# Questionable and Unquestionable in Quantum Mechanics

László E. Szabó(1), Márton Gömöri(1)(2) and Zalán Gyenis(3)

_(1) Department of Logic, Institute of Philosophy, Eötvös University,
Budapest_
_(2) Institute of Philosophy, Research Centre for the Humanities, Budapest_
_(3) Department of Logic, Jagiellonian University, Krakow_

( )

###### Abstract

We derive the basic postulates of quantum physics from a few very simple and
easily testable operational assumptions based exclusively on the relative
frequencies of observable events (measurement operations and measurement
outcomes). We isolate a notion which can be identified with the system’s own
state, in the sense that it characterizes the system’s probabilistic behavior
against all possible measurement operations. We investigate some important
features of the possible states of the system. All those investigations remain
within the framework of classical Kolmogorovian probability theory, meaning
that any physical system (traditionally categorized as classical or quantum)
that can be described in operational terms can be described within classical
Kolmogorovian probability theory.

In the second part of the paper we show that anything that can be described in
operational terms can, if we wish, be represented in the Hilbert space quantum
mechanical formalism. The outcomes of each measurement can be represented by a
system of pairwise orthogonal closed subspaces spanning the entire Hilbert
space; the states of the system can be represented by pure state operators,
and the probabilities of the outcomes can be reproduced by the usual trace
formula. Each real valued quantity can be associated with a suitable self-
adjoint operator, such that the possible measurement results are the
eigenvalues and the outcome events are represented by the eigenspaces,
according to the spectral decomposition of the operator in question.

This suggests that the basic postulates of quantum theory are in fact analytic
statements: they do not tell us anything about a physical system beyond the
fact that the system can be described in operational terms. This is almost
true. At the end of the paper we discuss a few subtle points where the
representation we obtained is not completely identical with standard quantum
mechanics.

##  1 Introduction

The main result of this paper is an entirely operational foundation of quantum
theory. Operational approach to quantum mechanics is of course not a novel
idea. In the past decades, a number of great works have appeared with similar
aims (e.g. Ludwig 1970; Foulis and Randall 1974; Davies 1976; Busch,
Grabowski, and Lahti 1995; Spekkens 2005; Barum _et al._ 2007; 2008; Aerts
2009; Abramsky and Heunen 2016; Schmid, Spekkens, and Wolfe 2018). The novelty
of our approach lies in the fact that our goal is not to translate the known
theory of quantum mechanics into operational terms, or to reconstruct the
theory in terms of operationally interpretable postulates. Such a
“reconstruction”, as we shall see, will be only a by-product—though not a
mathematically trivial one—of a general operational description. While the
operational scheme we are considering does not itself refer in any way to
quantum theory. It is based exclusively on the notion of observable events
(measurement operations and measurement outcomes) and on general, empirically
established simple laws concerning their relative frequencies. These
regularities are so simple and fundamental that they are equally valid whether
the physical system under study is traditionally considered to be a classical
or a quantum physical phenomenon.

We describe a typical empirical scenario in the following way: One can perform
different measurement operations on a physical system, each of which may have
different possible outcomes. The performance of a measuring operation is
regarded as a physical event on par with the measurement outcomes. Empirical
data are, exclusively, the observed relative frequencies of how many times
different measurement operations are performed and how many times different
outcome events occur, including the joint performances of two or more
measurements and the conjunctions of their outcomes. In terms of the observed
relative frequencies we stipulate two empirical conditions, (E1) and (E2),
which are simple, plausible, and empirically testable.

Of course, the observed relative frequencies essentially depend on the
frequencies with which the measurement operations are performed; that is, on
circumstances external to the physical system under consideration; for
example, on the free choice of a human. Under a further empirically testable
assumption about the observed frequencies, (E3), we can isolate a notion which
is independent of the relative frequencies of the measurement operations and
can be identified with the system’s own state; in the sense that it
characterizes the system’s probabilistic behavior against all possible
measurement operations. The largest part of our further investigation is at
the level of generality defined by assumptions (E1)–(E3).

In Section 3, we derive important theorems, solely from conditions (E1)–(E3),
concerning the possible states of the system. In Section 4, we characterize
the time evolution of these states on the basis of a further, empirically
testable assumption (E4). Section 5 considers various possible ontological
pictures consistent with our probabilistic notion of state.

All these investigations are expressed in terms of relative frequencies, which
by definition satisfy the Kolmogorovian axioms of classical probability
theory. This means that any physical system—traditionally categorized as
classical or quantum—that can be described in operational terms can be
described within classical Kolmogorovian probability theory—whether we talk
about the system’s probabilistic state, time evolution or ontology. In the
second part of the paper, at the same time, we will show that anything that
can be described in these operational terms can, if we wish, be represented in
the Hilbert space quantum mechanical formalism. It will be proved that there
always exists:

  * •

a suitable Hilbert space, such that

  * •

the outcomes of each measurement can be represented by a system of pairwise
orthogonal closed subspaces, spanning the whole Hilbert space,

  * •

the states of the system can be represented by pure state operators with
suitable state vectors, and

  * •

the probabilities of the measurement outcomes can be reproduced by the usual
trace formula of quantum mechanics.

Moreover, if appropriate, one can label the possible outcomes of a measurement
with numbers, and talk about them as the measured values of a physical
quantity. Each such quantity

  * •

can be associated with a suitable self-adjoint operator, such that

  * •

the expectation value of the quantity, in all states of the system, can be
reproduced by the usual trace formula applied to the associated self-adjoint
operator,

  * •

the possible measurement results are exactly the eigenvalues of the operator,
and

  * •

the corresponding outcome events are represented by the eigenspaces pertaining
to the eigenvalues respectively, according to the spectral decomposition of
the operator in question.

This suggests that the basic postulates of quantum theory are in fact analytic
statements: they do not tell us anything about a physical system beyond the
fact that the system can be described in operational terms. This is almost
true. Nevertheless, it must be mentioned that the quantum-mechanics-like
representation we will obtain is not completely identical with standard
quantum mechanics. The interesting fact is that most of the deviations from
the quantum mechanical folklore, discussed in Section 8, are related with
exactly those issues in the foundations of quantum mechanics that have been
hotly debated for long decades.

##  2 The General Operational Schema

Consider a general experimental scenario: we can perform different measurement
operations denoted by $a_{1},a_{2},\ldots a_{r},\ldots a_{m}$ on a physical
system. We shall use the same notation $a_{r}$ for the physical event that the
measurement operation $a_{r}$ happened. Each measurement $a_{r}$ may have
different outcomes denoted by $X_{1}^{r},X_{2}^{r},\ldots X_{n_{r}}^{r}$. Let
$M=\sum_{r=1}^{m}n_{r}$, and let $I^{M}$ denote the following set of indices:

$\displaystyle I^{M}$ $\displaystyle=$
$\displaystyle\left\\{{}_{i}^{r}\,\left|\,1\leq r\leq m,1\leq i\leq
n_{r}\right.\right\\}$ (1)

Sometimes we perform two or more measurement operations simultaneously—that
is, in the same run of the experiment. So we also consider the double, triple,
and higher conjunctions of measurement operations and the possible outcome
events. In general, we consider the free Boolean algebra $\mathcal{A}$
generated by the set of all measurement operation and measurement outcome
events

$\displaystyle G$ $\displaystyle=$
$\displaystyle\left\\{a_{r}\right\\}_{r=1,2,\ldots
m}\cup\left\\{X_{i}^{r}\right\\}_{{}_{i}^{r}\in I^{M}}$ (2)

with the usual Boolean operations, denoted by $\wedge$, $\vee$ and $\neg$.
Introduce the following concise notation: let $S_{max}^{M}$ denote the set of
the indices of all double, triple, and higher conjunctions of the outcome
events in $G$. That is, for example, ${}_{i_{1}i_{2}\ldots
i_{L}}^{r_{1}r_{2}\ldots r_{L}}\in S_{max}^{M}$ will stand for the conjunction
$X_{i_{1}}^{r_{1}}\wedge X_{i_{2}}^{r_{2}}\ldots\wedge X_{i_{L}}^{r_{L}}$,
etc.

The event algebra $\mathcal{A}$ has $2^{M+m}$ atoms, each having the form of

$\displaystyle\varDelta_{\vec{\varepsilon},\vec{\eta}}$ $\displaystyle=$
$\displaystyle\left(\underset{{}_{i}^{r}\in
I^{M}}{\wedge}\left[X_{i}^{r}\right]^{\varepsilon_{i}^{r}}\right)\wedge\left(\stackrel{{\scriptstyle[}}{{s}}=1]{m}{\wedge}\left[a_{s}\right]^{\eta_{s}}\right)$
(3)

where
$\vec{\varepsilon}=\left(\varepsilon_{i}^{r}\right)\in\left\\{0,1\right\\}^{M}$,
$\vec{\eta}=\left(\eta_{s}\right)\in\left\\{0,1\right\\}^{m}$, and

$\displaystyle\left[X_{i}^{r}\right]^{\varepsilon_{i}^{r}}$ $\displaystyle=$
$\displaystyle\begin{cases}X_{i}^{r}&\text{if }\varepsilon_{i}^{r}=1\\\ \neg
X_{i}^{r}&\text{if }\varepsilon_{i}^{r}=0\end{cases}$
$\displaystyle\left[a_{s}\right]^{\eta_{s}}$ $\displaystyle=$
$\displaystyle\begin{cases}a_{s}&\text{if }\eta_{s}=1\\\ \neg a_{s}&\text{if
}\eta_{s}=0\end{cases}$

And, of course, all events in algebra $\mathcal{A}$ can be uniquely expressed
as a disjunction of atoms.

Assume that we can repeat the same experimental situation as many times as
needed; that is, we can prepare the same (or identical) physical system in the
same way and we can repeat the same measuring operations with the same (or
identical) measuring devices, etc. In every run of the experiment we observe
which measurement operations are performed and which outcome events occur,
including the joint performances of two or more measurements and the
conjunctions of their outcomes. In this way, we observe the _relative
frequencies_ of all elements of the event algebra $\mathcal{A}$. Let $\pi$
denote this relative frequency function on $\mathcal{A}$. Obviously,
$\left(\mathcal{A},\pi\right)$ constitutes a classical probability model
satisfying the Kolmogorovian axioms. Since the relative frequencies on the
whole event algebra are uniquely determined by the relative frequencies of the
atoms, $\pi$ can be uniquely given by

$\displaystyle\pi(\varDelta_{\vec{\varepsilon},\vec{\eta}})$
$\displaystyle\vec{\varepsilon}\in\left\\{0,1\right\\}^{M};\,\vec{\eta}\in\left\\{0,1\right\\}^{m}$
(4)

The observed relative frequencies on $\mathcal{A}$ are considered _the_
empirical data, exclusively.

We do not make _a priori_ assumptions about these relative frequencies. Any
truth about them will be regarded as empirical fact observed in the
experiment. Including for example the fact that two or more measurements
$a_{r_{1}},a_{r_{2}},\ldots a_{r_{L}}$ cannot be performed simultaneously;
which reveals in the observed fact that $\pi\left(a_{r_{1}}\wedge
a_{r_{2}}\ldots\wedge a_{r_{L}}\right)$ _always_ equals $0$. Though, this
“always” needs some further explanation. For, it is obviously true that the
frequencies $\pi(a_{r})$ sensitively depend on the will of the experimenter.
Therefore, it can be the case that $\pi\left(a_{r_{1}}\wedge
a_{r_{2}}\ldots\wedge a_{r_{L}}\right)=0$ simply because the experimenter
never chooses to perform the measurements $a_{r_{1}},a_{r_{2}},\ldots
a_{r_{L}}$ simultaneously. At least at first sight this seems to significantly
differ from the situation when a certain combination of experiments are never
performed due to objective reasons; because the simultaneous performance of
the measurement operations is—as we usually express—impossible. Without
entering into the metaphysical disputes about possibility–impossibility, we
only say that the impossibility of a combination of measurements is a
contingent fact of the world; the measuring devices and the measuring
operations are such that the joint measurement $a_{r_{1}}\wedge
a_{r_{2}}\ldots\wedge a_{r_{L}}$ never occurs. Let us denote by
$\mathfrak{I}\subset\mathcal{P}\left(\left\\{1,2,\ldots m\right\\}\right)$
(where $\mathcal{P}\left(A\right)$ is the power set of set $A$) the set of
indices of such “impossible” conjunctions. That is, for all $2\leq L\leq m$,

$\displaystyle\pi\left(a_{r_{1}}\wedge a_{r_{2}}\ldots\wedge a_{r_{L}}\right)$
$\displaystyle=$ $\displaystyle 0\,\,\,\,\,\,\,\,\,\,\,\,\,\text{if
}\left\\{r_{1},r_{2},\ldots r_{L}\right\\}\in\mathfrak{\mathrm{\mathfrak{I}}}$
(5)

In contrast, let $\mathfrak{P}\subset\mathcal{P}\left(\left\\{1,2,\ldots
m\right\\}\right)$ denote the set of indices of the “possible” conjunctions:

$\displaystyle\mathfrak{P}$ $\displaystyle=$
$\displaystyle\Bigl{\\{}\left\\{r_{1},r_{2},\ldots
r_{L}\right\\}\in\mathcal{P}\left(\left\\{1,2,\ldots
m\right\\}\right)\Bigl{|}\,2\leq L\leq m;\left\\{r_{1},r_{2},\ldots
r_{L}\right\\}\notin\mathfrak{I}\Bigr{\\}}$

  * (E1)

We assume, as empirically observed fact, that every conjunction of
measurements that is possible does occur with some non-zero frequency:

$\displaystyle\pi\left(a_{r_{1}}\wedge a_{r_{2}}\ldots\wedge a_{r_{L}}\right)$
$\displaystyle>$ $\displaystyle 0\,\,\,\,\,\,\,\,\,\,\text{if
}\left\\{r_{1},r_{2},\ldots r_{L}\right\\}\in\mathfrak{P}$ (6)

We also assume that for all $1\leq r\leq m$,

$\displaystyle\pi\left(a_{r}\right)$ $\displaystyle>$ $\displaystyle 0$ (7)

Similarly to (1), we introduce the following sets of indices:

$\displaystyle S$ $\displaystyle=$ $\displaystyle\left\\{{}_{i_{1}i_{2}\ldots
i_{L}}^{r_{1}r_{2}\ldots r_{L}}\in S_{max}^{M}\,|\,\left\\{r_{1},r_{2},\ldots
r_{L}\right\\}\in\mathfrak{P}\right\\}$ $\displaystyle S_{\mathfrak{I}}$
$\displaystyle=$ $\displaystyle\left\\{{}_{i_{1}i_{2}\ldots
i_{L}}^{r_{1}r_{2}\ldots r_{L}}\in S_{max}^{M}\,|\,\left\\{r_{1},r_{2},\ldots
r_{L}\right\\}\in\mathfrak{I}\right\\}$

  * (E2)

The following assumptions are also regarded as empirically observed
regularities: for all ${}_{i}^{r},{}_{i^{\prime}}^{r^{\prime}}\in I^{M}$ and
$\left\\{r_{1},r_{2},\ldots r_{L}\right\\}\in\mathfrak{P}$,

$\displaystyle\pi\left(a_{r}\wedge X_{i}^{r}\right)$ $\displaystyle=$
$\displaystyle\pi\left(X_{i}^{r}\right)$ (8) $\displaystyle\text{if
}r=r^{\prime}\text{ and }i\not=i^{\prime}\text{ then }\pi\left(X_{i}^{r}\wedge
X_{i^{\prime}}^{r^{\prime}}\right)$ $\displaystyle=$ $\displaystyle 0$ (9)
$\displaystyle\sum_{\begin{array}[]{c}k\\\ \left({}_{k}^{r}\in
I^{M}\right)\end{array}}\pi\left(X_{k}^{r}|a_{r}\right)$ $\displaystyle=$
$\displaystyle 1$ (12) $\displaystyle\sum_{\begin{array}[]{c}k_{1}\ldots
k_{L}\\\ \left({}_{k_{1}\ldots k_{L}}^{r_{1}\ldots r_{L}}\in
S\right)\end{array}}\pi\left(X_{k_{1}}^{r_{1}}\wedge\ldots\wedge
X_{k_{L}}^{r_{L}}|a_{r_{1}}\wedge\ldots\wedge a_{r_{L}}\right)$
$\displaystyle=$ $\displaystyle 1$ (15)

where $\pi\left(\,|\,\right)$ denotes the usual conditional relative frequency
defined by the Bayes rule—$\pi\left(a_{r}\right)\neq 0$ and
$\pi\left(a_{r_{1}}\wedge a_{r_{2}}\ldots\wedge a_{r_{L}}\right)\neq 0$, due
to (6)–(7). That is to say, an outcome event does not occur without the
performance of the corresponding measurement operation; it is never the case
that two different outcomes of the same measurement occur simultaneously;
whenever a measurement operation is performed, one of the possible outcomes
occurs; whenever a conjunction of measurement operations is performed, one of
the possible outcome combinations occurs.

In the picture we suggest, an outcome of a measurement is, primarily, a
physical event, an occurrence of a certain state of affairs at the end of the
measuring process; rather than obtaining a numeric value of a quantity. To
give an example, the state of affairs when the rotated coil of a voltmeter
takes a new position of equilibrium with the distorted spring is ontologically
prior to the number on the scale to which its pointer points at that moment.
Nevertheless, in some cases the measurement outcomes are labeled by real
numbers that are interpreted as the “measured value” of a real-valued physical
quantity:

$\alpha_{r}:X_{i}^{r}\mapsto\alpha_{i}^{r}\in\mathbb{R}$ (16)

In this case, at least formally, it may make sense to talk about conditional
expectation value, that is the average of the measured values, given that the
measurement is performed:

$\displaystyle\left\langle\alpha_{r}\right\rangle$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{r}}\alpha_{i}^{r}\pi\left(X_{i}^{r}|a_{r}\right)$

About all labelings $\alpha_{r}$ we will assume that
$\alpha_{i}^{r}\neq\alpha_{j}^{r}$ for $i\neq j$.

##  3 The State of the System

Of course, the relative frequency $\pi$ in $\left(\mathcal{A},\pi\right)$
depends not only on the behavior of the physical system after a certain
physical preparation but also on the autonomous decisions of the experimenter
to perform this or that measurement operation. One can hope a scientific
description of the system only if the two things can be separated. Whether
this is possible is a contingent fact of the empirically observed reality,
reflected in the observed relative frequencies.

Let $\left|S\right|$ denote the number of elements of $S$. Consider the
following vector:

$\displaystyle\vec{Z}$ $\displaystyle=$
$\displaystyle\left(Z_{i}^{r},Z_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\right)\in\mathbb{R}^{M+\left|S\right|}$ (17)

where

$Z_{i}^{r}=\pi\left(X_{i}^{r}|a_{r}\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,{}_{i}^{r}\in
I^{M}$ (18)

and

$Z_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}=\pi\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}|a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)\,\,\,\,\,\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S$
(19)

In general, even if the physical preparation of the system is identical in
every run of the experiment, the conditional relative frequencies on the right
hand sides of (18)–(19), hence the values of $Z_{i}^{r}$ and $Z_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}$, may vary if the actual frequencies
$\left\\{\pi\left(a_{r}\right)\right\\}_{1\leq r\leq m}$ and
$\left\\{\pi\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)\right\\}_{\left\\{r_{1},\ldots
r_{L}\right\\}\in\mathfrak{P}}$ vary, for example, upon the experimenter’s
decisions.

However, we make the following stipulation as observed empirical fact:

  * (E3)

For all physical preparations, keeping the preparation fixed, $\vec{Z}$ is
independent of the actual nonzero values of
$\left\\{\pi\left(a_{r}\right)\right\\}_{1\leq r\leq m}$ and
$\left\\{\pi\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)\right\\}_{\left\\{r_{1},\ldots
r_{L}\right\\}\in\mathfrak{P}}$.

In other words, what (E3) says is that for all fixed physical preparations,
the observed relative frequencies are such that

$\displaystyle\pi\left(X_{i}^{r}\right)$
$\displaystyle=Z_{i}^{r}\pi\left(a_{r}\right)\,\,\,\,\,\,\,\,\,\,\,{}_{i}^{r}\in
I^{M}$ (20) $\displaystyle\pi\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}\right)$ $\displaystyle=\begin{cases}Z_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\pi\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)&{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S\\\
0&{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S_{\mathfrak{I}}\end{cases}$
(21)

with _one and the same_ $\vec{Z}=\left(Z_{i}^{r},Z_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\right)$. $\vec{Z}$ is therefore determined only by
the physical preparation. Notice that (E3) does not exclude that different
physical preparations lead to the same $\vec{Z}$.

$\vec{Z}$ can be regarded as a characterization of the system’s _state_ after
the given physical preparation, in the sense that it characterizes the
system’s future probabilistic behavior against all possible measurement
operations. This characterization is complete in the following sense:

######  Theorem 1.

State $\vec{Z}$ together with arbitrary relative frequencies of measurements,
$\left\\{\pi\left(a_{r}\right)\right\\}_{1\leq r\leq m}$ and
$\left\\{\pi\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)\right\\}_{\left\\{r_{1},\ldots
r_{L}\right\\}\in\mathfrak{P}}$, uniquely determine the relative frequency
function $\pi$ on the whole event algebra $\mathcal{A}$.

###### Proof.

Using the same notations we introduced in (3), each atom has the form of

$\varDelta_{\vec{\varepsilon},\vec{\eta}}=\underbrace{\left(\underset{{}_{i}^{r}\in
I^{M}}{\wedge}\left[X_{i}^{r}\right]^{\varepsilon_{i}^{r}}\right)}_{\varGamma_{\vec{\varepsilon}}}\wedge\left(\stackrel{{\scriptstyle[}}{{s}}=1]{m}{\wedge}\left[a_{s}\right]^{\eta_{s}}\right)$
(22)

Notice that the part $\varGamma_{\vec{\varepsilon}}=\underset{{}_{i}^{r}\in
I^{M}}{\wedge}\left[X_{i}^{r}\right]^{\varepsilon_{i}^{r}}$ in (22) uniquely
determines the whole $\varDelta_{\vec{\varepsilon},\vec{\eta}}$, whenever
$\pi\left(\varDelta_{\vec{\varepsilon},\vec{\eta}}\right)\neq 0$. Namely, due
to (8) and (12),

$\pi\left(\varDelta_{\vec{\varepsilon},\vec{\eta}}\right)\neq 0\text{ implies
that for all $1\leq r\leq m$}\text{,
}\sum_{i=1}^{n_{r}}\varepsilon_{i}^{r}=0\text{ iff }\eta_{r}=0$ (23)

In other words, for each $\vec{\varepsilon}\in\left\\{0,1\right\\}^{M}$ there
is exactly one $\vec{\eta}\in\left\\{0,1\right\\}^{m}$ for which (8) and (12)
do not imply that
$\pi\left(\varDelta_{\vec{\varepsilon},\vec{\eta}}\right)=0$. Let us denote it
by $\vec{\eta}\left(\vec{\varepsilon}\right)$; and, for the sake of brevity,
introduce the following notation:
$\delta_{\vec{\varepsilon}}=\pi\left(\varDelta_{\vec{\varepsilon},\vec{\eta}\left(\vec{\varepsilon}\right)}\right)$.
(It is not necessarily the case that $\delta_{\vec{\varepsilon}}\neq 0$. For
example, the empirical fact (9) will be accounted for in terms of the values
on the right hand side of (27) below.)

It must be also noticed that
$\left\\{\varGamma_{\vec{\varepsilon}}\right\\}_{\vec{\varepsilon}\in\left\\{0,1\right\\}^{M}}$
constitute the atoms of the free Boolean algebra $\mathcal{A}^{M}$ generated
by the set $\left\\{X_{i}^{r}\right\\}_{{}_{i}^{r}\in I^{M}}$. Events
$X_{i}^{r}$ and $X_{i_{1}}^{r_{1}}\wedge\ldots\wedge X_{i_{L}}^{r_{L}}$ on the
right hand sides of (18)–(19) are elements of $\mathcal{A}^{M}$, and have
therefore a unique decomposition into disjunction of atoms of
$\mathcal{A}^{M}$. Accordingly, taking into account (23), we have

$\displaystyle\sum_{\vec{\varepsilon}\in\left\\{0,1\right\\}^{M}}\delta_{\vec{\varepsilon}}$
$\displaystyle\,\,=\,\,$ $\displaystyle 1$ (24)
$\displaystyle\sum_{\vec{\varepsilon}\in\left\\{0,1\right\\}^{M}}{}_{i}^{r}R_{\vec{\varepsilon}}\,\,\delta_{\vec{\varepsilon}}$
$\displaystyle\,\,=\,\,$
$\displaystyle\pi\left(X_{i}^{r}\right)=Z_{i}^{r}\pi\left(a_{r}\right)$
${}_{i}^{r}\in I^{M}$ (25)
$\displaystyle\sum_{\vec{\varepsilon}\in\left\\{0,1\right\\}^{M}}{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots
r_{L}}R_{\vec{\varepsilon}}\,\,\delta_{\vec{\varepsilon}}\,$
$\displaystyle\,=\,\,$
$\displaystyle\pi\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}\right)$ $\displaystyle=Z_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\pi\left(a_{r_{1}}\wedge\ldots\wedge a_{r_{L}}\right)$ ${}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\in S$ (26)
$\displaystyle\sum_{\vec{\varepsilon}\in\left\\{0,1\right\\}^{M}}{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}R_{\vec{\varepsilon}}\,\,\delta_{\vec{\varepsilon}}$
$\displaystyle\,\,=\,\,$
$\displaystyle\pi\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}\right)=0$ ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in
S_{\mathfrak{I}}$ (27)

with

${}_{i}^{r}R_{\vec{\varepsilon}}$ $\displaystyle=$
$\displaystyle\begin{cases}1&\text{if }\varGamma_{\vec{\varepsilon}}\subseteq
X_{i}^{r}\\\ 0&\text{if }\varGamma_{\vec{\varepsilon}}\nsubseteq
X_{i}^{r}\end{cases}$ ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}R_{\vec{\varepsilon}}$ $\displaystyle=$
$\displaystyle\begin{cases}1&\text{if }\varGamma_{\vec{\varepsilon}}\subseteq
X_{i_{1}}^{r_{1}}\wedge\ldots\wedge X_{i_{L}}^{r_{L}}\\\ 0&\text{if
}\varGamma_{\vec{\varepsilon}}\nsubseteq X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}\end{cases}$

where $\subseteq$ is meant in the sense of the partial ordering in
$\mathcal{A}^{M}$.

Now, (24)–(27) constitute a system of $1+M+\left|S_{max}^{M}\right|=2^{M}$
linear equations with $2^{M}$ unknowns
$\delta_{\vec{\varepsilon}},\vec{\varepsilon}\in\left\\{0,1\right\\}^{M}$. The
equations are linearly independent due to the uniqueness of decomposition into
disjunction of atoms of $\mathcal{A}^{M}$, and due to the fact that there are
only conjunctions on the right hand side. (A similar equation for, say,
$X_{i_{1}}^{r_{1}}\vee X_{i_{2}}^{r_{2}}$ could be expressed as the sum of
equations for $X_{i_{1}}^{r_{1}}$ and $X_{i_{2}}^{r_{2}}$ minus the one for
$X_{i_{1}}^{r_{1}}\wedge X_{i_{2}}^{r_{2}}$.) Therefore, the system has a
unique solution for all $\delta_{\vec{\varepsilon}}$, that is, for the
relative frequencies of
$\left\\{\varDelta_{\vec{\varepsilon},\vec{\eta}\left(\vec{\varepsilon}\right)}\right\\}_{\vec{\varepsilon}\in\left\\{0,1\right\\}^{M}}$.
The rest of the atoms of $\mathcal{A}$ have zero relative frequency. ∎

Thus, the notion of state we introduced complies with the widespread intuition
about the notion of state within a probabilistic and operational context. As
Lucien Hardy puts it:

> The state associated with a particular preparation is defined to be (that
> thing represented by) any mathematical object that can be used to determine
> the probability associated with the outcomes of any measurement that may be
> performed on a system prepared by the given preparation. (2008, p. 2)

Such a typical formulation of the notion of state, however, results in the
possibility of misunderstanding. It must be clear that the state, in itself,
does not determine the probabilities of the measurement outcome events; only
the state of the system $\vec{Z}$ and the relative frequencies of the
measurements $\left\\{\pi\left(a_{r}\right)\right\\}_{1\leq r\leq m}$ and
$\left\\{\pi\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)\right\\}_{\left\\{r_{1},\ldots
r_{L}\right\\}\in\mathfrak{P}}$ together. And the fact that the frequencies of
the measurements in (20)–(21) can be arbitrary does not imply that the
components of $\vec{Z}$

$\left\\{Z_{i}^{r},Z_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\right\\}_{\begin{array}[]{l}{}_{i}^{r}\in I^{M};\,{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\in S\end{array}}$

constitute relative frequencies of the corresponding outcome events

$\left\\{X_{i}^{r},X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}\right\\}_{\begin{array}[]{l}{}_{i}^{r}\in
I^{M};\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S\end{array}}$

(or events whatsoever), as will be shown in Section 5.

In contrast, it is essential in our present analysis that the measurement
operations are treated on par with the outcome events; they belong to the
ontology. However, as it is clearly seen from (24)–(27), the notion of
$\vec{Z}$ detaches the “system’s contribution” to the totality of statistical
facts observed in the measurements from the “experimenter’s contribution”.

Still, the state of the system depends not only on the features intrinsic to
the system in itself, but also on the content of $\mathfrak{I}$, i.e., which
combinations of measuring operations cannot be performed simultaneously. This
means that the measuring devices and measuring operations, by means of which
we establish the empirically meaningful semantics of our physical description
of the system, play a _constitutive_ role in the notion of state _attributed
to the system_. This kind of constitutive role of the semantic conventions is
however completely natural in all empirically meaningful physical theories
(Szabó 2020).

The following lemma will be important for our further investigations:

######  Lemma 2.

For all states,

$\displaystyle Z_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}$ $\displaystyle\leq$
$\displaystyle\min\left\\{Z_{i_{\gamma_{1}}\ldots
i_{\gamma_{L-1}}}^{r_{\gamma_{1}}\ldots
r_{\gamma_{L-1}}}\right\\}_{\left\\{\gamma_{1},\ldots\gamma_{L-1}\right\\}\subset\left\\{1,\ldots
L\right\\}}$ (28)

where ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S$.

###### Proof.

It is known that similar inequality holds for arbitrary relative frequencies.
Therefore,

$\displaystyle\pi\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}\right)\leq\min\left\\{\pi\left(X_{i_{\gamma_{1}}}^{r_{\gamma_{1}}}\wedge\ldots\wedge
X_{i_{\gamma_{L-1}}}^{r_{\gamma_{L-1}}}\right)\right\\}_{\left\\{\gamma_{1},\ldots\gamma_{L-1}\right\\}\subset\left\\{1,\ldots
L\right\\}}$ (29)

for all ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S_{max}^{M}$, and

$\displaystyle\pi\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)\leq\min\left\\{\pi\left(a_{r_{\gamma_{1}}}\wedge\ldots\wedge
a_{r_{\gamma_{L-1}}}\right)\right\\}_{\left\\{\gamma_{1},\ldots\gamma_{L-1}\right\\}\subset\left\\{1,\ldots
L\right\\}}$ (30)

for all $2\leq L\leq m$, $1\leq r_{1},\ldots r_{L}\leq m$. It follows from the
definition of state that

$\displaystyle\pi\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}\right)$ $\displaystyle=$ $\displaystyle Z_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\pi\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)$ (31)
$\displaystyle\pi\left(X_{i_{\gamma_{1}}}^{r_{\gamma_{1}}}\wedge\ldots\wedge
X_{i_{\gamma_{L-1}}}^{r_{\gamma_{L-1}}}\right)$ $\displaystyle=$
$\displaystyle Z_{i_{\gamma_{1}}\ldots i_{\gamma_{L-1}}}^{r_{\gamma_{1}}\ldots
r_{\gamma_{L-1}}}\pi\left(a_{r_{\gamma_{1}}}\wedge\ldots\wedge
a_{r_{\gamma_{L-1}}}\right)$ (32)

for all ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S$ and
$\left\\{\gamma_{1},\ldots\gamma_{L-1}\right\\}\subset\left\\{1,\ldots
L\right\\}$. Consequently, from (29) we have

$\displaystyle\frac{Z_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}}{Z_{i_{\gamma_{1}}\ldots i_{\gamma_{L-1}}}^{r_{\gamma_{1}}\ldots
r_{\gamma_{L-1}}}}$ $\displaystyle\leq$
$\displaystyle\frac{\pi\left(a_{r_{\gamma_{1}}}\wedge\ldots\wedge
a_{r_{\gamma_{L-1}}}\right)}{\pi\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)}$ (33)

Since, according to the definition of state, (31)–(32) hold for all possible
relative frequencies $\left\\{\pi\left(a_{r}\right)\right\\}_{1\leq r\leq m}$
and $\left\\{\pi\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)\right\\}_{\left\\{r_{1},\ldots
r_{L}\right\\}\in\mathfrak{P}}$, inequality (33) must hold for the minimum
value of the right hand side, which is equal to $1$, due to (30). And this is
the case for all ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S$ and
$\left\\{\gamma_{1},\ldots\gamma_{L-1}\right\\}\subset\left\\{1,\ldots
L\right\\}$. ∎

It is of course an empirical question what states a system has after different
physical preparations. In what follows, we will answer the question: what can
we say about the “space” of theoretically possible states of a system? Where
by “theoretically possible states” we mean all vectors constructed by means of
definition (17)–(19) from arbitrary relative frequencies satisfying (6)–(15)
and (20)–(21). Here we should note that the general probabilistic description
includes the possibility—again, as an eventual empirical fact observed from
the frequencies (4)—that the system is deterministic, meaning that
$\vec{Z}\in\left\\{0,1\right\\}^{M+\left|S\right|}$, or at least it behaves
deterministically in some states.

We will show that the possible state vectors constitute a closed convex
polytope in $\mathbb{R}^{M+\left|S\right|}$, which we will denote by
$\varphi\left(M,S\right)$. First we will prove an important lemma.

######  Lemma 3.

If $\vec{Z}_{1}$ and $\vec{Z}_{2}$ are possible states then their convex
linear combination $\vec{Z}_{3}=\lambda_{1}\vec{Z}_{1}+\lambda_{2}\vec{Z}_{2}$
($\lambda_{1},\lambda_{2}\geq 0\,\,\,\,\lambda_{1}+\lambda_{2}=1$) also
constitutes a possible state.

###### Proof.

According to the definition of state, the observed relative frequencies of the
measurement outcomes in the two states are

$\displaystyle\pi_{1}\left(X_{i}^{r}\right)$
$\displaystyle=Z_{1}{}_{i}^{r}\pi_{1}\left(a_{r}\right)$ (34)
$\displaystyle\pi_{1}\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}\right)$ $\displaystyle=Z_{1}{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\pi_{1}\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)$ (35)

and

$\displaystyle\pi_{2}\left(X_{i}^{r}\right)$
$\displaystyle=Z_{2}{}_{i}^{r}\pi_{2}\left(a_{r}\right)$ (36)
$\displaystyle\pi_{2}\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}\right)$ $\displaystyle=Z_{2}{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\pi_{2}\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)$ (37)

for all ${}_{i}^{r}\in I^{M}$ and ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\in S$. Due to (E3), $\pi_{1}\left(a_{r}\right)$ and
$\pi_{1}\left(a_{r_{1}}\wedge\ldots\wedge a_{r_{L}}\right)$ as well as
$\pi_{2}\left(a_{r}\right)$ and $\pi_{2}\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)$ can be arbitrary relative frequencies satisfying (5)–(7).
Therefore, without loss of generality, we can take the case of

$\displaystyle\pi_{1}\left(a_{r}\right)=\pi_{2}\left(a_{r}\right)=\pi_{0}\left(a_{r}\right)$
$\displaystyle\pi_{1}\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)=\pi_{2}\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)=\pi_{0}\left(a_{r_{1}}\wedge\ldots\wedge a_{r_{L}}\right)$

Now, consider the convex linear combination
$\pi_{3}=\lambda_{1}\pi_{1}+\lambda_{2}\pi_{2}$. Obviously, $\pi_{3}$
satisfies (6)–(12), and

$\displaystyle\pi_{3}\left(a_{r}\right)$ $\displaystyle=$
$\displaystyle\pi_{0}\left(a_{r}\right)$
$\displaystyle\pi_{3}\left(a_{r_{1}}\wedge\ldots\wedge a_{r_{L}}\right)$
$\displaystyle=$ $\displaystyle\pi_{0}\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)$

Accordingly, we have

$\displaystyle\pi_{3}\left(X_{i}^{r}\right)$
$\displaystyle=\lambda_{1}\pi_{1}\left(X_{i}^{r}\right)+\lambda_{2}\pi_{2}\left(X_{i}^{r}\right)=\left(\lambda_{1}Z_{1}{}_{i}^{r}+\lambda_{2}Z_{2}{}_{i}^{r}\right)\pi_{3}\left(a_{r}\right)$
$\displaystyle\pi_{3}\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}\right)$
$\displaystyle=\lambda_{1}\pi_{1}\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}\right)+\lambda_{2}\pi_{2}\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}\right)$ $\displaystyle=\left(\lambda_{1}Z_{1}{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}+\lambda_{2}Z_{2}{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\right)\pi_{3}\left(a_{r_{1}}\wedge\ldots\wedge a_{r_{L}}\right)$

This means that $\vec{Z}_{3}=\lambda_{1}\vec{Z}_{1}+\lambda_{2}\vec{Z}_{2}$
satisfies condition (20)–(21), as $\pi_{3}\left(a_{r}\right)$ and
$\pi_{3}\left(a_{r_{1}}\wedge\ldots\wedge a_{r_{L}}\right)$ can be arbitrary
frequencies satisfying (5)–(7). That is, $\vec{Z}_{3}$ complies with the
definition of state, meaning that $\vec{Z}_{3}$ is a possible state of the
system. ∎

Now we turn to the question of the “space” of possible states. Let
$\vec{e}\,_{i}^{r},\vec{e}\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\in\mathbb{R}^{M+\left|S\right|}$ denote the ${}_{i}^{r}$-th and
${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}$-th coordinate base vector of
$\mathbb{R}^{M+\left|S\right|}$, where ${}_{i}^{r}\in I^{M}$ and
${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S$, and let
$\vec{f}=\left(f_{i}^{r},f_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\right)\in\mathbb{R}^{M+\left|S\right|}$ denote an arbitrary vector.

The empirical facts (E1)–(E3), partly through Lemmas 2–3, imply that the
possible state vectors constitute a closed convex polytope
$\varphi\left(M,S\right)\subset\mathbb{R}^{M+\left|S\right|}$ defined by the
following system of linear inequalities:

$\displaystyle f_{i}^{r}$ $\displaystyle\geq$ $\displaystyle 0$ (38)
$\displaystyle f_{i}^{r}$ $\displaystyle\leq$ $\displaystyle 1$ (39)
$\displaystyle f_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}$ $\displaystyle\geq$
$\displaystyle 0$ (40) $\displaystyle f_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}-f_{i_{\gamma_{1}}\ldots i_{\gamma_{L-1}}}^{r_{\gamma_{1}}\ldots
r_{\gamma_{L-1}}}$ $\displaystyle\leq$ $\displaystyle
0\,\,\,\,\,\,\,\,\,\,\left\\{\gamma_{1},\ldots\gamma_{L-1}\right\\}\subset\left\\{1,\ldots
L\right\\}$ (41) $\displaystyle\sum_{\begin{array}[]{c}k\\\
\left({}_{k}^{r}\in I^{M}\right)\end{array}}f_{k}^{r}$ $\displaystyle=$
$\displaystyle 1$ (44) $\displaystyle\sum_{\begin{array}[]{c}k_{1},k_{2}\ldots
k_{L}\\\ \left({}_{k_{1}\ldots k_{L}}^{r_{1}\ldots r_{L}}\in
S\right)\end{array}}f_{k_{1}\ldots k_{L}}^{r_{1}\ldots r_{L}}$
$\displaystyle=$ $\displaystyle 1$ (47) $\displaystyle f_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}$ $\displaystyle=$
$\displaystyle 0\,\,\,\,\,\,\,\,\,\,{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}\in S_{0}$ (48)

for all ${}_{i}^{r}\in I^{M}$, ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in
S$, and

$\displaystyle S_{0}$ $\displaystyle=$ $\displaystyle\left\\{{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\in
S\Bigl{|}r_{\gamma_{1}}=r_{\gamma_{2}},i_{\gamma_{1}}\neq
i_{\gamma_{2}},\left\\{\gamma_{1},\gamma_{2}\right\\}\subset\left\\{1,\ldots
L\right\\}\right\\}$

Denote by $l\left(M,S\right)\subset\mathbb{R}^{M+\left|S\right|}$ the closed
convex polytope defined by the first group of inequalities (38)–(41). As is
well known (Pitowsky 1989, pp. 51 and 65), the vertices of $l\left(M,S\right)$
are all the vectors $\vec{v}\in\mathbb{R}^{M+\left|S\right|}$ such that

  * (a)

$v_{i}^{r},v_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in\left\\{0,1\right\\}$
for all ${}_{i}^{r}\in I^{M}$ and ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\in S$.

  * (b)

$v_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\leq\underset{{}^{{}_{\left\\{\gamma_{1},\gamma_{2},\ldots\gamma_{L-1}\right\\}\subset\,\left\\{1,2,\ldots
L\right\\}}}}{\prod}v_{i_{\gamma_{1}}\ldots
i_{\gamma_{L-1}}}^{r_{\gamma_{1}}\ldots r_{\gamma_{L-1}}}$ for all
${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S$.

A vertex is called classical if the equality holds everywhere in (b), and non-
classical otherwise.

Obviously, $\varphi\left(M,S\right)\subseteq l\left(M,S\right)$. What can be
said about the vertices of $\varphi\left(M,S\right)$?

######  Lemma 4.

The vertices of $\varphi\left(M,S\right)$ are all the vectors
$\vec{f}\in\varphi\left(M,S\right)$ such that $f_{i}^{r},f_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\in\left\\{0,1\right\\}$ for all ${}_{i}^{r}\in
I^{M}$ and ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S$.

###### Proof.

One direction is trivial: if $\vec{f}\in\varphi\left(M,S\right)$ and
$f_{i}^{r},f_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in\left\\{0,1\right\\}$
for all ${}_{i}^{r}\in I^{M}$ and ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\in S$, then $\vec{f}$ is a vertex. For, if there exist
$\vec{f}^{\prime},\vec{f}^{\prime\prime}\in\varphi\left(M,S\right)$ such that
$\vec{f}=\lambda\vec{f}^{\prime}+(1-\lambda)\vec{f}^{\prime\prime}$ with some
$0<\lambda<1$, then obviously
$\vec{f}^{\prime}=\vec{f}^{\prime\prime}=\vec{f}$.

The proof of the other direction is quite involved. For a more concise
notation, introduce the following sets of indices:

$\displaystyle I$ $\displaystyle=$
$\displaystyle\Bigl{\\{}1|_{i}^{r},\,2|{}_{i}^{r},\,3|{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}},\,4|_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}|{}_{i_{\gamma_{1}}\ldots i_{\gamma_{L-1}}}^{r_{\gamma_{1}}\ldots
r_{\gamma_{L-1}}},\,5|r,\,6|r_{1}\ldots r_{L},\,7|{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}\,\Bigl{|}\,\text{for all
}$ ${}_{i}^{r}\in I^{M},\,\text{${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}$$\in S$,}{}_{i^{\prime}_{1}\ldots i^{\prime}_{L}}^{r^{\prime}_{1}\ldots
r^{\prime}_{L}}\in S_{0},\text{and
}\left\\{\gamma_{1},\ldots\gamma_{L-1}\right\\}\subset\left\\{1,\ldots
L\right\\}\Bigr{\\}}$ $\displaystyle I^{0}$ $\displaystyle=$
$\displaystyle\Bigl{\\{}1|_{i}^{r},\,2|{}_{i}^{r},\,3|{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}},\,4|_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}|{}_{i_{\gamma_{1}}\ldots i_{\gamma_{L-1}}}^{r_{\gamma_{1}}\ldots
r_{\gamma_{L-1}}}\,\Bigl{|}\,\text{for all }_{i}^{r}\in I^{M},{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\in S,$ $\displaystyle\text{and
}\left\\{\gamma_{1},\ldots\gamma_{L-1}\right\\}\subset\left\\{1,\ldots
L\right\\}\Bigr{\\}}$ $\displaystyle I^{+}$ $\displaystyle=$
$\displaystyle\Bigl{\\{}5|r,\,6|r_{1}\ldots r_{L},\,7|_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}\,\left|\,\text{for all
}1\leq r\leq m,\,\left\\{r_{1}\ldots r_{L}\right\\}\in\mathfrak{P},\right.$
$\displaystyle\text{and }{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}\in S_{0}\Bigr{\\}}$

Obviously, $I=I^{0}\cup I^{+}$ and $I^{0}\cap I^{+}=\emptyset$. Rewrite
(38)–(48) in the following standard form:

$\displaystyle\left\langle\vec{\omega}_{\mu},\vec{f}\right\rangle-b_{\mu}$
$\displaystyle\leq$ $\displaystyle 0\,\,\,\,\,\text{for all }\mu\in I^{0}$
(49) $\displaystyle\left\langle\vec{\omega}_{\mu},\vec{f}\right\rangle-
b_{\mu}$ $\displaystyle=$ $\displaystyle 0\,\,\,\,\,\text{for all }\mu\in
I^{+}$ (50)

with the following $\vec{\omega}_{\mu}\in\mathbb{R}^{M+\left|S\right|}$ and
$b_{\mu}\in\mathbb{R}$:

$\displaystyle\vec{\omega}_{1|_{i}^{r}}$ $\displaystyle=$
$\displaystyle(0\ldots
0\,\overset{\underset{\smallsmile}{{}_{i}^{r}}}{-1}\,0\ldots 0)$ (51)
$\displaystyle b_{1|_{i}^{r}}$ $\displaystyle=$ $\displaystyle 0$ (52)
$\displaystyle\vec{\omega}_{2|_{i}^{r}}$ $\displaystyle=$
$\displaystyle(0\ldots
0\,\overset{\underset{\smallsmile}{{}_{i}^{r}}}{1}\,0\ldots 0)$ (53)
$\displaystyle b_{2|_{i}^{r}}$ $\displaystyle=$ $\displaystyle 1$ (54)
$\displaystyle\vec{\omega}_{3|_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}}$
$\displaystyle=$ $\displaystyle(0\ldots
0\,\overset{\underset{\smallsmile}{{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}}}{-1}\,0\ldots 0)$ (55) $\displaystyle b_{3|_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}}$ $\displaystyle=$ $\displaystyle 0$ (56)
$\displaystyle\vec{\omega}_{4|_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}|{}_{i_{\gamma_{1}}\ldots i_{\gamma_{L-1}}}^{r_{\gamma_{1}}\ldots
r_{\gamma_{L-1}}}}$ $\displaystyle=$ $\displaystyle(0\ldots
0\,\overset{\underset{\smallsmile}{{}_{i_{\gamma_{1}}\ldots
i_{\gamma_{L-1}}}^{r_{\gamma_{1}}\ldots r_{\gamma_{L-1}}}}}{-1}\,0\ldots
0\,\overset{\underset{\smallsmile}{{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}}}{1}\,0\ldots 0)$ (57) $\displaystyle b_{4|_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}|{}_{i_{\gamma_{1}}\ldots
i_{\gamma_{L-1}}}^{r_{\gamma_{1}}\ldots r_{\gamma_{L-1}}}}$ $\displaystyle=$
$\displaystyle 0$ (58) $\displaystyle\vec{\omega}_{5|r}$ $\displaystyle=$
$\displaystyle(0\ldots
0\,\overset{\underset{\smallsmile}{{}_{1}^{r}}}{1}\,\overset{\underset{\smallsmile}{{}_{2}^{r}}}{1}\,\overset{\underset{\smallsmile}{{}_{3}^{r}}}{1}\ldots\overset{\underset{\smallsmile}{{}_{n_{r}}^{r}}}{1}\,0\ldots
0)$ (59) $\displaystyle b_{5|r}$ $\displaystyle=$ $\displaystyle 1$ (60)
$\displaystyle\vec{\omega}_{6|r_{1}\ldots r_{L}}$ $\displaystyle=$
$\displaystyle(0\ldots
0\,\overset{\underset{\smallsmile}{{}_{\,\,1\,\,\,1\,\,\ldots\,1}^{r_{1}r_{2}\ldots
r_{L}}}}{1}\,\,\overset{\underset{\smallsmile}{{}_{\,\,2\,\,\,1\,\,\ldots\,1}^{r_{1}r_{2}\ldots
r_{L}}}}{1}\ldots\overset{\underset{\smallsmile}{{}_{n_{r_{1}}\,n_{r_{2}}\ldots
n_{r_{L}}}^{\,r_{1}\,\,\,\,r_{2}\,\,\,\ldots r_{L}}}}{1}\,0\ldots 0)$ (61)
$\displaystyle b_{6|r_{1}\ldots r_{L}}$ $\displaystyle=$ $\displaystyle 1$
(62) $\displaystyle\vec{\omega}_{7|_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}}$ $\displaystyle=$
$\displaystyle(0\ldots
0\,\overset{\underset{\smallsmile}{{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}}}{1}\,0\ldots 0)$ (63)
$\displaystyle b_{7|_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}}$ $\displaystyle=$
$\displaystyle 0$ (64)

where ${}_{i}^{r}\in I^{M}$, ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in
S$, $\left\\{\gamma_{1},\ldots\gamma_{L-1}\right\\}\subset\left\\{1,\ldots
L\right\\}$, and ${}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}\in S_{0}$. Notice that
$l\left(M,S\right)$ is defined by (49).

For an arbitrary $\vec{f}\in l\left(M,S\right)$ we define the following sets:

$\displaystyle I_{\vec{f}}$ $\displaystyle=$ $\displaystyle\left\\{\mu\in
I\left|\,\left\langle\vec{\omega}_{\mu},\vec{f}\right\rangle-
b_{\mu}=0\right.\right\\}$ $\displaystyle I_{\vec{f}}^{0}$ $\displaystyle=$
$\displaystyle\left\\{\mu\in
I^{0}\left|\,\left\langle\vec{\omega}_{\mu},\vec{f}\right\rangle-
b_{\mu}=0\right.\right\\}$

Notice that if $\vec{f}\in\varphi\left(M,S\right)$, then
$I_{\vec{f}}=I_{\vec{f}}^{0}\cup I^{+}$, due to the fact that (44)–(48) can be
satisfied only with equality.

$I_{\vec{f}}$ constitutes the so called ‘active index set’ for
$\vec{f}\in\varphi\left(M,S\right)$; and according to a known theorem (see
Theorem 12 in Appendix), $\vec{f}$ is a vertex of $\varphi\left(M,S\right)$ if
and only if

$\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}}=\mathbb{R}^{M+\left|S\right|}$ (65)

Similarly, a vector $\vec{f}\in l\left(M,S\right)$ is a vertex of
$l\left(M,S\right)$ if and only if

$\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}^{0}}=\mathbb{R}^{M+\left|S\right|}$ (66)

For all $\vec{f}\in l\left(M,S\right)$ define

$\displaystyle J_{\vec{f}}$ $\displaystyle=$
$\displaystyle\left\\{{}_{i}^{r}\left|{}_{i}^{r}\in I^{M}\text{ and
}0<f_{i}^{r}<1\right.\right\\}$ $\displaystyle J^{\prime}_{\vec{f}}$
$\displaystyle=$ $\displaystyle\left\\{{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\left|{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S\text{ and
}0<f{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}<1\right.\right\\}$

Notice that for all ${}_{i}^{r}\in I^{M}$ and ${}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\in S$,

$\displaystyle\vec{e}\,_{i}^{r}\in\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}^{0}}$ if $\displaystyle f_{i}^{r}\in\\{0,1\\}$ (67)
$\displaystyle\vec{e}\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\in\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}^{0}}$ if $\displaystyle f_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\in\\{0,1\\}$ (68)

since the corresponding inequalities (38)–(41) must hold with equality. The
only case that requires a bit of reflection is when $f_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}=1$. For example, if $f_{i_{1}i_{2}}^{r_{1}r_{2}}=1$
then (41) is satisfied with equality, so that $f_{i_{1}}^{r_{1}}=1$, therefore
(39) is also satisfied with equality. Consequently,

$\displaystyle\vec{\omega}_{2|_{i_{1}}^{r_{1}}}$ $\displaystyle\in$
$\displaystyle\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}^{0}}$
$\displaystyle\vec{\omega}_{4|_{i_{1}i_{2}}^{r_{1}r_{2}}|{}_{i_{1}}^{r_{1}}}$
$\displaystyle\in$
$\displaystyle\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}^{0}}$

At the same time, as it can be seen from (53) and (57),

$\vec{e}\,_{i_{1}i_{2}}^{r_{1}r_{2}}=\vec{\omega}_{2|_{i_{1}}^{r_{1}}}+\vec{\omega}_{4|_{i_{1}i_{2}}^{r_{1}r_{2}}|{}_{i_{1}}^{r_{1}}}\in\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}^{0}}$

This can be recursively continued for the triple and higher conjunction
indices.

Assume now that $\vec{f}\in\varphi\left(M,S\right)$ is such that
$J_{\vec{f}}\cup J^{\prime}_{\vec{f}}\neq\emptyset$, and at the same time it
is a vertex of $\varphi\left(M,S\right)$, that is, (65) is satisfied. We are
going to show that this leads to contradiction.

Due to (67)–(68), the assumption that $\vec{f}$ is a vertex implies that all
base vectors of $\mathbb{R}^{M+\left|S\right|}$ must belong to
$\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in I_{\vec{f}}^{0}}$
save for some $\vec{e}\,_{i}^{r}$’s with ${}_{i}^{r}\in J_{\vec{f}}$ and/or
some $\vec{e}\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}$’s with
${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in J^{\prime}_{\vec{f}}$. On the
other hand, $\vec{f}$ being a vertex implies that (65) holds, therefore

$\displaystyle\vec{e}\,_{i}^{r}$ $\displaystyle\in$
$\displaystyle\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}}\,\,\,\,\,\,\,\,\,\text{for all }{}_{i}^{r}\in J_{\vec{f}}$
$\displaystyle\vec{e}\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}$
$\displaystyle\in$
$\displaystyle\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}}\,\,\,\,\,\,\,\,\,\text{for all }{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\in J^{\prime}_{\vec{f}}$

Taking into account that $I_{\vec{f}}=I_{\vec{f}}^{0}\cup I^{+}$, it means
that for all ${}_{i}^{r}\in J_{\vec{f}}$ and arbitrary $\tau_{i}^{r}\neq 0$
there exist vectors

${}_{i}^{r}\vec{v}$ $\displaystyle\in$
$\displaystyle\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}^{0}}$ (69)

such that,

$\displaystyle\tau_{i}^{r}\vec{e}\,_{i}^{r}$ $\displaystyle=$
${}_{i}^{r}\vec{v}+\sum_{s=1}^{m}{}_{i}^{r}\kappa_{s}\vec{\omega}_{5|s}+\sum_{\left\\{r_{1},\ldots
r_{L^{\prime}}\right\\}\in\mathfrak{P}}{}_{i}^{r}\kappa^{\prime}_{r_{1}\ldots
r_{L^{\prime}}}\vec{\omega}_{6|r_{1}\ldots r_{L^{\prime}}}$ (70)
$\displaystyle+\sum_{{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}\in
S_{0}}{}_{i}^{r}\lambda_{{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots
r^{\prime}_{L}}}\vec{\omega}_{7|_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}}$

with some real numbers ${}_{i}^{r}\kappa_{s}$,
${}_{i}^{r}\kappa^{\prime}_{r_{1}\ldots r_{L^{\prime}}}$, and
${}_{i}^{r}\lambda_{{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}}$. From the definitions
of $\vec{\omega}_{5|r}$, $\vec{\omega}_{6|r_{1}\ldots r_{L}}$, and
$\vec{\omega}_{7|_{i^{\prime}_{1}\ldots i^{\prime}_{L}}^{r^{\prime}_{1}\ldots
r^{\prime}_{L}}}$ in (51)–(64) we can write:

$\displaystyle\tau_{i}^{r}\vec{e}\,_{i}^{r}$
$\displaystyle=_{i}^{r}\vec{v}+\sum_{{}_{j}^{s}\in
I^{M}}{}_{i}^{r}\kappa_{s}\vec{e}\,_{j}^{s}+\sum_{{}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}\in
S}{}_{i}^{r}\kappa^{\prime}_{s_{1}\ldots
s_{L^{\prime}}}\vec{e}\,{}_{j_{1}\ldots j_{L^{\prime}}}^{s_{1}\ldots
s_{L^{\prime}}}$ $\displaystyle+\sum_{{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}\in
S_{0}}{}_{i}^{r}\lambda_{{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots
r^{\prime}_{L}}}\vec{e}\,{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}$ (71)

_Mutatis mutandis_ , we have the same equation for $\vec{e}\,{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}$ for all ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\in J^{\prime}_{\vec{f}}$ with arbitrary $\tau{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\neq 0$ and with some numbers ${}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\kappa_{s}$, ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\kappa^{\prime}_{r_{1}\ldots r_{L^{\prime}}}$, and ${}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\lambda_{{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}}$:

$\displaystyle\tau{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\vec{e}\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}$
$\displaystyle=\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\vec{v}+\sum_{{}_{j}^{s}\in I^{M}}{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\kappa_{s}\vec{e}\,_{j}^{s}+\sum_{{}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}\in S}{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\kappa^{\prime}_{s_{1}\ldots
s_{L^{\prime}}}\vec{e}\,{}_{j_{1}\ldots j_{L^{\prime}}}^{s_{1}\ldots
s_{L^{\prime}}}$ $\displaystyle+\sum_{{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}\in S_{0}}{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\lambda_{{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots
r^{\prime}_{L}}}\vec{e}\,{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}$ (72)

With some rearrangement, from (71) we have

$\displaystyle\sum_{\begin{array}[]{c}_{j}^{s}\in J_{\vec{f}}\\\
_{j}^{s}\neq_{i}^{r}\end{array}}{}_{i}^{r}\kappa_{s}\vec{e}\,_{j}^{s}+\left({}_{i}^{r}\kappa_{r}-\tau_{i}^{r}\right)\vec{e}\,_{i}^{r}+\sum_{{}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}\in
J^{\prime}_{\vec{f}}}{}_{i}^{r}\kappa^{\prime}_{s_{1}\ldots
s_{L^{\prime}}}\vec{e}\,{}_{j_{1}\ldots j_{L^{\prime}}}^{s_{1}\ldots
s_{L^{\prime}}}$ (75)
$\displaystyle\,\,\,\,\,\,\,\,=-\sum_{{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}\in
S_{0}}{}_{i}^{r}\lambda_{{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots
r^{\prime}_{L}}}\vec{e}\,{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots
r^{\prime}_{L}}-\sum_{\begin{array}[]{c}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}\in S\\\ _{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}\notin
J^{\prime}_{\vec{f}}\end{array}}{}_{i}^{r}\kappa^{\prime}_{s_{1}\ldots
s_{L^{\prime}}}\vec{e}\,{}_{j_{1}\ldots j_{L^{\prime}}}^{s_{1}\ldots
s_{L^{\prime}}}$ (78)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\sum_{\begin{array}[]{c}_{j}^{s}\in
I^{M}\\\ _{j}^{s}\notin
J_{\vec{f}}\end{array}}{}_{i}^{r}\kappa_{s}\vec{e}\,_{j}^{s}-{}_{i}^{r}\vec{v}$
(81)

for all ${}_{i}^{r}\in J_{\vec{f}}$.

Similarly, from (72) we have

$\displaystyle\sum_{{}_{j}^{s}\in J_{\vec{f}}}{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots
r_{L}}\kappa_{s}\vec{e}\,_{j}^{s}\,\,+\sum_{\begin{array}[]{c}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}\in J^{\prime}_{\vec{f}}\\\
_{j_{1}\ldots j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}\neq{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\end{array}}{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\kappa^{\prime}_{s_{1}\ldots s_{L^{\prime}}}\vec{e}\,{}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}$ (84)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+\left({}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\kappa^{\prime}_{r_{1}\ldots
r_{L^{\prime}}}-\tau{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\right)\vec{e}\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}$
$\displaystyle\,\,\,\,\,\,\,\,=-\sum_{{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}\in S_{0}}{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\lambda_{{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots
r^{\prime}_{L}}}\vec{e}\,{}_{i^{\prime}_{1}\ldots
i^{\prime}_{L}}^{r^{\prime}_{1}\ldots r^{\prime}_{L}}$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\sum_{\begin{array}[]{c}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}\in S\\\ _{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}\notin
J^{\prime}_{\vec{f}}\end{array}}{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\kappa^{\prime}_{s_{1}\ldots s_{L^{\prime}}}\vec{e}\,{}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}$ (87)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-\sum_{\begin{array}[]{c}_{j}^{s}\in
I^{M}\\\ _{j}^{s}\notin J_{\vec{f}}\end{array}}{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\kappa_{s}\vec{e}\,_{j}^{s}-{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\vec{v}$ (90)

for all ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in J^{\prime}_{\vec{f}}$.

Denote the right hand side of (94) by $\vec{B}_{{}_{i}^{r}}$ and the right
hand side of (90) by $\vec{B}_{{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}}$.
Notice that the vectors $\vec{B}_{{}_{i}^{r}}$ and $\vec{B}_{{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}}$ are contained in
$\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in I_{\vec{f}}^{0}}$,
due to (69), and (67)–(68). So, in (94)–(90), together, we have a system of
linear equations with vector-variables
$\left\\{\vec{e}\,_{j}^{s}\right\\}_{{}_{j}^{s}\in J_{\vec{f}}}$ and
$\left\\{\vec{e}\,{}_{j_{1}\ldots j_{L^{\prime}}}^{s_{1}\ldots
s_{L^{\prime}}}\right\\}_{{}_{j_{1}\ldots j_{L^{\prime}}}^{s_{1}\ldots
s_{L^{\prime}}}\in J^{\prime}_{\vec{f}}}$, which can be written in the
following form:

$\displaystyle\sum_{{}_{j}^{s}\in J_{\vec{f}},\,{}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}\in
J^{\prime}_{\vec{f}}}\beta_{\Bigl{(}{}_{i}^{r},\,{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\Bigr{)}\Bigl{(}{}_{j}^{s},\,{}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots
s_{L^{\prime}}}\Bigr{)}}\left(\vec{e}\,_{j}^{s},\,\vec{e}\,{}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots
s_{L^{\prime}}}\right)=\left(\vec{B}_{{}_{i}^{r}},\vec{B}_{{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}}\right)$ (91)

where $\beta_{\Bigl{(}{}_{i}^{r},\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\Bigr{)}\Bigl{(}{}_{j}^{s},\,{}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}\Bigr{)}}$ is a
$\left(|J_{\vec{f}}|+|J^{\prime}_{\vec{f}}|\right)\times\left(|J_{\vec{f}}|+|J^{\prime}_{\vec{f}}|\right)$
matrix with diagonal elements

$\displaystyle\beta_{{}_{i}^{r}{}_{i}^{r}}$
$\displaystyle=_{i}^{r}\kappa_{r}-\tau_{i}^{r}$
$\displaystyle\beta_{{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}}$
$\displaystyle=_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\kappa^{\prime}_{r_{1}\ldots r_{L}}-\tau{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}$

The off diagonal elements depend only on ${}_{i}^{r}\kappa_{s}$’s and
${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\kappa^{\prime}_{s_{1}\ldots
s_{L^{\prime}}}$’s. Since the numbers $\tau_{i}^{r}\neq 0$ and
$\tau{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\neq 0$ in the diagonal can be
chosen arbitrarily, we may assume that
$\det\beta_{\Bigl{(}{}_{i}^{r},\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\Bigr{)}\Bigl{(}{}_{j}^{s},\,{}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}\Bigr{)}}\neq 0$. Therefore, the
system of linear equations (91) has a unique solution for all vector-variables
$\vec{e}\,_{i}^{r}$ and $\vec{e}\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}$,
namely,

$\left(\vec{e}\,{}_{i}^{r},\,\vec{e}\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\right)=\sum_{{}_{j}^{s}\in J_{\vec{f}},\,{}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}\in
J^{\prime}_{\vec{f}}}\beta_{\Bigl{(}{}_{i}^{r},\,{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\Bigr{)}\Bigl{(}{}_{j}^{s},\,{}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots
s_{L^{\prime}}}\Bigr{)}}^{-1}\left(\vec{B}_{{}_{j}^{s}},\,\vec{B}_{{}_{j_{1}\ldots
j_{L^{\prime}}}^{s_{1}\ldots s_{L^{\prime}}}}\right)$

Taking into account that $\vec{B}_{{}_{i}^{r}},\vec{B}_{{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots
r_{L}}}\in\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}^{0}},$ this all means that for all ${}_{i}^{r}\in J_{\vec{f}}$,
the base vectors $\vec{e}\,_{i}^{r}$, and for all ${}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\in J^{\prime}_{\vec{f}}$, the base vectors
$\vec{e}\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}$ can be expressed as
linear combinations of vectors contained in
$\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in I_{\vec{f}}^{0}}$.
As all the rest of base vectors belong to
$\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in I_{\vec{f}}^{0}}$
(as we have already mentioned above), we have

$\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}^{0}}=\mathbb{R}^{M+\left|S\right|}$

meaning that $\vec{f}$ must be a vertex of $l\left(M,S\right)$. Due to the
fact that all components of a vertex of $l\left(M,S\right)$ are necessarily
$0$ or $1$, there cannot exists a vertex $\vec{f}\in\varphi\left(M,S\right)$
with $0<f_{i}^{r}<1$ and/or $0<f_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}<1$. ∎

All this means that the vertices of $\varphi\left(M,S\right)$ are those
vertices of $l\left(M,S\right)$ which satisfy the further restrictions
(44)–(48).

To sum up, the “space” of possible states is a closed convex polytope
$\varphi\left(M,S\right)\subset\mathbb{R}^{M+\left|S\right|}$ whose vertices
are the vectors $\vec{w}\in\mathbb{R}^{M+\left|S\right|}$ such that

  * (a)

$w_{i}^{r},w_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in\left\\{0,1\right\\}$
for all ${}_{i}^{r}\in I^{M}$ and ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\in S$.

  * (b)

$w_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\leq\underset{{}^{{}_{\left\\{\gamma_{1},\ldots\gamma_{L-1}\right\\}\subset\,\left\\{1,\ldots
L\right\\}}}}{\prod}w_{i_{\gamma_{1}}\ldots
i_{\gamma_{L-1}}}^{r_{\gamma_{1}}\ldots r_{\gamma_{L-1}}}$ for all
${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S$.

  * (c)

$w_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}=0$ for all ${}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\in S_{0}$.

  * (d)

For all $1\leq r\leq m$ there is exactly one $1\leq i_{*}^{r}\leq n_{r}$ such
that $w_{i_{*}^{r}}^{r}=1$.

  * (e)

For all $\left\\{r_{1},\ldots r_{L}\right\\}\in\mathfrak{P}$ there is exactly
one ${}_{i_{1}^{*}\ldots i_{L}^{*}}^{r_{1}\ldots r_{L}}\in S$ such that
$w_{i_{1}^{*}\ldots i_{L}^{*}}^{r_{1}\ldots r_{L}}=1$.

(We note in advance that property (d) will be crucial in the proof of Theorem
9.) Let
$\mathcal{W}=\left\\{\vec{w}_{\vartheta}\right\\}_{\vartheta\in\Theta}$ denote
the set of vertices of $\varphi\left(M,S\right)$.

Again, $\varphi\left(M,S\right)$ is the space of theoretically possible
states; and it may be that the empirically determined possible states of the
system for different physical preparations constitute only a subset of
$\varphi\left(M,S\right)$. In the absence of such an empirical restriction,
for the sake of generality, in what follows we assume that the space of
possible states is the whole $\varphi\left(M,S\right)$.

A closed convex polytope like
$\varphi\left(M,S\right)\subset\mathbb{R}^{M+\left|S\right|}$ is a
$\text{dim}\left(\varphi\left(M,S\right)\right)$-dimensional manifold with
boundary. Any coordinate system in the affine hull of
$\varphi\left(M,S\right)$ can be a natural coordination of
$\varphi\left(M,S\right)$.

Thus, $\varphi\left(M,S\right)$ as a manifold with boundary is a perfect
mathematical representation of the states of the system; in fact, it is the
most straightforward one, expressible directly in empirical terms. This is
however not the only one. As Hardy puts it in the above quotation, the state
of the system can be represented by “any mathematical object that can be used
to determine the probability associated with the outcomes of any measurement”.
For example, for our later purposes the convex decomposition

$\vec{Z}=\sum_{\vartheta\in\Theta}\lambda_{\vartheta}\vec{w}_{\vartheta}\,\,\,\,\,\,\,\lambda_{\vartheta}\geq
0,\sum_{\vartheta\in\Theta}\lambda_{\vartheta}=1$ (92)

will be a more suitable characterization of a point of the state space.
However, in general, this decomposition is not unique. In fact there are
continuum many ways of such decomposition for all
$\vec{Z}\in\text{Int}\,\varphi\left(M,S\right)$; and a unique one if $\vec{Z}$
is on the boundary. As we will show, there are various good solutions for
obtaining a unique representation of states in terms of their vertex
decomposition (92).

Introduce the following notation:
$\vec{\lambda}=\left(\lambda_{\vartheta}\right)_{\vartheta\in\Theta}\in\mathbb{R}^{\left|\Theta\right|}$.
Let

$\Lambda=\left\\{\vec{\lambda}\in\mathbb{R}^{\left|\Theta\right|}\left|\lambda_{\vartheta}\geq
0,\sum_{\vartheta\in\Theta}\lambda_{\vartheta}=1\right.\right\\}$

$\Lambda$ is the $\left(\left|\Theta\right|-1\right)$-dimensional standard
simplex in $\mathbb{R}^{\left|\Theta\right|}$. Obviously,

$D:\Lambda\rightarrow\varphi\left(M,S\right);\,\,\,\,D\left(\vec{\lambda}\right)=\sum_{\vartheta\in\Theta}\lambda_{\vartheta}\vec{w}_{\vartheta}$
(93)

is a continuous projection, and it preserves convex combination.

######  Lemma 5.

For all $\vec{Z}\in\varphi\left(M,S\right)$, $D^{-1}\left(\vec{Z}\right)$ is a
polytope contained in $\Lambda$.

###### Proof.

To satisfy (92), beyond being contained in $\Lambda$, $\vec{\lambda}$ has to
satisfy the following system of linear equations:

$\displaystyle\sum_{\vartheta\in\Theta}\lambda_{\vartheta}w_{\vartheta}{}_{i}^{r}$
$\displaystyle=$ $\displaystyle
Z_{i}^{r}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{}_{i}^{r}\in
I^{M}$ (94)
$\displaystyle\sum_{\vartheta\in\Theta}\lambda_{\vartheta}w_{\vartheta}{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}$ $\displaystyle=$ $\displaystyle Z_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\,\,\,\,\,\,\,\,\,\,\,\,\,{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\in S$ (95)

For a given $\vec{Z}$, the set of solutions constitute an affine subspace
$\mathfrak{a}_{\vec{Z}}\subset\mathbb{R}^{\left|\Theta\right|}$ with
difference space $\mathcal{B}\subset\mathbb{R}^{\left|\Theta\right|}$
constituted by the solutions of the homogeneous equations

$\displaystyle\sum_{\vartheta\in\Theta}\lambda_{\vartheta}w_{\vartheta}{}_{i}^{r}$
$\displaystyle=$ $\displaystyle 0\,\,\,\,\,\,\,\,\,\,\,\,\,{}_{i}^{r}\in
I^{M}$
$\displaystyle\sum_{\vartheta\in\Theta}\lambda_{\vartheta}w_{\vartheta}{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}$ $\displaystyle=$ $\displaystyle
0\,\,\,\,\,\,\,\,\,\,\,\,\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S$

Notice that $D^{-1}\left(\vec{Z}\right)=\Lambda\cap\mathfrak{a}_{\vec{Z}}$.
Due to the fact that an intersection of a polytope with an affine subspace is
a polytope (Henk _et al_. 2004), each $D^{-1}\left(\vec{Z}\right)$ is a
polytope contained in $\Lambda$. ∎

######  Lemma 6.

$D^{-1}\left(\vec{Z}\right)$, as a subset of
$\mathbb{R}^{\left|\Theta\right|}$, continuously depends on $\vec{Z}$ in the
following sense:

$\displaystyle\underset{\vec{Z}^{\prime}\rightarrow\vec{Z}}{\text{lim}}\,\,\,\underset{\vec{\lambda}\in
D^{-1}\left(\vec{Z}\right)}{\text{max}}\,d\left(\vec{\lambda},D^{-1}\left(\vec{Z}^{\prime}\right)\right)$
$\displaystyle=$ $\displaystyle 0$ (96)
$\displaystyle\underset{\vec{Z}^{\prime}\rightarrow\vec{Z}}{\text{lim}}\,\,\,\underset{\vec{\lambda}\in
D^{-1}\left(\vec{Z}^{\prime}\right)}{\text{max}}\,d\left(\vec{\lambda},D^{-1}\left(\vec{Z}\right)\right)$
$\displaystyle=$ $\displaystyle 0$ (97)

where $d\left(\,,\,\right)$ denotes the usual distance of a point from a set.

###### Proof.

We have to show that (96)–(97) hold approaching from all possible directions
to $\vec{Z}$. In other words, if $t\in[0,1]$ and
$\varDelta\vec{Z}\in\mathbb{R}^{M+\left|S\right|}$ is an arbitrary non-zero
vector such that $\vec{Z}-\varDelta\vec{Z}\in\varphi\left(M,S\right)$, then

$\displaystyle\underset{t\rightarrow
0}{\text{lim}}\,\,\,\underset{\vec{\lambda}\in
D^{-1}\left(\vec{Z}\right)}{\text{max}}\,d\left(\vec{\lambda},D^{-1}\left(\vec{Z}-t\varDelta\vec{Z}\right)\right)$
$\displaystyle=$ $\displaystyle 0$ (98) $\displaystyle\underset{t\rightarrow
0}{\text{lim}}\,\,\,\underset{\vec{\lambda}\in
D^{-1}\left(\vec{Z}-t\varDelta\vec{Z}\right)}{\text{max}}\,d\left(\vec{\lambda},D^{-1}\left(\vec{Z}\right)\right)$
$\displaystyle=$ $\displaystyle 0$ (99)

Let $\varDelta\vec{\lambda}$ be a solution of equations (94)–(95) with
$\varDelta\vec{Z}$:

$\displaystyle\sum_{\vartheta\in\Theta}\varDelta\lambda_{\vartheta}w_{\vartheta}{}_{i}^{r}$
$\displaystyle=$ $\displaystyle\varDelta
Z_{i}^{r}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{}_{i}^{r}\in
I^{M}$ (100)
$\displaystyle\sum_{\vartheta\in\Theta}\varDelta\lambda_{\vartheta}w_{\vartheta}{}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}$ $\displaystyle=$ $\displaystyle\varDelta
Z_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\,\,\,\,\,\,\,\,\,\,\,\,\,{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in
S$ (101)

$\varDelta\vec{\lambda}$ can be orthogonally decomposed as follows:

$\varDelta\vec{\lambda}=\varDelta\vec{\lambda}^{\bot}+\varDelta\vec{\lambda}^{\|}\,\,\,\,\,\,\,\,\,\varDelta\vec{\lambda}^{\bot}\in\mathcal{B}^{\bot}\text{
and }\varDelta\vec{\lambda}^{\|}\in\mathcal{B}$

Obviously, $\varDelta\vec{\lambda}^{\bot}$ is uniquely determined by
$\varDelta\vec{Z}$; accordingly, replacing $\varDelta\vec{Z}$ with
$t\varDelta\vec{Z}$ on the right hand side of (100)–(101) we get
$t\varDelta\vec{\lambda}^{\bot}$ in place of $\varDelta\vec{\lambda}^{\bot}$.
Notice that $\left|t\varDelta\vec{\lambda}^{\bot}\right|$ is the distance
between the affine subspaces of solutions $\mathfrak{a}_{\vec{Z}}$ and
$\mathfrak{a}_{\vec{Z}-t\varDelta\vec{Z}}$; tending to zero if $t\rightarrow
0$.

Let $\vec{\lambda}$ be an arbitrary point in $D^{-1}\left(\vec{Z}\right)$ and
let $\vec{\lambda}^{\prime}$ be the point in
$D^{-1}\left(\vec{Z}-\varDelta\vec{Z}\right)$ closest to $\vec{\lambda}$, that
is,

$\displaystyle
d\left(\vec{\lambda},D^{-1}\left(\vec{Z}-\varDelta\vec{Z}\right)\right)$
$\displaystyle=$
$\displaystyle\left|\vec{\lambda}^{\prime}-\vec{\lambda}\right|$

Consider the point

$\vec{\lambda}_{t}=\vec{\lambda}+t\left(\vec{\lambda}^{\prime}-\vec{\lambda}\right)$

Obviously, $\vec{\lambda}_{t}\in\Lambda$ and
$\vec{\lambda}_{t}\in\mathfrak{a}_{\vec{Z}-t\varDelta\vec{Z}}$ for all
$t\in[0,1]$, that is,

$\vec{\lambda}_{t}\in D^{-1}\left(\vec{Z}-t\varDelta\vec{Z}\right)$

Therefore,

$d\left(\vec{\lambda},D^{-1}\left(\vec{Z}-t\varDelta\vec{Z}\right)\right)\leq
t\left|\vec{\lambda}^{\prime}-\vec{\lambda}\right|$

which implies (98).

Also, notice that

$\underset{t\rightarrow 0}{\text{lim}}\,\,\,\underset{\vec{\lambda}\in
D^{-1}\left(\vec{Z}-t\varDelta\vec{Z}\right)}{\text{max}}\,d\left(\vec{\lambda},\mathfrak{a}_{\vec{Z}}\right)=0$

which implies (99), otherwise there would exist a convergent sequence of
points from different $D^{-1}\left(\vec{Z}-t\varDelta\vec{Z}\right)$ sets such
that the limiting point is not contained in $D^{-1}\left(\vec{Z}\right)$,
contradicting to the facts that $\Lambda$ is closed and
$D^{-1}\left(\vec{Z}\right)=\Lambda\cap\mathfrak{a}_{\vec{Z}}$. ∎

Lemma 5 and 6 mean that the states of the system can be represented in a
continuous way by a disjoint family of polytopes contained in $\Lambda$. This
is of course a very unusual and inconvenient way of representation. However,
we can easily make it more convenient by assigning a point in each
$D^{-1}\left(\vec{Z}\right)$ representing the entire polytope. There are
several possibilities: for example, the center of mass, or any other notion of
the center of a polytope. Here we will use the notion of the point of maximal
entropy, which is perhaps physically also meaningful (Pitowsky 1989, p. 47).

The point of maximal entropy of an arbitrary polytope
$\mathcal{S}\subset\Lambda$:

$\vec{c}(\mathcal{S})=\,\begin{cases}\text{maximize}&H\left(\vec{\lambda}\right)=-\sum_{\vartheta\in\Theta}\lambda_{\vartheta}\text{log}\lambda_{\vartheta}\\\
\text{subject to}&\vec{\lambda}\in\mathcal{S}\end{cases}$

Since $\mathcal{S}$ is contained in $\Lambda$, this maximization problem
always has a solution. Meaning that $\vec{c}(\mathcal{S})$ is uniquely
determined and always contained in $\mathcal{S}$.

######  Lemma 7.

Let us define the following section of the bundle projection (93):

$\displaystyle\sigma:\varphi\left(M,S\right)\rightarrow\Lambda$
$\displaystyle\,\,\,\,\,\,\,\,\,\sigma\left(\vec{Z}\right)=\vec{c}\left(D^{-1}\left(\vec{Z}\right)\right)\,\,\,\in
D^{-1}\left(\vec{Z}\right)$ (102)

Then, $\sigma\left(\vec{Z}\right)$ is continuous in $\vec{Z}$, that is, for
all $\vec{Z},\vec{Z}^{\prime}\in\varphi\left(M,S\right)$,

$\underset{\vec{Z}^{\prime}\rightarrow\vec{Z}}{\text{lim}}\,\sigma\left(\vec{Z^{\prime}}\right)=\sigma\left(\vec{Z}\right)$

###### Proof.

Consider a sufficiently fine division of the unit cube
$C^{\left|\Theta\right|}\subset\mathbb{R}^{\left|\Theta\right|}$ into equally
sized small cubes of volume $\varDelta V$. Denote the $i$-th such elementary
cube by $C_{i}$. The point of maximal entropy of a polytope
$\mathcal{S}\subset\Lambda\subset C^{\left|\Theta\right|}$ can be approximated
with arbitrary precision in the following way:

$\vec{c}\left(\mathcal{S}\right)\simeq\,\begin{cases}\text{maximize}&H\left({}^{i}\vec{\lambda}\right)=-\sum_{\vartheta\in\Theta}{}^{i}\lambda_{\vartheta}\text{log}{}^{i}\lambda_{\vartheta}\\\
\text{subject to}&i\in\left\\{j\,|\,\mathcal{S}\cap
C_{j}\neq\textrm{\O}\right\\}\end{cases}$ (103)

where ${}^{i}\vec{\lambda}$ is, say, the center of $C_{i}$. Due to Lemma 6,
for all $\varDelta V>0$ there is an $\varepsilon>0$ such that, for all
elementary cube $C_{i}$,

$D^{-1}\left(\vec{Z}^{\prime}\right)\cap
C_{i}\neq\textrm{\O}\,\,\Leftrightarrow\,\,D^{-1}\left(\vec{Z}\right)\cap
C_{i}\neq\textrm{\O}\text{\,\,\,\,\,\,\, if
$\left|\vec{Z}^{\prime}-\vec{Z}\right|<\varepsilon$}$

Meaning that, for a sufficiently small $\varepsilon$, approximation (103)
leads to the same result for $D^{-1}\left(\vec{Z}^{\prime}\right)$ and
$D^{-1}\left(\vec{Z}\right)$. Therefore,

$\underset{\vec{Z}^{\prime}\rightarrow\vec{Z}}{\text{lim}}\,\vec{c}\left(D^{-1}\left(\vec{Z}^{\prime}\right)\right)=\vec{c}\left(D^{-1}\left(\vec{Z}\right)\right)$

∎

By means of $\sigma$ (or any similar continuous section) the whole state space
$\varphi\left(M,S\right)$ can be lifted into a
$\text{dim}\left(\varphi\left(M,S\right)\right)$-dimensional submanifold with
boundary:

$\Lambda_{\sigma}=\sigma\left(\varphi\left(M,S\right)\right)\subset\Lambda$
(104)

##  4 Dynamics

So far, nothing has been said about the dynamics of the system, that is, about
the time evolution of the state $\vec{Z}$. First we have to introduce the
concept of time evolution in general operational terms. Let us start with the
most general case.

Imagine that the system is in state $\vec{Z}\left(t_{0}\right)$ after a
certain physical preparation at time $t_{0}$. According to the definition of
state, this means that the system responds to the various measurement
operations right after time $t_{0}$ in a way described in (20)–(21). Let then
the system evolve under a given set of circumstances until time $t$. Let
$\vec{Z}\left(t\right)$ be the system’s state at moment $t$. Again, this means
that the system responds to the various measurement operations right after
time $t$ in a way described in (20)–(21) with $\vec{Z}\left(t\right)$. Thus,
we have a temporal path of the system in the space of states
$\varphi\left(M,S\right)$. It is quite plausible to assume that
$\vec{Z}\left(t\right)$ is a continuous curve in $\varphi\left(M,S\right)$. By
means of a continuous cross section like (102), $\vec{Z}\left(t\right)$ can be
lifted and expressed as $\sigma\left(\vec{Z}\left(t\right)\right)$, a
continuous curve on $\Lambda_{\sigma}$.

Whether the time evolution of the system shows any regularity whatsoever, is a
matter of empirical facts reflected in the observed relative frequencies under
various circumstances. Next, as an empirically observed regularity, we assume
a typical situation when the time evolution $\vec{Z}\left(t\right)$ can be
generated by a one-parameter group of transformations of
$\varphi\left(M,S\right)$.

  * (E4)

The time evolutions of states are such that there exists a one-parameter group
of transformations of $\varphi\left(M,S\right)$, $F_{t}$, satisfying the
following conditions:

    * $F_{t}:\varphi\left(M,S\right)\rightarrow\varphi\left(M,S\right)$ is one-to-one

    * $F:\mathbb{R}\times\varphi\left(M,S\right)\rightarrow\varphi\left(M,S\right);\,\left(t,\vec{Z}\right)\mapsto F_{t}\left(\vec{Z}\right)$ is continuous

    * $F_{t+s}=F_{s}\circ F_{t}$

    * $F_{-t}=F_{t}^{-1}$; consequently, $F_{0}=id_{\varphi\left(M,S\right)}$

and the time evolution of an arbitrary initial state
$\vec{Z}(t_{0})\in\varphi\left(M,S\right)$ is
$\vec{Z}(t)=F_{t-t_{0}}\left(\vec{Z}(t_{0})\right)$.

It is worth mentioning that though the state space $\varphi\left(M,S\right)$
is closed under convex combination, the stipulated empirical facts do not
imply that $F_{t}$ should preserve convex combinations.

By means of the continuous cross section (102), $F_{t}$ generates a one-
parameter group of transformations on $\Lambda_{\sigma}$, $K_{t}=\sigma\circ
F_{t}\circ D$, with exactly the same properties:

  * $K_{t}:\Lambda_{\sigma}\rightarrow\Lambda_{\sigma}$ is one-to-one

  * $K:\mathbb{R}\times\Lambda_{\sigma}\rightarrow\Lambda_{\sigma};\,\left(t,\vec{Z}\right)\mapsto K_{t}\left(\vec{Z}\right)$ is continuous

  * $K_{t+s}=K_{s}\circ K_{t}$

  * $K_{-t}=K_{t}^{-1}$; consequently, $K_{0}=id_{\Lambda_{\sigma}}$

##  5 Ontology

Thus, the state of the system $\vec{Z}\in\varphi\left(M,S\right)$, together
with its time evolution, completely characterizes the system’s probabilistic
behavior in the sense of Theorem 1. In general, such a probabilistic
description can admit different underlying ontological pictures. Though, as we
will see, some of those underlying ontologies imply further conditions on the
observed relative frequencies. We will mention three important cases, but
various combinations are conceivable.

#### Case 1

In the most general case, without any further restriction on the observed
relative frequencies, the outcomes of the measurements are random events
produced in the measurement process itself. The state $\vec{Z}$ characterizes
the system in a dispositional sense: the system has a propensity to behave in
a certain way, that is, to produce a certain statistics of outcomes, if a
given combination of measurements is performed. In general, the produced
statistics is such that, for example,

$\pi\left(X_{i}^{r}|a_{r}\wedge
a_{r^{\prime}}\right)\neq\pi\left(X_{i}^{r}|a_{r}\right)\,\,\,\,\,\,\,\,\,\,\,\,\,\left\\{r,r^{\prime}\right\\}\in\mathfrak{P}$
(105)

meaning that the underlying process is “contextual” in the sense that the
system’s statistical behavior against measurement $a_{r}$ can be influenced by
the performance of another measurement $a_{r^{\prime}}$.

#### Case 2

In the second case we assume that there is no such cross-influence in the
underlying ontology. That is, the observed relative frequencies satisfy the
following general condition:

$\displaystyle\pi\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}|a_{r_{1}}\wedge\ldots\wedge a_{r_{L}}\wedge
a_{r^{\prime}_{1}}\wedge\ldots\wedge a_{r^{\prime}_{L^{\prime}}}\right)$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\pi\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}|a_{r_{1}}\wedge\ldots\wedge a_{r_{L}}\right)$ (106)

for all $L,L^{\prime}$, $2\leq L+L^{\prime}\leq m$, ${}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\in S$, and $\left\\{r_{1},\ldots
r_{L},r^{\prime}_{1},\ldots r^{\prime}_{L^{\prime}}\right\\}\in\mathfrak{P}$.
This does not mean that there cannot be correlation between the outcomes
$X_{i_{1}}^{r_{1}}\wedge\ldots\wedge X_{i_{L}}^{r_{L}}$ and the performance of
measurement $a_{r^{\prime}_{1}}\wedge\ldots\wedge a_{r^{\prime}_{L^{\prime}}}$
It only means that the correlation must be the consequence of the fact that
the measurement operations $a_{r^{\prime}_{1}}\wedge\ldots\wedge
a_{r^{\prime}_{L^{\prime}}}$ and $a_{r_{1}}\wedge\ldots\wedge a_{r_{L}}$ are
correlated; $a_{r_{1}}\wedge\ldots\wedge a_{r_{L}}$ must be the common cause
responsible for the correlation. Indeed, (106) is equivalent with the
following “screening off” condition:

$\displaystyle\pi\left(a_{r^{\prime}_{1}}\wedge\ldots\wedge
a_{r^{\prime}_{L^{\prime}}}\wedge X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}|a_{r_{1}}\wedge\ldots\wedge a_{r_{L}}\right)$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\pi\left(a_{r^{\prime}_{1}}\wedge\ldots\wedge
a_{r^{\prime}_{L^{\prime}}}|a_{r_{1}}\wedge\ldots\wedge a_{r_{L}}\right)$
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\times\,\pi\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}|a_{r_{1}}\wedge\ldots\wedge a_{r_{L}}\right)$ (107)

for all $L,L^{\prime}$, $2\leq L+L^{\prime}\leq m$, ${}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\in S$, and $\left\\{r_{1},\ldots
r_{L},r^{\prime}_{1},\ldots r^{\prime}_{L^{\prime}}\right\\}\in\mathfrak{P}$.

All this means that the state of the system $\vec{Z}$ reflects the
propensities of the system to produce a certain statistics of outcomes against
each possible measurement/measurement combination, separately. The observed
statistics reveals the propensity in question, but, in general, we are not
entitled to say that a single outcome (of a measurement/measurement
combination) reveals an element of reality existing independently of the
measurement(s). As we will see below, that would require a stronger
restriction on the observed frequencies.

#### Case 3

Assume that the underlying ontology contains such elements of reality. Let us
denote them by $\\#X_{i}^{r}$ (${}_{i}^{r}\in I^{M}$). More precisely, let
$\\#X_{i}^{r}$ denote the event that the element of reality revealed in the
outcome $X_{i}^{r}$ is present in the given run of the experiment. Certainly,
every such event $\\#X_{i}^{r}$, even if hidden to us, must have some relative
frequency. That is to say, there must exists a relative frequency function
$\pi^{\prime}$ on the extended free Boolean algebra $\mathcal{A}^{\prime}$
generated by the set

$G^{\prime}=\left\\{a_{r}\right\\}_{r=1,2,\ldots
m}\cup\left\\{X_{i}^{r}\right\\}_{{}_{i}^{r}\in
I^{M}}\cup\left\\{\\#X_{j}^{s}\right\\}_{{}_{j}^{s}\in I^{M}}$ (108)

such that

$\displaystyle\pi^{\prime}\,\bigl{|}_{\mathcal{A}\subset\mathcal{A}^{\prime}}$
$\displaystyle=$ $\displaystyle\pi$ (109)

The ontological assumption that $\\#X_{i}^{r}$ is revealed by the measurement
outcome $X_{i}^{r}$ means that

$\displaystyle\pi^{\prime}\left(X_{i}^{r}|a_{r}\wedge\\#X_{i}^{r}\right)$
$\displaystyle=$ $\displaystyle 1$ (110)
$\displaystyle\pi^{\prime}\left(X_{i}^{r}|a_{r}\wedge\neg\\#X_{i}^{r}\right)$
$\displaystyle=$ $\displaystyle 0$ (111)
$\displaystyle\pi^{\prime}\left(a_{r}\wedge\\#X_{i}^{r}\right)$
$\displaystyle=$
$\displaystyle\pi^{\prime}\left(a_{r}\right)\pi^{\prime}\left(\\#X_{i}^{r}\right)$
(112)

Similarly,

$\displaystyle\pi^{\prime}\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}|a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\wedge\\#X_{i_{1}}^{r_{1}}\wedge\ldots\wedge\\#X_{i_{L}}^{r_{L}}\right)=1$
(113) $\displaystyle\pi^{\prime}\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}|a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\wedge\neg\left(\\#X_{i_{1}}^{r_{1}}\wedge\ldots\wedge\\#X_{i_{L}}^{r_{L}}\right)\right)=0$
(114) $\displaystyle\pi^{\prime}\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\wedge\\#X_{i_{1}}^{r_{1}}\wedge\ldots\wedge\\#X_{i_{L}}^{r_{L}}\right)=\pi^{\prime}\left(a_{r_{1}}\wedge\ldots\wedge
a_{r_{L}}\right)$
$\displaystyle\times\,\pi^{\prime}\left(\\#X_{i_{1}}^{r_{1}}\wedge\ldots\wedge\\#X_{i_{L}}^{r_{L}}\right)$
(115)

for all ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S$.

Now, (110)–(115) and (109) imply that

$\displaystyle\pi^{\prime}\left(\\#X_{i}^{r}\right)$ $\displaystyle=$
$\displaystyle\pi\left(X_{i}^{r}|a_{r}\right)=Z_{i}^{r}$ (116)
$\displaystyle\pi^{\prime}\left(\\#X_{i_{1}}^{r_{1}}\wedge\ldots\wedge\\#X_{i_{L}}^{r_{L}}\right)$
$\displaystyle=$ $\displaystyle\pi\left(X_{i_{1}}^{r_{1}}\wedge\ldots\wedge
X_{i_{L}}^{r_{L}}|a_{r_{1}}\wedge\ldots\wedge a_{r_{L}}\right)$ (117)
$\displaystyle=Z_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}$

for all ${}_{i}^{r}\in I^{M}$ and ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\in S$.

Notice that on the right hand side of (116)–(117) we have the components of
$\vec{Z}$. At the same time, on the left hand side of (116)–(117) we have
numbers that are values of relative frequencies. Therefore the components of
$\vec{Z}$ must constitute values of relative frequencies (of the occurrences
of elements of reality $\\#X_{i}^{r}$ and
$\\#X_{i_{1}}^{r_{1}}\ldots\wedge\\#X_{i_{L}}^{r_{L}}$). Since values of
relative frequencies satisfy the Kolmogorovian laws of classical probability,
$\vec{Z}$ must be in the so-called classical correlation polytope (Pitowsky
1989, Ch. 2):

$\vec{Z}\in c\left(M,S\right)$ (118)

(Equivalently, the components of $\vec{Z}$ must satisfy the corresponding
Bell-type inequalities.) In this case the physical state of the system admits
a more fine-grained characterization than the probabilistic description
provided by $\vec{Z}$: in each run of the experiment the system can be thought
of as being in an underlying physical state (fixing whether the elements of
reality $\\#X_{i}^{r}$ and
$\\#X_{i_{1}}^{r_{1}}\ldots\wedge\\#X_{i_{L}}^{r_{L}}$ are present or not)
that predetermines the outcome of every possible measurement, given that the
measurement in question is performed.

Thus, as we have seen from the above examples, the probabilistic–operational
notion of state admits different underlying ontologies, depending on whether
some further conditions are met or not. Note that condition (106) in Case 2 is
sometimes called “no-signaling condition”; and Case 3 is usually interpreted
as “admitting deterministic non-contextual hidden variables”. In what follows,
we do not assume anything more about the observed relative frequencies than we
stipulated in (E1)–(E3). Meaning that we remain within the most general
framework of Case 1.

##  6 Quantum Representation

So far in the previous sections, we have stayed within the framework of
classical Kolmogorovian probability theory; including the notion of state,
which is a simple vector constructed from classical conditional probabilities.
Meaning that any physical system—traditionally categorized as classical or
quantum—that can be described in operational terms can be described within
classical Kolmogorovian probability theory. It is worth pointing out that this
is also the case when the system is traditionally described in terms of the
Hilbert space quantum mechanical formalism. That is, all the empirically
expressible content of the quantum mechanical description can be described in
the language of classical Kolmogorovian probabilities; including what we refer
to as “quantum probability”, given by the usual trace formula, which can be
expressed simply as classical conditional probability. All this is in perfect
alignment with the content of the so-called Kolmogorovian Censorship
Hypothesis (Szabó 1995; Bana and Durt 1997; Szabó 2001; Rédei 2010; Hofer-
Szabó _et al_. 2013, Ch. 9).

In the remainder of the paper we will show that the opposite is also true:
anything that can be described in operational terms can be represented in the
Hilbert space quantum mechanical formalism, if we wish. We will show that
there always exists:

  * (Q1)

a suitable Hilbert space, such that

  * (Q2)

the outcomes of each measurement can be represented by a system of pairwise
orthogonal closed subspaces, spanning the whole Hilbert space,

  * (Q3)

the states of the system can be represented by pure state operators with
suitable state vectors, and

  * (Q4)

the probabilities of the measurement outcomes, with arbitrarily high
precision, can be reproduced by the usual trace formula of quantum mechanics.

Moreover, in the case of real-valued quantities,

  * (Q5)

each quantity, if we wish, can be associated with a suitable self-adjoint
operator, such that

  * (Q6)

in all states of the system, the expectation value of the quantity can be
reproduced, with arbitrarily high precision, by the usual trace formula
applied to the associated self-adjoint operator,

  * (Q7)

the possible measurement results are the eigenvalues of the operator,

  * (Q8)

and the corresponding outcome events are represented by the eigenspaces
pertaining to the eigenvalues respectively, according to the spectral
decomposition of the operator in question.

In preparation for our quantum representation theorem, first we prove a lemma,
which is a straightforward consequence of previous results in Pitowsky’s
_Quantum Probability – Quantum Logic._

######  Lemma 8.

For each vector $\vec{f}\in l\left(M,S\right)$ there exists a Hilbert space
${}^{(\vec{f})}H$ and closed subspaces ${}^{(\vec{f})}E_{i}^{r}$ in the
subspace lattice $L\left({}^{(\vec{f})}H\right)$ and a pure state
$P_{\Psi_{\vec{f}}}$ with a suitable unit vector
$\Psi_{\vec{f}}\in{}^{(\vec{f})}H$, such that

$\displaystyle f_{i}^{r}$ $\displaystyle\simeq$ $\displaystyle
tr\left(P_{\Psi_{\vec{f}}}{}^{(\vec{f})}E_{i}^{r}\right)$ (119) $\displaystyle
f_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}$ $\displaystyle\simeq$
$\displaystyle
tr\left(P_{\Psi_{\vec{f}}}\left({}^{(\vec{f})}E_{i_{1}}^{r_{1}}\wedge\ldots\wedge^{(\vec{f})}E_{i_{L}}^{r_{L}}\right)\right)$
(120)

for all ${}_{i}^{r}\in I^{M}$ and ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\in S$.

###### Proof.

It follows from a straightforward generalization of a theorem in (Pitowsky
1989, p. 65) that the so called quantum polytope $q\left(M,S\right)$,
constituted by the vectors satisfying (119)–(120) with exact equality, is a
dense convex subset of $l\left(M,S\right)$; it is essentially
$l\left(M,S\right)$ save for some points on the boundary of
$l\left(M,S\right)$, namely the finite number of non-classical vertices.
$q\left(M,S\right)$ contains the interior of $l\left(M,S\right)$. This means
that arbitrary vector $\vec{f}\in l\left(M,S\right)$ can be regarded as “an
element of” $q\left(M,S\right)$ _with arbitrary precision_. That is, there
exists a Hilbert space ${}^{(\vec{f})}H$ and for each ${}_{i}^{r}\in I^{M}$ a
closed subspace/projector ${}^{(\vec{f})}E_{i}^{r}$ in the subspace/projector
lattice $L\left({}^{(\vec{f})}H\right)$ and a suitable unit vector
$\Psi_{\vec{f}}\in{}^{(\vec{f})}H$, such that the approximate equalities
(119)–(120) hold. ∎

######  Theorem 9.

There exists a Hilbert space $H$ and for each outcome event $X_{i}^{r}$ a
closed subspace/projector $E_{i}^{r}$ in the subspace/projector lattice
$L\left(H\right)$, such that for each state $\vec{Z}$ of the system there
exists a pure state $P_{\Psi_{\vec{Z}}}$ with a suitable unit vector
$\Psi_{\vec{Z}}\in H$, such that

$\displaystyle Z_{i}^{r}$ $\displaystyle\simeq$ $\displaystyle
tr\left(P_{\Psi_{\vec{Z}}}E_{i}^{r}\right)$ (121) $\displaystyle
Z_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}$ $\displaystyle\simeq$
$\displaystyle
tr\left(P_{\Psi_{\vec{Z}}}\left(E_{i_{1}}^{r_{1}}\wedge\ldots\wedge
E_{i_{L}}^{r_{L}}\right)\right)$ (122)

and

$\displaystyle E_{i}^{r}\bot\,E_{j}^{r}$ $\displaystyle\,\,\,\,\,\,i\neq j$
(123)
$\displaystyle\stackrel{{\scriptstyle[}}{{k}}=1]{n_{r}}{\vee}E_{k}^{r}=H$
(124)

for all ${}_{i}^{r},{}_{j}^{r}\in I^{M}$ and ${}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\in S$.

###### Proof.

The proof is essentially based on Lemma 4 and proceeds in two major steps.

### Step I

Consider the vertices of $\varphi\left(M,S\right)$,
$\left\\{\vec{w}_{\vartheta}\right\\}_{\vartheta\in\Theta}$. Each
$\vec{w}_{\vartheta}$ is a vector in $l\left(M,S\right).$ Therefore, due to
Lemma 8, for each $\vec{w}_{\vartheta}$ there exists a Hilbert space
${}^{\vartheta}\tilde{H}$ and closed subspaces
${}^{\vartheta}\tilde{E}_{i}^{r}$ in the subspace lattice
$L\left({}^{\vartheta}\tilde{H}\right)$ and a pure state
$P_{\tilde{\Psi}_{\vartheta}}$ with a suitable unit vector
$\tilde{\Psi}_{\vartheta}\in{}^{\vartheta}\tilde{H}$, such that

$\displaystyle w_{\vartheta}{}_{i}^{r}$ $\displaystyle\simeq$ $\displaystyle
tr\left(P_{\tilde{\Psi}_{\vartheta}}{}^{\vartheta}\tilde{E}_{i}^{r}\right)$
(125) $\displaystyle w_{\vartheta}{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}$
$\displaystyle\simeq$ $\displaystyle
tr\left(P_{\tilde{\Psi}_{\vartheta}}\left({}^{\vartheta}\tilde{E}_{i_{1}}^{r_{1}}\wedge\ldots\wedge{}^{\vartheta}\tilde{E}_{i_{L}}^{r_{L}}\right)\right)$
(126)

for all ${}_{i}^{r}\in I^{M}$ and ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots
r_{L}}\in S$.

Now, let

${}^{\vartheta}H$ $\displaystyle=$ $\displaystyle H^{n_{1}}\otimes
H^{n_{2}}\otimes\ldots\otimes H^{n_{m}}\otimes{}^{\vartheta}\tilde{H}$ (127)

where $H^{n_{1}},H^{n_{2}},\ldots H^{n_{m}}$ are Hilbert spaces of dimension
$n_{1},n_{2},\ldots n_{m}$. Let $e_{1}^{r},e_{2}^{r},\ldots e_{n_{r}}^{r}$ be
an orthonormal basis in $H^{n_{r}}$. Define the corresponding subspace for
each event $X_{i}^{r}$ as follows:

${}^{\vartheta}E_{i}^{r}=H^{n_{1}}\otimes\ldots
H^{n_{r-1}}\otimes\left[e_{i}^{r}\right]\otimes H^{n_{r+1}}\ldots\otimes
H^{n_{m}}\otimes{}^{\vartheta}\tilde{E}_{i}^{r}$ (128)

where $\left[e_{i}^{r}\right]$ stands for the one-dimensional subspace spanned
by $e_{i}^{r}$ in $H^{n_{r}}$. Notice that, for all $r$,

${}^{\vartheta}E_{i}^{r}\bot{}^{\vartheta}E_{j}^{r}\,\,\,\,\,\,\,\,\,\,\text{
if \,\,\,\,}i\neq j$ (129)

due to the fact that $e_{1}^{r},e_{2}^{r},\ldots e_{n_{r}}^{r}$ is an
orthonormal basis in $H^{n_{r}}$.

Due to Lemma 4, for all $1\leq r\leq m$ there is exactly one
$1\leq{}^{\vartheta}i_{*}^{r}\leq n_{r}$ such that
$w_{\vartheta}{}_{{}^{\vartheta}i_{*}^{r}}^{r}=1$. This makes it possible to
define the state vector in ${}^{\vartheta}H$ as the following unit vector:

$\Psi_{\vartheta}=e_{{}^{\vartheta}i_{*}^{1}}^{1}\otimes
e_{{}^{\vartheta}i_{*}^{2}}^{2}\otimes\ldots\otimes
e_{{}^{\vartheta}i_{*}^{r}}^{r}\otimes\ldots\otimes
e_{{}^{\vartheta}i_{*}^{m}}^{m}\otimes\tilde{\Psi}_{\vartheta}$

Now, it is easily verifiable that

$\displaystyle w_{\vartheta}{}_{i}^{r}$ $\displaystyle\simeq$ $\displaystyle
tr\left(P_{\Psi_{\vartheta}}{}^{\vartheta}E_{i}^{r}\right)$ (130)
$\displaystyle w_{\vartheta}{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}$
$\displaystyle\simeq$ $\displaystyle
tr\left(P_{\Psi_{\vartheta}}\left({}^{\vartheta}E_{i_{1}}^{r_{1}}\wedge\ldots\wedge{}^{\vartheta}E_{i_{L}}^{r_{L}}\right)\right)$
(131)

for all ${}_{i}^{r}\in I^{M}$, ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in
S$, and for all $\vartheta\in\Theta$. For example:

If $w_{\vartheta}{}_{i}^{r}=1$, and so $i=^{\vartheta}i_{*}^{r}$, then

$\displaystyle
tr\left(P_{\Psi_{\vartheta}}{}^{\vartheta}E_{i}^{r}\right)=\underbrace{tr\left(P_{e_{{}^{\vartheta}i_{*}^{1}}^{1}}H^{n_{1}}\right)}_{1}tr\left(P_{e_{{}^{\vartheta}i_{*}^{2}}^{2}}H^{n_{2}}\right)\ldots\underbrace{tr\left(P_{e_{{}^{\vartheta}i_{*}^{r}}^{r}}\left[e_{{}^{\vartheta}i_{*}^{r}}^{r}\right]\right)}_{1}\ldots$
$\displaystyle
tr\left(P_{e_{{}^{\vartheta}i_{*}^{m}}^{m}}H^{n_{m}}\right)\underbrace{tr\left(P_{\tilde{\Psi}_{\vartheta}}{}^{\vartheta}\tilde{E}_{i}^{r}\right)}_{\,\,\,\,\,\,\,\,\,\,\simeq\,w_{\vartheta}{}_{i}^{r}=1}\simeq
1$

If $w_{\vartheta}{}_{i}^{r}=0$, and so $i\neq{}^{\vartheta}i_{*}^{r}$, then

$\displaystyle
tr\left(P_{\Psi_{\vartheta}}{}^{\vartheta}E_{i}^{r}\right)=\underbrace{tr\left(P_{e_{{}^{\vartheta}i_{*}^{1}}^{1}}H^{n_{1}}\right)}_{1}tr\left(P_{e_{{}^{\vartheta}i_{*}^{2}}^{2}}H^{n_{2}}\right)\ldots\underbrace{tr\left(P_{e_{{}^{\vartheta}i_{*}^{r}}^{r}}\left[e_{i\neq{}^{\vartheta}i_{*}^{r}}^{r}\right]\right)}_{0}\ldots$
$\displaystyle
tr\left(P_{e_{{}^{\vartheta}i_{*}^{m}}^{m}}H^{n_{m}}\right)\underbrace{tr\left(P_{\tilde{\Psi}_{\vartheta}}{}^{\vartheta}\tilde{E}_{i}^{r}\right)}_{\,\,\,\,\,\,\,\,\,\,\,\simeq\,w_{\vartheta}{}_{i}^{r}=0}=0$

Similarly, if $w_{\vartheta}{}_{i_{1}}^{r_{1}}=0$,
$w_{\vartheta}{}_{i_{2}}^{r_{2}}=1$ then

$\displaystyle
tr\left(P_{\Psi_{\vartheta}}\left({}^{\vartheta}E_{i_{1}}^{r_{1}}\wedge{}^{\vartheta}E_{i_{2}}^{r_{2}}\right)\right)=\underbrace{tr\left(P_{e_{{}^{\vartheta}i_{*}^{1}}^{1}}(H^{n_{1}}\wedge
H^{n_{1}})\right)}_{1}\ldots$
$\displaystyle\underbrace{tr\left(P_{e_{{}^{\vartheta}i_{*}^{r_{1}}}^{r_{1}}}\left(\left[e_{i_{1}\neq^{\vartheta}i_{*}^{r_{1}}}^{r_{1}}\right]\wedge
H^{n_{r_{1}}}\right)\right)}_{0}\ldots\underbrace{tr\left(P_{e_{i_{*}^{r_{2}}}^{r_{2}}}\left(H^{n_{r_{2}}}\wedge\left[e_{i_{2}=^{\vartheta}i_{*}^{r_{2}}}^{r_{2}}\right]\right)\right)}_{1}\ldots$
$\displaystyle tr\left(P_{e_{{}^{\vartheta}i_{*}^{m}}^{m}}(H^{n_{m}}\wedge
H^{n_{m}})\right)\underbrace{tr\left(P_{\tilde{\Psi}_{\vartheta}}\left({}^{\vartheta}\tilde{E}_{i_{1}}^{r_{1}}\wedge{}^{\vartheta}\tilde{E}_{i_{2}}^{r_{2}}\right)\right)}_{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\simeq\,w_{\vartheta}{}_{i_{1}i_{2}}^{r_{1}r_{2}}=0}=0$

in accordance with that ${}^{\vartheta}w{}_{i_{1}i_{2}}^{r_{1}r_{2}}$ must be
equal to $0$, due to (41).

If $w_{\vartheta}{}_{i_{1}}^{r_{1}}=1$, $w_{\vartheta}{}_{i_{2}}^{r_{2}}=1$
then

$\displaystyle
tr\left(P_{\Psi_{\vartheta}}\left({}^{\vartheta}E_{i_{1}}^{r_{1}}\wedge{}^{\vartheta}E_{i_{2}}^{r_{2}}\right)\right)=\underbrace{tr\left(P_{e_{{}^{\vartheta}i_{*}^{1}}^{1}}(H^{n_{1}}\wedge
H^{n_{1}})\right)}_{1}\ldots$
$\displaystyle\underbrace{tr\left(P_{e_{{}^{\vartheta}i_{*}^{r_{1}}}^{r_{1}}}\left(\left[e_{i_{1}={}^{\vartheta}i_{*}^{r_{1}}}^{r_{1}}\right]\wedge
H^{n_{r_{1}}}\right)\right)}_{1}\ldots\underbrace{tr\left(P_{e_{i_{*}^{r_{2}}}^{r_{2}}}\left(H^{n_{r_{2}}}\wedge\left[e_{i_{2}=^{\vartheta}i_{*}^{r_{2}}}^{r_{2}}\right]\right)\right)}_{1}\ldots$
$\displaystyle tr\left(P_{e_{{}^{\vartheta}i_{*}^{m}}^{m}}(H^{n_{m}}\wedge
H^{n_{m}})\right)\underbrace{tr\left(P_{\tilde{\Psi}_{\vartheta}}\left({}^{\vartheta}\tilde{E}_{i_{1}}^{r_{1}}\wedge{}^{\vartheta}\tilde{E}_{i_{2}}^{r_{2}}\right)\right)}_{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\simeq\,w_{\vartheta}{}_{i_{1}i_{2}}^{r_{1}r_{2}}}\simeq
w_{\vartheta}{}_{i_{1}i_{2}}^{r_{1}r_{2}}$

### Step II

Consider an arbitrary state $\vec{Z}$. Since
$\vec{Z}\in\varphi\left(M,S\right)$, it can be decomposed in terms of the
vertices $\left\\{\vec{w}_{\vartheta}\right\\}_{\vartheta\in\Theta}$ in the
fashion of (92) with some coefficients
$\left\\{\lambda_{\vartheta}\right\\}_{\vartheta\in\Theta}$.

Now we construct the Hilbert space $H$ and the state vector $\Psi_{\vec{Z}}$:

$\displaystyle H$ $\displaystyle=$
$\displaystyle\underset{\vartheta\in\Theta}{\mathop{\oplus}}^{\vartheta}H$
(132) $\displaystyle\Psi_{\vec{Z}}$ $\displaystyle=$
$\displaystyle\underset{\vartheta\in\Theta}{\mathop{\oplus}}\sqrt{\lambda_{\vartheta}}\Psi_{\vartheta}$
(133)

Obviously,

$\left\langle\Psi_{\vec{Z}},\Psi_{\vec{Z}}\right\rangle=\sum_{\vartheta\in\Theta}\lambda_{\vartheta}\left\langle\Psi_{\vartheta},\Psi_{\vartheta}\right\rangle=1$

The subspaces $E_{i}^{r}$ representing the outcome events will be defined
further below. First we consider the following subspaces of $H$:

${}^{*}E_{i}^{r}$ $\displaystyle=$
$\displaystyle\underset{\vartheta\in\Theta}{\mathop{\oplus}}^{\vartheta}E_{i}^{r}$

Since

$\displaystyle tr\left(P_{\Psi_{\vec{Z}}}{}^{*}E_{i}^{r}\right)$
$\displaystyle=$
$\displaystyle\left\langle\Psi_{\vec{Z}},{}^{*}E_{i}^{r}\Psi_{\vec{Z}}\right\rangle=\sum_{\vartheta\in\Theta}\left\langle\sqrt{\lambda_{\vartheta}}\Psi_{\vartheta},^{\vartheta}E_{i}^{r}\sqrt{\lambda_{\vartheta}}\Psi_{\vartheta}\right\rangle$
$\displaystyle=\sum_{\vartheta\in\Theta}\lambda_{\vartheta}tr\left(P_{\Psi_{\vartheta}}{}^{\vartheta}E_{i}^{r}\right)$

$\displaystyle
tr\left(P_{\Psi_{\vec{Z}}}\left({}^{*}E_{i_{1}}^{r_{1}}\wedge\ldots\wedge{}^{*}E_{i_{L}}^{r_{L}}\right)\right)$
$\displaystyle=$
$\displaystyle\left\langle\Psi_{\vec{Z}},\left({}^{*}E_{i_{1}}^{r_{1}}\wedge\ldots\wedge{}^{*}E_{i_{L}}^{r_{L}}\right)\Psi_{\vec{Z}}\right\rangle$
$\displaystyle=$
$\displaystyle\sum_{\vartheta\in\Theta}\left\langle\sqrt{\lambda_{\vartheta}}\Psi_{\vartheta},\left({}^{\vartheta}E_{i_{1}}^{r_{1}}\wedge\ldots\wedge{}^{\vartheta}E_{i_{L}}^{r_{L}}\right)\sqrt{\lambda_{\vartheta}}\Psi_{\vartheta}\right\rangle$
$\displaystyle=$
$\displaystyle\sum_{\vartheta\in\Theta}\lambda_{\vartheta}tr\left(P_{\Psi_{\vartheta}}\left({}^{\vartheta}E_{i_{1}}^{r_{1}}\wedge\ldots\wedge{}^{\vartheta}E_{i_{L}}^{r_{L}}\right)\right)$

from (130)–(131) and (92) we have

$\displaystyle Z_{i}^{r}$ $\displaystyle\simeq$ $\displaystyle
tr\left(P_{\Psi_{\vec{Z}}}{}^{*}E_{i}^{r}\right)$ (134) $\displaystyle
Z_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}$ $\displaystyle\simeq$
$\displaystyle
tr\left(P_{\Psi_{\vec{Z}}}\left({}^{*}E_{i_{1}}^{r_{1}}\wedge\ldots\wedge{}^{*}E_{i_{L}}^{r_{L}}\right)\right)$
(135)

Also, as direct sum preserves orthogonality, from (129) we have

${}^{*}E_{i}^{r}\bot{}^{*}E_{j}^{r}\text{ \,\,\,\,\,if \,\,\,}i\neq j$ (136)

For all $1\leq r\leq m$, let
${}^{*}E_{i_{0}}^{r}\in\left\\{{}^{*}E_{1}^{r},{}^{*}E_{2}^{r},\ldots{}^{*}E_{n_{r}}^{r}\right\\}$
be arbitrarily chosen, and let
${}^{*}E_{\bot}^{r}=\left(\stackrel{{\scriptstyle[}}{{i}}=1]{n_{r}}{\vee}{}^{*}E_{i}^{r}\right)^{\bot}=\stackrel{{\scriptstyle[}}{{i}}=1]{n_{r}}{\wedge}\left({}^{*}E_{i}^{r}\right)^{\bot}$.
We define the subspaces representing the outcome events as follows:

$\displaystyle E_{i}^{r}$ $\displaystyle=$
$\displaystyle\begin{cases}{}^{*}E_{i}^{r}&i\neq i_{0}\\\
{}^{*}E_{i_{0}}^{r}\vee{}^{*}E_{\bot}^{r}&i=i_{0}\end{cases}$ (137)

Obviously, (136) implies
${}^{*}E_{i_{0}}^{r}\leq\stackrel{{\scriptstyle[}}{{i}}\neq
i_{0}]{}{\wedge}\left({}^{*}E_{i}^{r}\right)^{\bot}$. Due to the
orthomodularity of the subspace lattice $L(H)$, we have

${}^{*}E_{i_{0}}^{r}\vee\left(\underbrace{\left({}^{*}E_{i_{0}}^{r}\right)^{\bot}\wedge\left(\stackrel{{\scriptstyle[}}{{i}}\neq
i_{0}]{}{\wedge}\left({}^{*}E_{i}^{r}\right)^{\bot}\right)}_{{}^{*}E_{\bot}^{r}}\right)=\stackrel{{\scriptstyle[}}{{i}}\neq
i_{0}]{}{\wedge}\left({}^{*}E_{i}^{r}\right)^{\bot}$

meaning that

$E_{i_{0}}^{r}=\stackrel{{\scriptstyle[}}{{i}}\neq
i_{0}]{}{\wedge}\left({}^{*}E_{i}^{r}\right)^{\bot}$

Therefore, taking into account (136) and (137),

$E_{i}^{r}\bot E_{j}^{r}\text{ \,\,\,\,\,if \,\,\,}i\neq j$ (138)

Also, it is obviously true that

$\stackrel{{\scriptstyle[}}{{i}}=1]{n_{r}}{\vee}E_{i}^{r}=H$ (139)

Both (138) and (139) hold for all $1\leq r\leq m$. There remains to show
(121)–(122).

It follows from (137) that

$E_{i}^{r}\geq{}^{*}E_{i}^{r}$

for all ${}_{i}^{r}\in I^{M}$. Similarly,

$E_{i_{1}}^{r_{1}}\wedge\ldots\wedge
E_{i_{L}}^{r_{L}}\geq{}^{*}E_{i_{1}}^{r_{1}}\wedge\ldots\wedge{}^{*}E_{i_{L}}^{r_{L}}$

for all ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S$. Therefore, for all
$\vec{Z}\in\varphi\left(M,S\right)$,

$\left\langle\Psi_{\vec{Z}},E_{i}^{r}\Psi_{\vec{Z}}\right\rangle\geq\left\langle\Psi_{\vec{Z}},{}^{*}E_{i}^{r}\Psi_{\vec{Z}}\right\rangle$
(140)

and

$\left\langle\Psi_{\vec{Z}},E_{i_{1}}^{r_{1}}\wedge\ldots\wedge
E_{i_{L}}^{r_{L}}\Psi_{\vec{Z}}\right\rangle\geq\left\langle\Psi_{\vec{Z}},{}^{*}E_{i_{1}}^{r_{1}}\wedge\ldots\wedge{}^{*}E_{i_{L}}^{r_{L}}\Psi_{\vec{Z}}\right\rangle$
(141)

Now, (12) and (134) imply that

$\sum_{\begin{array}[]{c}i\\\ \left({}_{i}^{r}\in
I^{M}\right)\end{array}}\left\langle\Psi_{\vec{Z}},{}^{*}E_{i}^{r}\Psi_{\vec{Z}}\right\rangle\simeq
1$

At the same time, taking into account (138)–(139), we have

$1=\sum_{\begin{array}[]{c}i\\\ \left({}_{i}^{r}\in
I^{M}\right)\end{array}}\left\langle\Psi_{\vec{Z}},E_{i}^{r}\Psi_{\vec{Z}}\right\rangle\geq\sum_{\begin{array}[]{c}i\\\
\left({}_{i}^{r}\in
I^{M}\right)\end{array}}\left\langle\Psi_{\vec{Z}},{}^{*}E_{i}^{r}\Psi_{\vec{Z}}\right\rangle\simeq
1$ (142)

From (140) and (142), therefore,

$tr\left(P_{\Psi_{\vec{Z}}}E_{i}^{r}\right)\simeq
tr\left(P_{\Psi_{\vec{Z}}}{}^{*}E_{i}^{r}\right)$ (143)

Similarly, on the one hand, (15) and (135) imply that

$\sum_{\begin{array}[]{c}i_{1},i_{2}\ldots i_{L}\\\ \left({}_{i_{1}\ldots
i_{L}}^{r_{1}\ldots r_{L}}\in
S\right)\end{array}}\left\langle\Psi_{\vec{Z}},{}^{*}E_{i_{1}}^{r_{1}}\wedge\ldots\wedge{}^{*}E_{i_{L}}^{r_{L}}\Psi_{\vec{Z}}\right\rangle\simeq
1$ (144)

On the other hand, $\left\\{E_{i_{1}}^{r_{1}}\wedge\ldots\wedge
E_{i_{L}}^{r_{L}}\right\\}_{\begin{array}[]{c}i_{1},i_{2}\ldots i_{L}\\\
\left({}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S\right)\end{array}}$ is
an orthogonal system of subspaces. Therefore,

$\displaystyle 1\geq\sum_{\begin{array}[]{c}i_{1},i_{2}\ldots i_{L}\\\
\left({}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in
S\right)\end{array}}\left\langle\Psi_{\vec{Z}},E_{i_{1}}^{r_{1}}\wedge\ldots\wedge
E_{i_{L}}^{r_{L}}\Psi_{\vec{Z}}\right\rangle$ (147)
$\displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\geq\sum_{\begin{array}[]{c}i_{1},i_{2}\ldots
i_{L}\\\ \left({}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in
S\right)\end{array}}\left\langle\Psi_{\vec{Z}},{}^{*}E_{i_{1}}^{r_{1}}\wedge\ldots\wedge{}^{*}E_{i_{L}}^{r_{L}}\Psi_{\vec{Z}}\right\rangle\simeq
1$ (150)

From (141) and (150), for all ${}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in
S$, we have

$tr\left(P_{\Psi_{\vec{Z}}}E_{i_{1}}^{r_{1}}\wedge\ldots\wedge
E_{i_{L}}^{r_{L}}\right)\simeq
tr\left(P_{\Psi_{\vec{Z}}}{}^{*}E_{i_{1}}^{r_{1}}\wedge\ldots\wedge{}^{*}E_{i_{L}}^{r_{L}}\right)$
(151)

Thus, (134)–(135) together with (143) and (151) imply (121)–(122). ∎ With
Theorem 9 we have accomplished (Q1)–(Q4). The next two theorems cover
statements (Q5)–(Q8).

######  Theorem 10.

Let $a_{r}$ be the measurement of a real valued quantity with labeling (16).
On the Hilbert space $H$, there exists a self-adjoint operator $A_{r}$ such
that for every state of the system $\vec{Z}$,

$\left\langle\alpha_{r}\right\rangle_{\vec{Z}}\simeq
tr\left(P_{\Psi_{\vec{Z}}}A_{r}\right)$ (152)

###### Proof.

Let

$A_{r}=\sum_{i=1}^{n_{r}}\alpha_{i}^{r}E_{i}^{r}$ (153)

$A_{r}$ is obviously a self-adjoint operator, and

$\displaystyle\left\langle\alpha_{r}\right\rangle_{\vec{Z}}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n_{r}}\alpha_{i}^{r}\pi\left(X_{i}^{r}|a_{r}\right)=\sum_{i=1}^{n_{r}}\alpha_{i}^{r}Z_{i}^{r}\simeq\sum_{i=1}^{n_{r}}\alpha_{i}^{r}tr\left(P_{\Psi_{\vec{Z}}}E_{i}^{r}\right)$
$\displaystyle=tr\left(P_{\Psi_{\vec{Z}}}\sum_{i=1}^{n_{r}}\alpha_{i}^{r}E_{i}^{r}\right)=tr\left(P_{\Psi_{\vec{Z}}}A_{r}\right)$

∎

######  Theorem 11.

The possible measurement results of the $\alpha_{r}$-measurement are exactly
the eigenvalues of the associated operator $A_{r}.$ The subspace $E_{i}^{r}$
representing the outcome event labeled by $\alpha_{i}^{r}$ is the eigenspace
pertaining to eigenvalue $\alpha_{i}^{r}$. Accordingly, (153) constitutes the
spectral decomposition of $A_{r}$.

###### Proof.

First, let $\psi\in E_{i}^{r}$. Then, due to (123),
$A_{r}\psi=\left(\sum_{i=1}^{n_{r}}\alpha_{i}^{r}E_{i}^{r}\right)\psi=\alpha_{i}^{r}\psi$.
Meaning that every $\alpha_{i}^{r}$ is an eigenvalue of $A_{r}.$ Now consider
an arbitrary eigenvector of $A_{r}$, that is, a vector $\psi\in H$ such that

$A_{r}\psi=x\psi$ (154)

with some $x\in\mathbb{R}$. Due to (123)–(124),
$\left\\{E_{1}^{r},E_{2}^{r},\ldots E_{n_{r}}^{r}\right\\}$ constitutes an
orthogonal decomposition of $H$, meaning that arbitrary $\psi\in H$ can be
decomposed as

$\psi=\sum_{i=1}^{n_{r}}\psi_{i}\,\,\,\,\,\,\,\psi_{i}\in E_{i}^{r}$

From (153) we have

$\sum_{i=1}^{n_{r}}\alpha_{i}^{r}\psi_{i}=\sum_{i=1}^{n_{r}}x\psi_{i}$ (155)

About the labeling we have assumed that $\alpha_{i}^{r}\neq\alpha_{j}^{r}$ for
$i\neq j$, therefore, (155) implies that

$\displaystyle x$ $\displaystyle=$
$\displaystyle\alpha_{i}^{r}\,\,\,\,\,\,\,\text{ for one }\alpha_{i}^{r}$
$\displaystyle\psi_{j}$ $\displaystyle=$ $\displaystyle
0\,\,\,\,\,\,\,\,\,\text{ for all }j\neq i$

that is, $\psi\in E_{i}^{r}$. Meaning that (153) is the spectral decomposition
of $A_{r}$. ∎

A consequence of Theorems 10 and 11 is that if
$f:\mathbb{R}\rightarrow\mathbb{R}$ is an arbitrary injective function “re-
labeling” the outcomes, then

$\displaystyle\left\langle f(\alpha_{r})\right\rangle_{\vec{Z}}$
$\displaystyle=\sum_{i=1}^{n_{r}}f(\alpha_{i}^{r})\pi\left(X_{i}^{r}|a_{r}\right)=\sum_{i=1}^{n_{r}}f(\alpha_{i}^{r})Z_{i}^{r}\simeq\sum_{i=1}^{n_{r}}f(\alpha_{i}^{r})\,tr\left(P_{\Psi_{\vec{Z}}}E_{i}^{r}\right)$
$\displaystyle=tr\left(P_{\Psi_{\vec{Z}}}\sum_{i=1}^{n_{r}}f(\alpha_{i}^{r})E_{i}^{r}\right)=tr\left(P_{\Psi_{\vec{Z}}}f(A_{r})\right)$

##  7 Representation of Dynamics

Notice that not all unit vectors of $H$ are involved in the representation of
states. In order to specify the ones being involved, consider the following
subspace $\mathcal{H}\subset H$:

$\displaystyle\mathcal{H}$ $\displaystyle=$
$\displaystyle\mathsf{span}\left\\{\Psi_{\vartheta}\right\\}_{\vartheta\in\Theta}$

where $\left\\{\Psi_{\vartheta}\right\\}_{\vartheta\in\Theta}$ is the set of
vectors in the direct sum (133), understood as being pairwise orthogonal,
unit-length elements of $H$. Denote by $\mathcal{O}$ the closed first
hyperoctant (orthant) of the $\left(\left|\Theta\right|-1\right)$-dimensional
sphere of unit vectors in $\mathcal{H}$:

$\mathcal{O}=\left\\{\sum_{\vartheta\in\Theta}o_{\vartheta}\Psi_{\vartheta}\left|o_{\vartheta}\geq
0\,\,\,\,\sum_{\vartheta\in\Theta}o_{\vartheta}^{2}=1\right.\right\\}$

Obviously, there is a continuous one-to-one map between the $\Lambda$ and
$\mathcal{O}$:

$O:\Lambda\rightarrow\mathcal{O};\,\,O\left(\vec{\lambda}\right)=\sum_{\vartheta\in\Theta}\sqrt{\lambda_{\vartheta}}\Psi_{\vartheta}$

As we have shown, however, the states of the system actually are represented
on the $\text{dim}\left(\varphi\left(M,S\right)\right)$-dimensional slice
$\Lambda_{\sigma}\subset\Lambda$ (see (104)). Accordingly, the quantum
mechanical representation of states constitutes a
$\text{dim}\left(\varphi\left(M,S\right)\right)$-dimensional submanifold with
boundary:
$\mathcal{O}_{\sigma}=O\left(\Lambda_{\sigma}\right)\subset\mathcal{O}$.

Consequently, the time evolution of state $\vec{Z}\left(t\right)$ will be
represented by a path in $\mathcal{O}_{\sigma}$:

$\Psi(t)=O\circ\sigma\left(\vec{Z}\left(t\right)\right)$

The representation is of course not unique, as it depends on the choice of
cross section $\sigma$. This is however inessential; just like a choice of a
coordinate system.

If (E4) holds, that is the time evolution $\vec{Z}\left(t\right)$ can be
generated by a one-parameter group of transformations on
$\varphi\left(M,S\right)$,
$\vec{Z}(t)=F_{t-t_{0}}\left(\vec{Z}(t_{0})\right)$, then the same is true for
$\mathcal{O}_{\sigma}$. Let $G_{t}=O\circ\sigma\circ F_{t}\circ D\circ
O^{-1}$. Obviously, $G_{t}$ is a map
$\mathcal{O}_{\sigma}\rightarrow\mathcal{O}_{\sigma}$, such that

  * $G_{t}:\mathcal{O}_{\sigma}\rightarrow\mathcal{O}_{\sigma}$ is one-to-one

  * $G:\mathbb{R}\times\mathcal{O}_{\sigma}\rightarrow\mathcal{O}_{\sigma};\,\left(t,\Psi\right)\mapsto G_{t}\left(\Psi\right)$ is continuous

  * $G_{t+s}=G_{s}\circ G_{t}$

  * $G_{-t}=G_{t}^{-1}$; consequently, $G_{0}=id_{\mathcal{O}_{\sigma}}$

and the time evolution of an arbitrary initial state
$\Psi(t_{0})\in\mathcal{O}_{\sigma}$ is
$\Psi(t)=G_{t-t_{0}}\left(\Psi(t_{0})\right)$.

##  8 Questionable and Unquestionable in Quantum Mechanics

What we have _proved_ in the above theorems, that is, statements (Q1)–(Q8),
are nothing but the basic postulates of quantum theory. This means that the
basic postulates of quantum theory are in fact analytic statements: they do
not tell us anything about a physical system beyond the fact that the system
can be described in empirical/operational terms—even if this logical
relationship is not so evident. In this sense, of course, these postulates of
quantum theory are unquestionable. Though, as we have seen, the Hilbert space
quantum mechanical formalism is only an optional mathematical representation
of the probabilistic behavior of a system—empirical facts do not necessitate
this choice.

Nevertheless, it must be mentioned that the quantum-mechanics-like
representation, characterized by (Q1)–(Q8), is not completely identical with
standard quantum mechanics. There are several subtle deviations:

  * (D1)

There is no one-to-one correspondence between operationally meaningful
physical quantities and self-adjoint operators. First of all, it is not
necessarily true that every self-adjoint operator represents some
operationally meaningful quantity.

  * (D2)

There is no obvious connection between commutation of the associated self-
adjoint operators and joint measurability of the corresponding physical
quantities. In general, there is no obvious role of the mathematically
definable algebraic structures over the self-adjoint operators in the
operational context. First of all because those mathematically “natural”
structures are mostly meaningless in an operational sense. As we have already
mentioned, the outcome _events_ are ontologically prior to the labeling of the
outcomes by means of numbers; and the events themselves are well represented
in the subspace/projector lattice, prior to any self-adjoint operator
associated with a numerical coordination.

For example, consider three real-valued physical quantities with labelings
$\alpha_{r_{1}},\alpha_{r_{2}},\alpha_{r_{3}}$. The three physical quantities
reflect three different features of the system defined by three different
measurement operations. A functional relationship
$\alpha_{r_{1}}=f\left(\alpha_{r_{2}},\alpha_{r_{3}}\right)$ means that
whenever we perform the measurements $a_{r_{1}},a_{r_{2}},a_{r_{3}}$ in
conjunction (meaning that $\left\\{r_{1},r_{2},r_{3}\right\\}\in\mathfrak{P}$)
the outcomes $X_{i_{1}}^{r_{1}},X_{i_{2}}^{r_{2}},X_{i_{3}}^{r_{3}}$ are
strongly correlated: if $X_{i_{2}}^{r_{2}}$ and $X_{i_{3}}^{r_{3}}$ are the
outcomes of $a_{r_{2}}$ and $a_{r_{3}}$, labeled by $\alpha_{i_{2}}^{r_{2}}$
and $\alpha_{i_{3}}^{r_{3}}$, then the outcome of measurement $a_{r_{1}}$,
$X_{i_{1}}^{r_{1}}$, is the one labeled by
$\alpha_{i_{1}}^{r_{1}}=f\left(\alpha_{i_{2}}^{r_{2}},\alpha_{i_{3}}^{r_{3}}\right)$.
That is, in probabilistic terms:

$\displaystyle\pi\left(\alpha_{r_{1}}^{-1}\left(f\left(\alpha_{i_{2}}^{r_{2}},\alpha_{i_{3}}^{r_{3}}\right)\right)\wedge\alpha_{r_{2}}^{-1}\left(\alpha_{i_{2}}^{r_{2}}\right)\wedge\alpha_{r_{3}}^{-1}\left(\alpha_{i_{3}}^{r_{3}}\right)\left|a_{r_{1}}\wedge
a_{r_{2}}\wedge a_{r_{3}}\right.\right)$
$\displaystyle=\pi\left(\alpha_{r_{2}}^{-1}\left(\alpha_{i_{2}}^{r_{2}}\right)\wedge\alpha_{r_{3}}^{-1}\left(\alpha_{i_{3}}^{r_{3}}\right)\left|a_{r_{1}}\wedge
a_{r_{2}}\wedge a_{r_{3}}\right.\right)$ (156)

This contingent fact of regularity in the observed relative frequencies of
physical events is what is a part of the ontology. And it is well reflected in
our quantum mechanical representation, in spite of the fact that the
relationship (156) is generally not reflected in some algebraic or other
functional relation of the associated self-adjoint operators $A_{r_{1}}$,
$A_{r_{2}}$ and $A_{r_{3}}$.

  * (D3)

It is worthwhile emphasizing that the Hilbert space of representation is
finite dimensional and real. It is of course no problem to embed the whole
representation into a complex Hilbert space of the same dimension. As it
follows from (123) and (127), the required minimal dimension increases with
increasing the number of possible measurements $m$, and/or increasing the
number of possible outcomes $n_{r}$. In any event, it is finite until we have
a finite operational setup. Employing complex Hilbert spaces is only necessary
if, in addition to the stipulated operational setup, we have some further
algebraic requirements, for example, in the form of commutation relations, and
the likes. How those further requirements are justified in operational terms,
of course, can be a question.

  * (D4)

There is no problem with the empirical meaning of the lattice-theoretic meet
of subspaces/projectors representing outcome events: the meet represents the
empirically meaningful conjunction of the outcome events, regardless whether
the corresponding projectors commute or not. Of course, by definition (21),
the conjunctions that do not belong to $S$ have zero probability in all states
of the system.

In contrast, the lattice-theoretic joins and orthocomplements, in general,
have nothing to do with the disjunctions and negations of the outcome events.
Nevertheless, as we have seen, the quantum state uniquely determines the
probabilities on the whole event algebra, including the conjunctions,
disjunctions and negations of all events—in the sense of Theorem 1.

  * (D5)

All possible states of the system, $\vec{Z}\in\varphi\left(M,S\right)$, are
represented by _pure_ states. That is to say, the quantum mechanical notion of
mixed state is not needed. The reason is very simple.
$\varphi\left(M,S\right)$ is a convex polytope being closed under convex
linear combinations. The state of the system intended to be represented by a
mixed state, say,

$W=\mu_{1}P_{\Psi_{\vec{Z}_{1}}}+\mu_{2}P_{\Psi_{\vec{Z}_{2}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mu_{1},\mu_{2}\geq
0;\,\,\,\mu_{1}+\mu_{2}=1$

is nothing but another element of $\varphi\left(M,S\right)$,

$\vec{Z}_{3}=\mu_{1}\vec{Z}_{1}+\mu_{2}\vec{Z}_{2}\in\varphi\left(M,S\right)$

However, in our representation theorem (Theorem 9) the Hilbert space and the
representations of the outcome events are constructed in a way that all states
$\vec{Z}\in\varphi\left(M,S\right)$ are represented by a suitable state vector
in one and the same Hilbert space. Therefore, $\vec{Z}_{3}$ is also
represented by a pure state $P_{\Psi_{\vec{Z}_{3}}}$ with a suitably
constructed state vector $\Psi_{\vec{Z}_{3}}$. Namely, given that

$\displaystyle\vec{Z}_{1}$ $\displaystyle=$
$\displaystyle\sum_{\vartheta\in\Theta}\lambda_{\vartheta}^{1}\vec{w}_{\vartheta}\,\,\,\,\,\,\,\lambda_{\vartheta}^{1}\geq
0,\sum_{\vartheta\in\Theta}\lambda_{\vartheta}^{1}=1$
$\displaystyle\vec{Z}_{2}$ $\displaystyle=$
$\displaystyle\sum_{\vartheta\in\Theta}\lambda_{\vartheta}^{2}\vec{w}_{\vartheta}\,\,\,\,\,\,\,\lambda_{\vartheta}^{2}\geq
0,\sum_{\vartheta\in\Theta}\lambda_{\vartheta}^{2}=1$

we have

$\displaystyle\vec{Z}_{3}$ $\displaystyle=$
$\displaystyle\sum_{\vartheta\in\Theta}\left(\mu_{1}\lambda_{\vartheta}^{1}+\mu_{2}\lambda_{\vartheta}^{2}\right)\vec{w}_{\vartheta}$

therefore, from (133),

$\displaystyle\Psi_{\vec{Z}_{3}}$ $\displaystyle=$
$\displaystyle\underset{\vartheta\in\Theta}{\mathop{\oplus}}\sqrt{\mu_{1}\lambda_{\vartheta}^{1}+\mu_{2}\lambda_{\vartheta}^{2}}\Psi_{\vartheta}$

To avoid a possible misunderstanding, it is worthwhile mentioning that all we
said above is not in contradiction with the mathematical fact that the density
operators $W$ and $P_{\Psi_{\vec{Z}_{3}}}$ generate different “quantum
probability” measures over the _whole_ subspace lattice $L\left(H\right)$. The
two measures will coincide on those elements of $L\left(H\right)$ that
represent operationally meaningful
events—$E_{i}^{r},E_{i_{1}}^{r_{1}}\wedge\ldots\wedge E_{i_{L}}^{r_{L}}$ for
${}_{i}^{r}\in I^{M},{}_{i_{1}\ldots i_{L}}^{r_{1}\ldots r_{L}}\in S$. This
reinforces the point in (D4) that there is no one-to-one correspondence
between the operationally meaningful events and the elements of
$L\left(H\right)$.

  * (D6)

We don’t need to invoke the entire Hilbert space for representing the totality
of operationally meaningful possible states of the system; subspace
$\mathcal{H}$ is sufficient. Even in this restricted sense, there is no one-
to-one correspondence between the rays of the subspace $\mathcal{H}\subset H$
and the states of the system. The unit vectors involved in the representation
are the ones pointing to $\mathcal{O}_{\sigma}$, a
$\text{dim}\left(\varphi\left(M,S\right)\right)$-dimensional submanifold with
boundary on the unit sphere of $\mathcal{H}$.

  * (D7)

The so called “superposition principle” does not hold. The ray determined by
the linear combination of two different vectors pointing to
$\mathcal{O}_{\sigma}$ does not necessarily intersect $\mathcal{O}_{\sigma}$;
meaning that such a linear combination, in general, has nothing to do with a
third state of the system. Neither has it anything to do with the
logical/probability theoretic notion of “disjunction” of events, of course.
Nevertheless, as we have already emphasized in (D4) and (D5), all possible
states of the system are well represented in $\mathcal{O}_{\sigma}$; and these
states uniquely determine the probabilities on the whole event algebra of
operationally meaningful events, including their disjunctions too.

  * (D8)

The dynamics of the system can be well represented in the usual way, by means
of $G_{t}$, a one-parameter group of transformations of the state manifold
$\mathcal{O}_{\sigma}$. These transformations are in no way related to the
unitary transformations of $H$ (or $\mathcal{H}$); because they do not respect
the linear structure of the Hilbert space or orthogonality; but they do
respect that the state space $\mathcal{O}_{\sigma}$ is a manifold with
boundary.

It is remarkable that most of the above mentioned deviations from the quantum
mechanical folklore are related with exactly those issues in the foundations
of quantum mechanics that have been hotly debated for long decades (e.g.
Strauss 1936; Reichenbach 1944; Popper 1967; Park and Margenau 1968; 1971;
Ross 1974; Bell 1987; Gudder 1988; Malament 1992; Leggett 1998; Griffiths
2013; Cassinelli and Lahti 2017; Fröhlich and Pizzo 2022). The fact that so
much of the core of quantum theory can be unquestionably deduced from three
elementary empirical conditions, equally true about all physical systems
whether classical or quantum, may shed new light on these old problems in the
foundations.

## Appendix

######  Theorem 12.

Let $P$ be a polytope in $\mathbb{R}^{d}$, defined by the following set of
linear inequalities:

$\displaystyle\left\langle\vec{\omega}_{\mu},\vec{f}\right\rangle-b_{\mu}$
$\displaystyle\leq$ $\displaystyle 0\,\,\,\,\,\text{for all }\mu\in I$ (157)

For each $\vec{f}\in P$, define the active index set:

$I_{\vec{f}}:=\left\\{\mu\in
I\left|\,\left\langle\vec{\omega}_{\mu},\vec{f}\right\rangle-
b_{\mu}=0\right.\right\\}$

$\vec{f}\in P$ is a vertex of $P$ if and only if

$\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}}=\mathbb{R}^{d}$ (158)

###### Proof.

First, suppose $\vec{f}$ is vertex of $P$, but
$\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}}\neq\mathbb{R}^{d}$. Then choose a non-zero
$\vec{g}\in\left(\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}}\right)^{\bot}$. Obviously, if $\mu\not\in I_{\vec{f}}$ then there
exists a neighborhood $U$ of $\vec{f}$ such that $\mu\not\in I_{\vec{f}_{*}}$
for all $\vec{f}_{*}\in U$. Consider the points $\vec{f}+\lambda\vec{g}$. If
$\lambda$ is small enough, both $\vec{f}+\lambda\vec{g}$ and
$\vec{f}-\lambda\vec{g}$ are in $P$, since (157) are satisfied. Now, we can
write

$\vec{f}=\frac{1}{2}\left(\left(\vec{f}+\lambda\vec{g}\right)+\left(\vec{f}-\lambda\vec{g}\right)\right)$

which contradicts the fact that $\vec{f}$ is vertex of $P$.

Second, now suppose that $\vec{f}\in P$ and
$\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}}=\mathbb{R}^{d}$. Suppose
$\vec{f}=\lambda\vec{f}_{*}+(1-\lambda)\vec{f}_{**}$ with some
$\vec{f}_{*},\vec{f}_{**}\in P$ and $0<\lambda<1$. We know that $\mu\in
I_{\vec{f}}$ implies

$\left\langle\vec{\omega}_{\mu},\vec{f}\right\rangle=\lambda\left\langle\vec{\omega}_{\mu},\vec{f}_{*}\right\rangle+(1-\lambda)\left\langle\vec{\omega}_{\mu},\vec{f}_{**}\right\rangle=b_{\mu}$

On the other hand, from (157) we have

$\displaystyle\left\langle\vec{\omega}_{\mu},\vec{f}_{*}\right\rangle$
$\displaystyle\leq$ $\displaystyle b_{\mu}$
$\displaystyle\left\langle\vec{\omega}_{\mu},\vec{f}_{**}\right\rangle$
$\displaystyle\leq$ $\displaystyle b_{\mu}$

which implies that
$\left\langle\vec{\omega}_{\mu},\vec{f}\right\rangle=\left\langle\vec{\omega}_{\mu},\vec{f}_{*}\right\rangle=\left\langle\vec{\omega}_{\mu},\vec{f}_{**}\right\rangle$
(for all $\mu\in I_{\vec{f}}$). Therefore,

$\left(\vec{f}-\vec{f}_{*}\right),\left(\vec{f}-\vec{f}_{**}\right)\in\left(\mathrm{span}\left\\{\vec{\omega}_{\mu}\right\\}_{\mu\in
I_{\vec{f}}}\right)^{\bot}=\textrm{\O}$

meaning that $\vec{f}=\vec{f}_{*}=\vec{f}_{**}$. Therefore, $\vec{f}$ is a
vertex. ∎

## Funding

This work was supported by Hungarian National Research, Development and
Innovation Office (Grant No. K134275 ).

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