# Contextual Measurement Model and Quantum Theory

Andrei Khrennikov
Linnaeus University, International Center for Mathematical Modeling
in Physics and Cognitive Sciences Växjö, SE-351 95, Sweden
email<EMAIL_ADDRESS>
###### Abstract

We develop the contextual measurement model (CMM) which is used for
clarification of the quantum foundations. This model matches with Bohr’s views
on the role of experimental contexts. CMM is based on contextual probability
theory which is connected with generalized probability theory. CMM covers
measurements in classical, quantum, and semi-classical physics. The CMM
formalism is illustrated by a few examples. We consider CMM framing of
classical probability, the von Neumann measurement theory, the quantum
instrument theory. CMM can also be applied outside of physics, in cognition,
decision making, and psychology, so called quantum-like modeling.

keywords: contextual measurement model, quantum foundations, contextual
probability, generalized probability, von Neumann measurements, quantum
instruments, quantum-like modeling

##  1 Introduction

Interrelation of quantum and classical probability theories is very complex
foundational issue, involving interpretational, mathematical, and philosophic
questions. Research in this area is characterized by the diversity of views,
opinions, and mathematical formalisms (see, e.g., [1]-[24]). We remark that
generally quantum mechanics (QM) is characterized by the diversity of
interpretations.

My own understanding is that quantum probability is a machinery for
probability update, analogous to classical Bayesian inference [25]-[34]. In
contrast to the latter, quantum probability inference is not based on the
Bayes formula for conditional probability. Quantum probability theory is a
theory of probability inference with a special class of probability update
transformations given by projections or quantum instruments. It is natural to
create a general probabilistic framework that covers both the classical and
quantum ones. Such generalization can come with up a global panorama as from
the top of a mountain one can enjoy a panorama of the whole city and through
this panorama connect districts which otherwise look as totally separated. In
this way it is easier to fiend similarities and differences in districts plans
and architecture of buildings. As just one of the possible machines for
probability update, the quantum probabilistic formalism would lose its
mystery.

One of such “panoramic frameworks” is the contextual measurement model (CMM)
based on contextual probability space. Its development was initiated in [25],
continued in a series of author’s works (e.g., [26]-[28]) and summarized in
monograph [29]. In these studies the main emphasis was to modification of the
formula of total probability (FTP) – its transformation into FTP with an
interference term, expressing interference of probabilities, e.g., in the two
slit experiment. In my previous studies contextual probability approach was
partially shadowed by appeal to von Mises frequency theory of probability [30,
22] and realization of experimental contexts as von Mises collectives.

In this paper CMM’s development is continued to towards the abstract
contextual formalization of other basic features of quantum probability as
order and response replicability effects in sequential measurements,
entanglement, the violation of the Bell inequalities and establishing coupling
with quantum instruments theory as well as with linear space representation
(LSR) of generalized probability theory.

CMC is the basis of the Växjö interpretation of QM [25], [31]-[32], [29] \-
one of the contextual probabilistic interpretations. Since probability update
is at the same time information update, the Växjö interpretation is part of
the information interpretation of QM. This paper presents CMM consistently in
the most general form by highlighting its basic properties, interference of
probabilities, order effect, entanglement, violation of the Bell inequalities.

The abstract CMM formalism is illustrated by a few examples. We start with CMM
framing of classical probability theory (Kolmogorov [35, 36]) serving as the
basis of classical statistical physics and thermodynamics. Then we consider
the von Neumann [1, 37] quantum measurement theory with observables given by
Hermitian operators and the state update of the projective type and represent
it as CMM. The quantum instrument theory is generalization of the von Neumann
theory permitting state updates of the non-projective type and it is also can
be represented as CMM. We also show connection of the generalized probability
theory with the state space consisting of probability measures with CMM.
Finally, LSR for contextual probability space is constructed by using the
construction going back to Mackey.

CMM can also be applied outside of physics, in so called quantum-like modeling
(see, e.g., books [38, 40]). This the rapidly developing area of research
stimulated by the recent quantum information revolution. In quantum-like
modeling the quantum methodology is applied to cognition, decision making,
psychology, game theory, economics and finance, and AI. Universally, quantum-
like models need not be based on the complex Hilbert space formalism. They can
employ other contextual probability calculi and CMMs [38].

###  1.1 Contextuality of probability

From the mathematical viewpoint, the essence of the problem is in
generalizations of conditional probability and probability update. On the way
to such rethinking of the interrelation between the classical and quantum
probability theories, I was strongly influenced by Ballentine who treated all
quantum probabilities as conditional probabilities [14, 15, 18, 19, 20]. Later
I learned that this was also Koopman’s viewpoint [4]. It is interesting that
Kolmogorov (who in 1933 formalized classical probability in the measure-
theoretic framework [35, 36]) advertised this conditional viewpoint even for
classical probability. This viewpoint was especially clearly described in his
early works preceding monograph [35]. Unfortunately, these works (in Russian
and published in proceedings) are practically unknown, see [22] for references
and details. I got to know about these “pre-axiomatic” studies of Kolmogorov
from Shiryaev and Bulinski, former students of Kolmogorov. But even monograph
[35] (see also [36]) contains the statement which in modern words can be
formulated as a statement about contextuality of probability. Kolmogorov’s
message is that it is meaningless to speak about probability without
determining a complex of experimental conditions, measurement context. This
Kolmogorov position matches well with Bohr’s statement on contextuality of
measurement’s outcomes that is the cornerstone of his complementarity
principle [41] (which is better to call contextuality-complementarity
principle [42, 43]).

Unfortunately, Kolmogorov’s original message on contextuality of probability
was practically ignored in further development of classical probability
theory. A mathematical work on probability is typically started with fixing
one probability measure, without mentioning that it corresponds to some
measurement context. The contextuality component of Bohr’s statements on
complementarity was neither emphasized in quantum foundational research;
typically the Bohr complementarity principle is reduced to the wave-particle
duality.

Following Bohr [41], Kolmogorov [35, 36], Koopman [4], Accardi [14, 15], and
Ballentine [17]-[20], I introduced the contextual probability space and CMM
based on it [29]. The main idea behind this approach is operating solely with
contexts and observables and to exclude physical systems from consideration.

A measurement context consists of a pre-measurement context $C,$ an observable
$A,$ and post-measurement context $C_{A=x}$ corresponding to the outcome
$A=x,$ i.e., a triple $(C,A,C_{A=x}).$ Transformation $C\to C_{A=x}$ can be
described as a map $T_{A}(x):{\cal C}\to{\cal C},$ where ${\cal C}$ is the set
of pre-measurement contexts. Alike theory of quantum instruments , we call the
pair $I_{A}=(A,T_{A})$ a contextual instrument. The latter is the basic
mathematical component of measurement theory. It is meaningless to formulate
it solely in terms of observables. The same observable $A$ can be a component
of a variety of instruments describing different measurement procedures for
$A.$ So, an observable is a theoretical quantity expressing some features of
pre-measurement contexts.

In QM one operates with the notion of “state”, not “context”. These notions
are similar, but have some inetrpretational differences (see Appendix 1 for
the discussion).

We also mention Feynman’s contextual analysis of the two slit experiment in
the book [2, 3]. He presented the purely probabilistic picture this
fundamental experiment of QM and expressed the interference phenomenon as
interference of probabilities. Mathematically he described this situation as
the violation of additivity of probability, the classical formula is disturbed
by an additional term, the interference term. In classical probability the
combination of additivity and the Bayes formula for conditional probability
leads to FTP, the formula of total probability, playing the important role in
probability inference. Following this line, Feynman’s conclusion can be
rewritten as the violation of classical FTP. The difference between classical
and quantum probability models can be moved from the violation of additivity
of probability to the violation of the Bayes formula for conditional
probability - quantum probability is additive, but conditioning is not
Bayesian. The quantum FTP is a perturbation of classical FTP with an
additional term, the interference term. The main distinguishing feature of
Feynman’s presentation of the two slit experiment and generally quantum
interference is its contextual structure, he operates with three contexts
$C_{1},C_{2},C_{12},$ the first slit is open and the second is closed, vice
verse, and both slits are open. Quantum probabilistic specialty is expressed
not via LSR of states and observables, but in the purely contextual
probabilistic way.

We make a remark on the notion of contextuality. In the modern quantum
information literature the notion of contextuality is reduced to contextuality
of the joint measurement of a few quantum observables. This sort of
contextuality was considered by Bell in [44, 45] in his analysis of the
violation of the Bell inequalities (although he did not used the term
“contextuality”). (It seems that this term was introduced in the book of
Beltrametti and Cassinelli [46].)

Feynman’s contexuality [2, 3] is more general and in fact coincides with
Bohr’s contextuality [41]. In my works including those on the Växjö
interpretation, I followed Bohr and Feynman: context as a complex of all
physical conditions involved in an experiment.

###  1.2 Linear space vs. contextual frameworks for probability

Our foundational pathway is opposite to the pathway leading to the generalized
probability theories [11], [5]-[11], [46, 21] (see, e.g., [24] for a review)
which are directed to creation of general LSR, linear space representation, of
probability and measurement process. LRS also provides a panoramic view which
covers both classical and quantum probabilities and observables. This is the
linear panorama illuminating the place of the quantum probability and
measurement formalism among other linear models.

We remark that there are several different approaches to generalized
probability theory and corresponding measurement theory, but all of them are
either equivalent or only slightly different. One of them is the Davies-Lewis
[7] operational probability theory grounded on LRS with the base norm spaces,
a class of partially ordered linear spaces. The corresponding measurement
theory is formulated within instrument theory [7, 8, 11, 12]; in particular,
observables are mathematically represented as positive operator valued
measures (POVMs). They are widely used in quantum information theory
[47]-[50], [51]\- [53]. (see also [39]-[40] for applications to cognition and
decision making). Another approach is to start with an abstract definition of
state space, a convex subset of a linear space. It goes back to Gudder’s work
[9] who constructed the operational representation of quantum states and
observables starting with pre-convex structure. Under natural condition, this
approach leads to a convex state space and LRS for the latter. As was shown by
Ozawa [11], these two formulations (Davies-Lewis and Gudder) are actually
equivalent. In section 6 we explore the Davies-Lewis approach for operational
LRS of measurement model with states given by classical probability measures.
Then we express this model in the form of CMM.

The most close to CMM is the model in that one starts with all possible
probabilities that can be generated in an experiment, conditioned on
preparation and measurement procedures (see, Mackey’s book [6]. Then one
proceeds to LSR. This construction can be employed to construct LSR for CMM
(section 7). However, this is done just to show connection with previously
developed theories. 111Another approach to generation of the complex Hilbert
space representation of CMM is developed in series of author’s works, see,
e.g., [54, 55, 29]. It is based on the contextual version of FTP, FTP with
interference term.

CMM development is important for quantum foundations. CMM demystifies quantum
theory by reducing its probabilistic counterpart to the tool for probability
update and inference (cf. with QBism [56] \- [59]); CMM diminishes the role of
pure states, in the complete agreement with the statistical (ensemble)
interpretation of QM; in CMM quantum interference is just an additive
perturbation of classical FTP due to interplay between a few measurement
contexts; the violation of the Bell inequality has the same origin; contextual
entanglement is naturally coupled to classical dependence of random variables.
The latter demystifies entanglement. This is very important for resolution of
the one century long debate on quantum nonlocality. See Appendix 2 for
comparison of CMMs with and without LSRs.

On the other hand, LSR is very convenient from the mathematical viewpoint
(simply linear algebra) and operating within the LSR framework is useful in
concrete mathematical calculations. However, the calculations should be
completed by the critical analysis on connection of the mathematical LSR-
constructions with physics. In quantum-like modeling a similar problem arises
- the problem of matching between the output of the Hilbert space formalism
and some psychological effects in decision making [60]-[64].

Our CMM can be considered as the most general probabilistic framework for
measurement, in particular, the notion of contextual probability space is
based on the first three axioms of Mackey’s theory [6]. Then Mackey moves
towards quantum logic by constraining the model with additional axioms. This
path makes theory mathematically elegant, but at the same time more complex
and the basic probabilistic components are blurred by additional mathematical
constructions.

##  2 Contextual Measurement Model (CMM)

###  2.1 Contextual probability space

Definition 1. A contextual probability space is a triple $\Sigma=({\cal
C},{\cal O},{\cal P}),$ where ${\cal C}$ and ${\cal O}$ are sets of pre-
measurement contexts and observables and ${\cal P}$ is the space of the
corresponding probability distributions.

In physics pre-measurement contexts can be associated with preparation
procedures.222In applications outside of physics, so called quantum-like
modeling, not all pre-measurement contexts can be straightforwardly
represented in the form of a preparation procedure; here we operate with
mental, social, financial pre-measurement contexts. Within the statistical
(ensemble) interpretation of QM, contexts are represented as ensembles of
similarly prepared systems. Each observable $A$ has its range of values
$X_{A};$ for simplicity, consider discrete observables, i.e., having finite
ranges of values. The following considerations are straightforwardly extended
to observables with arbitrary ranges of values. For an observable $A$ and pre-
measurement context $C,$ denote the probability of an outcome $x\in X_{A}$ as
$P_{C}^{A}(x)\equiv P_{C}(A=x).$ By definition of a probability distribution

$P_{C}^{A}(x)\geq 0,\sum_{x\in X_{A}}P_{C}^{A}(x)=1.$ (1)

The range set $X_{A}$ can be endowed with the algebra of all its subsets
${\cal F}_{A}.$ We set $P_{C}^{A}(G)=\sum_{x\in G}P_{C}^{A}(x),G\in{\cal
F}_{A}.$ This is a probability measure on ${\cal F}_{A}.$ In the definition of
$\Sigma,$ the symbol ${\cal P}$ denotes the collection of such probability
measures

${\cal P}=\\{P_{C}^{A}:C\in{\cal C},A\in{\cal O}\\}$

(see Axiom 1 in Mackey’s book [6]). Elements of ${\cal P}$ are called
contextual probabilities. These are the analogs of the conditional
probabilities in the classical (Kolmogorov [36]) probability model. But we
reserve the term “conditional probability” for a special class of contextual
probabilities generated by context’s updates.

It is natural to assume (see Axiom 2 in Mackey’s book [6]) that two
observables having the same probability distribution for all contexts should
coincide, i.e.,

$P_{C}^{A_{1}}=P_{C}^{A_{2}}\;\mbox{ for any }C\in{\cal C}\Rightarrow
A_{1}=A_{2}.$ (2)

We also assume (see Axiom 2 in Mackey’s book [6]) that two contexts having the
same probability distribution for all observables should coincide, i.e.,

$P_{C_{1}}^{A}=P_{C_{2}}^{A}\;\mbox{ for any }A\in{\cal O}\Rightarrow
C_{1}=C_{2}.$ (3)

The average of an observable $A\in{\cal O}$ (with $X_{A}\subset\mathbb{R})$
w.r.t. a pre-measurement context $C\in{\cal C}$ is defined as

$\langle A\rangle_{C}\equiv E[A|C]=\sum_{x\in X_{A}}xP_{C}(A=x).$ (4)

###  2.2 Context update and conditional probability

Measurement of an observable $A$ with the concrete outcome $x$ in a pre-
measurement context $C$ updates this pre-measurement context:

$C\to C_{A=x}.$ (5)

In terms of preparation procedures, we can consider a measurement as a
subsequent preparation procedure; context $C_{A=x}$ is measurement of $A$ and
filtering w.r.t. the fixed outcome $x.$333In terms of ensembles,
$A$-measurement is performed for systems of initially prepared ensemble and
then systems generating the outcome $A=x$ form new ensemble.  It is natural to
consider this map only for contexts belonging to the set

${\cal C}_{A}(x)=\\{C:P_{C}(A=x)>0\\}.$ (6)

If $P_{C}(A=x)=0,$ then the post-measurement context is not well defined.
Thus, each observable $A$ and its outcome $x$ determine a map

$T_{A}(x):{\cal C}\to{\cal C},\;C\to C_{A=x}=T_{A}(x)C,$ (7)

with the domain of definition ${\cal C}_{A}(x).$

The delicate point of measurement theory is that generally an observable does
not determine the context update map unequally. An observable $A$ can be
measured via different measurement procedures and each procedure generates its
own context update map. A pair $I_{A}$= (observable, context update
map)=($A,T_{A})$ is called a contextual instrument (cf. section 5); a pair
$(C,I_{A}),$ or a triple $(C,A,T_{A}),$ is called a measurement context. We
stress once again that a variety of instruments can be associated with the
same observable
$A:\;I_{A}=(A,T_{A}),I_{A}^{\prime}=(A,T_{A}^{\prime}),I_{A}^{\prime\prime}=(A,T_{A}^{\prime\prime}),....\;.$
We stress that all these update maps have the same domain of definition
determined by the observer $A,$ see (6).

Typically one fixes some class of context update maps. In the von Neumann [1,
37] measurement theory (section 4), this is the class of normalized
projections. In quantum instrument theory (section 5) these are quantum
channels or more generally in the theory of Davis-Levis instruments, these are
positive trace preserving maps.

We emphasize that von Neumann’s measurement theory is very special: here by
fixing an observable, a Hermitian operator $\hat{A},$ we automatically fix the
update map – via operator’s spectral family. This special situation leads to
the illusion that an observable determines the update map. We repeat that
generally this is not the case.

Definition 2. Let $\Sigma=({\cal C},{\cal O},{\cal P})$ be a contextual
probability space. A contextual measurement model (CMM) is a pair
$M=(\Sigma,{\cal I}),$ where $\Sigma$ is a contextual probability space and
${\cal I}$ is a collection of contextual instruments.

CMM is a set of measurement contexts, i.e., triples $(C,A,T_{A}).$ CMM
generates the notion of the conditional probability:

Definition 3. Consider a measurement context $(C,A,T_{A}).$ Let it generate
the output $A=x$ and the corresponding context update, $C\to
C_{A=x}=T_{A}(x)C.$ Consider measurement of another observable $B$ under
condition $A=x,$ i.e., w.r.t. context $C_{A=x}.$ The conditional probability
is given by the formula:

$P_{C,I_{A}}(B=y|A=x)\equiv P_{C_{A=x}}(B=y)=P_{T_{A}(x)C}(B=y),\;C\in{\cal
C}_{A}(x).$ (8)

We note that this definition involves context update only for the
$A$-observable; different contextual instruments $I_{A},I^{\prime}_{A},...,$
induce their own probabilistic conditioning,
$P_{C,I_{A}}(B=y|A=x),P_{C,I^{\prime}_{A}}(B=y|A=x),...\;.$ For simplicity, we
shall typically omit the index $I_{A}$ of dependence on the concrete
instrument.

###  2.3 Contextual formula of total probability

Now we point out that generally contextual probability differs from the
classical Kolmogorov probability [36]. One of the basic classical laws of
probability is the law of total probability formulated in the form of FTP,
formula of total probability,

$P(B=y)=\sum_{x\in X_{A}}P(B=y|A=x)P(A=x).$ (9)

In classical probability theory contextual probability is identified with
conditional one (section 3) and the contextual-conditional analog of FTP has
the form,

$P_{C}(B=y)=\sum_{x\in X_{A}}P_{C}(B=y|A=x)P_{C}(A=x).$ (10)

see (48). However, in a general contextual probability space this formula can
be violated,

$P_{C}(B=y)\not=\sum_{x\in X_{A}}P_{C}(B=y|A=x)P_{C}(A=x)=$ (11) $\sum_{x\in
X_{A}}P_{C_{A=x}}(B=y)P_{C}(A=x).$

The difference between the LHS and RHS determines the degree of context-
disturbance due to its update; it can serve as a measure of nonclassicality of
a contextual model,

$\delta_{C}(B=y|A)=P_{C}(B=y)-\sum_{x\in X_{A}}P_{C}(B=y|A=x)P_{C}(A=x).$ (12)

We call this quantity the interference term [25, 29]. This consideration can
be formalized in the contextual formula of total probability (FTP with an
interference term):

$P_{C}(B=y)=\sum_{x\in X_{A}}P_{C}(B=y|A=x)P_{C}(A=x)+\delta_{C}(B=y|A).$ (13)

The equality of the interference term to zero is the necessary condition of
the classical probabilistic representation of a contextual probability model,
but it is not the sufficient condition [29].

Let us jump for the moment to section 4 in that the von Neumann quantum
measurement model is treated as CMM. Consider dichotomous observables
$A=x_{1},x_{2}$ and $B=y_{1},y_{2}.$ In this case the interference term has
the form

$\delta_{C}(B=y|A)=$
$2\cos\theta\sqrt{P_{C}(B=y|A=x_{1})P_{C}(A=x_{1})P_{C}(B=y|A=x_{2})P_{C}(A=x_{2})},$
(14)

where context $C$ is identified with quantum state $\psi$ (for simplicity
consider contexts corresponding to pure states) and the angle
$\theta=\theta(B=y|A;\psi).$ For dichotomous observables, even in general CMM
it is useful to write the interference term as

$\delta_{C}(B=y|A)=$
$2\lambda\sqrt{P_{C}(B=y|A=x_{1})P_{C}(A=x_{1})P_{C}(B=y|A=x_{2})P_{C}(A=x_{2})},$
(15)

where $\lambda=\lambda(B=y|A;C).$ If $|\lambda|\leq 1,$ then this is the
trigonometric interference and the interference term has the form (14). If
$|\lambda|\geq 1,$ then this is the hyperbolic interference and the
interference term can be represented as

$\delta_{C}(B=y|A)=$ $\pm
2\cosh\sqrt{P_{C}(B=y|A=x_{1})P_{C}(A=x_{1})P_{C}(B=y|A=x_{2})P_{C}(A=x_{2})}.$
(16)

In the quantum framework such interference can be generated by quantum
instruments [55]. In general CMM we can employ the hyperbolic version of QM
[29].

###  2.4 Conditional JPD and order effect

For observables $A_{1},A_{2}\in{\cal O},$ the conditional joint probability
distribution (JPD) is defined by

$P_{C}(A_{1}=x_{1},A_{2}=x_{2})=P_{C}(A_{1}=x_{1})P_{C}(A_{2}=x_{2}|A_{1}=x_{1}).$
(17)

We remark that this is really a probability distribution, i.e.,
$\sum_{x_{1},x_{2}}P_{C}(A_{1}=x_{1},A_{2}=x_{2})=1.$ We can also define JPD
for inverse order of measurements,
$P_{C}(A_{2}=x_{2},A_{1}=x_{1})=P_{C}(A_{2}=x_{2})P_{C}(A_{1}=x_{1}|A_{2}=x_{2}).$
Observables $A_{1},A_{2}$ show the order effect (OE) w.r.t. context $C,$ if

$P_{C}(A_{1}=x_{1},A_{2}=x_{2})\not=P_{C}(A_{2}=x_{2},A_{1}=x_{1}),$ (18)

for at least one pair of outcomes $(x_{1},x_{2});$ otherwise, there is no OE
in context $C.$

We remark OE was actively investigated in decision making and psychology, both
theoretically and experimentally; in particular, within quantum-like modeling
– the applications of quantum methodology and formalism to decision making and
psychology [60]-[64].

###  2.5 Conditional Compatibility

In the absence of OE we have:

$P_{C}(A_{1}=x_{1},A_{2}=x_{2})=P_{C}(A_{2}=x_{2},A_{1}=x_{1}),\;x_{j}\in
X_{A_{j}},$ (19)

i.e.,

$P_{C}(A_{1}=x_{1})P_{C}(A_{2}=x_{2}|A_{1}=x_{1})=P_{C}(A_{2}=x_{2})P_{C}(A_{1}=x_{1}|A_{2}=x_{2}).$
(20)

In this case we call the observables conditionally compatible for context
$C\in{\cal C}$ and their JPD is defined by (19). We remark that the marginals
of JPD coincide with the probability distributions $P_{C}^{A_{i}}.$

In the von Neumann CMM $M_{\rm{QVN}}$ (section 4) with observables and state
updates of the projection type, conditional compatibility for all possible
pre-measurement contexts (given by density operators) is equivalent to
commonly considered compatibility of observables and their representation by
commutative Hermitian operators.

By considering conditional JPD, we do not assume that the observables $A_{1}$
and $A_{2}$ are jointly measurable. We consider sequential measurements, say
first $A_{1}$ then $A_{2}$ or vice verse. We remark that, in fact, precisely
this experimental setup is realized in the Bell experiments. Here the
instances of time for measurement’s outputs on subsystems coincide with zero
probability; always the click of the photo-detector for the subsytem $S_{1}$
is before the click of the photo-detector for the subsystem $S_{2}$ or vice
verse and the time window serves for clicks pairing.

Conditional compatibility implies the Bayes formula for the conditional
probability:

$P_{C}(A_{2}=x_{2}|A_{1}=x_{1})=\frac{P_{C}(A_{1}=x_{1},A_{2}=x_{2})}{P_{C}(A_{1}=x_{1})},$
(21)
$P_{C}(A_{1}=x_{1}|A_{2}=x_{2})=\frac{P_{C}(A_{1}=x_{1},A_{2}=x_{2})}{P_{C}(A_{2}=x_{2})}.$
(22)

The equality (20) implies the Bayes theorem for the probability inference. Let
the outcomes of the observable $A_{2}$ label some hypotheses,
$H_{1},...,H_{m}.$ Then (20) is written as

$P_{C}(A_{2}=H_{j}|A_{1}=x_{1})=\frac{P_{C}(A_{2}=H_{j})P_{C}(A_{1}=x_{1}|A_{2}=H_{j})}{P_{C}(A_{1}=x_{1})}.$
(23)

The Bayes formula for conditional probability (21),(22) implies the validity
of the classical FTP, i.e., the Bayes theorem can be written in the standard
form:

$P_{C}(A_{2}=H_{j}|A_{1}=x_{1})=\frac{P_{C}(A_{2}=H_{j})P_{C}(A_{1}=x_{1}|A_{2}=H_{j})}{\sum_{H_{i}}P_{C}(A_{1}=x_{1}|H_{i})P_{C}(A_{2}=H_{i})}.$
(24)

###  2.6 Replicability and response replicability

Observable $A$ shows replicability for context $C,$ if

$P_{C}(A=x,A=x)=P_{C}(A=x),$ (25)

or

$P_{C_{A=x}}(A=x)=1.$ (26)

Observable $A$ shows replicability if (25) holds for any $C\in{\cal
C}_{A}(x),x\in X_{A}.$

In quantum-like modeling, the following effect plays the important role.
Observables $A_{1}$ and $A_{2}$ show the response replicability effect (RRE)
w.r.t. context $C,$ if

$P_{C}(A_{1}=x_{1},A_{2}=x_{2},A_{1}=x_{1})=P_{C}(A_{1}=x_{1},A_{2}=x_{2}),$
(27)
$P_{C}(A_{2}=x_{2},A_{1}=x_{1},A_{2}=x_{2})=P_{C}(A_{2}=x_{2},A_{1}=x_{1})$
(28)

for all pairs of outcomes $(x_{1},x_{2}).$ This is a kind of the memory
effect. The challenging problem for quantum-like modeling was the combination
of OE and RRE [62]. It was solved in articles [63, 64] within quantum
instrument theory.

###  2.7 Correlations and Bell type inequalities

Consider a pair of conditionally compatible observables $A,B\in{\cal O}$ (with
$X_{A},X_{B}\subset\mathbb{R}).$ Their correlation w.r.t. a context $C\in{\cal
C}$ is defined as

$\langle AB\rangle_{C}\equiv E[AB|C]=\sum_{x,y}xyP_{C}(A=x,B=y).$ (29)

The most popular Bell type inequality is the CHSH inequality. We consider this
inequality within CMM. There are given four observables $A_{i},B_{j},i,j=1,2,$
valued in $[-1,1];$ observables in the pairs $(A_{i},B_{j})$ are conditionally
compatible for some context $C$ with JPDs $P_{C}^{A_{i},B_{j}}.$ The CHSH
inequality has the form:

$|\langle A_{1}B_{1}\rangle_{C}+\langle A_{2}B_{1}\rangle_{C}+\langle
A_{1}B_{2}\rangle_{C}-\langle A_{2}B_{2}\rangle_{C}|\leq 2,$ (30)

i.e.,

$|\sum_{x,y}xy(P_{C}^{A_{1},B_{1}}(x,y)+P_{C}^{A_{2},B_{1}}(x,y)+P_{C}^{A_{1},B_{2}}(x,y)-P_{C}^{A_{2},B_{2}}(x,y))|\leq
2.$ (31)

If there exists a probability measure $P_{C}$ such that JPDs
$P_{C}^{A_{i},B_{j}}$ can be obtained as its marginals, e.g.,

$P_{C}^{A_{1},B_{1}}(x,y)=\sum_{x_{2},y_{2}}P_{C}(x,x_{2},y,y_{2}),$

then the CHSH inequality holds true. However, if such $P_{C}$ does not exist,
then this inequality can be violated and the maximum of its left-hand side
w.r.t. contexts $C\in{\cal C}$ and observables
$A_{1},A_{2},B_{1},B_{2}\in{\cal O}$ valued in $[-1,1]$ can approach the value
four, $\max_{\rm{CHSH}}=4.$ This maximum depends on CMM. For von Neumann CMM
$M_{\rm{QVN}},$ this is $\max_{\rm{CHSH}}=2\sqrt{2}.$

###  2.8 Functions of observables

Suppose that all observables are valued in multidimensional real space and we
remove (for the moment) the restriction that observables’ ranges of values are
finite. We consider probability measures on the $\sigma$-algebra ${\cal B}$ of
the Borel sets; it is generated by all semi-open intervals,
$(\alpha_{1},\beta_{1}]\times...\times(\alpha_{n},\beta_{n}].$ So, $P_{C}^{A}$
is a probability measure on ${\cal B}.$ A function
$f:\mathbb{R}^{n}\to\mathbb{R}^{m}$ is called ${\cal B}$-measurable if for any
Borel subset $G$ of $\mathbb{R}^{m}$ its preimage $f^{-1}(G)$ is a Borel
subset of $\mathbb{R}^{n}.$ Only such functions are considered.

Following Mackey [6] (Axiom 3), we assume that for each $A\in{\cal O}$ with
the range of values $\mathbb{R}^{n}$ and a Borel function
$f:\mathbb{R}^{n}\to\mathbb{R}^{m}$ there exists $B=B_{f}\in{\cal O}$ such
that, for any $C\in{\cal C}$ and Borel set $G\subset\mathbb{R}^{m},$

$P_{C}^{B}(G)=P_{C}^{A}(f^{-1}(G)).$ (32)

Such observable is uniquely defined, due to condition (2) (Mackey’s Axiom 2)
and it can be denoted as $B=f(A).$

Two observables $A_{1}$ and $A_{2}$ are called functionally compatible
(jointly measurable) if there exists an observable $A$ and functions $f_{i}$
such that $A_{i}=f_{i}(A).$ For the von Neumann CMM $M_{\rm{QVN}}$ (section 4)
functional compatibility is equivalent to compatibility and hence conditional
compatibility. Generally in CMM interrelation between these two notions is
complex and we shall not proceed to detailed comparison.

###  2.9 Entanglement of contextual instruments

Entanglement is typically considered as one of the distinguished features of
LSR; from my viewpoint the association of entanglement with LSR and the tensor
product structures shadows its physical nature; its mathematical description
is identified with physics. As was shown in articles [65, 66], the notion of
entanglement can be formalized in the purely probabilistic framework and
dissociated from the tensor product and generally from LSR.

By starting with such probabilistic approach to the notion of entanglement,
the authors of [65, 66] proceed towards its complex Hilbert space realization.
Now we present the purely probabilistic picture of entanglement. The main
value of the contextual probabilistic realization of entanglement is in
clarification of its foundational meaning. At the same time, the use of LSR
can essentially simplify the concrete calculations. However, one should be
careful with connection of the mathematical structures of LSR with physics (or
in quantum-like modeling with e.g. psychology and decision making).

Consider CMM $M=(\Sigma,{\cal I}),$ where $\Sigma=({\cal C},{\cal O},{\cal
P})$ is a contextual probability space and ${\cal I}$ is a collection of
contextual instruments of this model, i.e., pairs (observable, state update
map). Consider two contextual instruments $I_{A}=(A,T_{A})$ and
$I_{B}=(B,T_{B}).$

Definition 4. In pre-measurement context $C\in{\cal C},$ the outcome $B=\beta$
depends on the outcomes of $A$ if for at least two values of
$A,\alpha=\alpha_{i},\alpha_{j},$ the corresponding conditional probabilities
don’t coincide,

$P_{C}(B=\beta|A=\alpha_{i})\not=P_{C}(B=\beta|A=\alpha_{j}).$ (33)

Thus, the probability to get the outcome $B=\beta$ if the preceding
$A$-measurement had the outcome $A=\alpha_{i}$ differs from the probability to
get the same outcome $B=\beta$ if the preceding $A$-measurement had the
outcome $A=\alpha_{j}.$ We remark that the update map $T_{B}$ is not involved
in this definition, i.e., one can consider just an observable $B$ without
referring to the corresponding instrument $I_{B}.$ We consider two instruments
by symmetry reason.

We note that the outcome $B=\beta$ does not depend on the outcomes of the
observable $A$ iff

$P_{C}(B=\beta|A=\alpha_{i})=P_{C}(B=\beta|A=\alpha_{j}),\;\mbox{for all
pairs}\;\alpha_{i},\alpha_{j},$ (34)

i.e. the conditional probability for this outcome is constant w.r.t. the
outcomes of $A.$ Denote it $P_{C}(B=\beta|A).$ The following natural question
arises: Does the probability $P_{C}(B=\beta|A)$ coincide with unconditional
probability $P_{C}(B=\beta)?$

Definition 5. Two instruments $I_{A}$ and $I_{B}$ are called $AB$-entangled in
$C\in{\cal C}$ or $C$ is $AB$-entangled, if all outcomes of the $B$-observable
depend on outcomes of $A$-observable, i.e. for all $\beta$ condition (33)
holds for some $\alpha_{i},\alpha_{j}.$

Concerning the notation “$AB$-entangled”, it would be better to write
“$I_{A}I_{B}$-entangled”, to emphasize that this is entanglement of
instruments and not simply the observables, but to make the notation compact,
we proceed with “$AB$-entangled”. The order of observables is important.
Generally $AB$-entanglement does not imply $BA$-entanglement. This is the
purely probabilistic definition, it does not involve LSR, it can be applied to
any statistical physical theory. This definition formalizes dependence of
observables. We introduce the following quantitative measure of entanglement:

Definition 6. For contextual instruments $I_{A}$ and $I_{B}$ and pre-
measurement context $C,$ $AB$-concurrence of conditional probabilities is
defined as

$\lambda_{AB}(\psi)=\sum_{\beta}\sum_{\alpha\not=\alpha^{\prime}}|P_{C}(B=\beta|A=\alpha)-P_{C}(B=\beta|A=\alpha^{\prime})|.$
(35)

The crucial issue is that $AB$-concurrence depends on a pair of instruments.

Proposition 1. For dichotomous observables $A,B=\pm,$ dependencies of the
values $B=-$ and $B=+$ on the outcomes of $A$ are equivalent. Thus, each
dependence is equivalent to $AB$-entanglement.

Proof. In the state context $C,$ the value $B=-$ depends on the outcomes of
$A$ if

$P_{C}(B=-|A=+)\not=P_{C}(B=-|A=-),$ (36)

This automatically implies that even the value $B=+$ depends on the outcomes
of $A,$

$P_{C}(B=+|A=+)=1-P_{C}(B=-|A=+)\not=$ $1-P_{C}(B=-|A=-)=P_{C}(B=+|A=-),$

i.e., instruments $I_{A}$ and $I_{B}$ are entangled in context $C.$

As was already pointed out, in articles [65, 66] contextual probabilistic
entanglement can be realized in the complex Hilbert space and in this way
connected with the ordinary notion of entanglement. In the LSR representation,
the main distinguishing feature of $AB$-entanglement (Definition 5) is that it
is associated with a pair of instruments, $I_{A},I_{B}.$ The standard
definition of entanglement is coupled with the tensor product structure and
not with two concrete instruments (observables).

For simplicity consider CMM $M_{\rm{QVN}}$ (section 4) with von Neuman
observables [37]; in this CMM an observable $\hat{A},$ a Hermitian operator,
automatically determines the state update map, via its spectral family. So,
there is no need to operate with instruments, one can operate solely with
observables. In this CMM (which is typically used in entanglement studies),
contextual probabilistic entanglement is associated with the pairs of
observables, i.e., two observables are entangled or disentangled in some state
(context) $C=\hat{\rho}.$ The main mathematical features of $AB$-entanglement
and ordinary, tensor product based, entanglement are similar, but some
essential differences can be found [65, 66].

The probabilistic viewpoint on the “EPR-paradox” [67] is presented in
Schrödinger’s papers [68, 69] that initiated the modern theory of
entanglement. However, this theory ignores the important message of
Schrödinger: entanglement characterizes probability update for the outcomes of
observable $B$ conditioned on the outcomes of observable $A.$ In the framework
of [68, 69], it is meaningless to speak about entanglement without specifying
the observables. The state update – the Hilbert space representation of the
probability update – encodes the procedure of conditional prediction. For
Schrödinger, the quantum formalism is a mathematical machinery for probability
prediction (as in the Växjö interpretation or QBism) and a quantum state is a
part of such machinery. We can say that Schrödinger interpreted quantum
probabilities as conditional (contextual) probabilities and entanglement as
contextual probabilistic entanglement. But this is my private interpretation
of Schrödinger’s views and many experts in quantum foundations may disagree
with me.

By following Schrödinger [68, 69], in article [65] we considered a special
sort of the contextual probabilistic entanglement that matches perfectly the
Schrödinger analysis of the EPR-argument.

Definition 7. For $C\in{\cal C},$ instruments $I_{A}$ and $I_{B}$ are
perfectly conditionally correlated for values $(A=\alpha,B=\beta)$ if the
conditional probability to get the outcome $B=\beta$ if the preceding
$A$-measurement had the outcome $A=\alpha$ equals to 1,

$P_{C}(B=\beta|A=\alpha)=1.$ (37)

More generally consider observables with values $(\alpha_{i})$ and
$(\beta_{i})$ and some set $\Gamma$ of pairs $(\alpha_{i},\beta_{j}).$

Definition 7a. (EPR-entanglement) Let $C\in{\cal C}.$ If instruments $I_{A}$
and $I_{B}$ are perfectly conditionally correlated for all pairs belonging to
$\Gamma,$ then they are called EPR-entangled w.r.t. set $G$ in context $C.$

We are interested in sets $\Gamma$ such that each of $\alpha$ and $\beta$
values appears in the pairs once and only once. We call such EPR-entanglement
complete.

For example, for two dichotomous observables with $\alpha,\beta=\pm,$ we
consider, e.g., the set of the pairs $(A=+,B=-),(A=-,B=+),$ in short, $A=-B$
EPR-entanglement, or the pairs $(A=+,B=+),(A=-,B=-),$ $A=B$ EPR-entanglement.
We analyze such EPR-entanglements.

Let us start with $A=-B$ EPR-entanglement, i.e., $P_{C}(B=-|A=+)=1$ and
$P_{C}(B=+|A=-)=1.$ Thus, $P_{C}(B=+|A=+)=0$ and $P_{C}(B=-|A=-)=0,$ and
$P_{C}(B=-|A=+)=1\not=P_{C}(B=-|A=-)=0$ and
$P_{C}(B=+|A=-)=1\not=P_{C}(B=+|A=+)=0.$ In this case EPR-entangled
instruments are automatically entangled in the sense of Definition 5.

Now turn to $A=B$ EPR-entanglement, i.e., $P_{C}(B=+|A=+)=1$ and
$P_{C}(B=-|A=-)=1.$ Thus, $P_{C}(B=-|A=+)=0$ and $P_{C}(B=+|A=-)=0,$ and
$P_{C}(B=+|A=+)=1\not=P_{C}(B=+|A=-)=0$ and
$P_{C}(B=-|A=-)=1\not=P_{C}(B=-|A=+)=0.$ And again EPR-entangled instruments
are automatically entangled in the sense of Definition 5.

Thus, EPR-entanglement is just the very special case of $AB$-entanglement.

For dichotomous observables, $AB$-concurrence of conditional probabilities
(35) has the form ${\cal
Con}_{AB}(\psi)=|P(B=+|A=-)-P(B=+|A=+)|+|P(B=-|A=-)-P(B=-|A=+)|,$ hence it can
written as

$\lambda_{AB}(\psi)=2|P(B=+|A=-)-P(B=+|A=+)|.$ (38)

From this formula, we immediately obtain the following characterization of
maximally $AB$-entangled states:

Proposition 4. $AB$-concurrence of conditional probabilities approaches its
maximal value, $\lambda_{AB}(\psi)=2,$ if and only if the instruments are EPR-
entangled in pre-measurement context $C.$

###  2.10 Distinguishing features of contextual measurement models

We list the probabilistic constraints which can be used to distinguish
different CMMs (theoretically and experimentally):

  * •

violation of FTP

  * •

OE

  * •

RRE

  * •

OE+RRE

  * •

violation of Bell inequalities

###  2.11 Interpretations of contextual probability

Probability is characterized by the diversity of interpretations [22]. We now
discuss the interpretations of contextual probability. We start with the
remark that mathematically a contextual probability space $\Sigma=({\cal
C},{\cal O},{\cal P})$ cannot be described as the single Kolmogorov
probability space: $\Sigma$ is a bunch of such spaces However, fixed
$C\in{\cal C}$ and $A\in{\cal O}$ can be realized within some probability
space $K=(\Omega,{\cal F},P)$ with realization of observable $A$ by a random
variable $a:\Omega\to X_{A};$ its probability distribution coincides with
$P_{C}^{A},$ i.e.,

$P_{C}^{A}(\alpha)=P(\omega\in\Omega:a(\omega)=\alpha),\;\alpha\in X_{A}.$

We note that a contextual probability space $\Sigma$ can be represented as

$\Sigma=\cup_{C,A}K_{C,A},$ (39)

where $K_{C,A}$ is a Kolmogorov probability space for describing measurement
of the observable $A$ in the pre-measurement context $C.$

Therefore one can assign to the contextual probability any interpretation used
for probability defined in the measure-theoretic framework. The main
interpretation employed in classical and quantum physics is the frequency
interpretation. In the Kolmogorov theory [35, 36], this interpretation is
mathematically rooted to the strong law of large numbers. Another basic
interpretation of probability in physics is the statistical (ensemble)
interpretation. By this interpretation $\Omega=\Omega_{C}$ represents an
ensemble of systems prepared for measurement and the probability measure
$P=P_{A}$ depends on the observable $A.$ Finally, we mention the subjective
interpretation. It is widely employed in decision making and psychology, but
was not used in physics until QBism was invented.

In contextual probability, we need not represent a pre-measurement context $C$
by an ensemble of systems. Instead, we can consider a sequence of measurements
of an observable $A$ in the same pre-measurement context $C,$

${\bf x}\equiv{\bf x}_{C,A}=(x_{1},...,x_{N},...,...),$ (40)

where $x_{j}(\in X_{A}=\\{\alpha_{1},...,\alpha_{m}\\})$ are measurement’s
outcomes. Such sequence determines the frequencies of realizations of the
concrete values,

$\nu_{N}(\alpha_{j})=n_{N}(\alpha_{j})/N,$ (41)

where $\nu_{N}(\alpha_{j})$ is the number of measurement’s outcomes with the
fixed value $\alpha_{j}.$ The probability to obtain the value $\alpha_{j}$ in
a sequence of measurements ${\bf x}$ is defined as the limit

$P_{C}^{A}(\alpha_{j})\equiv\lim_{N\to\infty}\nu_{N}(\alpha_{j}).$ (42)

This is the straightforward frequency introduction of probability. The deep
mathematical theory of frequency probability was developed by von Mises [30]
(see also [22] for an introduction). A sequence ${\bf x}$ generated in by
observations is called a collective. Von Mises’ theory is a theory of
collectives. Instead of operations on sets, as done in the Kolmogorov measure-
theoretic theory, von Mises constructed a probability theory based on
operations with collectives. We remark that $P_{C}(A=\alpha_{j})\equiv
P_{C}^{A}(\alpha_{j})$ can be considered as the probability generated by the
collective ${\bf x}\equiv{\bf x}_{C,A},$ i.e., $P_{C}^{A}(\alpha_{j})=P_{{\bf
x}_{C,A}}(\alpha_{j}).$ It is important to note that not all collectives are
compatible - combinable in von Mises’ terminology. Two collectives ${\bf x}$
and ${\bf y},$ are combinable if their combination

${\bf z}=(z_{1},...,z_{N},...),\;z_{j}=(x_{j},y_{j}),$ (43)

is also a collective and the probability distributions $P_{{\bf x}}$ and
$P_{{\bf y}}$ are marginal for the probability distribution $P_{{\bf z}},$
i.e.,

$P_{{\bf x}}(\alpha_{j})=\sum_{\beta_{i}}P_{{\bf
z}}(\alpha_{j},\beta_{i}),\;P_{{\bf z}}(\beta_{i})=\sum_{\alpha_{j}}P_{{\bf
z}}(\alpha_{j},\beta_{i}).$ (44)

The frequency probability theory contains the notion of conditional
probability that is similar to CMM’s conditioning, the post-measurement
context $C_{A=a}$ corresponds to the post-measurement collective ${\bf
x}_{C,A=a}$ (see [30, 22] for the details).

In contrast to the Kolmogorov measure-theoretic probability theory, in the von
Mises frequency probability theory classical FTP is violated [22, 29], the
probabilities can interfere and generate the aditive perturbation of FTP in
the form of the interference term. The presence of incombinable collectives
leads to the violation of the Bell type inequalities [22, 29].

The notion of collective was the seed for future growing theory of random
sequences. Besides the existence of the limits for frequencies, (42), a
collective is characterized by stability of these limits w.r.t. place
selections within a sequence ${\bf x}$, i.e., the limit-probability is the
same for all subsequences of ${\bf x}$ for a special class of place
selections. However, von Mises’ definition of place selection was criticized
for non-rigorousness. Its critical analysis was very fruitful and led to the
modern theory of randomness, see,e.g., monograph [70]). In particular, the
monograph presents a “light version” of the von Mises theory. In physics (at
least in quantum physics) one does not analyze in the random structure of the
sequences of measurement’s outcomes. In principle, in QM one can proceed with
“light-collectives” determined solely by the existence of limits (42). The
calculus of such “light-collectives” can be explored for the description of
the probabilistic structure of QM [70]. The first version of the contextual
probability was presented in such “light frequency” framework.

Finally, we note that von Neumann [1, 37] pointed to the von Mises [30]
frequency probability as the probabilistic foundation for QM. This is the
complex foundational issue.

##  3 Contextual Measurement Model for Kolmogorov Theory

Let $K=(\Omega,{\cal F},P)$ be a Kolmogorov probability space [36]. Here
$\Omega$ is a set of any origin, ${\cal F}$ is a collection its subsets
forming $\sigma$-algebra, i.e., ${\cal F}$ is closed w.r.t. countable unions
and intersections, and the operation of complement. (If $\Omega$ is finite,
then ${\cal F}$ is collection of all its subsets.) And $P$ is a probability
measure on ${\cal F}.$

Set ${\cal C}=\\{C\in{\cal F}:P(C)\not=0\\}.$ This is the set of contexts. For
each context $C,$ the Bayes formula defines the conditional probability
measure

$P_{C}(G)=P(G\cap C)/P(C),G\in{\cal F}.$

We highlight that the statistical mixtures of contexts are not determined,
i.e., for subsets $C_{1},C_{2}$ of $\Omega$ and weights $p_{1},p_{2}\geq
0,p_{1}+p_{2}=1,$ there is no a subset $C$ of $\Omega$ which can be identified
with the weighted sum $p_{1}C_{1}+p_{2}C_{2}.$

As an illustrative example, consider some agricultural region $\Omega$ and as
contexts consider its sub-fields (some areas). Generally there is no field of
the form $p_{1}C_{1}+p_{2}C_{2}.$ In applications to decision making and
cognition, one can meet the situations such that $p_{1}C_{1}+p_{2}C_{2}$ is
determined only for a few pairs of weights $p_{1},p_{2}.$ This situation is
related to poorness of the set of possible experimental contexts.

The set of observables ${\cal O}$ is the set of (discrete) random variables,
$a:\Omega\to X_{a},$ where $X_{a}$ is a finite set. (Discrete random variables
are considered for simplicity.) Denote this set by the symbol ${\cal R}_{d}.$
For $x\in X_{a},$ we set $\Omega_{a=x}=\\{\omega\in\Omega:a(\omega)=x\\}.$
Contextual probability coincides with conditional probability given by the
Bayes formula:

$P_{C}^{a}(x)\equiv P_{C}(a=x)=P(\Omega_{a=x}\cap C)/P(C).$ (45)

Thus, the set of probability distributions ${\cal P}=\\{P_{C}^{a}:C\in{\cal
C},a\in{\cal O}\\}.$

For any set $D\in{\cal F}$ and random variable $a\in{\cal R}_{d},x\in X_{a},$
we define set

$D_{a=x}=\\{\omega\in D:a(\omega)=x\\}=D\cap\Omega_{a=x}$

and the map

$T_{a}(x):{\cal F}\to{\cal F},\;D\to D_{a=x}=T_{a}(x)D.$ (46)

For any context $C\in{\cal C}$ and random variable $a\in{\cal R}_{d},x\in
X_{a},$ we define the family of contexts

${\cal C}_{a}(x)=\\{C\in{\cal C}:P_{C}(a=x)>0\\}=\\{C\in{\cal
C}:P(C_{a=x})>0\\}.$

Each random variable $a$ and its outcome $x$ determine a map

$T_{a}(x):{\cal C}\to{\cal C},\;C\to C_{a=x}=T_{a}(x)C,$ (47)

with the domain of definition ${\cal C}_{a}(x).$

Thus, classical CMM $M_{\rm{cl}}$ consists of measurement contexts composed of
pre-measurement contexts - elements of ${\cal F}$ with non-zero probabilities,
observables - (discrete) random variables, and context update maps, ${\cal
T}=\\{T_{a}(x)\\}.$ Here each observable $a,$ random variable, determines
uniquely the context update maps $T_{a}(x),$ and, hence, the contextual
instrument.

The conditional probability is given by the Bayes formula:

$P_{C}(b=y|a=x)=P(\Omega_{b=y}\cap\Omega_{a=x}\cap C)/P(\Omega_{a=x}\cap C)$
$=P(\Omega_{b=y}\cap C_{a=x})/P(C_{a=x})=P_{C_{a=x}}(b=y).$

Since, for each $C\in{\cal C},$ $P_{C}$ is a probability measure, for any pair
of random variables $a,b$, we have the following version of FTP, formula of
total probability, section 2.3,

$P_{C}(b=y)=\sum_{x\in X_{a}}P_{C}(b=y|a=x)P_{C}(a=x)=$ (48) $\sum_{x\in
X_{a}}P_{C_{a=x}}(b=y)P_{C}(a=x).$

In this measurement model all observables are compatible, conditional JPD
coincides with JPD (again by Bayes formula); no pair of observables show OE,
since

$P_{C}(a_{1}=x_{1},a_{2}=x_{2})=P_{C}(a_{2}=x_{2},a_{1}=x_{1})=$
$P_{C}(\omega\in\Omega:a_{1}(\omega)=x_{1},a_{2}(\omega)=x_{2}).$

All observables show repeatability, since $P_{C}(a=x,a=x)=P_{C}(a=x),$ and
RRE, since

$P_{C}(a_{1}=x_{1},a_{2}=x_{2},a_{1}=x_{1})=P_{C}(\omega\in\Omega:a_{1}(\omega)=x_{1},a_{2}(\omega)=x_{2},a_{1}(\omega)=x_{1})=$
$P_{C}(a_{1}=x_{1},a_{2}=x_{2}).$

The Bell inequalities are not violated, since their derivation is based on the
existence of JPD.

We can summarize properties of classical CMM by referring to the
aforementioned list of possibilities:

  * •

violation of FTP - no

  * •

OE - no

  * •

violation of replicability - no

  * •

RRE - yes

  * •

OE+RRE - yes

  * •

violation of Bell inequalities - no

One of the problems of the above contextual representation of the classical
probability is that the uniqueness conditions (2), (3), Mackey’s axiom 2, can
be violated, i.e., generally

$P_{C}^{a_{1}}=P_{C}^{a_{2}}\;\mbox{ for any }C\in{\cal C}\not\Rightarrow
a_{1}=a_{2}.$ (49) $P_{C_{1}}^{a}=P_{C_{2}}^{a}\;\mbox{ for any }a\in{\cal
O}\not\Rightarrow C_{1}=C_{2}.$ (50)

This problem can be easily resolved in the standard way, see below.

Let us consider a Kolmogorov probability space $K=(\Omega,{\cal F},P)$ with a
complete probability measure, i.e., any subset of a set $D\in{\cal F},P(D)=0,$
also belongs to ${\cal F}.$ We recall that the symmetric difference of two
sets $D_{1}$ and $D_{2}$ is defined as

$D_{1}\Delta D_{2}=(D_{1}\setminus D_{2})\cup(D_{2}\setminus D_{1})=(D_{1}\cup
D_{2})\setminus(D_{1}\cap D_{2}).$

For $D_{1},D_{2}\in{\cal F},$ we set $D_{1}\sim D_{2}$ if $P(D_{1}\Delta
D_{2})=0.$ This is an equivalence relation on ${\cal F};$ it splits ${\cal F}$
into disjoint equivalence classes. Denote the set of equivalence classes by
the symbol $\tilde{{\cal F}};$ denote the equivalence class of zero
probability sets by the symbol $\tilde{Z}.$ The set of pre-measurement
contexts is $\tilde{{\cal C}}=\tilde{{\cal F}}\setminus\tilde{Z},$ i.e., all
equivalent classes of sets from ${\cal F}$ of non-zero measure.

We also modify the class of observables. Two random variables are equivalent,
$a_{1}\sim a_{2},$ if $P(\omega\in\Omega:a_{1}(\omega)\not=a_{2}(\omega))=0.$
This is the equivalence relation on the space of random variables, in our
consideration these are discrete random variables ${\cal R}_{d}.$ So, ${\cal
R}_{d}$ is split into disjoint classes of equivalent random variables, denote
the set of these classes by the symbol $\tilde{{\cal R}}_{d}$ and set
$\tilde{{\cal O}}=\tilde{{\cal R}}_{d}.$

For a pre-measurement context $\tilde{C}\in\tilde{\cal C}$ and observable
$\tilde{a}\in\tilde{\cal R}_{d}$ we define the probability distribution
$P_{\tilde{C}}^{\tilde{a}}(x)=P_{C}(a=x)$ for some representatives
$C\in\tilde{\cal C}$ and $a\in\tilde{\cal R}_{d}$ (the correctness of this
definition is proved below), and set $\tilde{\cal
P}=\\{P_{\tilde{C}}^{\tilde{a}}\\}.$ The modified classical contextual
probability space is the triple $\tilde{\Sigma}=(\tilde{\cal C},\tilde{\cal
R}_{d},\tilde{\cal P}).$

The map $T_{a}(x),$ see (47), generates a map of $\tilde{{\cal F}}$ into
itself

$\tilde{T}_{a}(x):\tilde{{\cal F}}\to\tilde{{\cal F}}.$ (51)

Set $\tilde{\cal C}_{a}(x)=\\{\tilde{C}\in\tilde{\cal
C}:\tilde{T}_{a}(x)\tilde{C}\in\tilde{\cal C}\\}.$ Then
$\tilde{T}_{a}(x):\tilde{\cal C}_{a}(x)\to\tilde{\cal C}.$ A measurement
context is a triple $(\tilde{C},\tilde{a},\tilde{T}_{a}).$ Modified classical
CMM $\tilde{M}_{\rm{cl}}$ is given by the set of such measurement contexts.

Now we demonstrate that in $\tilde{M}_{\rm{cl}}$ the uniqueness conditions
(2), (3), Mackey’s axiom 2, hold true. We assume that the ranges of values of
random variables are subsets of a set $X;$ for simplicity let $X=\mathbb{R}.$
First we note that if $D_{1},D_{2}\in\tilde{D},$ then $P(D_{1})=P(D_{2}).$ We
have $P(D_{1})=P((D_{1}\cap D_{2})\cup(D_{1}\setminus D_{2}))=P(D_{1}\cap
D_{2})=P(D_{2}).$ Let now $C_{1},C_{2}\in\tilde{C},$ then, as we have seen
$P(C_{1})=P(C_{2});$ also, for any $G\in{\cal F},P(G\cap C_{1})=P(G\cap
C_{1}\cap C_{2})=P(G\cap C_{2}).$ Hence, $P_{C_{1}}(G)=P_{C_{2}}(G).$ So,
conditional probability measure $P_{C}$ does not depend on the choice of a
representative $C\in\tilde{C}$ and it can denoted as $P_{\tilde{C}}.$

We show that implication (2) holds. Let for random variables $a_{1},a_{2},$
$P_{C}^{a_{1}}=P_{C}^{a_{2}}$ for any context $C.$ Let, for some
$x,P(\Omega_{a_{1}=x})>0.$ Take $C=\Omega_{a_{1}=x},$ then

$1=P_{\Omega_{a_{1}=x}}(\Omega_{a_{1}=x})=P(\Omega_{a_{2}=x}\cap\Omega_{a_{1}=x})/P(\Omega_{a_{1}=x}),$

i.e.,

$P(\Omega_{a_{2}=x}\cap\Omega_{a_{1}=x})=P(\Omega_{a_{1}=x})=P((\Omega_{a_{2}=x}\cap\Omega_{a_{1}=x})\cup(\Omega_{a_{1}=x}\setminus\Omega_{a_{2}=x}).$

Hence, $P(\Omega_{a_{1}=x}\setminus\Omega_{a_{2}=x})=0$ and symmetrically
$P(\Omega_{a_{2}=x}\setminus\Omega_{a_{1}=x})=0.$ The sets
$\Omega_{a_{i}=x},i=1,2,$ belong to the same equivalence class. This implies
that the random variables also belong to the same equivalence class.

Now we turn to implication (3). Let, for any random variable $a,$
$P_{C_{1}}^{a}=P_{C_{2}}^{a}.$ Select $a$ as the characteristic function of
the set $C_{1}.$ Then $P_{C_{1}}^{a}(1)=1=P(C_{1}\cap C_{2})/P(C_{2}),$ i.e.,
$P(C_{1}\cap C_{2})=P(C_{2})=P((C_{1}\cap C_{2})\cup(C_{2}\setminus C_{1}),$
i.e., $P(C_{2}\setminus C_{1})=0$ and symmetrically $P(C_{1}\setminus
C_{2})=0,$ i.e., contexts belong to the same equivalence class.

##  4 Contextual Measurement Model for von Neumann Observables

We restrict consideration to finite dimensional Hilbert state spaces. The
space of pre-measurement contexts is mathematically represented as the space
of density operators ${\cal D}$, i.e., ${\cal C}={\cal D},$ observables are
Hermitian operators (von Neumann observables [1, 37]). Denote the space of
Hermitian operators by the symbol ${\cal L}_{H},$ i.e., ${\cal O}={\cal
L}_{H}.$ This real linear space is endowed the with scalar product
$\langle\hat{A}|\hat{B}\rangle=\rm{Tr}\hat{A}\hat{B}.$

Operator $\hat{A}\in{\cal L}_{H}$ has the spectral decomposition:
$\hat{A}=\sum_{x\in X_{A}}x\;\hat{E}_{A}(x),$ where $\hat{E}_{A}(x)$ is the
orthogonal projection on the subspace ${\cal H}_{A}(x)$ composed of
eigenvectors with eigenvalue $x,$ and $X_{A}$ is operator’s spectral set. Then

$P_{\rho}^{A}(x)\equiv
P_{\rho}(A=x)=\rm{Tr}\hat{E}_{A}(x)\hat{\rho}=\rm{Tr}\hat{E}_{A}(x)\hat{\rho}\hat{E}_{A}(x),$
(52) $T_{A}(x):{\cal D}\to{\cal D},\hat{\rho}_{A=x}\equiv
T_{A}(x)\hat{\rho}=\frac{\hat{E}_{A}(x)\hat{\rho}\hat{E}_{A}(x)}{\rm{Tr}\hat{E}_{A}(x)\hat{\rho}\hat{E}_{A}(x)},$
(53)

with the domain of definition ${\cal C}_{A}(x)=\\{\hat{\rho}\in
D:P_{\rho}(A=x)>0\\}.$

Thus, ${\cal P}=\\{P_{\rho}^{A}:\rho\in{\cal D},A\in{\cal L}_{H}\\}$ and
${\cal T}=\\{T_{A}(x):A\in{\cal L}_{H},x\in X_{A}\\}.$ We remark that
observable $A$ uniquely determines the family of maps $T_{A}(x),x\in X_{A}$ by
equality (53). So, measurement contexts can be represented by pairs
$(\hat{\rho},\hat{A}),\hat{\rho}\in{\cal D},\hat{A}\in{\cal L}_{H}.$ Denote
this CMM by the symbol $M_{\rm{QVN}}.$

In this CMM one need not define separately context update maps, they are
automatically encoded in observables. On the one hand, this simplifies theory.
On the other hand, this is the misleading path in measurement theory, cf. with
quantum instrument theory.

In CMM $M_{\rm{QVN}},$ the conditional probability is given by the formula:

$P_{\rho}(B=y|A=x)=\rm{Tr}\hat{E}_{B}(y)T_{A}(x)\rho=\frac{\rm{Tr}\hat{E}_{B}(y)\hat{E}_{A}(x)\hat{\rho}\hat{E}_{A}(x)}{\rm{Tr}\hat{E}_{A}(x)\hat{\rho}\hat{E}_{A}(x)}.$
(54)

It can be rewritten as

$P_{\rho}(B=y|A=x)=\frac{\rm{Tr}\hat{E}_{B}(y)\hat{E}_{A}(x)\hat{\rho}\hat{E}_{A}(x)\hat{E}_{B}(y)}{\rm{Tr}\hat{E}_{A}(x)\hat{\rho}\hat{E}_{A}(x)}.$
(55)

In this LSR-based CMM, it is convenient to introduce the maps:

${\cal I}_{A}(x)\hat{\rho}=\hat{E}_{A}(x)\hat{\rho}\hat{E}_{A}(x)$ (56)

Then the above formulas can be rewritten as

$P_{\rho}(A=x)=\rm{Tr}{\cal I}_{A}(x)\hat{\rho},$ (57)
$T_{A}(x)\hat{\rho}=\frac{1}{P_{\rho}(A=x)}{\cal I}_{A}(x)\hat{\rho}.$ (58)

or

${\cal I}_{A}(x)\hat{\rho}=P_{\rho}(A=x)T_{A}(x)\hat{\rho};$ (59)

and the conditional probability is written in the form

$P_{\rho}(B=y|A=x)=\frac{\rm{Tr}{\cal I}_{B}(y){\cal
I}_{A}(x)\hat{\rho}}{\rm{Tr}{\cal I}_{A}(x)\hat{\rho}},$ (60)

and conditional JPD as

$P_{\rho}(A=x,B=y)=\rm{Tr}{\cal I}_{B}(y){\cal I}_{A}(x)\hat{\rho}.$ (61)

These formulas lead to quantum instrument theory (section 5): $(A,{\cal
I}_{A}(x))$ is a special quantum instrument and $(A,T_{A}(x))$ is the
corresponding contextual instrument.

We note that in CMM $M_{\rm{QVN}}$ probabilities determine contexts (states)
and observables (operators), i.e., (2) and (3) hold (Mackey’s axiom 2).

Let $P_{\rho}^{A_{1}}(x)=P_{\rho}^{A_{1}}(x)$ for all $\hat{\rho}\in{\cal D}$
and real $x.$ Then $\rm{Tr}(E_{A_{1}}(x)-E_{A_{2}}(x))\hat{\rho}=\langle
E_{A_{1}}(x)-E_{A_{2}}(x)|\hat{\rho}\rangle=0.$ Hence,
$E_{A_{1}}(x)=E_{A_{2}}(x)$ for any $x;$ so $\hat{A}_{1}=\hat{A}_{2}.$ Thus,
an observable can be identified with the set of probability distributions
${\cal P}_{A}=\\{P_{\rho}^{A}:\rho\in{\cal D}\\}.$

Now let $P_{\rho_{1}}^{A}(x)=P_{\rho_{2}}^{A}(x)$ for all $\hat{A}\in{\cal
L}_{H}$ and real $x.$ Then
$\rm{Tr}E_{A}(x)(\hat{\rho}_{1}-\hat{\rho}_{2})=\langle
E_{A}(x)|\hat{\rho}_{1}-\hat{\rho}_{2}\rangle=0.$ Hence,
$\hat{\rho}_{1}=\hat{\rho}_{2}.$ Thus, a quantum state can be identified with
the set of probability distributions ${\cal
P}_{\rho}=\\{P_{\rho}^{A}:A\in{\cal L}_{H}\\}.$

In the von Neumann measurement theory, two observables are compatible if they
are represented by commuting operators
$\hat{A},\hat{B}:\;[\hat{A},\hat{B}]=0.$ Compatibility is interpreted as
guarantying the possibility of joint measurement of these observables; their
JPD is given by the formula:

$P_{\rho}^{A,B}=\rm{Tr}\hat{E}_{A}(x)\hat{E}_{B}(y)\rho=\rm{Tr}\hat{E}_{B}(y)\hat{E}_{A}(x)\rho.$
(62)

In fact, this is the separate axiom - a complement to the Born rule [37]. For
compatible observables, JPD and conditional JPD coincide. The conditional JPD
is given by

$P_{\rho}(A=x,B=y)=P_{\rho}(A=x)P_{\rho}(B=y|A=x)=$ (63)
$P_{\rho}(A=x)P_{\rho_{A=x}}(B=y)==\rm{Tr}\hat{E}_{B}(y)\hat{E}_{A}(x)\rho\hat{E}_{A}(x)=$
$\rm{Tr}\hat{E}_{A}(x)\hat{E}_{B}(y)\hat{E}_{A}(x)\rho=\rm{Tr}\hat{E}_{B}(y)\hat{E}_{A}(x)\rho.$

In particular, for compatible observables there is no OE for any state $\rho.$
If operators $\hat{A}_{1},\hat{A}_{2},$ do not commute, then there exists a
state $\hat{\rho}$ showing OE for these observables, i.e.,
$P_{\rho}(A=x,B=y)\not=P_{\rho}(B=y,A=x).$

Each observable shows replicability, e.g.,

$P_{\rho}(A=x,A=x)=\rm{Tr}\hat{E}_{A}(x)\hat{E}_{A}(x)\hat{\rho}\hat{E}_{A}(x)=\rm{Tr}\hat{E}_{A}(x)\hat{\rho}=P_{\rho}(A=x).$

If observables are compatible, then for any $\hat{\rho}$ they show RRE, e.g.,

$P_{\rho}(A_{1}=x_{1},A_{2}=x_{2},A_{1}=x_{1})=\rm{Tr}\hat{E}_{A_{1}}(x_{1})\hat{E}_{A_{2}}(x_{2})\hat{E}_{A_{1}}(x_{1})\hat{\rho}\hat{E}_{A_{1}}(x)\hat{E}_{A_{2}}(x_{2})=$
$\rm{Tr}\hat{E}_{A_{2}}(x_{2})\hat{E}_{A_{1}}(x_{1})\hat{\rho}\hat{E}_{A_{1}}(x)=P_{\rho}(A_{1}=x_{1},A_{2}=x_{2}).$

We highlight that it is impossible to combine OE and RRE within CMM
$M_{\rm{QVN}}$ [62].

FTP (the formula of total probability) can be violated; classical FTP is
additively perturbed by the interference term; consider a pure state
$|\psi\rangle,$ then

$P_{\psi}(B=\beta)=\sum_{\alpha,\alpha^{\prime}}\langle\psi|\hat{E}_{A}(\alpha)\hat{E}_{B}(\beta)\hat{E}_{A}(\alpha^{\prime})|\psi\rangle$
(64)
$=\sum_{\alpha}P_{\psi}(B=\beta|A=\alpha)P_{\psi}(A=\alpha)+\delta_{\psi}(B=y|A),$

where

$\delta_{\psi}(B=y|A)=\sum_{\alpha\not=\alpha^{\prime}}\langle\psi|\hat{E}_{A}(\alpha)\hat{E}_{B}(\beta)\hat{E}_{A}(\alpha^{\prime})|\psi\rangle.$
(65)

In RHS, the first summand corresponds to classical FTP and the second one is
the interference term; it quantifies the degree of non-classicality for this
CMM. See section 2.3 for FTP in the general contextual probabilistic
framework.

Consider dichotomous observables, $A=x_{1},x_{2}$ and $B=y_{1},y_{2},$ of the
von Neumann type. In this case the interference term has the form

$\delta_{\psi}(B=y|A)=$
$2\cos\theta\sqrt{P_{\psi}(B=y|A=x_{1})P_{\psi}(A=x_{1})P_{\psi}(B=y|A=x_{2})P_{\psi}(A=x_{2})},$
(66)

where the angle $\theta=\theta(B=y|A;\psi).$

We can summarize properties of von Neumann CMM with the list of possibilities
presented above:

  * •

violation of FTP - yes

  * •

OE - yes

  * •

violation of replicability - yes

  * •

RRE - no

  * •

OE+RRE - no

  * •

violation of Bell inequalities - yes

##  5 Contextual Measurement Model for Quantum Instruments

In this section we present CMM $M_{\rm{QI}}$ in that a measurement process is
mathematically described by quantum instrument theory [7, 8], [11, 12],
[47]-[53]. This CMM extends CMM $M_{\rm{QVN}}$ in that a measurement process
is mathematically described by a Hermitian operator (von Neumann observable).

The space of linear Hermitian operators ${\cal L}_{H}$ is a real Hilbert
space. We consider linear operators acting in it to be superoperators. A
superoperator $T$ is called positive if it maps the set of positive
semidefinite operators onto itself: for $g\geq 0,\;T(g)\geq 0.$

Consider an observable $A$ with finite range of values $X.$ Its measurements
can be performed with various apparatuses; for each apparatus, the
corresponding measurement procedure is mathematically described in the
following way.

Any map $x\to{\cal I}(x),$ where for each $x\in X,$ the map ${\cal I}(x)$ is a
positive superoperator and

${\cal I}(X)\equiv\sum_{x\in X}{\cal I}(x):{\cal D}\to{\cal D}$ (67)

is called a quantum instrument. It determines some observable; denote it by
the symbol $A,$ see mathematical formula (75).

The probability of the output $A=x$ is given by the generalized Born rule in
the form

$P_{\rho}(A=x)=\rm{Tr}\;[{\cal I}(x)\rho].$ (68)

We note that measurement with the output $A=x$ generates the state-update by
transformation

$\rho\to\rho_{A=x}=T_{A}(x)\rho\equiv\frac{{\cal I}(x)\rho}{Tr{\cal
I}(x)\rho},$ (69)

with the domain of definition ${\cal C}_{A}(x)=\\{\hat{\rho}\in
D:P_{\rho}(A=x)>0\\}.$

Let

${\cal I}(x)\rho=\hat{E}(x)\rho\hat{E}(x),$ (70)

where $(\hat{E}(x))$ are projections giving the orthogonal decomposition of
$I.$ Such an instrument is called a projection instrument.

The most natural generalization of projection instruments is an atomic
instrument. Let $(\hat{V}(x))$ be a family of linear operators contrained by
the normalization condition:

$\sum_{x}\hat{V}(x)\hat{V}^{\star}(x)=I.$ (71)

An atomic quantum instrument is a super-operator of the form:

$\rho\to{\cal I}(x)\rho=\hat{V}(x)\rho\hat{V}^{\star}(x).$ (72)

Applications of the quantum instrument theory to quantum information are
typically restricted by the use of atomic instruments.

The space of Hermitian operators ${\cal L}_{H}$ is the real Hilbert space,
i.e., for each linear operator acting in ${\cal L}_{H}$ (superoperator) its
adjoint is well defined. The generalized Born rule can written as

$P_{\rho}(A=x)=\langle{\cal I}(x)\rho|I\rangle=\langle\rho|{\cal
I}^{\star}(x)I\rangle=\langle{\cal I}^{\star}(x)I|\rho\rangle,$ (73)

where $I$ is the unit operator and ${\cal I}^{\star}(x)$ is superoperator that
is adjoint to ${\cal I}(x)$ in Hilbert space ${\cal L}_{H}.$ Hence, the
generalized Born rule has the form:

$P_{\rho}(A=x)=\rm{Tr}\hat{A}(x)\rho,$ (74)

where

$\hat{A}(x)={\cal I}^{\star}(x)I.$ (75)

Operators $\hat{A}(x),x\in X,$ are called effects; they are positive semi-
definite Hermitian and sum up to the unit operator:

$\sum_{x\in X}\hat{A}(x)=I.$

The family of operators $A=(\hat{A}(x),x\in X)$ is called a positive operator
valued measure (POVM): for a subset $\Delta$ of $X,$ we set

$\hat{A}(\Delta)=\sum_{x\in\Delta}\hat{A}(x)\geq 0;$

this is an additive operator-valued measure, i.e., for
$\Delta_{1},\Delta_{2}\subset X,\Delta_{1}\cap\Delta_{2}=\emptyset,$

$\hat{A}(\Delta_{1}\cup\Delta_{2})=\hat{A}(\Delta_{1})+\hat{A}(\Delta_{2}).$

Instruments of the projection type, see (70), determine the special class of
POVMs, projection valued measures (PVMs).

Two POVMs $A=(\hat{A}(x),x\in X)$ and $B=(\hat{B}(y),y\in Y)$ are called
compatible if there exists a POVM $C=(\hat{C}(x,y),(x,y)\in X\times Y)$ such
that

$\hat{A}(x)=\sum_{y\in Y}\hat{C}(x,y),\;\hat{B}(x)=\sum_{x\in X}\hat{C}(x,y).$
(76)

Compatibility is interpreted as guarantying the possibility of the joint
measurement of these observables, their JPD is given by generalization of the
Born rule for compatible von Neumann observables:

$P_{\rho}^{A,B}(x,y)=\rm{Tr}\hat{C}(x,y)\hat{\rho}.$ (77)

In contextual probability space, contexts are mathematically represented by
density operators (quantum states), ${\cal C}={\cal D},$ observables by POVMs
(also known as generalized quantum observables), the probability distributions
are determined by the generalized Born rule (68). CMM $M_{\rm{QI}}$ is endowed
by quantum instrument maps updating quantum states (contexts) due to the
measurement feedback (69).

Consider POVM $\hat{A}=(\hat{A}(x))$ and all quantum instruments generating it
via (75). Then the corresponding state (context) update maps are defined by
equality (69). The same POVM, generalized observable, can be coupled to a
variety of such maps. Therefore the commonly used approach highlighting POVMs
as generalized observables is ambiguous. POVMs are just biproducts generated
by quantum instruments.

Quantum instruments considered above were invented in article [7] (see also
monograph [8]) and these are Davies-Levis instruments, so
$M_{\rm{QI}}=M_{\rm{QI;DL}}.$ In quantum information theory, one uses the
special class of quantum instruments given by completely positive maps ${\cal
I}(x);$ denote the corresponding CMM $M_{\rm{QI;O}},$ where I use the index
“O” to mention Masanao Ozawa who contributed so much into theory of such
quantum instruments [11, 12], [47]-[50]. It is commonly assumed that the
instruments belonging to $M_{\rm{QI;DL}}\setminus M_{\rm{QI;O}}$ are non-
physical. I debated this question with Masanao Ozawa and he firmly stays on
this position. As was proved by him only completely positive instruments can
be realized via the indirect measurement scheme [11]. This scheme is adequate
to quantum measurement processes and any deviation from this scheme is non-
physical. Nevertheless, it might be that in quantum-like modeling, even
instruments which are not completely positive can find applications. Such
applications would lead to modification of the indirect measurement scheme,
amy be via consideration of non-unitary interactions.

We remark that instrument maps ${\cal I}(x)$ are linear in the Hilbert space
${\cal L}_{H}.$ In terms of context (state) update these maps they can be
written as

${\cal I}(x)=P_{\rho}(A=x)T_{A}(x).$ (78)

Hence, in quantum instrument CMM scaling of the update map $T_{A}(x)$ by
probability $P_{\rho}(A=x)$ is a linear map. Generally, LSR-CMM with the
context (state) space ${\cal C}={\cal D}$ can be endowed with context (state)
update maps such that scaling (78) can be nonlinear. CMM $M_{\rm{QI;NL}}$ with
nonlinear context update maps might be useful for quantum-like modeling. It is
interesting to find concrete applications of $M_{\rm{QI;NL}}.$ Of course, such
applications would lead to modification of the indirect measurement scheme.

We can summarize properties of quantum instrument CMM, both models
$M_{\rm{QI;DL}}$ and $M_{\rm{QI;O}}$:

  * •

violation of FTP - yes

  * •

OE - yes

  * •

violation of replicability - yes

  * •

RRE - no

  * •

OE+RRE - generally no, but can be realized by special instruments

  * •

violation of Bell inequalities - yes

It is interesting to find a property distinguishing $M_{\rm{QI;DL}}$ and
$M_{\rm{QI;O}}$ via an experimental test, i.e., some experimentally testable
property such that only completely positive instruments have it.

One of the important features of the von Neumann model is coincidence of JPD
and conditional JPD for compatible observables. In contrast, the instrument
model shows that generally the situation is not simple at all. Consider two
instruments ${\cal I}_{A}(x)$ and ${\cal I}_{B}(y)$ such that their
observables are POVMs of PVM type, i.e., $\hat{A}=(\hat{E}_{A}(x))$ and
$\hat{B}=(\hat{E}_{B}(y)).$ They are jointly measurable and the JPD is given
by formula (62). The conditional JPD is given by

$P_{\rho}(A=x,B=y)=P_{\rho}(A=x)P_{\rho}(B=y|A=x)=$ (79)
$P_{\rho}(A=x)P_{\rho_{A=x}}(B=y)=\rm{Tr}{\cal I}_{B}(x){\cal I}_{A}(x)\rho.$

The right hand sides of (62) and (79) coincide only if the instrument
superoperators are of the projection type, i.e., ${\cal
I}(x)\rho=\hat{E}(x)\rho\hat{E}(x).$

Moreover, in this case two projection type observables, PVMs, can have a
variety of conditional probability distributions corresponding different
instruments generating them by the rule $\hat{E}(x)={\cal I}^{\star}(x).$

##  6 Ordered Space Measurement Model with Probability Measure States

In this section we connect the generalized probability theory (the Davies-
Lewis approach [7]) for probability measures with CMM. Here we use the ordered
linear space approach. This is the concrete application of the universal
scheme based on the abstract framework of ordered linear spaces.

Consider the space ${\cal M}$ of all real valued measures on some set $\Omega$
with a $\sigma$-algebra of subsets ${\cal F},$ i.e., ${\cal M}\equiv{\cal
M}(\Omega,{\cal F}).$ Real linear space ${\cal M}$ has the natural order
structure and the positive cone ${\cal M}^{+}$ consisting of non-negative
measures. Consider the elements of this cone given by probability measures,
i.e., $\mu\geq 0$ and $\mu(\Omega)=1;$ denote this set by the symbol ${\cal
S};$ this is the set of states; ${\cal S}$ is a convex subset of ${\cal M}.$
The latter is endowed with the variation norm, $||\mu||=\rm{var}(\mu)$ and it
is a Banach space. Consider its dual space ${\cal M}^{\prime},$ the space of
continuous linear functionals $f:{\cal M}\to\mathbb{R}.$ Denote by ${\cal A}$
the subset of ${\cal M}^{\prime},$ consisting of functionals mapping ${\cal
S}$ into $[0,1].$ Elements of ${\cal A}$ are called effects, these are basic
observables. They can be described solely in the terms of the state space
${\cal S}$ as affine functionals valued in $[0,1],$ i.e., ${\cal A}\equiv{\cal
A}({\cal S}).$

Consider the functional $u\in{\cal M}^{\prime}$ defined as $\mu\to\langle
u|\mu\rangle=\mu(\Omega).$ Its characteristic property is that $\langle
u|\mu\rangle=1$ for any state $\mu\in{\cal S}.$

Let $X=\\{x_{1},..,x_{m}\\}$ be a finite set and let $A=(A(x_{i}),i=1,...,m)$
where $A(x_{i})\in{\cal A}({\cal S})$ and $A(X)\equiv\sum_{x\in X}A(x)=u.$
Such vectors of functionals are analogs of POVMs; we call them ${\cal
M}$-POVMs. These are observables of the contextual probability space
$\Sigma_{\rm{measure}}$ with contexts ${\cal C}={\cal S}$ and the set of
probability distributions ${\cal P}$ defined as

$P_{\mu}^{A}(x)\equiv P_{C}(A=x)=\langle A(x)|\mu\rangle.$

As we learn from the quantum instrument theory, the basic elements of
measurement procedures are not observables, but instruments. Let ${\cal
L}({\cal M})$ denote the space of continuous linear operators, $J:{\cal
M}\to{\cal M}.$ The ${\cal M}$-instrument with the range of values $X$ is a
map ${\cal I}:X\to{\cal L}({\cal M})$ such that ${\cal I}(x){\cal
M}^{+}\subset{\cal M}^{+}$ and

${\cal I}(X)\equiv\sum_{x}{\cal I}(x):{\cal S}\to{\cal S}.$

Each instrument determines the state update map

$\mu\to T_{A}(x)\mu\equiv\frac{1}{{\cal I}(x)\mu(\Omega)}{\cal
I}(x)\mu=\frac{1}{\langle u|{\cal I}(x)\mu\rangle}{\cal I}(x)\mu,$

and the probability distribution

$P_{\mu}(A=x)=\langle u|{\cal I}(x)\mu\rangle$

The domain of definition of the state update map $T_{A}(x)$ is given by the
set of probability measures ${\cal C}_{A}(x)=\\{\mu\in S:P_{\mu}(A=x)>0\\}.$

Let $J:{\cal M}\to{\cal M}$ be a continuous linear operator. Then its adjoint
operator $J^{\star}$ is well defined, $J^{\star}:{\cal M}^{\prime}\to{\cal
M}^{\prime},$ and

$\langle J^{\star}f|\mu\rangle=\langle f|J\mu\rangle.$

Set

$A(x)={\cal I}^{\star}(x)u.$

Then, for each $x,A(x)$ is an effect, i.e. $A(x)\in{\cal A}({\cal S}).$ So,
each ${\cal M}$-instrument determines a ${\cal M}$-POVM.

The ${\cal M}$-CMM consists of context (states) given by probability measures
and POVM-observables with state updates given by instruments.

##  7 Linear Space Representation for Contextual Probability Space

The state space is given by the set $S,$ the set of possible measurement
outcomes of an observable quantity is denoted by $X.$ Let a system be in a
state $s\in S.$ A probability $p(x,s)$ is assigned to any possible outcome
$x\in X.$ Thus, we have a function

$p:X\times S\to[0,1].$

To each outcome $x\in X$ and state $s\in S,$ this function is a probability of
the outcome $x$ for the system is in the state $s.$ Generalized probability
model is a triple $(S,p(\cdot,\cdot),X).$ Denote by $\Phi_{[0,1]}$ the space
of function from $X$ to $[0,1].$ By considering state $s$ as a variable, we
obtain the map

$S\to\Phi_{[0,1]},\;s\to s(\cdot)=p(\cdot,s).$ (80)

It is natural to assume that each state $s\in S$ determines the probability
distribution uniquely, i.e.,

$p(x,s_{1})=p(x,s_{2})\;\mbox{ for any }x\in X\Rightarrow s_{1}=s_{2}.$ (81)

Under this assumption the map (80) is injection. Thus, each state $s$ can be
mapped to a function belonging to space $\Phi_{[0,1]},$ it will be denoted by
the same symbol $s.$ Consider now the vector space $\Phi$ of all real valued
functions on $X.$ So, $S$ is identified with a subset of this functional
space. Consider its closed convex hull $\bar{S}.$ The vectors from it are all
possible probabilistic mixtures (convex combinations) of states in $S.$

Each $x\in X$ defines a linear functional on $\Phi,\;\phi\to
f_{x}(\phi)=\phi(x).$ If $\phi=s\in\bar{S},$ then $f_{x}(s)=s(x)\in[0,1],$
i.e., $f_{x}:\bar{S}\to[0,1].$ This is an affine functional on the convex set
$\bar{S}.$ It describes a measurement outcome and $f_{x}(s)=p(x,s)$ is the
probability for this outcome in state $s.$

Denote by $A(\bar{S})$ to the space of all affine functionals

$f:\bar{S}\to[0,1].$

In particular, for any $x\in X,\;f_{x}\in A(\bar{S}).$ Any functional $f\in
A(\bar{S})$ describes an outcomes of some observable, and thus $f(s)$ is the
probability for that outcome in state $s.$

In QM, $\bar{S}$ is the set of density operators and elements of $A(\bar{S})$
are effects - components of POVMs, $f(\rho)=\rm{Tr}\hat{E}_{f}\hat{\rho},$
where $\hat{E}_{f}$ is the effect corresponding to the affine functional $f.$

The elements of $A(\bar{S})$ are called effects. It is typically assumed that
there exists an element $u$ of $A(\bar{S})$ such that $u(s)=1$ for any
$s\in\bar{S}.$ It is an analog of quantum observable given by the unit
operator $I.$ Consider the point wise order structure on $A(\bar{S}),f\leq g$
iff $f(s)\leq g(s)$ for any state $s.$ Thus, any observable $f\in A(\bar{S})$
is majorated by $u,0\leq f\leq u.$ A discrete measurement is represented by a
set of effects $(f_{i})$ such that $\sum_{i}f_{i}=u.$

We now connect LSR for the contextual probabilistic model. We assume that all
observables have the same range of values $X.$ The straightforward intention
is to set $S={\cal C}\times{\cal O}$. Let, as above, $\Phi_{[0,1]}$ denotes
the space of functions from $X$ to $[0,1].$ We map $S$ into
$\Phi_{[0,1]},\;s=(C,A)\to P_{C}^{A}.$ However, generally this map is not
injection: $P_{C_{1}}^{A_{1}}(x)=P_{C_{2}}^{A_{2}}(x)$ for all $x\in X$ does
not imply that $C_{1}=C_{2}$ and $A_{1}=A_{2}.$ So, such straightforward
construction seems to be non-proper for our aim.

We modify it by setting ${\bf X}={\cal O}\times X,$ the elements of ${\bf X}$
are pairs ${\bf x}=$(observable, outcome)$=(A,x).$ We now use the symbols
$\Phi_{[0,1]}$ and $\Phi$ for functions from ${\bf X}\to[0,1]$ and to real
line respectively. Each context $C$ can be represented as a vector belonging
to $\Phi_{[0,1]},\;C({\bf x})=C(A,x)=P_{C}^{A}(x).$ Due to (3), embedding of
the set of contexts ${\cal C}$ into $\Phi_{[0,1]}$ is injection. Again denote
by $\bar{\cal C}$ the convex hull of ${\cal C}.$ Each point ${\bf
x}=$(observable, outcome)$=(A,x)$ determines the affine functional $C\to
f_{A,x}(C)=P_{C}^{A}(x)\in[0,1].$ Now fix $A\in{\cal O}$ and consider the
family of functionals $F=(F_{A}(x)=f_{A,x}:x\in X).$ This is representation of
observable $A.$

So, any contextual probability model can be realized alike COM – an
observational COM.

##  8 Concluding remarks

As was emphasized in introduction, CMM can be considered as the most general
probabilistic model for measurement. It also can be considered as the
minimalist restructuring of Mackey’s project [6]. Mackey proceeded to quantum
logic and this makes the mathematical construction more complicated. One may
even say that mathematics shadowed measurement theory. Surprisingly, even this
minimalist model (CMM) has a complex structure and represents the basic
elements of quantum probability and measurement theory,e.g., interference of
probability, order effect, entanglement, the violation of the Bell
inequalities.

CMM can be employed not only in quantum foundations, but also in quantum-like
modeling that can employ contextual probability calculi and CMMs which are not
based on the complex Hilbert space formalism.

## Acknowledgments

This paper is the completion of the long project on contextual probability and
quantum physics which started with my discussions with Kolmogorov and Mackey
and later with Accardi, Ballentine, Gudder, Mittelstaedt, Ohya, Shiryaev,
Volovich. During the recent years, I discussed with Plotnitsky the Bohr’s
complementarity principle and with Ozawa quantum instrument realization of
measurement theory. Since year 2000, I was involved in critical and stimulated
debates with Fuchs on QBism and subjective probability in QM. During my visits
to Vienna, I had exciting conversations with Rauch and Zeilinger on the
(non)realism, (non)locality, and (non)contextuality of QM. All these
discussions stimulated my thinking on contextual measurement theory and
probability.

## Apepndix 1. Terminology: Context vs. State

We make the following remark about the terminology “context vs. state.” Since
QM operates with the notion “state”, generalized probability theory also
employes this terminology. However, even in QM using the term “state” is
ambiguous. It matches the orthodox Copenhagen interpretation by which a state
is treated as the state of an individual quantum system, say the state of an
electron - one concrete electron. Many experts consider this interpretation of
the quantum state as leading to paradoxes and mismatching with the statistical
nature of quantum phenomena. This is a complicated foundational issue, since
the leading supporters of the orthodox Copenhagen interpretation also consider
QM as a statistical theory in that the state of an individual system encodes
the statistics of the coming experimental runs.

For example, Einstein, Koopman, Margenau, Blohintzev, Ballentine, and
nowadays, e.g., Ballian, Nieuwenhuizen, Khrennikov use the so called
statistical (or ensemble) interpretation of QM. By this interpretation a
quantum state represents statistical properties of an ensemble of identically
prepared systems. So, whose state? The state of an ensemble? In the
operational approach “state” corresponds to a preparation procedure. It seems
that the term “state” borrowed from the orthodox Copenhagen interpretation
does not match to the statistical and operational interpretations of QM. In
the generalized probability theory the term “state” is typically associated
with a preparation procedure or a class of equivalent preparation procedures.
However, this meaning of the state is not highlighted and the output of the
generalized probability theory is often projected onto the the orthodox
Copenhagen interpretation, i.e., this theory interpreted as a theory about the
structure of the state space of individual quantum systems. Therefore in the
Växjö interpretation we prefer to use the notion of context as a complex of
experimental conditions, pre-measurement context can be associated with a
class of equivalent preparation procedures (as is done in the consistent
presentation of the generalized probability theory), measurement context is
the combination of the preparation, measurement, and state update generated by
measurement feedback with the fixed outcome.

In contrast to the generalized probability theory employing LSR, we do not
assume that the set of pre-measurement contexts ${\cal C}$ contains contexts
generated by statistical mixtures (see Axiom 4 in Mackey’s book [6]), i.e.,
for $C_{1},C_{2}\in{\cal C}$ and $p_{1},p_{2}\geq 0,p_{1}+p_{2}=1,$ the set
${\cal C}$ need not contain a context which can be identified with
$p_{1}C_{1}+p_{2}C_{2}.$ Proceeding without the mixture axiom illuminates the
difference between state and context; consider e.g., the “basic contextual
probability representation” of the classical Kolmogorov probability space
(section 3). Here contexts are not probability distributions, but elements of
the ($\sigma$-)algebra. Generally a context provides a finer description of
the measurement setup than a probability distribution.

## Apepndix 2. Contextual measurement model with vs. without linear space
representation

Why is it useful to proceed in contextual probabilistic framework as far as
possible without appealing to LSR?

I start with some remarks on uncritical using of LSR:

  * •

LSR shadows the essence of the quantum probability formalism as the machinery
for probability inference.

  * •

LSR for classical probability via the use of the linear space of measures with
the positive cone of non-negative measures and convex state space of
probability measures seems to be inadequate to Kolmogorov’s theory [35, 36]
based on conditioning (contextualization) with Bayes’ formula (section 3).

  * •

LSR generates (via creation of convex linear hull and its closure) a plenty of
unphysical states and observables (see Ballentine [18]), operating with them
led e.g. to von Neumann’s no-go theorem [37].444 Generally some outputs of
quantum information theory obtained in the abstract LSR framework might be its
artifacts, without coupling to physical reality. The critical analysis of
connection of LSR mathematics and physics is needed.

  * •

The picture that quantum probability theory is just one of LSR of probability
diminishes the exclusiveness of linearity of QM. One losses the physical
ground for the latter, LSR becomes just a part of the mathematical apparatus
of QM (see section LABEL:QB on my views on the physical ground for QM-
linearity).

  * •

Linking of entanglement to the LSR tensor product structure shadows its
contextual probabilistic nature and supports the ambiguous statements on
quantum nonlocality.

  * •

Recently the mathematical formalism of quantum theory, especially probability,
started to be widely applied outside of physics, e.g., in cognition,
psychology, social and political sciences, economics and finance, so called
quantum-like modeling (see, e.g., recent monograph [40]). In such models the
set of possible states (pre-measurement contexts) is not so rich as in
physics. In quantum-like modeling even the possibility to prepare statistical
mixtures is not evident, i.e., proceeding towards convex structures might be
misleading.

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