^{*}\geq
0.$

  4. (Q4)

bi-consistency:

$\sum_{f^{+}_{j},f^{-}_{j}\in\Omega}Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})=Q_{t_{n},\ldots,\cancel{t_{j}},\ldots,t_{1}}(f^{+}_{n},f^{-}_{n};\ldots;\cancel{f^{+}_{j},f^{-}_{j}};\ldots;f^{+}_{1},f^{-}_{1}).$

###### Proof.

Property (Q1) is an immediate consequence of the normalization of the PVM,
$\sum_{f}\hat{P}_{t}(f)=\hat{1}$ for any $t\in I$; (Q2) follows from the
cyclic property of the trace and orthogonality of projector operators,
$\hat{P}_{t_{n}}(f_{n}^{-})\hat{P}_{t_{n}}(f_{n}^{+})\propto\delta_{f_{n}^{+},f_{n}^{-}}$.
Let us show (Q3),

$\displaystyle\sum_{\bm{f}_{n}^{+},\bm{f}_{n}^{-}}Z(\bm{f}^{+}_{n})Q_{\bm{t}_{n}}(\bm{f}^{+}_{n},\bm{f}^{-}_{n})Z(\bm{f}_{n}^{-})^{*}$
$\displaystyle\phantom{=}=\operatorname{tr}\left[\left(\sum_{\bm{f}_{n}^{+}}Z(\bm{f}_{n}^{+})\Big{(}\prod_{j=n}^{1}\hat{P}_{t_{j}}(f^{+}_{j})\Big{)}\hat{\rho}^{1/2}\right)\left(\sum_{\bm{f}_{n}^{-}}\hat{\rho}^{1/2}\Big{(}\prod_{j=1}^{n}\hat{P}_{t_{j}}(f^{-}_{j})\Big{)}Z(\bm{f}_{n}^{-})^{*}\right)\right]$
$\displaystyle\phantom{=}=\operatorname{tr}\left[\left(\sum_{\bm{f}_{n}}Z(\bm{f}_{n})\Big{(}\prod_{j=n}^{1}\hat{P}_{t_{j}}(f_{j})\Big{)}\hat{\rho}^{1/2}\right)\left(\sum_{\bm{f}_{n}}Z(\bm{f}_{n})\Big{(}\prod_{j=n}^{1}\hat{P}_{t_{j}}(f_{j})\Big{)}\hat{\rho}^{1/2}\right)^{\dagger}\right]\geq
0.$

Finally, the key bi-consistency (Q4) results from a direct calculation,

$\displaystyle\sum_{f_{j}^{+},f_{j}^{-}}Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})$
$\displaystyle=\operatorname{tr}\Big{[}\hat{P}_{t_{n}}(f_{n}^{+})\cdots\Big{(}\sum_{f_{j}^{+}}\hat{P}_{t_{j}}(f_{j}^{+})\Big{)}\cdots\hat{P}_{t_{1}}(f_{1}^{+})\hat{\rho}$
$\displaystyle\phantom{=\operatorname{tr}\Big{[}}\times\hat{P}_{t_{1}}(f_{1}^{-})\cdots\Big{(}\sum_{f_{j}^{-}}\hat{P}_{t_{j}}(f_{j}^{-})\Big{)}\cdots\hat{P}_{t_{n}}(f_{n}^{-})\Big{]}$
$\displaystyle=\operatorname{tr}\Big{[}\hat{P}_{t_{n}}(f_{n}^{+})\cdots\cancel{\hat{P}_{t_{j}}(f_{j}^{+})}\cdots\hat{P}_{t_{1}}(f_{1}^{+})\hat{\rho}\hat{P}_{t_{1}}(f_{1}^{-})\cdots\cancel{\hat{P}_{t_{j}}(f_{j}^{-})}\cdots\hat{P}_{t_{n}}(f_{n}^{-})\Big{]}$
$\displaystyle=Q_{t_{n},\ldots,\cancel{t_{j}},\ldots,t_{1}}(f_{n}^{+},f_{n}^{-};\ldots;\cancel{f_{j}^{+},f_{j}^{-}};\ldots;f_{1}^{+},f_{1}^{-}).$
(3.8)

∎

######  Proposition 3.3.

The family of functions $\mathbb{P}_{I}^{F}$ as in Eq. (3.4) has the following
properties:

  1. (P1)

joint probability distributions:

$\displaystyle P_{\bm{t}_{n}}(\bm{f}_{n})\geq
0,\qquad\sum_{\bm{f}_{n}}P_{\bm{t}_{n}}(\bm{f}_{n})=1;$

  2. (P2)

bi-probability bounding:

$|Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}^{-}_{n})|\leq\sqrt{P_{\bm{t}_{n}}(\bm{f}^{+}_{n})}\,\sqrt{P_{\bm{t}_{n}}(\bm{f}^{-}_{n})};$

  3. (P3)

causality of measurements:

$\sum_{f_{n+1}}P_{t_{n+1},t_{n},\dots,t_{1}}(f_{n+1};f_{n};\dots;f_{1})=P_{\bm{t}_{n}}(\bm{f}_{n});$

  4. (P4)

measurement inconsistency:

$\displaystyle
P_{t_{n},\ldots,\cancel{t_{j}},\ldots,t_{1}}(f_{n},\ldots,\cancel{f_{j}},\ldots,f_{1})-\sum_{f_{j}}P_{\bm{t}_{n}}(\bm{f}_{n})=\sum_{f_{j}^{+}\neq
f_{j}^{-}}Q_{\bm{t}_{n}}(f_{n},f_{n};\ldots;f_{j}^{+},f_{j}^{-};\ldots;f_{1},f_{1}).$

###### Proof.

The estimate (P2) comes from the Cauchy–Schwarz inequality for the Frobenius
inner product $(\hat{A},\hat{B})=\operatorname{tr}[\hat{A}^{\dagger}\hat{B}]$,

$\displaystyle|Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})|^{2}$
$\displaystyle=\Big{|}\Big{(}\hat{\rho}^{1/2}\prod_{j=1}^{n}\hat{P}_{t_{j}}(f_{j}^{+})\
,\
\hat{\rho}^{1/2}\prod_{j=1}^{n}\hat{P}_{t_{j}}(f^{-}_{j})\Big{)}\Big{|}^{2}$
$\displaystyle\leq\Big{(}\hat{\rho}^{1/2}\prod_{j=1}^{n}\hat{P}_{t_{j}}(f_{j}^{+})\
,\
\hat{\rho}^{1/2}\prod_{j=1}^{n}\hat{P}_{t_{j}}(f^{+}_{j})\Big{)}\Big{(}\hat{\rho}^{1/2}\prod_{j=1}^{n}\hat{P}_{t_{j}}(f_{j}^{-})\
,\ \hat{\rho}^{1/2}\prod_{j=1}^{n}\hat{P}_{t_{j}}(f^{-}_{j})\Big{)}$
$\displaystyle=Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{+})Q_{\bm{t}_{n}}(\bm{f}_{n}^{-},\bm{f}_{n}^{-}).$
(3.9)

Property (P3) follows from the causality (Q2) and the bi-consistency (Q4) of
bi-probabilities,

$\displaystyle\sum_{f_{n}}P_{\bm{t}_{n}}(\bm{f}_{n})$
$\displaystyle=\sum_{f_{n}}Q_{\bm{t}_{n}}(\bm{f}_{n},\bm{f}_{n})=\sum_{f_{n}^{+},f_{n}^{-}}Q_{\bm{t}_{n}}(f_{n}^{+},f_{n}^{-};\bm{f}_{n-1},\bm{f}_{n-1})=P_{\bm{t}_{n-1}}(\bm{f}_{n-1}),$
(3.10)

Then, the normalization of $P_{\bm{t}_{n}}$ is obtained by a repeated
application of (P3) and the non-negativity follows from positive
semidefiniteness (Q3) with
$Z(\bm{f}_{n}^{\pm})=\prod_{j=1}^{n}\delta_{f_{j}^{\pm},f_{j}}$. Finally, (P4)
is simply the bi-consistency (Q4) for the diagonal part of bi-probabilities. ∎

The relation (P4) explains why the single-trajectory picture is inadequate for
the description of quantum measurements [9]: The remainder on the right-hand-
side, that consists of a combination of “off-diagonal” parts of bi-
probabilities, is the cause for violation of the classical Kolmogorov
consistency (1.1). Consequently, the (classical) extension theorem for
probabilities $\mathbb{P}_{I}^{F}$ does not work here, and without it the
(single) trajectory interpretation for the observed measurements cannot be
valid.

###  3.2 Uniform bound for quantum bi-probabilities

While bi-consistency is trivially satisfied for $\mathbb{Q}_{I}^{F}$, see
property (Q4), uniform boundedness,

$\displaystyle\sup\big{\\{}\|Q_{\bm{t}_{n}}\|_{1}\,:\,Q_{\bm{t}_{n}}\in\mathbb{Q}_{I}^{F}\big{\\}}<\infty,$
(3.11)

is not obvious. It would be tempting to guess that the uniform bound can be
somehow derived from the normalization (Q1) coupled with some of the other
already identified properties of $\mathbb{Q}_{I}^{F}$. This turns out to not
be the case, however. By only relying on the previous properties, the best one
can do is the following (non-uniform) bound,

$\displaystyle\|Q_{\bm{t}_{n}}\|_{1}$
$\displaystyle=\sum_{\bm{f}^{+}_{n},\bm{f}^{-}_{n}}|Q_{\bm{t}_{n}}(\bm{f}^{+}_{n},\bm{f}^{-}_{n})|\leq\left(\sum_{\bm{f}_{n}}\sqrt{P_{\bm{t}_{n}}(\bm{f}_{n})}\right)^{2}\leq\left(|\Omega|^{\frac{n}{2}}\sum_{\bm{f}_{n}}P_{\bm{t}_{n}}(\bm{f}_{n})\right)^{2}=|\Omega|^{n},$
(3.12)

where we used Hölder’s inequality and properties (P1) and (P2), and
$|\Omega|\leq d$ is the number of elements in $\Omega$. As this bound depends
explicitly on the number of arguments $n$, it cannot be used for our purpose.

To show that the family of quantum bi-probabilities $\mathbb{Q}_{I}^{F}$ is
uniformly bounded, one has to supplement its mathematical properties with two
additional physical constraints. The first one is to restrict bi-probabilities
to a finite time window, i.e., hereafter, $I=[0,T]$, $0<T<\infty$. The second
necessary constraint—or rather a natural feature of quantum dynamics—is the
strong continuity of quantum evolution. The significance of both will soon
become apparent.

######  Definition 3.4 (Time mesh refinement).

Let $n,N\in\mathbb{N}$ and $\bm{t}_{n}\in I^{n}$, $\bm{\tau}_{N}\in I^{N}$ two
ordered tuples of times. We say that $\bm{\tau}_{N}$ is a refinement of
$\bm{t}_{n}$ if the following conditions hold: $N\geq n$, and there exists a
map $\iota:\\{1,2,\dots,n\\}\rightarrow\\{1,2,\dots,N\\}$ such that
$\iota(n)=N$, and

$\tau_{\iota(j)}=t_{j},\qquad j=1,\dots,n.$ (3.13)

######  Lemma 3.5.

Let $n,N\in\mathbb{N}$ with $n\leq N$, $\bm{t}_{n}\in I^{n}$, and
$\bm{\tau}_{N}\in I^{N}$ such that $\bm{\tau}_{N}$ is a refinement of
$\bm{t}_{n}$, then

$\|Q_{\bm{t}_{n}}\|_{1}\leq\|Q_{\bm{\tau}_{N}}\|_{1}.$ (3.14)

###### Proof.

Denote, for all $j=1,\dots,n$, $\phi^{\pm}_{\iota(j)}\equiv f^{\pm}_{j}$,
where $\tau_{\iota(j)}=t_{j}$ as per Definition 3.4. Then, using bi-
consistency (Q4) we write

$\displaystyle Q_{\bm{t}_{n}}(\bm{f}^{+}_{n},\bm{f}^{-}_{n})$
$\displaystyle=Q_{\tau_{\iota(n)},\dots,\tau_{\iota(1)}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})=Q_{\tau_{\iota(n)},\ldots,\tau_{\iota(1)}}(\phi^{+}_{\iota(n)},\phi^{-}_{\iota(n)};\ldots;\phi^{+}_{\iota(1)},\phi^{-}_{\iota(1)})$
$\displaystyle=\sum_{\phi^{\pm}_{j}:j\notin\iota(\\{1,\dots,n\\})}Q_{\bm{\tau}_{N}}(\bm{\phi}_{N}^{+},\bm{\phi}_{N}^{-}),$
(3.15)

which leads to the claimed property,

$\displaystyle\|Q_{\bm{t}_{n}}\|_{1}$
$\displaystyle=\sum_{\bm{f}^{\pm}_{n}}|Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})|=\sum_{\phi_{j}^{\pm}:j\in\iota(\\{1,\ldots,n\\})}\bigg{|}\sum_{\phi^{\pm}_{j}:j\notin\iota(\\{1,\ldots,n\\})}Q_{\bm{\tau}_{N}}(\bm{\phi}_{N}^{+},\bm{\phi}_{N}^{-})\bigg{|}$
$\displaystyle\leq\sum_{\phi_{j}^{\pm}:j\in\iota(\\{1,\ldots,n\\})}\sum_{\phi^{\pm}_{j}:j\notin\iota(\\{1,\ldots,n\\})}|Q_{\bm{\tau}_{N}}(\bm{\phi}_{N}^{+},\bm{\phi}_{N}^{-})|=\sum_{\bm{\phi}_{N}^{\pm}}|Q_{\bm{\tau}_{N}}(\bm{\phi}_{N}^{+},\bm{\phi}_{N}^{-})|=\|Q_{\bm{\tau}_{N}}\|_{1}.$

∎

######  Proposition 3.6.

Let $\mathbb{Q}^{F}_{[0,T]}$ be the family of quantum bi-probabilities
associated with the time interval $I=[0,T]$ and the observable $F$ in the
$d$-dimensional Hilbert space $\mathcal{H}$. Then

$\sup\left\\{\|Q_{\bm{t}_{n}}\|_{1}\,:\,Q_{\bm{t}_{n}}\in\mathbb{Q}^{F}_{[0,T]}\right\\}\leq
d^{2}\,{\exp}\Big{[}2(d-1)\int_{0}^{T}\\!\\!\|\hat{H}(s)\|_{\mathrm{op}}\,\mathrm{d}s\Big{]}<\infty,$
(3.16)

where $\hat{H}(t)$ is the Hamiltonian of the system, and
$\|\cdot\|_{\mathrm{op}}$ the operator norm. That is, the family is uniformly
bounded in the sense of assumption (B2).

###### Proof.

To show that the whole family is bounded, we will proceed by first finding a
uniform bound for its individual members. Without loss of generality, we will
hereafter assume that the PVM $\\{\hat{P}(f)\\}_{f\in\Omega}$ representing the
observable $F$ is composed entirely of rank-$1$ projectors,
$\hat{P}_{t}(f)=\hat{U}_{0,t}|f\rangle\langle f|\hat{U}_{t,0}$, where
$\\{\ket{f}\\}_{f\in\Omega}$ is a complete orthonormal basis of $\mathcal{H}$
and $|\Omega|=d$. This can be done without loss of generality. Indeed, if
$\mathbb{Q}_{I}^{F}$ associated with such a non-degenerate observable is
bounded, then the family associated with any degenerate observable $G$
represented by a PVM $\\{\hat{P}(g)\\}_{g\in\Omega^{\prime}}$ such that
$\hat{P}(g)=\sum_{f\in\omega(g)}|f\rangle\langle f|$,

$\displaystyle\mathbb{Q}_{I}^{G}$
$\displaystyle=\Big{\\{}Q^{G}_{\bm{t}_{n}}(\bm{g}_{n}^{+},\bm{g}_{n}^{-})=\sum_{f_{n}^{\pm}\in\omega(g_{n}^{\pm})}\cdots\sum_{f_{1}^{\pm}\in\omega(g_{1}^{\pm})}Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})\
:\ n\in\mathbb{N},\,t_{1}<\ldots<t_{n},t_{1},\ldots,t_{n}\in I\Big{\\}},$
(3.17)

will also be uniformly bounded,

$\displaystyle\sup_{Q_{\bm{t}_{n}}^{G}\in\mathbb{Q}_{I}^{G}}\|Q_{\bm{t}_{n}}^{G}\|_{1}$
$\displaystyle=\sup_{Q_{\bm{t}_{n}}^{G}\in\mathbb{Q}_{I}^{G}}\sum_{\bm{g}_{n}^{\pm}}\Big{|}\sum_{f_{n}^{\pm}\in\omega(g_{n}^{\pm})}\cdots\sum_{f_{1}^{\pm}\in\omega(g_{1}^{\pm})}Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})\Big{|}$
$\displaystyle\leq\sup_{Q_{\bm{t}_{n}}\in\mathbb{Q}_{I}^{F}}\sum_{\bm{g}_{n}^{\pm}}\sum_{f_{n}^{\pm}\in\omega(g_{n}^{\pm})}\cdots\sum_{f_{1}^{\pm}\in\omega(g^{\pm}_{1})}|Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})|$
$\displaystyle=\sup_{Q_{\bm{t}_{n}}\in\mathbb{Q}_{I}^{F}}\sum_{\bm{f}_{n}^{\pm}}|Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})|=\sup_{Q_{\bm{t}_{n}}\in\mathbb{Q}_{I}^{F}}\|Q_{\bm{t}_{n}}\|_{1}.$
(3.18)

Therefore, in what follows we will be dealing with quantum bi-probabilities
for which Eq. (3.3) reads

$Q_{\bm{t}_{n}}(\bm{f}^{+}_{n},\bm{f}^{-}_{n})=\delta_{f_{n}^{+}f_{n}^{-}}\sum_{f_{0}}\langle
f_{0}^{+}|\hat{\rho}|f_{0}^{-}\rangle\prod_{j=0}^{n-1}\langle
f^{+}_{j+1}|\hat{U}_{t_{j+1},t_{j}}|f_{j}^{+}\rangle\,\langle
f^{-}_{j+1}|\hat{U}_{t_{j+1},t_{j}}|f_{j}^{-}\rangle^{*}.$ (3.19)

In the next step, starting from a given time partition
$\bm{t}_{n}=(t_{n},\ldots,t_{1})$ designating the chosen member bi-probability
$Q_{\bm{t}_{n}}$, we define a sequence of its refinements
$(\bm{\tau}_{N})_{N=N_{0}}^{\infty}$. The initial $N_{0}>n$ is chosen in such
a way that, for every $j=1,\ldots,n-1$, there exists $k_{j}\in\mathbb{N}$ for
which

$\displaystyle t_{j-1}<\frac{(k_{j}-1)}{N_{0}}t_{n}\leq
t_{j}\leq\frac{k_{j}}{N_{0}}t_{n}<t_{j+1}.$ (3.20)

This $N_{0}$ always exists because the rational numbers form a dense set of
the real axis. Then, define the time partitioning scheme $\bm{\tau}_{N}$ for
$N\geq N_{0}$ with the following procedure: (1) Assign a uniform partition,

$\displaystyle\tau_{k}$ $\displaystyle=t_{n}k/N,\quad k=1,\ldots,N.$ (3.21)

(2) Identify the times that are closest to the original partitioning,

$\displaystyle K(N)=\\{k_{j}\ :\ t_{n}(k_{j}-1)/N\leq t_{j}\leq
t_{n}k_{j}/N\\}.$ (3.22)

(3) “Snap” the identified times to the originals they approximate,

$\displaystyle\tau_{k_{j}}=t_{n}k_{j}/N\,\to\,\tau_{k_{j}}=t_{j},\quad
k_{j}\in K(N),$ (3.23)

which makes the resultant $\bm{\tau}_{N}$ a refinement of $\bm{t}_{n}$. The
refinements defined this way are mostly uniform:
$\Delta\tau_{k}=\tau_{k+1}-\tau_{k}=t_{n}/N$ for all $k\notin K(N)$, except
for all $k\in K(N)$ such that

$\displaystyle\Delta\tau_{k}=\frac{t_{n}}{N}+h_{k}\leq\frac{2t_{n}}{N},\quad\Delta\tau_{k-1}=\frac{t_{n}}{N}-h_{k}\leq\frac{t_{n}}{N},\quad
0\leq h_{k}\leq\frac{t_{n}}{N}.$ (3.24)

We then apply the refinements in a sequence according to Lemma 3.5,

$\displaystyle\|Q_{\bm{t}_{n}}\|_{1}\leq\|Q_{\bm{\tau}_{N_{0}}}\|_{1}\leq\|Q_{\bm{\tau}_{2N_{0}}}\|_{1}\leq\cdots\leq\|Q_{\bm{\tau}_{2^{k}N_{0}}}\|_{1}\leq\cdots\leq\lim_{N\to\infty}\|Q_{\bm{\tau}_{N}}\|_{1}.$
(3.25)

In the process we have determined that $Q_{\bm{t}_{n}}$ is bounded by the
limit of infinitely refined time mesh. Since for any pair of sequences
$(a_{N})_{N=N_{0}}^{\infty}$ and $(b_{N})_{N=N_{0}}^{\infty}$ such that
$a_{N}\leq b_{N}$ for all $N\geq N_{0}$ one has
$\lim_{N\to\infty}a_{N}\leq\lim_{N\to\infty}b_{N}$, we are now allowed to
estimate $\|Q_{\bm{\tau}_{N}}\|_{1}$ before taking the limit. To this end, we
define the transition probabilities,

$\displaystyle p^{\text{stay}}_{t,t^{\prime}}(f)$ $\displaystyle:=|\langle
f|\hat{U}_{t,t^{\prime}}|f\rangle|^{2},\qquad
p^{\text{leave}}_{t,t^{\prime}}(f):=1-p^{\text{stay}}_{t,t^{\prime}}(f)=\sum_{f^{\prime}\neq
f}|\langle f^{\prime}|\hat{U}_{t,t^{\prime}}|f\rangle|^{2}.$ (3.26)

The two functions satisfy $0\leq p^{\text{stay/leave}}_{t,t^{\prime}}(f)\leq
1$, and their square roots can be estimated from above (using the Taylor
theorem in the case of the “leave” probability),

$\displaystyle\delta_{f_{j+1}^{\pm},f_{j}^{\pm}}|\langle
f_{j+1}^{\pm}|\hat{U}_{\tau_{j+1},\tau_{j}}|f_{j}^{\pm}\rangle|$
$\displaystyle=\delta_{f_{j+1}^{\pm},f_{j}^{\pm}}\sqrt{p_{\tau_{j+1},\tau_{j}}^{\text{stay}}(f_{j}^{\pm})}\leq\delta_{f_{j+1}^{\pm},f_{j}^{\pm}};$
(3.27a) $\displaystyle(1\\!-\\!\delta_{f_{j+1}^{\pm},f_{j}^{\pm}})|\langle
f_{j+1}^{\pm}|\hat{U}_{\tau_{j+1},\tau_{j}}|f_{j}^{\pm}\rangle|$
$\displaystyle\leq(1\\!-\\!\delta_{f_{j+1}^{\pm},f_{j}^{\pm}})\sqrt{p^{\text{leave}}_{\tau_{j+1},\tau_{j}}(f_{j}^{\pm})}$
$\displaystyle=(1\\!-\\!\delta_{f_{j+1}^{\pm},f_{j}^{\pm}})\Big{[}\Delta\tau_{j}\sqrt{\langle
f_{j}^{\pm}|\hat{H}(\tau_{j})^{2}|f_{j}^{\pm}\rangle-\langle
f_{j}^{\pm}|\hat{H}(\tau_{j})|f_{j}^{\pm}\rangle^{2}}$
$\displaystyle\phantom{=(1\\!-\\!\delta_{f_{j}^{\pm},f_{j-1}^{\pm}})}+R_{\tau_{j+1}}^{(1)}(f_{j}^{\pm})\Big{]}$
$\displaystyle\leq(1\\!-\\!\delta_{f_{j+1}^{\pm},f_{j}^{\pm}})\Big{(}\Delta\tau_{j}\|\hat{H}(\tau_{j})|f_{j}^{\pm}\rangle\|+\big{|}R_{\tau_{j+1}}^{(1)}(f_{j}^{\pm})\big{|}\Big{)}$
$\displaystyle\leq(1\\!-\\!\delta_{f_{j+1}^{\pm},f_{j}^{\pm}})\left(v(\tau_{j})\Delta\tau_{j}+\frac{1}{2}M\Delta\tau_{j}^{2}\right)$
$\displaystyle\leq(1\\!-\\!\delta_{f_{j+1}^{\pm},f_{j}^{\pm}})\left(v(\tau_{j})\Delta\tau_{j}+\frac{2Mt_{n}^{2}}{N^{2}}\right),$
(3.27b)

where $0<M<\infty$,
$v(t)=\sup_{|f\rangle}\|\hat{H}(t)|f\rangle\|=\|\hat{H}(t)\|_{\text{op}}$, and
$\Delta\tau_{j}^{2}\leq 4t_{n}^{2}/N^{2}$ which covers the possibility $j\in
K(N)$ and $\Delta\tau_{j}=t_{n}/N+h_{j}$. These estimates lead to

$\displaystyle\|Q_{\bm{\tau}_{N}}\|_{1}$
$\displaystyle\leq\sum_{f_{0}^{\pm}}|\langle
f_{0}^{+}|\hat{\rho}|f_{0}^{-}\rangle|\prod_{j=0}^{N-1}\sum_{f_{j+1}^{+}}\left[\delta_{f_{j+1}^{+},f_{j}^{+}}+(1-\delta_{f_{j+1}^{+},f_{j}^{+}})\left(v(\tau_{j})\Delta\tau_{j}+\frac{2Mt_{n}^{2}}{N^{2}}\right)\right]$
$\displaystyle\phantom{\sum_{f_{0}^{\pm}}|\langle
f_{0}^{+}|\hat{\rho}|f_{0}^{-}\rangle|}\times\prod_{k=0}^{N-1}\sum_{f_{k+1}^{-}}\left[\delta_{f_{k+1}^{-},f_{k}^{-}}+(1-\delta_{f_{k+1}^{-},f_{k}^{-}})\left(v(\tau_{k})\Delta\tau_{k}+\frac{2Mt_{n}^{2}}{N^{2}}\right)\right]$
$\displaystyle=\left(\sum_{f_{0}^{\pm}}|\langle
f_{0}^{+}|\hat{\rho}|f_{0}^{-}\rangle|\right)\prod_{j=0}^{N-1}\left[1+(d-1)\left(v(\tau_{j})\Delta\tau_{j}+\frac{2Mt_{n}^{2}}{N^{2}}\right)\right]^{2}$
$\displaystyle\leq
d^{2}\prod_{j=0}^{N-1}\left[1+(d-1)\left(v(\tau_{j})\Delta\tau_{j}+\frac{2Mt_{n}^{2}}{N^{2}}\right)\right]^{2}$
$\displaystyle\leq
d^{2}\left[1+(d-1)\frac{2Mt_{n}^{2}}{N^{2}}\right]^{2N}\prod_{j=0}^{N-1}\left[1+(d-1)v(\tau_{j})\Delta\tau_{j}\right]^{2}.$
(3.28)

In this form, the limit $N\to\infty$, which simultaneously implies
$\Delta\tau_{j}\to 0$, results in

$\displaystyle\|Q_{\bm{t}_{n}}\|_{1}$
$\displaystyle\leq\lim_{N\to\infty}\|Q_{\bm{\tau}_{N}}\|_{1}\leq
d^{2}{\exp}\Bigg{[}2(d-1)\lim_{\begin{subarray}{c}N\to\infty;\\\
\Delta\tau_{j}\to
0\end{subarray}}\sum_{j=0}^{N-1}v(\tau_{j})\Delta\tau_{j}\Bigg{]}=d^{2}\,\mathrm{e}^{2(d-1)\int_{0}^{t_{n}}v(s)\mathrm{d}s}.$
(3.29)

Since the above bound for individual bi-probabilities $Q_{t_{n},\ldots,t_{1}}$
depends only on the latest time $t_{n}$, the bound for the whole family is
identified without further difficulty,

$\displaystyle\sup\left\\{\|Q_{\bm{t}_{n}}\|_{1}\,:\,Q_{\bm{t}_{n}}\in\mathbb{Q}_{[0,T]}^{F}\right\\}$
$\displaystyle\leq
d^{2}\,{\exp}\Big{[}2(d-1)\int_{0}^{T}\\!\\!\|\hat{H}(s)\|_{\text{op}}\,\mathrm{d}s\Big{]}<\infty,$
(3.30)

so that the claim is proven. ∎

We have therefore shown that the family of quantum bi-probabilities introduced
in Eq. (3.3) satisfies both assumptions (B1) (by Proposition 3.2) and (B2) (by
Proposition 3.6) of the abstract extension result proven in Section 2, i.e.
Theorem 2.2; this concludes the proof of Theorem 3.1.

##  4 Discussion

###  4.1 The extension theorem and the choice of time interval

We have shown that, in quantum mechanics, the extension theorem works for
families of bi-probabilities confined to a specific time interval $I=[0,T]$.
Note, however, that the members of the family associated with one interval
also belong, in general, to other families associated with different
intervals; in particular, if a given bi-probability $Q_{\bm{t}_{n}}$ is a
member of $\mathbb{Q}_{[0,T]}^{F}$, then so is
$Q_{\bm{t}_{n}}\in\mathbb{Q}_{[0,T^{\prime}]}^{F}$ when $T<T^{\prime}$.
Therefore, this $Q_{\bm{t}_{n}}$ is a restriction of the master bi-measure
$\mathcal{Q}_{[0,T]}$ that extends $\mathbb{Q}_{[0,T]}^{F}$, and
simultaneously, of $\mathcal{Q}_{[0,T^{\prime}]}$ that extends
$\mathbb{Q}_{[0,T^{\prime}]}^{F}$,

$\displaystyle Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})$
$\displaystyle=\iint\Big{(}\prod_{j=1}^{n}\delta_{f^{+}(t_{j}),f_{j}^{+}}\,\delta_{f^{-}(t_{j}),f_{j}^{-}}\Big{)}\mathcal{Q}_{[0,T]}[f^{+},f^{-}][\mathcal{D}f^{+}][\mathcal{D}f^{-}]$
$\displaystyle=\iint\Big{(}\prod_{j=1}^{n}\delta_{f^{+}(t_{j}),f_{j}^{+}}\,\delta_{f^{-}(t_{j}),f_{j}^{-}}\Big{)}\mathcal{Q}_{[0,T^{\prime}]}[f^{+},f^{-}][\mathcal{D}f^{+}][\mathcal{D}f^{-}].$
(4.1)

This begs the question: what is the relation between the two master bi-
measures $\mathcal{Q}_{[0,T]}$ and $\mathcal{Q}_{[0,T^{\prime}]}$?

We find the answer by considering the relations between the corresponding
families of bi-probabilities. Observe that the bi-consistent family associated
with a shorter interval is a subset of the family corresponding to the longer
interval, $\mathbb{Q}^{F}_{[0,T]}\subseteq\mathbb{Q}^{F}_{[0,T^{\prime}]}$.
Moreover, the union of both families is a bi-consistent family itself and it
is simply equal to the larger family,

$\displaystyle\mathbb{Q}_{[0,T]}^{F}\cup\mathbb{Q}_{[0,T^{\prime}]}^{F}$
$\displaystyle=\\{Q_{\bm{\tau}_{k}}(\bm{f}_{k}^{+},\bm{f}^{-}_{k})\ :\
\forall_{k\geqslant 1}\
T\geqslant\tau_{k}\geqslant\cdots\geqslant\tau_{1}\geqslant
0\,\lor\,T^{\prime}\geqslant\tau_{k}\geqslant\cdots\geqslant\tau_{1}\geqslant
0\\}$ $\displaystyle=\\{Q_{\bm{\tau}_{k}}(\bm{f}_{k}^{+},\bm{f}^{-}_{k})\ :\
\forall_{k\geqslant 1}\
T^{\prime}\geqslant\tau_{k}\geqslant\cdots\geqslant\tau_{1}\geqslant
0\\}=\mathbb{Q}^{F}_{[0,T^{\prime}]},$ (4.2)

implying that the master bi-measure $\mathcal{Q}_{[0,T^{\prime}]}$ is an
extension to $\mathbb{Q}_{[0,T^{\prime}]}^{F}$ as well as
$\mathbb{Q}_{[0,T]}^{F}$. Therefore, given that the extension of a family is
unique, it follows that $\mathcal{Q}_{[0,T]}$ has to be the restriction of
$\mathcal{Q}_{[0,T^{\prime}]}$.

This reasoning can be iterated any number of times; given a sequence of
increasing time intervals, $I_{1}\subset I_{2}\subset\cdots\subset I_{N}$,
with $I_{n}=[0,T_{n}]$ and $0<T_{1}<\cdots<T_{N}$, we have an increasing
sequence of bi-consistent families,
$\mathbb{Q}^{F}_{I_{1}}\subseteq\cdots\subseteq\mathbb{Q}^{F}_{I_{N}}$, such
that $\bigcup_{n=1}^{N}\mathbb{Q}^{F}_{I_{n}}=\mathbb{Q}^{F}_{I_{N}}$. As long
as the largest time $T_{N}$ is a finite number, the bi-consistent families are
uniformly bounded and they extend to the same master bi-measure
$\mathcal{Q}_{I_{N}}$ (modulo an appropriate restriction to the time interval
corresponding to a given family). Therefore, since $N$ can be arbitrary
large—and thus, so can $T_{N}$—we can state that a family
$\mathbb{Q}^{F}=\lim_{N\to\infty}\bigcup_{n=1}^{N}\mathbb{Q}^{F}_{I_{n}}$
extends to the master bi-measure $\mathcal{Q}$ such that any
$\mathcal{Q}_{[0,T]}$ with a finite $T$ is its restriction.

Note that this result cannot be interpreted to mean that
$\mathbb{Q}^{F}=\mathbb{Q}^{F}_{[0,\infty)}$ or
$\mathcal{Q}=\mathcal{Q}_{[0,\infty)}$. Indeed, a bi-consistent family
associated with an interval $I$ is extendable only if it is uniformly bounded,
which is not the case when $I$ is not compact, e.g., for $I=[0,\infty)$ the
bound
$d^{2}{\exp}[2(d-1)\int_{0}^{\infty}\|\hat{H}(s)\|_{\mathrm{op}}\mathrm{d}s]$
generally diverges. In fact, the inability to bound, and thus to extend,
quantum bi-probabilities on $[0,\infty)$ in the general case could be regarded
as a desirable feature rather than some limitation of the theory. If the
master bi-probability $\mathcal{Q}_{[0,\infty)}$ would exist for any quantum
system, then its single-time restriction,

$\displaystyle Q_{\infty}(f,f^{\prime})$
$\displaystyle=\delta_{f,f^{\prime}}P_{\infty}(f)=\delta_{f,f^{\prime}}\lim_{T\to\infty}\operatorname{tr}\big{[}\hat{P}_{T}(f)\hat{\rho}\big{]},$
(4.3)

would also be sensible. However, a simple physical analysis shows that such a
thing cannot be the case. We can illustrate this point with a basic example.
Consider a $2$-dimensional system (qubit) with $\hat{F}=\hat{\sigma}_{z}$,
$\hat{H}=\omega\hat{\sigma}_{x}/2$ and $\hat{\rho}=|{+1}\rangle\langle{+1}|$:
then, for as long as time $T$ is a finite number (no matter how large or small
it is), one gets the probability that has a well-defined physical and
mathematical meaning,

$\displaystyle Q_{T}(\pm 1,\pm 1)$ $\displaystyle=\frac{1\pm\cos\omega T}{2};$
(4.4)

however, of course, this ceases to be the case when $T\to\infty$, as
$\lim_{T\to\infty}Q_{T}$ does not exist. To make an infinite time a viable
consideration, one would have to postulate some additional physical
constraints on the system.

To sum up, as far as the extension theorem goes, a subtle but fundamental
distinction between “arbitrarily large time” and “infinite time” holds. On the
other hand, this implies that every property of quantum system which is
formulated in terms of those of the underlying master measure only makes sense
on finite time windows. For example, if one decides to deem a quantum system
“classical” depending on whether the corresponding master measure
$\mathcal{Q}_{I}[f^{+},f^{-}]$ reduces to an actual, positive-valued measure
on single trajectories (i.e.,
$\mathcal{Q}_{I}[f,f^{\prime}]=\delta[f-f^{\prime}]\,\mathcal{P}_{I}[f]$),
then one can only speak about classicality in finite times. Broadly speaking,
this is reminiscent of the fact that quantum non-Markovianity, in its diverse
declinations, is undecidable unless a finite time interval is selected a
priori. [15, 16]

###  4.2 Quantum bi-probabilities associated with multiple non-commuting
observables

Up to this point we have considered quantum bi-probabilities associated with a
single observable. As stated previously, $\mathbb{Q}_{I}^{F}$ suffices for the
purpose of employing the bi-probability parameterization in contexts involving
only $F$ (and observables that commute with it), including the sequential
measurements of $F$. A natural question is whether the bi-probability
formalism can be generalized to handle multiple non-commuting observables as
well, so that it could parameterize, among others, multi-observable
measurements.

Let $\\{F\\}$ be a collection of observables, each represented by a
corresponding PVM,

$\displaystyle\big{\\{}\hat{P}^{F}(f)\big{\\}}_{f\in\Omega(F)}\quad\text{where
$F\in\\{F\\}$ and $|\Omega(F)|\leq d$.}$ (4.5)

Then define the family associated with time interval $I$ and the collection
$\\{F\\}$,

$\displaystyle\mathbb{Q}_{I|\rho}^{\\{F\\}}$
$\displaystyle=\big{\\{}Q^{\bm{F}_{n}}_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-}|\rho)\
:\ n\in\mathbb{N},\,0=t_{0}<t_{1}<\cdots<t_{n},\,\forall_{j=1,\ldots,n}\
t_{j}\in I,\,F_{j}\in\\{F\\}\big{\\}},$ (4.6)

with its members given by

$\displaystyle
Q_{\bm{t}_{n}}^{\bm{F}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-}|\rho):=\operatorname{tr}\Big{[}\Big{(}\prod_{j=n}^{1}\hat{P}^{F_{j}}_{t_{j}}(f_{j}^{+})\Big{)}\hat{\rho}\Big{(}\prod_{j=1}^{n}\hat{P}^{F_{j}}_{t_{j}}(f_{j}^{-})\Big{)}\Big{]}.$
(4.7)

These functions can be considered as bi-probability distributions, as it is
straightforward to check that they share all the properties 3.2 of the single-
observable quantum bi-probabilities; crucially, this includes the bi-
consistency of $\mathbb{Q}_{I|\rho}^{\\{F\\}}$. Moreover, the family
$\mathbb{P}_{I|\rho}^{\\{F\\}}$ composed of the diagonal parts of
$Q_{\bm{t}_{n}}^{\bm{F}_{n}}$,

$\displaystyle P_{\bm{t}_{n}}^{\bm{F}_{n}}(\bm{f}_{n}|\rho)$
$\displaystyle:=Q_{\bm{t}_{n}}^{\bm{F}_{n}}(\bm{f}_{n},\bm{f}_{n}|\rho),$
(4.8)

has the properties 3.3, and thus, it constitutes the standard quantum
mechanical description for the sequential projective measurements of multiple
observables. Finally, it can also be shown that the bi-probabilities belonging
to $\mathbb{Q}_{I|\rho}^{\\{F\\}}$ are sufficient for parameterizing all other
contexts involving observables $\\{F\\}$ [10, 12]. Therefore, the simple
answer to the question from before is that, indeed, the bi-probability
formalism is not restricted to the case of single observable.

However, even though $\mathbb{Q}_{I|\rho}^{\\{F\\}}$ is a bi-consistent
family, the matter of its extension to a multi-observable master bi-measure is
not fully resolved at this moment—the issue of uniform boundedness turns out
to be problematic. The method used in the single-observable case cannot be
directly applied to the multi-observable one, at least not without introducing
some additional structure on top of $\mathbb{Q}^{\\{F\\}}_{I|\rho}$. To better
explain why this is the case, we first introduce the collection of unitary
basis transformations $\\{\hat{U}_{F}\\}$ corresponding to each observable in
$\\{F\\}$. That is, given the eigenbasis of $F\in\\{F\\}$,
$\\{|f_{k}\rangle\\}_{k=1}^{d}$, the operators of the PVM representing this
observable can be written in the following form:

$\displaystyle\hat{P}_{t}^{F}(f)$
$\displaystyle=\sum_{k=1}^{d}\delta\big{(}\langle
f_{k}|\hat{F}|f_{k}\rangle-f\big{)}\,\hat{U}_{0,t}|f_{k}\rangle\langle
f_{k}|\hat{U}_{t,0}=\sum_{k=1}^{d}\delta\big{(}\langle
k|\hat{U}_{F}^{\dagger}\hat{F}\hat{U}_{F}|k\rangle-f\big{)}\hat{U}_{0,t}\hat{U}_{F}|k\rangle\langle
k|\hat{U}_{F}^{\dagger}\hat{U}_{t,0},$ (4.9)

where the unitary operator $\hat{U}_{F}$ transforms the chosen reference basis
$\\{|k\rangle\\}_{k=1}^{d}$ into $\\{|f_{k}\rangle\\}_{k=1}^{d}$. Then, making
use of our previous methods whenever possible, we arrive at the following
estimation for the norm of a multi-observable family of bi-probabilities:

$\displaystyle\|Q_{\bm{t}_{n}}^{\bm{F}_{n}}\|_{1}$
$\displaystyle\leq\left(\sum_{k_{n},\ldots,k_{0}=1}^{d}\sqrt{\rho(k_{0})}\prod_{j=1}^{n}|\langle
k_{j}|\hat{U}_{F_{j}}^{\dagger}\hat{U}_{t_{j},t_{j-1}}\hat{U}_{F_{j-1}}|k_{j-1}\rangle|\right)^{2}$
$\displaystyle=\left(\sum_{k_{n},\ldots,k_{0}=1}^{d}\sqrt{\rho(k_{0})}\prod_{j=1}^{n}|\langle
k_{j}|\hat{U}_{F_{j}}^{\dagger}\hat{U}_{t_{j},t_{j-1}}\Big{(}\hat{U}_{F_{j}}\sum_{k^{\prime}=1}^{d}|k^{\prime}\rangle\langle
k^{\prime}|\hat{U}_{F_{j}}^{\dagger}\Big{)}\hat{U}_{F_{j-1}}|k_{j-1}\rangle|\right)^{2}$
$\displaystyle\leq\left(\sum_{k_{n},\ldots,k_{0}=1}^{d}\sum_{k_{n}^{\prime},\ldots,k_{1}^{\prime}=1}^{d}\sqrt{\rho(k_{0})}\prod_{j=1}^{n}|\langle
k_{j}|\hat{U}_{F_{j}}^{\dagger}\hat{U}_{t_{j},t_{j-1}}\hat{U}_{F_{j}}|k_{j}^{\prime}\rangle|\,|\langle
k_{j}^{\prime}|\hat{U}_{F_{j}}^{\dagger}\hat{U}_{F_{j-1}}|k_{j-1}\rangle|\right)^{2}$
$\displaystyle\leq\left(\sum_{k_{n},\ldots,k_{0}=1}^{d}\sqrt{\rho(k_{0})}\prod_{j=1}^{n}\mathrm{e}^{(d-1)\int_{t_{j-1}}^{t_{j}}\|\hat{H}(s)\|_{\mathrm{op}}\mathrm{d}s}|\langle
k_{j}|\hat{U}_{F_{j}}^{\dagger}\hat{U}_{F_{j-1}}|k_{j-1}\rangle|\right)^{2}$
$\displaystyle\leq
d^{2}\mathrm{e}^{2(d-1)\int_{I}\|\hat{H}(s)\|_{\mathrm{op}}\mathrm{d}s}\left(\sum_{k_{n},\ldots,k_{0}=1}^{d}\prod_{j=1}^{n}|\langle
k_{j}|\hat{U}_{F_{j}}^{\dagger}\hat{U}_{F_{j-1}}|k_{j-1}\rangle|\right)^{2},$
(4.10)

where $\hat{U}_{F_{0}}=\hat{U}_{\rho}$ such that
$\hat{\rho}\hat{U}_{\rho}|k\rangle=\rho(k)\hat{U}_{\rho}|k\rangle$. If there
were only one observable (or if all observables commuted with other), then
$\hat{U}_{F_{j}}^{\dagger}\hat{U}_{F_{j-1}}=\hat{1}$ for all $j$ and one would
recover the single-observable uniform bound (3.16). In general, $|\langle
k_{j}|\hat{U}_{F_{j}}^{\dagger}\hat{U}_{F_{j-1}}|k_{j-1}\rangle|$ is greater
than $\delta_{k_{j},k_{j-1}}$, and in extreme cases it may even reach a
$k_{j},k_{j-1}$-independent value $1/\sqrt{d}$, which then leads to an
additional factor $d^{2}$. Consequently, the above estimate depends explicitly
on the number of arguments $n$ of the distribution, and it is thus unfit for
obtaining a uniform bound for the family.

Of course, if one cannot demonstrate that $\mathbb{Q}_{I|\rho}^{\\{F\\}}$ is
uniformly bounded, then one cannot apply here the extension theorem to find
the corresponding master bi-measure. Nevertheless, there is still a way to
show that every multi-observable bi-probability
$Q_{\bm{t}_{n}}^{\bm{F}_{n}}\in\mathbb{Q}_{I|\rho}^{\\{F\\}}$ is a restriction
of a certain form of master bi-measure (compare with Section 4.1). To this end
we introduce two new notions. First are the generic bi-probabilities,

$\displaystyle Q_{\bm{\hat{U}}_{n}}(\bm{k}^{+}_{n},\bm{k}^{-}_{n})$
$\displaystyle:=\delta_{k_{n}^{+},k_{n}^{-}}\delta_{k_{1}^{+},k_{1}^{-}}\operatorname{tr}\Big{[}\Big{(}\prod_{j=n}^{1}\hat{U}_{j}|k_{j}^{+}\rangle\langle
k_{j}^{+}|\hat{U}_{j}^{\dagger}\Big{)}\Big{(}\prod_{j=1}^{n}\hat{U}_{j}|k_{j}^{-}\rangle\langle
k_{j}^{-}|\hat{U}_{j}^{\dagger}\Big{)}\Big{]},$ (4.11)

These are indeed bi-probabilities because the family of such functions,
together with the family of their diagonal parts, fulfills the properties 3.2
and 3.3. Also, this is the most generic form of quantum bi-probability, as any
multi-observable bi-probability (including the single-observable variants) can
be expressed as a linear combination of $Q_{\bm{\hat{U}}_{n}}$,

$\displaystyle
Q_{\bm{t}_{n}}^{\bm{F}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-}|\rho)$
$\displaystyle=\sum_{\bm{k}_{n+1}^{\pm}}\prod_{j=1}^{n}\delta\big{(}\langle
k_{j+1}^{+}|\hat{U}_{F_{j}}^{\dagger}\hat{F}_{j}\hat{U}_{F_{j}}|k_{j+1}^{+}\rangle-
f_{j}^{+}\big{)}\delta\big{(}\langle
k_{j+1}^{-}|\hat{U}_{F_{j}}^{\dagger}\hat{F}_{j}\hat{U}_{F_{j}}|k_{j+1}^{-}\rangle-
f_{j}^{-}\big{)}$
$\displaystyle\phantom{=}\times\sqrt{\rho(k_{1}^{+})\rho(k_{1}^{-})}\,Q_{\hat{U}_{0,t_{n}}\hat{U}_{F_{n}},\,\ldots\,,\hat{U}_{0,t_{1}}\hat{U}_{F_{1}},\,\hat{U}_{\rho}}(\bm{k}_{n+1}^{+},\bm{k}_{n+1}^{-}),$
(4.12)

where $\hat{\rho}\hat{U}_{\rho}|k\rangle=\rho(k)\hat{U}_{\rho}|k\rangle$. The
second is a path in the space of unitary transformations (i.e., the Lie group
$\operatorname{SU}(d)$),

$\displaystyle\hat{\gamma}:\
[0,1]\in\tau\mapsto\mathcal{T}\mathrm{e}^{-\mathrm{i}\int_{0}^{\tau}\hat{V}(s)\mathrm{d}s}\hat{U}_{0},$
(4.13)

where $\hat{V}(s)$ is a Hermitian operator-valued function on $[0,1]$, and
$\hat{U}_{0}$ is a fixed unitary operator. We combine the two into the family
of generic bi-probabilities associated with a path,

$\displaystyle\mathbb{Q}_{\gamma}$
$\displaystyle:=\big{\\{}Q_{\hat{\bm{U}}_{n}}(\bm{k}^{+},\bm{k}^{-})\ :\
n\in\mathbb{N},\,\forall_{j=1,\ldots,n}\
\tau_{j}\in[0,1],\,\tau_{1}<\cdots<\tau_{n},\,\hat{U}_{j}=\hat{\gamma}(\tau_{j})\big{\\}}.$
(4.14)

The family is clearly bi-consistent, and it is also uniformly bounded,

$\displaystyle\sup\Big{\\{}\big{\|}Q_{\hat{\bm{U}}_{n}}\big{\|}_{1}\ :\
Q_{\hat{\bm{U}}_{n}}\in\mathbb{Q}_{\gamma}\Big{\\}}\leqslant
d^{2}\mathrm{e}^{2(d-1)\operatorname{Length}(\gamma)},$ (4.15)

where the length of path $\gamma$ is given by

$\displaystyle\operatorname{Length}(\gamma)$
$\displaystyle:=\int_{0}^{1}\bigg{\|}\frac{\mathrm{d}}{\mathrm{d}\tau}\hat{\gamma}(\tau)\bigg{\|}_{\mathrm{op}}\mathrm{d}\tau=\int_{0}^{1}\|\hat{V}(\tau)\|_{\mathrm{op}}\,\mathrm{d}\tau.$
(4.16)

This bound is immediately obtained by noticing that $\mathbb{Q}_{\gamma}$
equals $\mathbb{Q}_{[0,T]}^{F}$ provided that $\hat{V}(s)=T\hat{H}(Ts)$,
$\hat{U}_{0}=\hat{U}_{F}$, and $\hat{\rho}=\hat{U}_{F}|k_{0}\rangle\langle
k_{0}|\hat{U}_{F}^{\dagger}$. Therefore, any family of generic bi-
probabilities associated with a path of finite length extends to a master bi-
measure $\mathcal{Q}_{\gamma}$ on the space of field pairs,

$\displaystyle\operatorname{SU}(d)\supset\operatorname{Img}(\gamma)\ni\hat{U}\mapsto\big{(}k^{+}(\hat{U}),k^{-}(\hat{U})\big{)}\in\\{1,\ldots,d\\}\times\\{1,\ldots,d\\}.$
(4.17)

Conversely, when a given generic bi-probability $Q_{\bm{\hat{U}}_{n}}$ is
found to be a member of a family associated with a certain path $\gamma$—i.e.,
there exist $\tau_{1}<\ldots<\tau_{n}$ such that
$\hat{U}_{j}=\hat{\gamma}(\tau_{j})$ for all $j=1,\ldots,n$—then this bi-
probability is a restriction of the master bi-measure $\mathcal{Q}_{\gamma}$
that extends $\mathbb{Q}_{\gamma}$,

$\displaystyle Q_{\hat{\bm{U}}_{n}}(\bm{k}^{+}_{n},\bm{k}^{-}_{n})$
$\displaystyle=\iint\Big{(}\prod_{j=1}^{n}\delta_{k^{+}(\hat{\gamma}(\tau_{j})),k^{+}_{j}}\,\delta_{k^{-}(\hat{\gamma}(\tau_{j})),k^{-}_{j}}\Big{)}\mathcal{Q}_{\gamma}[k^{+},k^{-}][\mathcal{D}k^{+}][\mathcal{D}k^{-}].$
(4.18)

Using this construction we can conclude that each multi-observable bi-
probability is a restriction of a master bi-measure. A given bi-probability
$Q_{\bm{t}_{n}}^{\bm{F}_{n}}$ decomposes into the linear combination of
generic bi-probabilities, as demonstrated in Eq. (4.12). For every constituent
bi-probability $Q_{\bm{\hat{U}}_{n}}$, there exists a path
$\gamma_{\bm{\hat{U}}_{n}}$ such that
$Q_{\bm{\hat{U}}_{n}}\in\mathbb{Q}_{\gamma_{\bm{\hat{U}}_{n}}}$. Then, the
paths $\gamma_{\bm{\hat{U}}_{n}}$ can be “stitched” together into a master
path $\Gamma$ (or, equivalently, they can be viewed as segments of $\Gamma$),
so that each
$\mathbb{Q}_{\gamma_{\bm{\hat{U}}_{n}}}\subset\mathbb{Q}_{\Gamma}$. And so,
each bi-probability $Q_{\bm{\hat{U}}_{n}}$ becomes a restriction of the
corresponding master bi-measures $\mathcal{Q}_{\gamma_{\bm{\hat{U}}_{n}}}$,
that in turn are all restrictions of the one master bi-measure
$\mathcal{Q}_{\Gamma}$ which extends $\mathbb{Q}_{\Gamma}$. Since each
constituent generic bi-probability is a restriction of $\mathcal{Q}_{\Gamma}$,
it follows that so is $Q_{\bm{t}_{n}}^{\bm{F}_{n}}$. This reasoning is a
direct generalization of the result obtained in Sec. 4.1.

The above line of argumentation is however incomplete; there are still several
yet unresolved issues and open questions that, hopefully, shall be addressed
in the future. For example, one such question is whether it is possible to
construct a single grandmaster path $\Sigma$ such that all
$Q_{\bm{t}_{n}}^{\bm{F}_{n}}\in\mathbb{Q}_{I|\rho}^{\\{F\\}}$ are restrictions
of the same $\mathcal{Q}_{\Sigma}$. Attempting to find an answer, one could
opt for a “brute force” solution where $\Sigma$ is simply obtained by
stitching together all master paths $\Gamma$ considered previously. It is
however unclear whether the corresponding $\mathbb{Q}_{\Sigma}$ would actually
be bounded, and thus, be extendable to $\mathcal{Q}_{\Sigma}$—it all depends
on whether the length of such a $\Sigma$ is finite. Moreover, even if this
solution is correct, then one still has to grapple with the problem of
uniqueness of the grandmaster bi-measure $\mathcal{Q}_{\Sigma}$. Indeed, note
that in the path-stitching method we have employed here so liberally, the
order at which the constituent paths are being connected is arbitrary.
Therefore, one can stitch together a possibly infinite number of grandmaster
paths that only differ by the ordering of master paths $\Gamma$. What are the
relations between grandmaster bi-measures corresponding to those different
orderings is yet another open question.

###  4.3 Connection to the quantum comb formalism

As mentioned in the introduction, there are several studies concerning
generalized counterparts of stochastic processes in quantum theory [2, 3, 17,
18], together with generalizations of the Kolmogorov extension theorem adapted
to such different descriptions of quantum processes. For instance, Accardi et
al. prove a generalized Kolmogorov theorem for quantum stochastic process in
the language of operator algebras, where a probability (or a correlation
function) is defined as a linear functional over sequential products of
quantum instruments. Similarly, a Kolmogorov extension theorem for the so-
called quantum combs222 From now on, the term “quantum comb” will refer both
to the quantum comb formalism and other descriptions for quantum stochastic
processes in the aforementioned references.  or quantum process matrices, was
recently shown, with a more general index set, in [2]. Those are probability
functionals on a sequence, or a mixture of sequences, of completely positive
and trace non-increasing maps on some state spaces.

In principle, there are similarities and differences between the bi-
probability and comb formulations, in both fundamental and technical aspects,
which we shall elucidate in the following. We stress, however, that the
extension theorem for combs presented in [2] and the extension theorem
presented in the current paper are different. Essentially, the extension
theorem for combs states that the consistent family of $n$-step combs (see
below) can be derived from a master comb. Equivalently, it states that there
exists a quantum stochastic process [17, 18] which reproduces the family of
$n$-step combs (equivalently, correlation kernels [18]). Here, the quantum
process is already defined in terms of evolution operator $\hat{U}_{t,0}$
generated by the Hamiltonian $\hat{H}(t)$, that is, given $\hat{U}_{t,0}$ (or
$\hat{H}(t)$) one defines a family of bi-probabilities $\mathbb{Q}_{I}^{F}$.
We proved that this family can be reconstructed from a master, complex-valued
bi-probability measure
$\mathcal{Q}[f^{+},f^{-}][\mathcal{D}f^{+}][\mathcal{D}f^{-}]$ on the space of
trajectory $t\mapsto(f^{+}(t),f^{-}(t))$.

Let us briefly discuss the construction of quantum combs and the corresponding
extension theorem. Let $\mathcal{H}$ be a finite-dimensional Hilbert space,
and $\mathcal{B}(\mathcal{H})$ the Banach space of bounded linear operators on
$\mathcal{H}.$ The state space $\mathcal{S}$ is defined as the subset
$\mathcal{S}\subset\big{\\{}\hat{\rho}\in\mathcal{B}(\mathcal{H}):\hat{\rho}=\hat{\rho}^{\dagger},\,\operatorname{tr}(\hat{\rho})=1,\,\hat{\rho}\geq
0\big{\\}}$. A completely positive (CP) and trace non-increasing map is a map
$\mathcal{M}(m):\mathcal{S}\rightarrow\mathcal{S}$ such that
$\mathcal{M}(m)\otimes\mathcal{I}$ is a positive map, where $\mathcal{I}$ is
the identity on a Hilbert space $\mathbb{C}^{d}$ of arbitrary dimension $d.$ A
sequence of CP maps can be represented by an operator
$\mathcal{M}_{t_{n},\ldots,t_{1}}(m_{n},\ldots,m_{1})=\bigotimes_{j=n}^{1}\mathcal{M}_{t_{j}}(m_{j})$
on a product space $\mathcal{S}^{n},$ where the subscript $t_{j}$ denotes the
time at which the map $\mathcal{M}_{t_{j}}(m_{j})$ acts. For a given time step
$j$, if $\mathcal{M}_{t_{j}}:=\sum_{m_{j}}\mathcal{M}_{t_{j}}(m_{j})$ is
trace-preserving, we will call it an instrument.

In this setting, a quantum comb is a positive linear functional $\mathcal{T}$
on a set of all CP maps $\mathcal{M}_{t_{n},\ldots,t_{1}}(m_{n},\ldots,m_{1})$
taking values in $[0,1]$ and such that
$\mathcal{T}[\mathcal{M}_{t_{n},\ldots,t_{1}}(m_{n},\ldots,m_{1})]=1$ if and
only if all $\mathcal{M}_{t_{j}}(m_{j})$ are identity maps. For instance, for
a sequential measurement, one can assign $\mathcal{M}_{t_{j}}(m_{j})$ as a
measurement apparatus acting at a time step $j$ and associating with an
outcome $m_{j}.$ In this sense, a quantum comb can describe the joint
probability of a sequence of outcomes $(m_{n},\ldots,m_{1}),$ given a sequence
of measurement apparati
$(\mathcal{M}_{t_{n}}(m_{n}),\ldots,\mathcal{M}_{t_{1}}(m_{1}))$: namely,

$P_{t_{n},\ldots,t_{1}}(m_{n},\ldots,m_{1})=\mathcal{T}\Big{[}\mathcal{M}_{t_{n},\ldots,t_{1}}(m_{n},\ldots,m_{1})\Big{]}.$
(4.19)

A similar idea can be extended to a probabilistic measurement by considering a
mixture of different operators
$\mathcal{M}_{t_{n},\ldots,t_{1}}(m_{n},\ldots,m_{1}).$

Since we assume finite dimensions, by Riesz’ representation theorem one can
find a trace-$1$ positive semidefinite matrix $\mathbf{Q}$ in
$M_{d^{2n}}(\mathbb{C})$ such that

$\mathcal{T}\Big{[}\mathcal{M}_{t_{n},\ldots,t_{1}}(m_{n},\ldots,m_{1})\Big{]}=(\mathbf{Q},\mathbf{M}_{t_{n},\ldots,t_{1}}(m_{n},\ldots,m_{1})),$
(4.20)

where $\mathbf{M}_{t_{n},\ldots,t_{1}}(m_{n},\ldots,m_{1})$ is a matrix
representation of $\mathcal{M}_{t_{n},\ldots,t_{1}}\\!(m_{n},\ldots,m_{1})$ in
$M_{d^{2n}}(\mathbb{C})$ and $(\cdot,\cdot)$ is an inner product on that
matrix space. Recalling the sequential measurement picture outlined above, one
can now define an observable

$\displaystyle\mathbf{X}=\sum_{m_{n},\ldots,m_{1}}x(m_{n},\ldots,m_{1})\mathbf{M}_{t_{n},\ldots,t_{1}}(m_{n},\ldots,m_{1}),$
(4.21)

where $x(m_{n},\ldots,m_{1})$ is a complex-valued function such that the
matrix $\mathbf{X}$ is Hermitian. In other words,
$\mathbf{M}_{t_{n},\ldots,t_{1}}(m_{n},\ldots,m_{1})$ represents a measurement
of the observable $\mathbf{X}$, and the average of $\mathbf{X}$ can written as

$\operatorname{E}_{\mathbf{Q}}\big{[}\mathbf{X}\big{]}=(\mathbf{Q},\mathbf{X}).$
(4.22)

This is similar to the average functional Eq. (2.8) used in the sketch of the
proof for Theorem 2.2.

The connection between the formulation of quantum combs and the bi-probability
one is made manifest by the average function (4.22): one can consider Eq.
(2.8) in a different way by defining a matrix $\mathbf{Q}$ through
$[\mathbf{Q}]_{\bm{f}_{n}^{-},\bm{f}_{n}^{+}}:=Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})^{*}$
using the trajectory set as an index set for the canonical coordinate, and
also defining $\mathbf{X}$ through
$[\mathbf{X}]_{\bm{f}_{n}^{+},\bm{f}_{n}^{-}}:=X_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-}).$
The average (2.8) can be represented as an inner product of these two
matrices, i.e.
$\operatorname{E}_{\bm{t}_{n}}[X_{\bm{t}_{n}}]=\operatorname{tr}(\mathbf{Q}^{\dagger}\cdot\mathbf{X}),$
in a similar fashion as in Eq. (4.22) in the quantum comb formulation.

On the other hand, it is worth noting that the quantum comb formalism outlined
above and the one developed in the present paper exhibit a crucial difference
in the way they pursue a generalization of the trajectory picture in quantum
theory. Let us elaborate on that point. In classical stochastic processes, one
only has single trajectories which, in a standard quantum description, can be
represented by a sequence of projection operators, possibly alternated by
evolution maps. These elementary trajectories and their mixture (and the union
of all sets of mixed trajectories) can describe all possible classical
stochastic processes; however, quantum mechanics allows processes that cannot
be represented in this way [19]. To accommodate such processes, one may relax
the projection operators to CP maps and define the so-called coherent
trajectories. This is the generalization customarily employed in the
literature, e.g. in [17, 18, 3], including the comb formulation in [2]. The
present work explores an alternative direction—to keep the notion of
elementary trajectories by keeping the intervention of the projective type,
and “doubling” the trajectories to a bi-trajectory picture. Within this
picture, one still preserves the fundamental aspect of classical mechanics but
one needs two trajectories to describe quantum phenomena. Let us remark that
in the first generalization, one can obtain two (coherent) trajectories by a
polarization technique, but they are not necessarily elementary in the
classical sense, i.e., each trajectory in the pair does not need to correspond
to a sequence of projective measurements.

Although the origins of both formulations are motivated by different
generalizations of trajectory pictures in quantum mechanics, on the collective
level they represent the same physical scenarios. This can be seen from the
definitions of the average functions, cf. Eq. (2.8) and Eq. (4.22), which show
no notions of representation dependence. Indeed, one can extend the quantum
comb formulations to cover the bi-probabilities and vice versa. For instance,
from the coherent trajectory picture, one relaxes the CP condition and allows
the interventions to be non-CP. In this sense, one can employ a bi-instrument

$\mathcal{N}_{t_{j}}(f^{+}_{j},f^{-}_{j})\hat{A}:=\hat{P}_{t_{j}}(f^{+}_{j})\hat{A}\hat{P}_{t_{j}}(f^{-}_{j}),$
(4.23)

which is non-CP but satisfies the consistency property
$\sum_{f^{+}_{j},f^{-}_{j}}\mathcal{N}_{t_{j}}(f^{+}_{j},f^{-}_{j})=\mathcal{I}$,
where $\mathcal{I}$ is an identity channel. Using this type of maps in a
similar construction to that of quantum combs, it follows that

$\mathcal{T}[\mathcal{N}_{t_{n}}(f^{+}_{n},f^{-}_{n})\otimes\cdots\otimes\mathcal{N}_{t_{1}}(f^{+}_{1},f^{-}_{1})]=Q_{t_{n},\ldots,t_{1}}(f_{n}^{+},f_{n}^{-};\ldots;f_{1}^{+},f_{1}^{-}).$
(4.24)

Note that, by allowing non-CP maps, one will lose the notion of coherent
trajectories, resolving in the appearance of bi-trajectories.

Apart from the arguments above, the corresponding generalizations of the
Kolmogorov extension theorem show clear similarities. One can observe that bi-
consistency for quantum bi-probabilities is the same as the consistency
condition for quantum combs in Ref. [2]. This simply follows from the fact
that the sum of projective operators both on the right and the left-hand sides
is an identity channel, as explicitly written in Eq. (4.23). As for the
uniform boundedness, it can be seen that the constraint
$(\mathbf{Q},\mathbf{M}_{t_{n},\ldots,t_{1}}(m_{n},\ldots,m_{1}))\leq 1$ for
the generalized extension theorem (GET) in Ref. [2] can also imply our
assumption (B2). In particular, we know from the Hölder inequality that

$0\leq(\mathbf{Q},\mathbf{M}_{t_{n},\ldots,t_{1}}(m_{n},\ldots,m_{1}))\leq\|\mathbf{Q}\|_{1}\|\mathbf{M}_{t_{n},\ldots,t_{1}}(m_{n},\ldots,m_{1})\|_{\infty}.$
(4.25)

Up to some scaling, one may expect the connection between the uniform
boundedness and the constraint
$(\mathbf{Q},\mathbf{M}_{t_{n},\ldots,t_{1}}(m_{n},\ldots,m_{1}))\leq 1$ used
for GET, and hence the connection between our results on the extension theorem
for the bi-trajectory picture and GET for the coherent trajectory picture
would be established. We leave a detailed analysis of this aspect for further
studies.

##  5 Concluding remarks

We have shown that the multitime complex bi-probability distributions
$Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})$—and thus the corresponding
(positive-valued) probability distributions
$P_{\bm{t}_{n}}(\bm{f}_{n})$—associated with an observable $F$ of a finite-
dimensional quantum system are a discrete-time restrictions of a unique
complex master bi-measure
$\mathcal{Q}[f^{+},f^{-}][\mathcal{D}f^{+}][\mathcal{D}f^{-}]$ on the space of
pairs of trajectories traced through the spectrum of $F$. Mathematically, this
follows from a conceptually simple, yet nontrivial generalization of
Kolmogorov’s extension theorem to the complex-valued scenario. Consequently,
the multitime probabilities $P_{t_{n},\ldots,t_{1}}(f_{n},\ldots,f_{1})$ used
to describe sequential projective measurements of quantum observable do not
correspond to a sampling of an underlying trajectory (akin to classical
theories), but rather they result from a superposition of trajectory pairs,
$t\mapsto(f^{+}(t),f^{-}(t))$, such that $f^{+}(t_{j})=f^{-}(t_{j})=f_{j}$.
Therefore, in the bi-trajectory picture, the classical limit is achieved when
the trajectory pairs that do not overlap completely have vanishing measure.

Our results may pave the way to diverse research directions. The most natural
one would be the extension of our results to the infinite-dimensional
scenario, in which the spectrum of the observable $F$ may be unbounded and/or
admit continuous components; while the generalization of Kolmogorov’s
extension theorem does not forbid, in principle, either possibility (see
Theorem A.8 in the appendix), a nontrivial refinement of the bounds presented
in Section 3 is needed to this purpose.

Besides, it is our firm belief that the proof of the existence of a master bi-
measure—and thus, of the bi-trajectory picture—must entail some physical
ramifications for the quantum theory. Admittedly, the analysis of these
potential physical implications is outside the scope of the present work, but
it is a natural direction for future investigations. However, it is likely
that the comprehensive assessment of the impact of our result is pending
further research; to see why, let us review the “program” we were following in
this work: The starting point was the phenomenology of quantum mechanics that
gives us families of inconsistent multitime probability distributions. Taking
the inspiration from classical theories, we hypothesized that the individual
probabilities are connected by being components of some master object that
describe to some extent the physical system in question. The hypothesis was
then confirmed with the constructive mathematical proof of the object’s
existence. This last step, the mathematical inference, is the only real way to
uncover the master object’s form—the object itself cannot be directly “seen”,
only its components, the probabilities, are accessible. Here, we were able to
carry out this program to its conclusion—and thus, we have found the master
bi-measure—but only in the case of single observable. As indicated in Section
4.2, we were unable to do the same for the multi-observable scenario.
Consequently, we cannot be sure what is the exact nature of the true master
object, the single object that is the source of all phenomenologically
accessible multi-observable probabilities (including the single observable as
a special case). We believe that this true master object does exist, but the
question of its form remains; only a well-constructed mathematical proof will
be able to provide the answer.

## Acknowledgments

D.L. acknowledges financial support from the PNRR MUR project CN00000013 -
National Centre for HPC, Big Data and Quantum Computing, and partial support
by Istituto Nazionale di Fisica Nucleare (INFN) through the project “QUANTUM”
and the Italian National Group of Mathematical Physics (GNFM-INdAM). F.S.
acknowledges support by the Foundation for Polish Science (IRAP project,
ICTQT, contract no. 2018/MAB/5, co-financed by EU within Smart Growth
Operational Programme). D.C. was supported by the Polish National Science
Center project No. 2018/30/A/ST2/00837. P.S. acknowledges support from the
Polish National Science Centre through the project MAQS under QuantERA, which
has received funding from the European Union’s Horizon 2020 research and
innovation program under Grant Agreement No. 731473, Project No.
2019/32/Z/ST2/00016. P.S. also extends sincere gratitude to J. Korbicz and J.
Wehr for their invaluable advice and insightful discussions; they have been
immensely helpful during the development of this work.

##  Appendix A Kolmogorov extension theorem for complex measures

In this appendix we will collect some basic notions about complex measures,
and we will ultimately state and prove an extension theorem (Theorem A.8) for
consistent families of complex measures, which also reduces to the “standard”
Kolmogorov extension theorem in the case in which all measures are positive-
valued. Theorem 2.2 in the main text will henceforth follow as a particular
case of Theorem A.8. Further details on measure theory can be found in any of
the standard monographs on the subjects—among them, we mainly refer to Rudin’s
classic textbook [20] for the main concepts, and Bhattacharya and Waymire’s
book [14] for Kolmogorov’s extension theorem and related concepts.

The aim of this discussion (which, needless to say, has no pretense of
completeness) is twofold. From one hand, this appendix could be regarded as a
quick “handbook” of measure-theoretical notions of immediate use for the
purpose of the paper—incidentally, also providing a relatively simple,
functional analysis–oriented proof of the Kolmogorov extension theorem which
should be of interest for quantum physicists even beyond the scope of this
paper. From the other hand, to prove the desired extension theorem, we will
need to explicitly state some properties of complex measures which, in the
literature, are often only discussed in the positive-valued scenario. For the
convenience of the reader, we therefore decided to provide an explicit, if
concise, discussion of complex measures.

###  A.1 Basic notions on complex measures

######  Definition A.1 (Measurable space).

A measurable space is a pair $(M,\mathcal{F})$, where $M$ is a set and
$\mathcal{F}$ is a $\sigma$-algebra on $M$, that is, a family of subsets of
$M$ such that

  * •

$M\in\mathcal{F}$;

  * •

$A\in\mathcal{F}$ iff $M\setminus A\in\mathcal{F}$;

  * •

given a countable family $\\{A_{i}\\}_{i\in\mathbb{N}}\subset\mathcal{F}$, we
have $\bigcup_{i\in\mathbb{N}}A_{i}\in\mathcal{F}$.

$M$ is called sample space, and the elements of $\mathcal{F}$ are its
measurable subsets.

It is easy to see that $\mathcal{F}$ is also closed with respect to countable
intersections, and that the null set $\emptyset$ also belongs to
$\mathcal{F}$. By construction, the intersection of $\sigma$-algebras is also
a $\sigma$-algebra itself. An obvious example of $\sigma$-algebra on $M$ is
the class of all its subsets—its power set: this is indeed the standard choice
when dealing with spaces of finite cardinality. Nevertheless, to define
measures on spaces of trajectories, we shall also need to work with spaces
with infinite cardinality, in which case, for technical reasons, other choices
of $\sigma$-algebras are to be made.

Given a family $F$ of subsets of $M$, there always exists a $\sigma$-algebra
containing all elements of $F$ (e.g. the power set itself). Then the
intersection of all $\sigma$-algebras containing $F$ is still a
$\sigma$-algebra containing $F$—by construction, the smallest one. We will
call it the $\sigma$-algebra generated by $F$. A particularly useful example
is the following: whenever $M$ is endowed with a topological space structure
(e.g., $\mathbb{C}$ or $\mathbb{R}$), the Borel $\sigma$-algebra of $M$, which
we shall denote as $\mathfrak{B}(M)$, is the $\sigma$-algebra generated by the
open subsets of $M$. In such a case, all elements of $\mathfrak{B}(M)$ are
denoted as the Borel (sub)sets of $M$.

The notion of $\sigma$-algebra generated by a family of subsets is also
involved in the following definition, of primary importance for our purposes:

######  Definition A.2 (Product space and product $\sigma$-algebra).

Let $(M_{t})_{t\in I}$ an arbitrary collection of sets, each endowed with a
$\sigma$-algebra $\mathcal{F}_{t}$. The product space of $(M_{t})_{t\in I}$ is
defined as

$\prod_{t\in I}M_{t}=\left\\{f=(f_{t})_{t\in I}:\;f_{t}\in M_{t}\right\\},$
(A.1)

that is, the space of functions from $I$ to $\cup_{t\in I}M_{t}$. Besides, the
product $\sigma$-algebra of $(\mathcal{F}_{t})_{t\in I}$, denoted by
$\bigotimes_{t\in I}\mathcal{F}_{t}$, is the $\sigma$-algebra on $\prod_{t\in
I}M_{t}$ generated by all finite-dimensional rectangles, that is, all subsets
of $\prod_{t\in I}M_{t}$ in the form

$\left\\{f\in\prod_{t\in I}M_{t}:\;(f_{t_{1}},\dots,f_{t_{n}})\in
A_{t_{1}}\times\cdots\times
A_{t_{n}},\;A_{t_{1}}\in\mathcal{F}_{t_{1}},\ldots,A_{t_{n}}\in\mathcal{F}_{t_{n}}\right\\}.$
(A.2)

With these definitions, the pair $(\prod_{t\in I}M_{t},\otimes_{t\in
I}\mathcal{F}_{t})$ is itself a measurable space. In the definition above, $I$
(which will eventually play the role of time in our applications) can either
be finite, countably finite, or having the cardinality of the continuum.

######  Definition A.3 (Complex-valued measure).

Let $(M,\mathcal{F})$ a measurable space. A complex-valued measure (or complex
measure) on $(M,\mathcal{F})$ is a function
$\nu:\mathcal{F}\rightarrow\mathbb{C}$ which satisfies the following
properties:

  * (i)

grade-1 additivity: given two disjoint sets $A_{1},A_{2}\in\mathcal{F}$,

$\nu(A_{1}\sqcup A_{2})=\nu(A_{1})+\nu(A_{2});$ (A.3)

  * (ii)

continuity: given an increasing sequence of sets
$\\{A_{i}\\}_{i\in\mathbb{N}}$ (that is, $A_{i}\subset A_{i+1}$),

$\lim_{i\to\infty}\nu(A_{i})=\nu\left(\bigcup_{i\in\mathbb{N}}A_{i}\right);$
(A.4)

If $M$ is a topological space and $\mathcal{F}:=\mathfrak{B}(M)$ its
corresponding Borel $\sigma$-algebra, we will refer to $\nu$ as a Borel
measure.

Note that we are using the symbol $\sqcup$ for the union of disjoint sets:
this should be understood anytime such symbols appear. By iterating (i), one
obtains, for all $n\geq 2$,

$\nu\left(\bigsqcup_{i=1}^{n}A_{i}\right)=\sum_{i=1}^{n}\nu(A_{i}),$ (A.5)

whence the measure of the union of any family of disjoint sets can be entirely
reconstructed by knowing the measure associated with each separate set;
furthermore, (ii) allows us to take the limit $n\to\infty$ in the expression
above ($\sigma$-additivity).

If $\nu(A)$ is positive for all $A\in\mathcal{F}$, one recovers the familiar
definition of finite positive-valued measures—in fact, without further
specification, one often reserves the noun “measure” for the positive-valued
case. The adjective “finite” is here due to the fact that, differently from
the complex case, positive-valued measures may admit infinite values. In the
remainder of this work we shall always take for granted that all measures are
finite and possibly complex-valued.

The following definition is of major importance for complex measures:

######  Definition A.4.

Let $(M,\mathcal{F})$ be a measurable space, and $\nu$ a complex-valued
measure on it. Its variation $|\nu|$ is the positive measure defined as
follows:

$|\nu|(A)=\sup\left\\{\sum_{k}|\nu(A_{k})|:(A_{k})_{k}\text{ pairwise
disjoint, }\bigsqcup_{k}A_{k}=A\right\\}.$ (A.6)

Its total variation is the quantity $|\nu|(M)$.

It can be proven that this is indeed a positive measure on $(M,\mathcal{F})$,
and $|\nu(A)|\leq|\nu|(A)$ [20, Theorem 6.2]; besides, of course $|\nu|=\nu$
whenever $\nu$ is positive. As such, the definition above only acquires
importance when dealing with non-positive measures—indeed, it will be of
primary importance for obtaining a Kolmogorov extension theorem beyond the
positive scenario.

Proceeding in analogy with positive measures, we can likewise define a notion
of integral with respect to a complex measure. Recall that a function
$f:M\rightarrow\mathbb{C}$ is said to be measurable if, for every Borel subset
$A\subset\mathbb{C}$, its preimage

$f^{-1}(A):=\\{\omega\in M:\;f(\omega)\in A\\}$ (A.7)

satisfies $f^{-1}(A)\in\mathcal{F}$. For such functions, we can define the
Lebesgue integral with respect to a complex measure $\nu$, denoted by

$\int_{M}f(x)\;\mathrm{d}\nu(x)\qquad\text{or}\qquad\int_{M}f\;\mathrm{d}\nu$
(A.8)

via the same procedure that is used for positive measures, that is, by
approximating a measurable function with simple functions (i.e., linear
combinations of indicator functions), defining the integral of such functions
in the usual way, and taking the limit.

An important relation between a complex measure $\nu$ and its variation
$|\nu|$ is the polar decomposition of $\nu$, see e.g. [20, Theorem 6.12]:
there always exists a measurable function $h:M\rightarrow\mathbb{C}$ with
$|h(x)|\leq 1$ such that $\mathrm{d}\nu(x)=h(x)\;\mathrm{d}|\nu|(x)$, which
implies

$\int_{M}f\;\mathrm{d}\nu=\int_{M}fh\;\mathrm{d}|\nu|$ (A.9)

and, as a direct consequence,

$\left|\int_{M}f\;\mathrm{d}\nu\right|\leq\int_{M}|f|\;\mathrm{d}|\nu|.$
(A.10)

Eq. (A.9) can be used to show that the dominated convergence theorem also
holds for complex measures. We shall state and prove this fact explicitly.

######  Theorem A.5 (Dominated convergence theorem).

Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of complex-valued measurable
functions on $(M,\mathcal{F})$, pointwise converging to a complex-valued
function $f$. Suppose that there exists a positive-valued function $g$ which
is integrable with respect to a complex measure $\nu$ and such that
$|f_{n}(x)|\leq g(x)$. Then

$\lim_{n\to\infty}\int_{M}f_{n}\;\mathrm{d}\nu=\int_{M}f\;\mathrm{d}\nu.$
(A.11)

###### Proof.

If $\nu$ is a positive measure, this is the usual statement of the dominated
convergence theorem, see e.g. [20, Theorem 1.34]. In the general case, we know
that $\nu$ admits a polar decomposition
$\mathrm{d}\nu(x)=h(x)\;\mathrm{d}|\nu|(x)$, with $|h(x)|\leq 1$; then clearly
$f_{n}h$ converges pointwise to $fh$ and $|f_{n}(x)h(x)|\leq|f_{n}(x)|\leq
g(x)$, whence, applying the dominated convergence theorem for positive
measures,

$\lim_{n\to\infty}\int_{M}f_{n}h\;\mathrm{d}|\nu|=\int_{M}fh\;\mathrm{d}|\nu|,$
(A.12)

which, by Eq. (A.9), is the desired equality. ∎

###  A.2 Kolmogorov extension theorem for complex measures

We will now provide an extension theorem for consistent families of complex-
valued Borel measures, akin to the “standard” Kolmogorov extension theorem for
positive ones. The result hereby presented can be regarded as an immediate
generalization of Theorem 2.2 (cf. Remark A.9).

Given an arbitrary set $I$, we consider a family $(M_{t})_{t\in I}$ of compact
metric spaces, each endowed with the corresponding Borel $\sigma$-algebra
$\mathcal{F}_{t}:=\mathfrak{B}(M_{t})$. Recalling Definition A.2, for every
$n\in\mathbb{N}$ and every (not necessarily ordered)
$\bm{t}_{n}=(t_{1},\dots,t_{n})\in I^{n}$, with $t_{j}\neq t_{\ell}$ for all
$j\neq\ell$, we are automatically given a product space (cf. Definition A.2)

$\left(M_{\bm{t}_{n}},\mathcal{F}_{\bm{t}_{n}}\right),\qquad
M_{\bm{t}_{n}}=M_{t_{1}}\times\cdots\times
M_{t_{n}},\qquad\mathcal{F}_{\bm{t}_{n}}=\mathcal{F}_{t_{1}}\otimes\cdots\otimes\mathcal{F}_{t_{n}},$
(A.13)

with each $M_{\bm{t}_{n}}$ being a compact metric space, and
$\mathcal{F}_{\bm{t}_{n}}$ being the corresponding Borel $\sigma$-algebra.
Besides, we can also consider the “total” product space

$\left(\prod_{t\in I}M_{t},\;\;\bigotimes_{t\in I}\mathcal{F}_{t}\right),$
(A.14)

whose elements, cf. Definition A.2, can be either regarded as (possibly
uncountable) families $(f_{t})_{t\in I}$, with $f_{t}\in M_{t}$, or
equivalently as functions $f:I\rightarrow\cup_{t}M_{t}$.

######  Definition A.6 (Consistent family of measures).

Consider a family $(\nu_{\bm{t}_{n}})_{\bm{t}_{n}\in I^{n}}$, with
$\nu_{\bm{t}_{n}}$ being a complex-valued Borel measure on
$(M_{\bm{t}_{n}},\mathcal{F}_{\bm{t}_{n}})$. The family is said to be
consistent if it satisfies the following properties:

  * (a)

for every $n\in\mathbb{N}$, every permutation
$\sigma:\\{1,\dots,n\\}\rightarrow\\{1,\dots,n\\}$, and every
$A_{1}\in\mathcal{F}_{t_{1}},\ldots,A_{n}\in\mathcal{F}_{t_{n}}$,

$\nu_{t_{n},\dots,t_{1}}(A_{n}\times\ldots\times
A_{1})=\nu_{t_{\sigma(n)},\dots,t_{\sigma(1)}}(A_{\sigma(n)}\times\ldots\times
A_{\sigma(1)});$ (A.15)

  * (b)

for every $n\in\mathbb{N}$, every distinct $t_{1},\dots,t_{n},t_{n+1}\in I$
and every $A\in\mathcal{F}_{t_{1}}\times\ldots\times\mathcal{F}_{t_{n}}$,

$\nu_{t_{1},\ldots,t_{n}}(A)=\nu_{t_{1},\dots,t_{n},t_{n+1}}(A\times
M_{t_{n+1}}).$ (A.16)

######  Remark A.7.

Eq. (A.16) can be equivalently rephrased as follows: for every
$n\in\mathbb{N}$, every distinct $t_{1},\dots,t_{n},t_{n+1}\in I$ and every
$B\in\mathcal{F}_{t_{1}}\times\ldots\times\mathcal{F}_{t_{n}}$,

$\int 1_{B}\;\mathrm{d}\nu_{t_{1},\dots,t_{n}}=\int 1_{B\times
M_{t_{n+1}}}\;\mathrm{d}\nu_{t_{1},\dots,t_{n},t_{n+1}},$ (A.17)

which therefore implies the equality

$\int X_{t_{1},\dots,t_{n}}\mathrm{d}\nu_{t_{1},\dots,t_{n}}=\int
X_{t_{1},\dots,t_{n}}\;\mathrm{d}\nu_{t_{1},\dots,t_{n},t_{n+1}}$ (A.18)

for every simple function
$X_{t_{1},\dots,t_{n}}:M_{\bm{t}_{n}}\rightarrow\mathbb{C}$ and thus, via a
standard approximation argument, for every measurable function
$X_{t_{1},\dots,t_{n}}$. Besides, by Eq. (A.15), the same holds if the role of
$t_{n+1}$ is replaced by any other coordinate:

$\int
X_{t_{1},\ldots,\cancel{t_{j}},\ldots,t_{n+1}}\mathrm{d}\nu_{t_{1},\ldots,\cancel{t_{j}},\ldots,t_{n+1}}=\int
X_{t_{1},\ldots,\cancel{t_{j}},\ldots,t_{n+1}}\mathrm{d}\nu_{t_{1},\ldots,t_{j},\ldots,t_{n+1}}$
(A.19)

and so on; that is, “adding times” to the integration measure does not modify
the value of the integral.

######  Theorem A.8.

Let $(M_{t})_{t\in I}$ a family of compact metric spaces, with
$\mathcal{F}_{t}$ being the corresponding Borel $\sigma$-algebra, and
$(\nu_{\bm{t}_{n}})_{\bm{t}_{n}\in I}$ a family of complex-valued Borel
measures on $(M_{\bm{t}_{n}},\mathcal{F}_{\bm{t}_{n}})$ satisfying the
following assumptions:

  * •

consistency in the sense of Definition A.6;

  * •

uniformly bounded total variation:

$\sup_{n\in\mathbb{N},\bm{t}_{n}\in
I^{n}}\left|\nu_{\bm{t}_{n}}\right|(M_{\bm{t}_{n}})<\infty,$ (A.20)

with $|\nu_{\bm{t}_{n}}|$ as in Definition A.4.

Then there exists a unique complex-valued measure $\nu$ on the product space
$(\mathcal{M},\otimes_{t}\mathcal{F}_{t})$ such that, for all
$n\in\mathbb{N},\bm{t}_{n}\in I^{n}$ and $A\in\mathcal{F}_{\bm{t}_{n}}$,

$\nu_{\bm{t}_{n}}(A)=\nu\left(\left\\{f\in\prod_{t\in
I}M_{t}:(f(t_{1}),\ldots,f(t_{n}))\in A\right\\}\right).$ (A.21)

The proof of the theorem, postponed to Section A.3, is an adaptation to the
complex-valued case of the “Functional Analysis Version” of the proof of the
standard Kolmogorov extension theorem [14, Theorem 9.1] (also see [13, Theorem
11.1]). The assumption of compactness of the spaces $M_{t}$ is not strictly
necessary and may be removed by following a similar argument as the one
presented in both referenced proofs to cover the non-compact case.

Eq. (A.21) states the following: given any Borel set
$A\in\mathcal{F}_{\bm{t}_{n}}$, the value of $\nu_{\bm{t}_{n}}$ on it
corresponds to the value of $\nu$ on the corresponding cylinder—the set of all
functions $f$ whose values at the times $t_{1},\dots,t_{n}$ are in $A$. As
such, $\nu$ unambiguously generates the full consistent family of measures
$\nu_{\bm{t}_{n}}$. Theorem A.8 reduces to the “standard” Kolmogorov extension
theorem (at least in the case of compact metric spaces) when all measures
involved are positive-valued—importantly, in such a case, the request (A.20)
becomes trivial for probability measures since
$|\nu_{\bm{t}_{n}}|=\nu_{\bm{t}_{n}}$ and all quantities in Eq. (A.20) are
simply equal to one by definition. In this sense, the request of uniformly
bounded total variation is the additional ingredient required in order to
extend the result to complex-valued measures.

######  Remark A.9.

We can easily check that Theorem 2.2 in the main text is a corollary of
Theorem A.8 corresponding to the case in which $M_{t}=\Omega\times\Omega$ (and
thus $\prod_{t}M_{t}=(\Omega\times\Omega)^{I}$) for all $t\in I$, where
$\Omega$ is a finite set of real numbers (and thus a compact metric space), in
which case each $\nu_{\bm{t}_{n}}$ is thus given by

$\nu_{\bm{t}_{n}}(A^{+}\times A^{-})=\sum_{\bm{f}^{\pm}_{n}\in
A^{\pm}}B_{\bm{t}_{n}}(\bm{f}^{+}_{n},\bm{f}^{-}_{n})$ (A.22)

for all couple of Borel sets $A^{\pm}$. This is shown by taking into account
the following facts:

  * •

the Borel $\sigma$-algebras on $\Omega\times\Omega$ and all products of
finitely many copies of it simply coincide with the respective power sets;

  * •

the integral with respect to each measure $\nu_{\bm{t}_{n}}$ reduces to a
finite sum:

$\int
X_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})\;\mathrm{d}\nu_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})=\sum_{\bm{f}_{n}^{+},\bm{f}_{n}^{-}}X_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-});$
(A.23)

  * •

having Eq. (A.23) into account, the consistency condition (A.19) (up to
permutations) reduces to Eq. (1.5);

  * •

similarly, the uniform boundedness condition (A.20) reduces to Eq. (3.16).

Finally, setting $A=\\{\bm{f}_{n}^{+}\times\bm{f}_{n}^{-}\\}$ in Eq. (A.21),
one obtains

$\displaystyle Q_{\bm{t}_{n}}(\bm{f}_{n}^{+},\bm{f}_{n}^{-})$
$\displaystyle=\nu\Bigl{(}\bigl{\\{}(f^{+},f^{-})\in(\Omega\times\Omega)^{I}:f^{\pm}(t_{j})=f^{\pm}_{j},\;j=1,\dots,n\bigr{\\}}\Bigr{)},$
(A.24)

and the latter quantity can be formally expressed as the integral over a
product of Dirac distributions as in Eq. (2.7), where
$\mathcal{Q}_{I}[f^{+},f^{-}]\,[\mathcal{D}f^{+}][\mathcal{D}f^{-}]$ stands
for the measure element $\mathrm{d}\nu[f^{+},f^{-}]$.

Some final comments are in order. The quantity $\nu_{\bm{t}_{n}}$ can be
equivalently interpreted, with an obvious abuse of notation, as a bi-measure
$(A^{+},A^{-})\mapsto\nu_{\bm{t}_{n}}(A^{+},A^{-})$, that is, a function of
pairs of sets that is additive in both arguments. If, in particular, we assume
that the functions $B_{\bm{t}_{n}}$ satisfy the property (Q3) (as is always
the case for those associated with a quantum mechanical system, cf. Section
3), then for any finite collection $(A_{j})_{j}$, the matrix
$\\{\nu_{\bm{t}_{n}}(A_{j},A_{k})\\}_{j,k}$ is positive semidefinite. Bi-
measures satisfying this property are sometimes called decoherence functionals
(or functions). In particular, their “diagonal” part of such measures,
$\mu_{\bm{t}_{n}}(A):=\nu_{\bm{t}_{n}}(A,A)$, are positive-valued functions
which, however, are not additive (and thus, not measures). Instead, they
satisfy a weaker condition, known as grade-2 additivity:

$\displaystyle\mu_{\bm{t}_{n}}\left(A_{1}\sqcup A_{2}\sqcup A_{3}\right)$
$\displaystyle=\mu_{\bm{t}_{n}}(A_{1}\sqcup
A_{2})+\mu_{\bm{t}_{n}}(A_{2}\sqcup A_{3})+\mu_{\bm{t}_{n}}(A_{1}\sqcup
A_{3})$
$\displaystyle\phantom{=}-\mu_{\bm{t}_{n}}(A_{1})-\mu_{\bm{t}_{n}}(A_{2})-\mu_{\bm{t}_{n}}(A_{3}),$
(A.25)

and are thus known as grade-2 or quantum measures, cf. [21, 22, 23, 24].

###  A.3 Proof of Theorem A.8

In order to prove Theorem A.8, we will need to state some intermediate results
that hold for regular complex measures. To begin with, a complex Borel measure
is said to be regular if, for every Borel set $A$,

$|\nu|(A)=\sup\\{|\nu|(A^{\prime}):A^{\prime}\subset A\text{ is
closed}\\}=\inf\\{|\nu|(A^{\prime}):A^{\prime}\supset A\text{ is open}\\};$
(A.26)

in particular, all Borel measures on a compact metric space are regular.

If $M$ is a compact topological space and is also Hausdorff (that is, every
couple of points admit disjoint neighborhoods), the space $\mathrm{C}(M)$ of
complex-valued continuous functions on $M$, endowed with the supremum norm, is
a Banach space. The following result, often referred to as the Riesz–Markov
representation theorem, establishes an important connection between bounded
functionals on $\mathrm{C}(M)$ and complex-valued measures.

######  Theorem A.10 (Riesz–Markov theorem).

Let $M$ be a compact Hausdorff space. Then, for every bounded linear
functional $\mathcal{E}:\mathrm{C}(M)\rightarrow\mathbb{C}$, there exists a
unique regular complex-valued Borel measure $\nu$ such that

$\mathcal{E}(f)=\int_{M}f\;\mathrm{d}\nu;$ (A.27)

besides, $\mathcal{E}$ is a positive functional iff $\nu$ is a positive-valued
measure. The operator norm of $\mathcal{E}$ equals the total variation of
$\nu$:

$\left\|\mathcal{E}\right\|:=\sup_{\|f\|_{\infty}=1}|\mathcal{E}(f)|=|\nu|(M).$
(A.28)

This is a standard result in measure theory, see e.g. [20, Theorem 6.19] (in a
slightly more general setting). Of course, the converse of the theorem is also
true—every functional in the form (A.27), for a given complex measure $\nu$,
has the desired properties. This establishes a one-to-one correspondence
between bounded linear functionals on $\mathrm{C}(M)$ and regular complex
Borel measures on $M$.

We will also need the following property which, while elementary, is usually
stated for positive measures only—as such, we shall provide an explicit proof
for it in the general case:

######  Lemma A.11.

Let $M$ be a metric space, and $\nu_{1}$, $\nu_{2}$ two complex regular Borel
measures such that the equality

$\int_{M}f\;\mathrm{d}\nu_{1}=\int_{M}f\;\mathrm{d}\nu_{2}$ (A.29)

holds for all real-valued continuous functions $f$ on it. Then
$\nu_{1}=\nu_{2}$.

The proof of this statement is analogous to the proof of [14, Proposition 1.6]
(for positive measures); we shall sketch it below.

###### Proof.

Following the construction in the proof of [14, Proposition 1.6], one shows
that, given any closed Borel set $A\subset M$, then there exists a sequence of
nonnegative continuous functions $(f_{n})_{n\in\mathbb{N}}$ which converge
pointwise to the indicator function $1_{A}$ of $A$. By hypothesis, for all
$n\in\mathbb{N}$ we have

$\int_{M}f_{n}\;\mathrm{d}\nu_{1}=\int_{M}f_{n}\;\mathrm{d}\nu_{2};$ (A.30)

applying the dominated convergence theorem yields $\nu_{1}(A)=\nu_{2}(A)$,
whence the two measures $\nu_{1}$ and $\nu_{2}$ agree on all closed Borel
sets.

To obtain the desired equality for an arbitrary Borel set $A$, we note that,
since the two measures are regular, for every $\epsilon>0$ there exists
$A^{\prime}\subset A$ closed such that

$|\nu_{1}|(A\setminus
A^{\prime})<\frac{\epsilon}{2},\qquad|\nu_{2}|(A\setminus
A^{\prime})<\frac{\epsilon}{2},$ (A.31)

whence

$\displaystyle|\nu_{1}(A)-\nu_{2}(A)|$
$\displaystyle=\left|\nu_{1}(A^{\prime})+\nu_{1}(A\setminus
A^{\prime})-\nu_{2}(A^{\prime})-\nu_{2}(A\setminus A^{\prime})\right|$
$\displaystyle=\left|\nu_{1}(A\setminus A^{\prime})-\nu_{2}(A\setminus
A^{\prime})\right|$ $\displaystyle=|\nu_{1}|(A\setminus
A^{\prime})+|\nu_{2}|(A\setminus A^{\prime})<\epsilon.$ (A.32)

Since $\epsilon$ is arbitrary, we can finally conclude
$\nu_{1}(A)=\nu_{2}(A)$. ∎

Below, after two preliminary remarks, we will finally present the proof of
Theorem A.8, which—like the one presented in the main text—essentially follows
step-by-step the second version of the proof of [14, Theorem 9.1].

######  Remark A.12.

When considering complex-valued functions on the product space,
$\mathcal{X}:\prod_{t}M_{t}\rightarrow\mathbb{C}$, we shall use the following
notation:

$\mathcal{X}[f]:=\mathcal{X}\left((f_{t})_{t\in I}\right),$ (A.33)

i.e. with the square brackets reminding that we are considering a function,
$\mathcal{X}$, of functions (the elements $f\in\prod_{t}M_{t}$).

######  Remark A.13.

A more compact way to state Eq. (A.21), which we will use in the proof, is the
following. Given $n\in\mathbb{N},\bm{t}_{n}\in I^{n}$, let
$\pi_{\bm{t}_{n}}:\prod_{t}M_{t}\rightarrow M_{\bm{t}_{n}}$ be defined (with
the notation of Remark A.12) by

$\pi_{\bm{t}_{n}}[f]:=(f_{t_{1}},\ldots,f_{t_{n}}).$ (A.34)

Then for every $A\in\mathcal{F}_{\bm{t}_{n}}$ one has

$\displaystyle\pi_{\bm{t}_{n}}^{-1}(A)$
$\displaystyle=\left\\{f\in\prod_{t}M_{t}:\pi_{\bm{t}_{n}}[f]\in A\right\\}$
$\displaystyle=\left\\{f\in\prod_{t}M_{t}:(f_{t_{1}},\ldots,f_{t_{n}})\in
A\right\\},$ (A.35)

and thus the validity of Eq. (A.21) for all Borel sets of $M_{\bm{t}_{n}}$
corresponds to the equality

$\nu_{\bm{t}_{n}}=\nu\circ\pi_{\bm{t}_{n}}^{-1}.$ (A.36)

The right-hand side above is the pushforward of $\nu$, i.e. the measure on
$M_{\bm{t}_{n}}$ which, to every Borel set $A$ of $M_{\bm{t}_{n}}$, associates
the quantity $\nu(\pi_{\bm{t}_{n}}^{-1}(A))$.

Also notice that $\pi_{\bm{t}_{n}}$ allows us to “lift” any
$X_{\bm{t}_{n}}\in\mathrm{C}(M_{\bm{t}_{n}})$ to an element of
$\mathrm{C}(\prod_{t}M_{t})$ defined by
$\mathcal{X}:=X_{\bm{t}_{n}}\circ\pi_{\bm{t}_{n}}$, that is,

$\mathcal{X}[f]:=X_{\bm{t}_{n}}(\pi_{\bm{t}_{n}}[f])=X_{\bm{t}_{n}}(f_{t_{1}},\ldots,f_{t_{n}});$
(A.37)

in fact, the set of all elements of $\mathrm{C}(\prod_{t}M_{t})$ defined this
way will play a fundamental role in the proof below.

###### Proof of Theorem A.8.

To simplify the notation, hereafter we will set

$\mathcal{M}:=\prod_{t\in I}M_{t}.$ (A.38)

To begin with, since each space $M_{t}$, $t\in I$, is a compact metric space,
their product space $\mathcal{M}=\prod_{t}M_{t}$, endowed with the product
topology, is also compact (and thus a compact Hausdorff space) by Tychonoff’s
theorem; as such, the space $\mathrm{C}\left(\mathcal{M}\right)$ of continuous
functions, endowed with the uniform norm

$\|\mathcal{X}\|_{\infty}:=\max_{f\in\mathcal{M}}\left|\mathcal{X}[f]\right|,$
(A.39)

is a Banach space. We will construct a bounded linear functional $\mathcal{E}$
on $\mathrm{C}\left(\mathcal{M}\right)$ defined as follows. Let
$\mathrm{C}_{\rm fin}\left(\mathcal{M}\right)$ be the space of elements of
$\mathrm{C}\left(\mathcal{M}\right)$ that only depend on finitely many
coordinates, that is: $\mathcal{X}\in\mathrm{C}_{\rm
fin}\left(\mathcal{M}\right)$ if and only if there exist $n\in\mathbb{N}$,
$\bm{t}_{n}\in I^{n}$, and a function
$\hat{X}_{\bm{t}_{n}}\in\mathrm{C}(M_{\bm{t}_{n}})$ such that

$\mathcal{X}[f]=\tilde{X}_{\bm{t}_{n}}(f_{t_{1}},\ldots,f_{t_{n}}),$ (A.40)

or, equivalently, such that
$\mathcal{X}=\tilde{X}_{\bm{t}_{n}}\circ\pi_{\bm{t}_{n}}$ for some
$n\in\mathbb{N}$, $\bm{t}_{n}\in I^{n}$, and
$\tilde{X}_{\bm{t}_{n}}\in\mathrm{C}(M_{\bm{t}_{n}})$. By the
Stone–Weierstrass theorem, $\mathrm{C}_{\rm fin}\left(\mathcal{M}\right)$ is a
dense subspace of $\mathrm{C}\left(\mathcal{M}\right)$. We now define a linear
functional $\mathcal{E}:\mathrm{C}_{\rm
fin}\left(\mathcal{M}\right)\rightarrow\mathbb{C}$ as follows: for any
$\mathcal{X}\in\mathrm{C}_{\rm fin}\left(\mathcal{M}\right)$ admitting a
representation as in Eq. (A.40), we define

$\mathcal{E}(\mathcal{X}):=\int\tilde{X}_{\bm{t}_{n}}(f_{t_{1}},\ldots,f_{t_{n}})\;\mathrm{d}\nu_{\bm{t}_{n}}(f_{t_{1}},\ldots,f_{t_{n}}).$
(A.41)

We will prove that this functional is (a) well-defined, that is, its value
does not depend on the particular representation (A.40); and (b) bounded.
Point (a) arises from the following observation: a function
$\mathcal{X}\in\mathrm{C}_{\rm fin}\left(\mathcal{M}\right)$ admitting the
$n$-time representation (A.40) also admits the $(n+1)$-time representation

$\mathcal{X}[f]=\tilde{X}_{t_{1},\dots,t_{n},t_{n+1}}(f_{t_{1}},\ldots,f_{t_{n}},f_{t_{n+1}}),$
(A.42)

for any $t_{n+1}\in I$, by simply defining
$\tilde{X}_{t_{1},\dots,t_{n},t_{n+1}}(f_{t_{1}},\ldots,f_{t_{n}},f_{t_{n+1}}):=\tilde{X}_{t_{1},\dots,t_{n}}(f_{t_{1}},\ldots,f_{t_{n}})$.
Using this representation, we have

$\displaystyle\mathcal{E}(\mathcal{X})$
$\displaystyle=\int\tilde{X}_{\bm{t}_{n+1}}(f_{t_{1}},\ldots,f_{t_{n}},f_{t_{n+1}})\;\mathrm{d}\nu_{\bm{t}_{n+1}}(f_{t_{1}},\ldots,f_{t_{n}},f_{t_{n+1}})$
$\displaystyle=\int\tilde{X}_{\bm{t}_{n}}(f_{t_{1}},\ldots,f_{t_{n}})\;\mathrm{d}\nu_{\bm{t}_{n+1}}(f_{t_{1}},\ldots,f_{t_{n}},f_{t_{n+1}})$
$\displaystyle=\int\tilde{X}_{\bm{t}_{n}}(f_{t_{1}},\ldots,f_{t_{n}})\;\mathrm{d}\nu_{\bm{t}_{n}}(f_{t_{1}},\ldots,f_{t_{n}}),$
(A.43)

where the latter equality follows by consistency, see Remark A.7. This proves
that the functional is well-defined. As for boundedness, we have (cf. Eq.
(A.10))

$\displaystyle\left|\mathcal{E}(\mathcal{X})\right|$
$\displaystyle=\left|\int\tilde{X}_{\bm{t}_{n}}(f_{t_{1}},\ldots,f_{t_{n}})\;\mathrm{d}\nu_{\bm{t}_{n}}(f_{t_{1}},\ldots,f_{t_{n}})\right|$
$\displaystyle\leq\int\left|\tilde{X}_{\bm{t}_{n}}(f_{t_{1}},\ldots,f_{t_{n}})\right|\;\mathrm{d}\left|\nu_{\bm{t}_{n}}\right|(f_{t_{1}},\ldots,f_{t_{n}})$
$\displaystyle\leq\|\mathcal{X}\|_{\infty}\left|\nu_{\bm{t}_{n}}\right|(M_{\bm{t}_{n}}),$
(A.44)

whence, maximizing,

$\sup_{\mathcal{X}\in\mathrm{C}_{\rm
fin}(\mathcal{M})}\frac{\left|\mathcal{E}(\mathcal{X})\right|}{\|\mathcal{X}\|_{\infty}}\leq\sup_{n\in\mathbb{N},\bm{t}_{n}\in
I^{n}}\left|\nu_{\bm{t}_{n}}\right|(M_{\bm{t}_{n}})<\infty.$ (A.45)

This shows that $\mathcal{E}$ is a bounded functional on $\mathrm{C}_{\rm
fin}\left(\mathcal{M}\right)$. Since the latter, by the Stone–Weierstrass
theorem, is a dense subset of the Banach space
$\mathrm{C}\left(\mathcal{M}\right)$, by the continuous linear extension
theorem $\mathcal{E}$ admits a unique extension to a bounded functional on
$\mathrm{C}\left(\mathcal{M}\right)$, which we will denote by the same symbol.
By the Riesz–Markov theorem (Theorem A.10), this implies the existence of a
unique regular Borel measure $\nu$ on the product space such that, for every
$\mathcal{X}\in\mathrm{C}\left(\mathcal{M}\right)$,

$\mathcal{E}(\mathcal{X})=\int\mathcal{X}[f]\;\mathrm{d}\nu[f].$ (A.46)

To conclude the proof, we will need to prove Eq. (A.21). By construction, the
functional $\mathcal{E}$ has the following property: with the notation
introduced in Remark A.13, for any
$X_{\bm{t}_{n}}\in\mathrm{C}(M_{\bm{t}_{n}})$ we have

$\mathcal{E}(X_{\bm{t}_{n}}\circ\pi_{\bm{t}_{n}})=\int
X_{\bm{t}_{n}}(f_{t_{1}},\ldots,f_{t_{n}})\;\mathrm{d}\nu_{\bm{t}_{n}}(f_{t_{1}},\ldots,f_{t_{n}}).$
(A.47)

But, by Eq. (A.46), the left-hand side of the above equation also reads:

$\displaystyle\mathcal{E}(X_{\bm{t}_{n}}\circ\pi_{\bm{t}_{n}})$
$\displaystyle=\int(X_{\bm{t}_{n}}\circ\pi_{\bm{t}_{n}})[f]\;\mathrm{d}\nu[f]$
$\displaystyle=\int
X_{\bm{t}_{n}}(f_{t_{1}},\ldots,f_{t_{n}})\;\mathrm{d}(\nu\circ\pi_{\bm{t}_{n}}^{-1})(f_{t_{1}},\ldots,f_{t_{n}}).$
(A.48)

That is, the two Borel measures $\nu_{\bm{t}_{n}}$ and
$\nu\circ\pi_{\bm{t}_{n}}^{-1}$ on $M_{\bm{t}_{n}}$ have the following
property: their integral over any continuous function $X_{\bm{t}_{n}}$ is
equal. Besides, both measures $\nu_{\bm{t}_{n}}$ and
$\nu\circ\pi_{\bm{t}_{n}}^{-1}$ are regular (the former because it is defined
on a compact metric space, the latter by the regularity of $\nu$ ensured by
the Riesz–Markov theorem); by Lemma A.11, this is sufficient to conclude that
$\nu_{\bm{t}_{n}}=\nu\circ\pi_{\bm{t}_{n}}^{-1}$, and thus (see Remark A.13)
Eq. (A.21) holds, thus concluding the proof. ∎

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