Contextual measurement model and quantum theory

We develop a contextual measurement model (CMM) that is used for the clarification of the quantum foundations. This model matches Bohr’s views on the role of experimental contexts. CMM is based on a contextual probability theory that 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 the CMM framing of classical probability, the von Neumann measurement theory and the quantum instrument theory. CMM can also be applied outside of physics, e.g. in cognition, decision-making and psychology, the so-called quantum-like modelling.


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 quantumlike 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].

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.
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 → C A=x can be described as a map T A (x) : C → C, where 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 inter-ference 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.

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 general-ized 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. 1MM 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.

Contextual Measurement Model (CMM)
2.1 Contextual probability space Definition 1.A contextual probability space is a triple Σ = (C, O, P), where C and O are sets of pre-measurement contexts and observables and P is the space of the corresponding probability distributions.
In physics pre-measurement contexts can be associated with preparation procedures. 2 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 ∈ X A as P A C (x) ≡ P C (A = x).By definition of a probability distribution The range set X A can be endowed with the algebra of all its subsets F A .We set This is a probability measure on F A .In the definition of Σ, the symbol P denotes the collection of such probability measures (see Axiom 1 in Mackey's book [6]).Elements of 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., 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., The average of an observable A ∈ O (with X A ⊂ R) w.r.t. a pre-measurement context C ∈ C is defined as

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: 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. 3 It is natural to consider this map only for contexts belonging to the set 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 with the domain of definition 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 : 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 Â, 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 Σ = (C, O, P) be a contextual probability space.A contextual measurement model (CMM) is a pair M = (Σ, I), where Σ is a contextual probability space and 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 → 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: We note that this definition involves context update only for the A-observable; different contextual instruments I A , I ′ A , ..., induce their own probabilistic conditioning, P C,I A (B = y|A = x), P C,I ′ A (B = y|A = x), ... .For simplicity, we shall typically omit the index I A of dependence on the concrete instrument.

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, In classical probability theory contextual probability is identified with conditional one (section 3) and the contextual-conditional analog of FTP has the form, see (48).However, in a general contextual probability space this formula can be violated, 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, 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): 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 where context C is identified with quantum state ψ (for simplicity consider contexts corresponding to pure states) and the angle θ = θ(B = y|A; ψ).For dichotomous observables, even in general CMM it is useful to write the interference term as where λ = λ(B = y|A; C).If |λ| ≤ 1, then this is the trigonometric interference and the interference term has the form (14).If |λ| ≥ 1, then this is the hyperbolic interference and the interference term can be represented as 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].

Conditional JPD and order effect
For observables A 1 , A 2 ∈ O, the conditional joint probability distribution (JPD) is defined by We remark that this is really a probability distribution, i.e., We can also define JPD for inverse order of measurements, 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].

Conditional Compatibility
In the absence of OE we have: i.e., In this case we call the observables conditionally compatible for context C ∈ C and their JPD is defined by (19).We remark that the marginals of JPD coincide with the probability distributions P A i C .In the von Neumann CMM M 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: 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 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:

Replicability and response replicability
Observable A shows replicability for context C, if or Observable A shows replicability if (25) holds for any 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 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.

Correlations and Bell type inequalities
Consider a pair of conditionally compatible observables A, B ∈ O (with X A , X B ⊂ R).Their correlation w.r.t. a context C ∈ C is defined as 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 The CHSH inequality has the form: i.e.,

|
x,y (31) If there exists a probability measure P C such that JPDs P A i ,B j C can be obtained as its marginals, e.g., 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 lefthand side w.r.t.contexts C ∈ C and observables

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 σ-algebra B of the Borel sets; it is generated by all semi-open intervals, (α Following Mackey [6] (Axiom 3), we assume that for each A ∈ O with the range of values R n and a Borel function 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 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.

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 = (Σ, I), where Σ = (C, O, P) is a contextual probability space and 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 ∈ C, the outcome B = β depends on the outcomes of A if for at least two values of A, α = α i , α j , the corresponding conditional probabilities don't coincide, Thus, the probability to get the outcome B = β if the preceding A-measurement had the outcome A = α i differs from the probability to get the same outcome B = β if the preceding A-measurement had the outcome A = α 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 = β does not depend on the outcomes of the observable A iff P C (B = β|A = α i ) = P C (B = β|A = α j ), for all pairs α i , α j , (34) i.e. the conditional probability for this outcome is constant w.r.t. the outcomes of A. Denote it P C (B = β|A).The following natural question arises: Does the probability P C (B = β|A) coincide with unconditional probability P C (B = β)?Definition 5. Two instruments I A and I B are called AB-entangled in C ∈ C or C is AB-entangled, if all outcomes of the B-observable depend on outcomes of A-observable, i.e. for all β condition (33) holds for some α i , α 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 premeasurement context C, AB-concurrence of conditional probabilities is defined as The crucial issue is that AB-concurrence depends on a pair of instruments.
Proposition 1.For dichotomous observables A, B = ±, 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 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 ABentanglement (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 QVN (section 4) with von Neuman observables [37]; in this CMM an observable Â, 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 = ρ.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.We are interested in sets Γ such that each of α and β values appears in the pairs once and only once.We call such EPR-entanglement complete.
For example, for two dichotomous observables with α, β = ±, we consider, e.g., the set of the pairs 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, λ AB (ψ) = 2, if and only if the instruments are EPR-entangled in pre-measurement context C.

Distinguishing features of contextual measurement models
We list the probabilistic constraints which can be used to distinguish different CMMs (theoretically and experimentally):

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 Σ = (C, O, P) cannot be described as the single Kolmogorov probability space: Σ is a bunch of such spaces However, fixed C ∈ C and A ∈ O can be realized within some probability space K = (Ω, F, P ) with realization of observable A by a random variable a : Ω → X A ; its probability distribution coincides with P A C , i.e., We note that a contextual probability space Σ can be represented as 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 Ω = Ω 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, x ≡ x C,A = (x 1 , ..., x N , ..., ...), where x j (∈ X A = {α 1 , ..., α m }) are measurement's outcomes.Such sequence determines the frequencies of realizations of the concrete values, where ν N (α j ) is the number of measurement's outcomes with the fixed value α j .The probability to obtain the value α j in a sequence of measurements x is defined as the limit 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 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 = α j ) ≡ P A C (α j ) can be considered as the probability generated by the collective x ≡ x C,A , i.e., P A C (α j ) = P x C,A (α j ).It is important to note that not all collectives are compatible -combinable in von Mises' terminology.Two collectives x and y, are combinable if their combination z = (z 1 , ..., z N , ...), z j = (x j , y j ), is also a collective and the probability distributions P x and P y are marginal for the probability distribution P z , i.e., P x (α j ) = The frequency probability theory contains the notion of conditional probability that is similar to CMM's conditioning, the postmeasurement context C A=a corresponds to the post-measurement collective 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 x, i.e., the limit-probability is the same for all subsequences of 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 "lightcollectives" 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.

Contextual Measurement Model for Kolmogorov Theory
Let K = (Ω, F, P ) be a Kolmogorov probability space [36].Here Ω is a set of any origin, F is a collection its subsets forming σ-algebra, i.e., F is closed w.r.t.countable unions and intersections, and the operation of complement.(If Ω is finite, then F is collection of all its subsets.)And P is a probability measure on F.
Set C = {C ∈ F : P (C) = 0}.This is the set of contexts.For each context C, the Bayes formula defines the conditional probability measure We highlight that the statistical mixtures of contexts are not determined, i.e., for subsets C 1 , C 2 of Ω and weights p 1 , p 2 ≥ 0, p 1 + p 2 = 1, there is no a subset C of Ω which can be identified with the weighted sum As an illustrative example, consider some agricultural region Ω 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 O is the set of (discrete) random variables, a : Ω → X a , where X a is a finite set.(Discrete random variables are considered for simplicity.)Denote this set by the symbol R d .For x ∈ X a , we set Ω a=x = {ω ∈ Ω : a(ω) = x}.Contextual probability coincides with conditional probability given by the Bayes formula: Thus, the set of probability distributions P = {P a C : C ∈ C, a ∈ O}.For any set D ∈ F and random variable a ∈ R d , x ∈ X a , we define set and the map For any context C ∈ C and random variable a ∈ R d , x ∈ X a , we define the family of contexts Each random variable a and its outcome x determine a map with the domain of definition C a (x).Thus, classical CMM M cl consists of measurement contexts composed of pre-measurement contexts -elements of F with non-zero probabilities, observables -(discrete) random variables, and context update maps, 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: Since, for each C ∈ 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, In this measurement model all observables are compatible, conditional JPD coincides with JPD (again by Bayes formula); no pair of observables show OE, since All observables show repeatability, since P C (a = x, a = x) = P C (a = x), and RRE, since 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 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 This problem can be easily resolved in the standard way, see below.
Let us consider a Kolmogorov probability space K = (Ω, F, P ) with a complete probability measure, i.e., any subset of a set D ∈ F, P (D) = 0, also belongs to F. We recall that the symmetric difference of two sets D 1 and D 2 is defined as This is an equivalence relation on F; it splits F into disjoint equivalence classes.Denote the set of equivalence classes by the symbol F; denote the equivalence class of zero probability sets by the symbol Z.The set of pre-measurement contexts is C = F \ Z, i.e., all equivalent classes of sets from F of non-zero measure.
We also modify the class of observables.Two random variables are equivalent, a 1 ∼ a 2 , if P (ω ∈ Ω : a 1 (ω) = a 2 (ω)) = 0.This is the equivalence relation on the space of random variables, in our consideration these are discrete random variables R d .So, R d is split into disjoint classes of equivalent random variables, denote the set of these classes by the symbol Rd and set Õ = Rd .
For a pre-measurement context C ∈ C and observable ã ∈ Rd we define the probability distribution P ã C (x) = P C (a = x) for some representatives C ∈ C and a ∈ Rd (the correctness of this definition is proved below), and set P = {P ã C }.The modified classical contextual probability space is the triple Σ = ( C, Rd , P).
The map T a (x), see (47), generates a map of F into itself  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 = R.First we note that if D 1 , D 2 ∈ D, then P (D 1 ) = P (D 2 ).We have . So, conditional probability measure P C does not depend on the choice of a representative C ∈ C and it can denoted as P C .
We show that implication (2) holds.Let for random variables a 1 , a 2 , P a 1 C = P a 2 C for any context C. Let, for some x, P (Ω a 1 =x ) > 0. Take C = Ω a 1 =x , then Hence, P (Ω a 1 =x \Ω a 2 =x ) = 0 and symmetrically P (Ω a 2 =x \Ω a 1 =x ) = 0.The sets Ω 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 a C 1 = P a C 2 .Select a as the characteristic function of the set C 1 .Then , contexts belong to the same equivalence class.

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 D, i.e., C = D, observables are Hermitian operators (von Neumann observables [1,37]).Denote the space of Hermitian operators by the symbol L H , i.e., O = L H .This real linear space is endowed the with scalar product Â| B = Tr Â B.
Operator Â ∈ L H has the spectral decomposition: Â = x∈X A x ÊA (x), where ÊA (x) is the orthogonal projection on the subspace H A (x) composed of eigenvectors with eigenvalue x, and X A is operator's spectral set.Then with the domain of definition We remark that observable A uniquely determines the family of maps T A (x), x ∈ X A by equality (53).So, measurement contexts can be represented by pairs (ρ, Â), ρ ∈ D, Â ∈ L H . Denote this CMM by the symbol M 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 QVN , the conditional probability is given by the formula: It can be rewritten as In this LSR-based CMM, it is convenient to introduce the maps: Then the above formulas can be rewritten as or and the conditional probability is written in the form and conditional JPD as These formulas lead to quantum instrument theory (section 5): (A, I A (x)) is a special quantum instrument and (A, T A (x)) is the corresponding contextual instrument.We note that in CMM M QVN probabilities determine contexts (states) and observables (operators), i.e., (2) and ( 3) hold (Mackey's axiom 2). Let for any x; so Â1 = Â2 .Thus, an observable can be identified with the set of probability distributions P A = {P A ρ : ρ ∈ D}.
Hence, ρ1 = ρ2 .Thus, a quantum state can be identified with the set of probability distributions In the von Neumann measurement theory, two observables are compatible if they are represented by commuting operators Â, B : [ Â, B] = 0. Compatibility is interpreted as guarantying the possibility of joint measurement of these observables; their JPD is given by the formula: 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 Tr ÊA (x) ÊB (y) ÊA (x)ρ = Tr ÊB (y) ÊA (x)ρ.
In particular, for compatible observables there is no OE for any state ρ.If operators Â1 , Â2 , do not commute, then there exists a state ρ showing OE for these observables, i.e., P ρ (A = x, B = y) = P ρ (B = y, A = x).
Each observable shows replicability, e.g., If observables are compatible, then for any ρ they show RRE, e.g., We highlight that it is impossible to combine OE and RRE within CMM M QVN [62].FTP (the formula of total probability) can be violated; classical FTP is additively perturbed by the interference term; consider a pure state |ψ , then Applications of the quantum instrument theory to quantum information are typically restricted by the use of atomic instruments.The space of Hermitian operators L H is the real Hilbert space, i.e., for each linear operator acting in L H (superoperator) its adjoint is well defined.The generalized Born rule can written as where I is the unit operator and I ⋆ (x) is superoperator that is adjoint to I(x) in Hilbert space L H . Hence, the generalized Born rule has the form: this is an additive operator-valued measure, i.e., for Instruments of the projection type, see (70), determine the special class of POVMs, projection valued measures (PVMs).Two POVMs A = ( Â(x), x ∈ X) and B = ( B(y), y ∈ Y ) are called compatible if there exists a POVM C = ( Ĉ(x, y), 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: In contextual probability space, contexts are mathematically represented by density operators (quantum states), C = D, observables by POVMs (also known as generalized quantum observables), the probability distributions are determined by the generalized Born rule (68).CMM M QI is endowed by quantum instrument maps updating quantum states (contexts) due to the measurement feedback (69).
Consider POVM Â = ( Â(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 QI = M QI;DL .In quantum information theory, one uses the special class of quantum instruments given by completely positive maps I(x); denote the corresponding CMM M 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 QI;DL \ M 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 I(x) are linear in the Hilbert space L H .In terms of context (state) update these maps they can be written as 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 I A (x) and I B (y) such that their observables are POVMs of PVM type, i.e., Â = ( ÊA (x)) and B = ( ÊB (y)).They are jointly measurable and the JPD is given by formula (62).The conditional JPD is given by P ρ (A = x, B = y) = P ρ (A = x)P ρ (B = y|A = x) = (79) The right hand sides of ( 62) and (79) coincide only if the instrument superoperators are of the projection type, i.e., I(x)ρ = Ê(x)ρ Ê(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 Ê(x) = I ⋆ (x).

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 M of all real valued measures on some set Ω with a σ-algebra of subsets F, i.e., M ≡ M(Ω, F).Real linear space M has the natural order structure and the positive cone M + consisting of non-negative measures.Consider the elements of this cone given by probability measures, i.e., µ ≥ 0 and µ(Ω) = 1; denote this set by the symbol S; this is the set of states; S is a convex subset of M. The latter is endowed with the variation norm, ||µ|| = var(µ) and it is a Banach space.Consider its dual space M ′ , the space of continuous linear functionals f : M → R. Denote by A the subset of M ′ , consisting of functionals mapping S into [0, 1].Elements of A are called effects, these are basic observables.They can be described solely in the terms of the state space S as affine functionals valued in [0, 1], i.e., A ≡ A(S).
Let X = {x 1 , .., x m } be a finite set and let A = (A(x i ), i = 1, ..., m) where A(x i ) ∈ A(S) and A(X) ≡ x∈X A(x) = u.Such vectors of functionals are analogs of POVMs; we call them M-POVMs.These are observables of the contextual probability space Σ measure with contexts C = S and the set of probability distributions P defined as As we learn from the quantum instrument theory, the basic elements of measurement procedures are not observables, but instruments.Let L(M) denote the space of continuous linear operators, J : M → M. The M-instrument with the range of values X is a map Each instrument determines the state update map and the probability distribution The domain of definition of the state update map T A (x) is given by the set of probability measures C A (x) = {µ ∈ S : P µ (A = x) > 0}.
Let J : M → M be a continuous linear operator.Then its adjoint operator J ⋆ is well defined, J ⋆ : M ′ → M ′ , and Then, for each x, A(x) is an effect, i.e.A(x) ∈ A(S).So, each Minstrument determines a M-POVM.
The M-CMM consists of context (states) given by probability measures and POVM-observables with state updates given by instruments.

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 ∈ S. A probability p(x, s) is assigned to any possible outcome x ∈ X.Thus, we have a function p : X × S → [0, 1].
To each outcome x ∈ X and state s ∈ 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(•, •), X).Denote by Φ [0,1] the space of function from X to [0, 1].By considering state s as a variable, we obtain the map It is natural to assume that each state s ∈ S determines the probability distribution uniquely, i.e., Under this assumption the map (80) is injection.Thus, each state s can be mapped to a function belonging to space Φ [0,1] , it will be denoted by the same symbol s.Consider now the vector space Φ of all real valued functions on X.So, S is identified with a subset of this functional space.Consider its closed convex hull S. The vectors from it are all possible probabilistic mixtures (convex combinations) of states in S. Each x ∈ X defines a linear functional on Φ, φ → f x (φ) = φ(x).If φ = s ∈ S, then f x (s) = s(x) ∈ [0, 1], i.e., f x : S → [0, 1].This is an affine functional on the convex set 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( S) to the space of all affine functionals In particular, for any x ∈ X, f x ∈ A( S).Any functional f ∈ A( S) describes an outcomes of some observable, and thus f (s) is the probability for that outcome in state s.
In QM, S is the set of density operators and elements of A( S) are effects -components of POVMs, f (ρ) = Tr Êf ρ, where Êf is the effect corresponding to the affine functional f.
The elements of A( S) are called effects.It is typically assumed that there exists an element u of A( S) such that u(s) = 1 for any s ∈ S. It is an analog of quantum observable given by the unit operator I. Consider the point wise order structure on A( S), f ≤ g iff f (s) ≤ g(s) for any state s.Thus, any observable f ∈ A( S) is majorated by u, 0 ≤ f ≤ u.A discrete measurement is represented by a set of effects (f i ) such that 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 = C × O. Let, as above, Φ [0,1] denotes the space of functions from X to [0, 1].We map S into Φ [0,1] , s = (C, A) → P A C .However, generally this map is not injection: So, such straightforward construction seems to be non-proper for our aim.
We modify it by setting X = O × X, the elements of X are pairs x =(observable, outcome)= (A, x).We now use the symbols Φ [0,1] and Φ for functions from X → [0, 1] and to real line respectively.Each context C can be represented as a vector belonging to Φ

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.

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 C contains contexts generated by statistical mixtures (see Axiom 4 in Mackey's book [6]), i.e., for C 1 , C 2 ∈ C and p 1 , p 2 ≥ 0, p 1 + p 2 = 1, the set 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 (σ-)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]. 4 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 ?? 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.
This automatically implies that even the value B = + depends on the outcomes of A, P C (B = +|A = +) = 1 − P C (B = −|A = +) = 1 − P C (B = −|A = −) = P C (B = +|A = −), i.e., instruments I A and I B are entangled in context C.

Definition 7 .
For C ∈ C, instruments I A and I B are perfectly conditionally correlated for values (A = α, B = β) if the conditional probability to get the outcome B = β if the preceding A-measurement had the outcome A = α equals to 1,P C (B = β|A = α) = 1.(37)More generally consider observables with values (α i ) and (β i ) and some set Γ of pairs (α i , β j ).Definition 7a.(EPR-entanglement) Let C ∈ C. If instruments I A and I B are perfectly conditionally correlated for all pairs belonging to Γ, then they are called EPR-entangled w.r.t.set G in context C.

Set
Ca (x) = { C ∈ C : Ta (x) C ∈ C}.Then Ta (x) : Ca (x) → C. A measurement context is a triple ( C, ã, Ta ).Modified classical CMM Mcl is given by the set of such measurement contexts.Now we demonstrate that in Mcl the uniqueness conditions ( where Â(x) = I ⋆ (x)I.(75) Operators Â(x), x ∈ X, are called effects; they are positive semidefinite Hermitian and sum up to the unit operator: x∈X Â(x) = I.The family of operators A = ( Â(x), x ∈ X) is called a positive operator valued measure (POVM): for a subset ∆ of X, we set Â(∆) = x∈∆ Â(x) ≥ 0; x). Due to (3), embedding of the set of contexts C into Φ [0,1] is injection.Again denote by C the convex hull of C. Each point x =(observable, outcome)= (A, x) determines the affine functional C → f A,x (C) = P A C (x) ∈ [0, 1].Now fix A ∈ O and consider the familyof functionals F = (F A (x) = f A,x : x ∈ X).This is representation of observable A. So, any contextual probability model can be realized alike COMan observational COM.
1] can approach the value four, max CHSH = 4.This maximum depends on CMM.For von Neumann CMM M QVN , this is max CHSH = 2 √ 2.
OE+RRE -generally no, but can be realized by special instruments• violation of Bell inequalities -yesIt is interesting to find a property distinguishing M QI;DL and M QI;O via an experimental test, i.e., some experimentally testable property such that only completely positive instruments have it.