Contextual measurement model and quantum theory — TEI-tagged view

Andrei KhrennikovpersName (2024date). “Contextual measurement model and quantum theory.” Royal Society Open Sciencetitle·j.

Object of study — a contextual measurement modelterm·mth built on a contextual probability spaceterm·mth, covering classical, quantum and semi-classical measurement

Research question — how far quantum probabilityterm·dom can be framed as contextual probabilityterm·key update, without Hilbert-space structure

Contribution — the contextual measurement modelterm·mth, presented as the most general probabilistic framework for measurement

Full text — tagged

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Contextual measurement modelterm·mth and quantum theoryterm·mth Abstract We develop a contextual measurement modelterm·mth (CMMterm·mth) that is used for the clarification of the quantum foundations. This model matches Bohr’s views on the role of experimental contextsterm·dom. CMMterm·mth is based on a contextual probabilityterm·key theoryterm·mth that is connected with generalized probability theoryterm·mth. CMMterm·mth covers measurements in classical, quantum and semi-classical physics. The CMMterm·mth formalism is illustrated by a few examples. We consider the CMMterm·mth framing of classical probability, the von Neumann measurement theoryterm·mth and the quantum instrumentterm·key theoryterm·mth. CMMterm·mth can also be applied outside of physics, e.g. in cognition, decision-making and psychology, the so-called quantum-like modellingterm·key.

Introduction The interrelation of quantum and classical probability theories is a very complex foundational issue involving interpretational, mathematical and philosophical questions. Research in this area is characterized by the diversity of views, opinions and mathematical formalisms (e.g. [ 1 – 24 ]). We remark that, generally, quantum mechanicsterm·dom (QMterm·dom) is characterized by a diversity of interpretations.

My own understanding is that quantum probabilityterm·dom is a machinery for probability update, analogous to classical Bayesian inferenceterm·mth [ 25 – 34 ]. In contrast to the latter, quantum probabilityterm·dom inference is not based on the Bayes formula for conditional probabilityterm·key. Quantum probabilityterm·dom theoryterm·mth is a theoryterm·mth of probability inference with a special class of probability update transformations given by projections or quantum instrumentsterm·key. It is natural to create a general probabilistic framework that covers both the classical and quantum ones. Such generalization can come with 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 that otherwise look totally separated. In this way, it is easier to find similarities and differences in district plans and the architecture of buildings. As just one of the possible machines for probability update, the quantum probabilistic formalism would lose its mystery.

One such ‘panoramic framework’ is the contextual measurement modelterm·mth (CMMterm·mth) based on contextual probability spaceterm·mth. Its development was initiated in [ 25 ], continued in a series of authors’ works (e.g. [ 26 – 28 ]) and summarized in a monograph [ 29 ]. In these studies, the main emphasis was on the modification of the formula of total probabilityterm·mth (FTPterm·mth)—its transformation into FTPterm·mth with an interference term, expressing interference of probabilities, e.g. in the two-slit experiment. In my previous studies, the contextual probabilityterm·key approach was partially shadowed by an appeal to von Mises’ frequency theoryterm·mth of probability [ 22 , 30 ] and the realization of experimental contextsterm·dom as von Mises collectives.

In this paper, CMMterm·mth’s development is continued towards the abstract contextual formalization of other basic features of quantum probabilityterm·dom, such as order and response replicability effectsterm·dom (RREsterm·dom) in sequential measurements, entanglementterm·key, the violation of the Bell inequalitiesterm·key and establishing coupling with quantum instrumentsterm·key theoryterm·mth as well as with linear space representationterm·mth (LSRterm·mth) of generalized probability theoryterm·mth.

CMMterm·mth is the basis of the Växjö interpretationterm·key of QMterm·dom [ 25 , 29 , 31 , 32 ]—one of the contextual probabilistic interpretations. Since the probability update is also an information update, the Växjö interpretationterm·key is part of the information interpretation of QMterm·dom. This paper presents CMMterm·mth consistently in the most general form by highlighting its basic properties, such as interference of probabilities, order effectterm·dom (OEterm·dom), entanglementterm·key and violation of the Bell inequalitiesterm·key.

The abstract CMMterm·mth formalism is illustrated by a few examples. We start with the CMMterm·mth framing of classical probability theoryterm·mth [ 35 , 36 ], which serves as the basis of classical statistical physics and thermodynamics. Then, we consider the von Neumann [ 1 , 37 ] quantum measurement theoryterm·mth with observables given by Hermitian operators and the state update of the projective type and represent it as CMMterm·mth. The quantum instrumentterm·key theoryterm·mth is the generalization of the von Neumann theoryterm·mth permitting state updates of the non-projective type, and it can also be represented as CMMterm·mth. We also show the connection of generalized probability theoryterm·mth with the state space consisting of probability measures with CMMterm·mth. Finally, LSRterm·mth for contextual probability spaceterm·mth is constructed using the construction dating back to Mackey.

CMMterm·mth can also be applied outside of physics, in so-called quantum-like modellingterm·key (e.g. [ 38 , 39 ]). This is a rapidly developing area of research stimulated by the recent quantum information revolution. In quantum-like modellingterm·key, the quantum methodology is applied to cognition, decision-making, psychology, game theoryterm·mth, economics and finance, and AI. Universally, quantum-like models need not be based on the complex Hilbert space formalism. They can employ other contextual probabilityterm·key calculi and CMMsterm·mth [ 38 ].

Contextuality of probability From the mathematical viewpoint, the essence of the problem lies in generalizations of conditional probabilityterm·key and probability update. On the way to such rethinking of the interrelation between the classical and quantum probabilityterm·dom theories, I was strongly influenced by Ballentine who treated all quantum probabilities as conditional probabilities [ 14 , 15 , 18 – 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 the 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 ] ([ 36 ]) contains the statement that, in modern words, can be formulated as a statement about the contextuality of probability . Kolmogorov’s message is that it is meaningless to speak about probability without determining a complex of experimental conditions and measurement contextterm·dom. This Kolmogorov position matches well with Bohr’s statement on the contextuality of measurement outcomes that is the cornerstone of his complementarity principle [ 40 ] (which is better to call the contextuality–complementarity principle [ 41 , 42 ]).

Unfortunately, Kolmogorov’s original message on the contextuality of probability was practically ignored in the further development of classical probability theoryterm·mth. A mathematical work on probability is typically started by fixing one probability measure, without mentioning that it corresponds to some measurement contextterm·dom. The contextuality component of Bohr’s statements on complementarity was not emphasized in quantum foundational research; typically, the Bohr complementarity principle is reduced to the wave–particle duality.

Following Bohr [ 40 ], Kolmogorov [ 35 , 36 ], Koopman [ 4 ], Accardi [ 14 , 15 ] and Ballentine [ 17 – 20 ], I introduced the contextual probability spaceterm·mth and CMMterm·mth based on it [ 29 ]. The main idea behind this approach is to operate solely with contextsterm·dom and observables and to exclude physical systems from consideration.

A measurement contextterm·dom consists of a pre-measurement contextterm·dom C , an observable A and a post-measurement contextterm·dom 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 contextsterm·dom. Alike theoryterm·mth of quantum instrumentsterm·key, we call the pair I A = ( A , T A ) a contextual instrumentterm·mth. The latter is the basic mathematical component of measurement theoryterm·mth. It is meaningless to formulate it solely in terms of observables. The same observable A can be a component of various instruments describing different measurement procedures for A . So, an observable is a theoretical quantity expressing some features of pre-measurement contextsterm·dom.

In QMterm·dom, one operates with the notion of ‘state’ not ‘contextterm·dom’. These notions are similar but have some inetrpretational differences (see appendix A for the discussion).

We also mention Feynman’s contextual analysis of the two-slit experiment in [ 2 , 3 ]. He presented the purely probabilistic picture of this fundamental experiment of QMterm·dom 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 probabilityterm·key leads to FTPterm·mth, which plays an important role in probability inference. Following this line, Feynman’s conclusion can be rewritten as the violation of classical FTPterm·mth. The difference between classical and quantum probabilityterm·dom models can be moved from the violation of additivity of probability to the violation of the Bayes formula for conditional probabilityterm·key—quantum probabilityterm·dom is additive, but conditioning is not Bayesian. The quantum FTPterm·mth is a perturbation of classical FTPterm·mth 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 contextsterm·dom C 1 , C 2 , C 12 : the first slit is open and the second is closed, vice versa, and both slits are open. Quantum probabilistic specialty is expressed not via LSRterm·mth of states and observables but in a 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 the contextuality of the joint measurement of a few quantum observables. This sort of contextuality was considered by Bell in [ 43 , 44 ] in his analysis of the violation of the Bell inequalitiesterm·key (although he did not use the term ‘contextuality’; it seems that this term was introduced in the book of Beltrametti and Cassinelli [ 45 ]).

Feynman’s contextuality [ 2 , 3 ] is more general and, in fact, coincides with Bohr’s contextuality [ 40 ]. In my works including those on the Växjö interpretationterm·key, I followed Bohr and Feynman: contextterm·dom as a complex of all physical conditions involved in an experiment.

Linear space versus contextual frameworks for probability Our foundational pathway is opposite to the pathway leading to the generalized probability theories [ 5 – 11 , 21 , 45 ] (see [ 24 ] for a review), which is directed towards the creation of general LSRterm·mth of probability and the measurement process. The LRS also provides a panoramic view that covers both classical and quantum probabilities and observables. This is the linear panorama that illuminates the place of the quantum probabilityterm·dom and measurement formalism among other linear models.

We remark that there are several different approaches to generalized probability theoryterm·mth and corresponding measurement theoryterm·mth, but all of them are either equivalent or only slightly different. One of them is the Davies–Lewis [ 7 ] operational probability theoryterm·mth grounded on LRS with the base norm spaces, a class of partially ordered linear spaces. The corresponding measurement theoryterm·mth is formulated within instrument theoryterm·mth [ 7 , 8 , 11 , 12 ]; in particular, observables are mathematically represented as positive operator-valued measures (POVMs). They are widely used in quantum information theoryterm·mth [ 46 – 52 ] (also see [ 39 , 53 ] 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 a 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 §6, we explore the Davies–Lewis approach for operational LRS of the measurement model with states given by classical probability measures. Then, we express this model in the form of CMMterm·mth.

The closest to CMMterm·mth is the model in which one starts with all possible probabilities that can be generated in an experiment, conditioned on preparation and measurement procedures [ 6 ]. Then, one proceeds to LSRterm·mth. This construction can be employed to construct LSRterm·mth for CMMterm·mth (§7). However, this is done just to show the connection with previously developed theories. 1 CMMterm·mth development is important for quantum foundations. CMMterm·mth demystifies quantum theoryterm·mth by reducing its probabilistic counterpart to a tool for probability update and inference (cf. with QBism [ 54 – 57 ]); CMMterm·mth diminishes the role of pure states, in complete agreement with the statistical (ensemble) interpretation of QMterm·dom; in CMMterm·mth, quantum interference is just an additive perturbation of classical FTPterm·mth due to the interplay between a few measurement contextsterm·dom; the violation of the Bell inequality has the same origin; contextual entanglementterm·key is naturally coupled to classical dependence of random variables. The latter demystifies entanglementterm·key. This is very important for the resolution of the century-long debate on quantum non-locality; see appendix B for a comparison of CMMsterm·mth with and without LSRsterm·mth.

On the other hand, LSRterm·mth is very convenient from a mathematical viewpoint (simply linear algebra), and operating within the LSRterm·mth framework is useful in concrete mathematical calculations. However, the calculations should be completed by the critical analysis on the connection of the mathematical LSRterm·mth constructions with physics. In quantum-like modellingterm·key, a similar problem arises—the problem of matching between the output of the Hilbert space formalism and some psychological effects in decision-making [ 58 – 62 ].

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

Contextual measurement modelterm·mth Contextual probability spaceterm·mth Definition 2.1 . A contextual probability spaceterm·mth is a triple Σ = ( C , O , P ) , where C and O are sets of pre-measurement contextsterm·dom and observables and P is the space of the corresponding probability distributions.

In physics, pre-measurement contextsterm·dom 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 contextterm·dom C , denote the probability of an outcome x ∈ X A as P C A ( x ) ≡ P C ( A = x ) . By the definition of a probability distribution, The range set X A can be endowed with the algebra of all its subsets F A . We set P C A ( G ) = ∑ x ∈ G P C A ( x ) , G ∈ F A . 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 [ 6 ]). Elements of P are called contextual probabilities. These are the analogues of the conditional probabilities in the classical (Kolmogorov [ 36 ]) probability model. But, we reserve the term ‘conditional probabilityterm·key’ for a special class of contextual probabilities generated by contextterm·dom updates.

It is natural to assume (see Axiom 2 in [ 6 ]) that two observables having the same probability distribution for all contextsterm·dom should coincide, i.e.

We also assume (see Axiom 2 in [ 6 ]) that two contextsterm·dom having the same probability distribution for all observables should coincide, i.e.

The average of an observable A ∈ O (with X A ⊂ R ) with respect to a pre-measurement contextterm·dom C ∈ C is defined as Contextterm·dom update and conditional probabilityterm·key Measurement of an observable A with the concrete outcome x in a pre-measurement contextterm·dom C updates this pre-measurement contextterm·dom: In terms of preparation procedures, we can consider a measurement as a subsequent preparation procedure; contextterm·dom C A = x is the measurement of A and filtering with respect to the fixed outcome x . 3 It is natural to consider this map only for contextsterm·dom belonging to the set If P C ( A = x ) = 0 , then the post-measurement contextterm·dom 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 theoryterm·mth is that generally an observable does not determine the contextterm·dom update map unequally. An observable A can be measured via different measurement procedures, and each procedure generates its own contextterm·dom update map. A pair I A = (observable, contextterm·dom update map) = ( A , T A ) is called a contextual instrumentterm·mth (cf. §5); a pair ( C , I A ) or a triple ( C , A , T A ) is called a measurement contextterm·dom . We stress once again that a variety of instruments can be associated with the same observable A : I A = ( A , T A ) , I A ′ = ( A , T A ′ ) , I A ′ ′ = ( A , T A ′ ′ ) , . . . . . We stress that all these update maps have the same domain of definition determined by the observer A ; see equation (2.6).

Typically, one fixes some class of contextterm·dom update maps. In the von Neumann [ 1 , 37 ] measurement theoryterm·mth (§4), this is the class of normalized projections. In quantum instrumentterm·key theoryterm·mth (§5), these are quantum channels, or more generally, in the theoryterm·mth of Davis–Levis instruments, these are positive trace-preserving maps.

We emphasize that von Neumann’s measurement theoryterm·mth is very special: here, by fixing an observable and a Hermitian operator 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 2.2 . Let Σ = ( C , O , P ) be a contextual probability spaceterm·mth. A CMMterm·mth is a pair M = ( Σ , I ) , where Σ is a contextual probability spaceterm·mth and I is a collection of contextual instrumentsterm·mth.

CMMterm·mth is a set of measurement contextsterm·dom, i.e. triples ( C , A , T A ) . CMMterm·mth generates the notion of the conditional probabilityterm·key: Definition 2.3 . Consider a measurement contextterm·dom ( C , A , T A ) . Let it generate the output A = x and the corresponding contextterm·dom update, C → C A = x = T A ( x ) C . Consider the measurement of another observable B under the condition A = x , i.e. with respect to the contextterm·dom C A = x . The conditional probabilityterm·key is given by the formula We note that this definition involves contextterm·dom update only for the A observable; different contextual instrumentsterm·mth 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 probabilityterm·mth Now, we point out that generally, contextual probabilityterm·key 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 FTPterm·mth: In classical probability theoryterm·mth, contextual probabilityterm·key is identified with the conditional one (§3), and the contextual-conditional analogue of FTPterm·mth has the form see equation (3.4). However, in a general contextual probability spaceterm·mth, this formula can be violated: The difference between the left-hand side (l.h.s.) and right-hand side (r.h.s.) determines the degree of contextterm·dom 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 FTPterm·mth (with an interference term): The equality of the interference term to zero is a necessary condition of the classical probabilistic representation of a contextual probabilityterm·key model, but it is not a sufficient condition [ 29 ].

Let us, for the moment, jump to §4, where the von Neumann quantum measurement model is treated as CMMterm·mth. Consider dichotomous observables A = x 1 , x 2 and B = y 1 , y 2 . In this case, the interference term has the form where contextterm·dom C is identified with the quantum state ψ (for simplicity, consider contextsterm·dom corresponding to pure states) and the angle θ = θ ( B = y | A ; ψ ) . For dichotomous observables, even in general CMMterm·mth, 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 of equation (2.14). 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 instrumentsterm·key [ 63 ]. In general CMMterm·mth, we can employ the hyperbolic version of QMterm·dom [ 29 ].

Conditional joint probability distribution and order effectterm·dom 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. ∑ 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 OEterm·dom with respect to contextterm·dom C , if for at least one pair of outcomes ( x 1 , x 2 ) ; otherwise, there is no OEterm·dom in contextterm·dom C .

We remark that OEterm·dom was actively investigated in decision-making and psychology, both theoretically and experimentally; in particular, within quantum-like modellingterm·key—the applications of quantum methodology and formalism to decision-making and psychology [ 58 – 62 ].

Conditional compatibility In the absence of OEterm·dom, we have i.e.

In this case, we call the observables conditionally compatible for contextterm·dom C ∈ C , and their JPD is defined by equation (2.19). We remark that the marginals of JPD coincide with the probability distributions P C A i .

In the von Neumann CMMterm·mth M Q V N (§4) with observables and state updates of the projection type, conditional compatibility for all possible pre-measurement contextsterm·dom (given by density operators) is equivalent to the 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 versa. We remark that, in fact, precisely, this experimental setup is realized in the Bell experiments. Here, the instances of time for the 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 versa, and the time window serves for click pairing.

Conditional compatibility implies the Bayes formula for the conditional probabilityterm·key: The equality given in equation (2.20) implies the Bayes theorem for probability inference. Let the outcomes of the observable A 2 label some hypotheses, H 1 , . . . , H m . Then, equation (2.20) is written as The Bayes formula for conditional probabilityterm·key equations (2.21) and (2.22) implies the validity of the classical FTPterm·mth, i.e. the Bayes theorem can be written in the standard form: Replicability and response replicability Observable A shows replicability for the contextterm·dom C , if Observable A shows replicability if equation (2.25) holds for any C ∈ C A ( x ) , x ∈ X A .

In quantum-like modellingterm·key, the following effect plays an important role. Observables A 1 and A 2 show RREterm·dom with respect to contextterm·dom C , if for all pairs of outcomes ( x 1 , x 2 ) . This is a kind of memory effect. The challenging problem for quantum-like modellingterm·key was the combination of OEterm·dom and RREterm·dom [ 60 ]. It was solved in articles [ 61 , 62 ] within quantum instrumentterm·key theoryterm·mth.

Correlations and Bell-type inequalities Consider a pair of conditionally compatible observables A , B ∈ O (with X A , X B ⊂ R ) . Their correlation with respect to a contextterm·dom C ∈ C is defined as The most popular Bell-type inequality is the CHSH inequality. We consider this inequality within CMMterm·mth. 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 contextterm·dom C with JPDs P C A i , B j . The CHSH inequality has the form i.e.

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.

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 l.h.s. with respect to contextsterm·dom C ∈ C and observables A 1 , A 2 , B 1 , B 2 ∈ O valued in [ − 1 , 1 ] can approach the value of 4, max C H S H = 4. This maximum depends on CMMterm·mth. For von Neumann CMMterm·mth M Q V N , this is max C H S H = 2 2 .

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, which is generated by all semi-open intervals, ( α 1 , β 1 ] × ⋯ × ( α n , β n ] . So, P C A is a probability measure on B . A function f : R n → R m is called B -measurable if for any Borel subset G of R m its preimage f − 1 ( G ) is a Borel subset of R n . Only such functions are considered.

Following Mackey [ 6 ] (Axiom 3), we assume that for each A ∈ O with the range of values R n and a Borel function f : R n → R m , there exists B = B f ∈ O such that, for any C ∈ C and Borel set G ⊂ R m , 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 CMMterm·mth M Q V N (§4), functional compatibility is equivalent to compatibility and, hence, conditional compatibility. Generally, in CMMterm·mth, the interrelation between these two notions is complex, and we shall not proceed to a detailed comparison.

Entanglementterm·key of contextual instrumentsterm·mth Entanglementterm·key is typically considered as one of the distinguishing features of LSRterm·mth; from my viewpoint, the association of entanglementterm·key with LSRterm·mth and the tensor product structures shadow its physical nature; its mathematical description is identified with physics. As was shown in the articles [ 64 , 65 ], the notion of entanglementterm·key can be formalized in the purely probabilistic framework and dissociated from the tensor product and generally from LSRterm·mth.

By starting with such a probabilistic approach to the notion of entanglementterm·key, the authors of [ 64 , 65 ] proceed towards its complex Hilbert space realization. Now, we present the purely probabilistic picture of entanglementterm·key. The main value of the contextual probabilistic realization of entanglementterm·key is in the clarification of its foundational meaning. At the same time, the use of LSRterm·mth can essentially simplify concrete calculations. However, one should be careful with the connection of the mathematical structures of LSRterm·mth with physics (or in quantum-like modellingterm·key such as psychology and decision-making).

Consider CMMterm·mth M = ( Σ , I ) , where Σ = ( C , O , P ) is a contextual probability spaceterm·mth and I is a collection of contextual instrumentsterm·mth of this model, i.e. pairs (observable, state update map). Consider two contextual instrumentsterm·mth I A = ( A , T A ) and I B = ( B , T B ) .

Definition 2.4 . In pre-measurement contextterm·dom 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 do not 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 . For symmetry reason, we consider two instruments.

We note that the outcome B = β does not depend on the outcomes of the observable A if i.e. the conditional probabilityterm·key for this outcome is constant with respect to 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 the unconditional probability P C ( B = β ) ?

Definition 2.5 . Two instruments I A and I B are called A B -entangled in C ∈ C or C is A B -entangled, if all outcomes of the B -observable depend on outcomes of the A -observable, i.e. for all β condition (2.33) holds for some α i , α j .

Concerning the notation ‘ A B -entangled’, it would be better to write ‘ I A I B -entangled’ to emphasize that this is the entanglementterm·key of instruments and not simply the observables but to make the notation compact, we proceed with ‘ A B -entangled’. The order of observables is important. Generally, A B -entanglement does not imply B A -entanglement. This is a purely probabilistic definition that does not involve LSRterm·mth and can be applied to any statistical physical theoryterm·mth. This definition formalizes the dependence of observables. We introduce the following quantitative measure of entanglementterm·key.

Definition 2.6 . For contextual instrumentsterm·mth I A and I B and pre-measurement contextterm·dom C , A B -concurrence of conditional probabilities is defined as The crucial issue is that A B -concurrence depends on a pair of instruments.

Proposition 2.7 . 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 the A B -entanglement .

Proof . In the state contextterm·dom C , the value B = − depends on the outcomes of A if This automatically implies that even the value B = + depends on the outcomes of A , i.e. instruments I A and I B are entangled in the contextterm·dom C .

As was already pointed out, in articles [ 64 , 65 ], contextual probabilistic entanglementterm·key can be realized in the complex Hilbert space and, in this way, connected with the ordinary notion of entanglementterm·key. In the LSRterm·mth representation, the main distinguishing feature of A B -entanglement (definition 2.5) is that it is associated with a pair of instruments, I A , I B . The standard definition of entanglementterm·key is coupled with the tensor product structure and not with two concrete instruments (observables).

For simplicity, let us consider CMMterm·mth M Q V N (§4) with von Neuman observables [ 37 ]; in this CMMterm·mth, an observable A ^ , a Hermitian operator, automatically determines the state update map through its spectral family. So, there is no need to operate with instruments; one can solely operate with observables. In this CMMterm·mth (which is typically used in entanglementterm·key studies), contextual probabilistic entanglementterm·key is associated with the pairs of observables, i.e. two observables are entangled or disentangled in some state (contextterm·dom) C = ρ ^ , where ρ ^ denotes a density operator. The main mathematical features of A B -entanglement and ordinary tensor product-based entanglementterm·key are similar, but some essential differences can be found [ 64 , 65 ].

The probabilistic viewpoint on the ‘EPR-paradox’ [ 66 ] is presented in Schrödinger’s papers [ 67 , 68 ], which initiated the modern theoryterm·mth of entanglementterm·key. However, this theoryterm·mth ignores the important message of Schrödinger: entanglementterm·key characterizes the probability update for the outcomes of observable B conditioned on the outcomes of observable A . In the framework of [ 67 , 68 ], it is meaningless to speak about the entanglementterm·key without specifying the observables. The state update—the Hilbert space representation of the probability update—encodes the procedure of conditional prediction. For Schrödinger, quantum formalism is a mathematical machinery for probability prediction (as in the Växjö interpretationterm·key 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 entanglementterm·key as contextual probabilistic entanglementterm·key. However, this is my private interpretation of Schrödinger’s views, and many experts in quantum foundations may disagree with me.

By following Schrödinger [ 67 , 68 ], in article [ 64 ], we considered a special sort of contextual probabilistic entanglementterm·key that matches perfectly with the Schrödinger analysis of the EPR argument.

Definition 2.8 . For C ∈ C , instruments I A and I B are perfectly conditionally correlated for values ( A = α , B = β ) if the conditional probabilityterm·key to get the outcome B = β and if the preceding A -measurement had the outcome A = α equal to 1: More generally, consider observables with values ( α i ) and ( β i ) and some set Γ of pairs ( α i , β j ) .

Definition 2.8a (EPR entanglementterm·key). 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 with respect to set G in the contextterm·dom C .

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 entanglementterm·key complete.

For example, for two dichotomous observables with α , β = ± , we consider, e.g. the set of the pairs ( A = + , B = − ) , ( A = − , B = + ) , in short, A = − B EPR entanglementterm·key or the pairs ( A = + , B = + ) , ( A = − , B = − ) , A = B EPR entanglementterm·key. We analyse such EPR entanglementsterm·key.

Let us start with A = − B EPR entanglementterm·key, 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 ≠ P C ( B = − | A = − ) = 0 and P C ( B = + | A = − ) = 1 ≠ P C ( B = + | A = + ) = 0. In this case, EPR-entangled instruments are automatically entangled in the sense of definition 2.5.

Now turn to A = B EPR entanglementterm·key, 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 ≠ P C ( B = + | A = − ) = 0 and P C ( B = − | A = − ) = 1 ≠ P C ( B = − | A = + ) = 0. And again, EPR-entangled instruments are automatically entangled in the sense of definition 2.5.

Thus, EPR entanglementterm·key is just a very special case of A B -entanglement.

For dichotomous observables, the A B -concurrence of conditional probabilities, equation (2.35), has the form C ≀ ∖ A B ( ψ ) = | P ( B = + | A = − ) − P ( B = + | A = + ) | + | P ( B = − | A = − ) − P ( B = − | A = + ) | , and hence it can be written as From this formula, we immediately obtain the following characterization of maximally A B -entangled states: Proposition 2.9 . A B -concurrence of conditional probabilities approaches its maximal value, λ A B ( ψ ) = 2 , if and only if the instruments are EPR-entangled in the pre-measurement contextterm·dom C .

Distinguishing features of contextual measurement modelsterm·mth We list the probabilistic constraints that can be used to distinguish different CMMsterm·mth (theoretically and experimentally): – violation of FTPterm·mth – OEterm·dom – RREterm·dom – OEterm·dom+RREterm·dom – violation of Bell inequalitiesterm·key Interpretations of contextual probabilityterm·key Probability is characterized by the diversity of interpretations [ 22 ]. We now discuss the interpretations of contextual probabilityterm·key. We start with the remark that mathematically, a contextual probability spaceterm·mth Σ = ( C , O , P ) cannot be described as 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 the realization of observable A by a random variable a : Ω → X A ; its probability distribution coincides with P C A , i.e.

We note that a contextual probability spaceterm·mth Σ can be represented as where K C , A is a Kolmogorov probability space for describing the measurement of the observable A in the pre-measurement contextterm·dom C .

Therefore, one can assign any interpretation used for probability defined in the measure-theoretic framework to the contextual probabilityterm·key. The main interpretation employed in classical and quantum physics is the frequency interpretation. In the Kolmogorov theoryterm·mth [ 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 the contextual probabilityterm·key, we need not represent a pre-measurement contextterm·dom C by an ensemble of systems. Instead, we can consider a sequence of measurements of an observable A in the same pre-measurement contextterm·dom C , where x j ( ∈ X A = { α 1 , . . . , α m } ) are the measurement’s outcomes. Such sequence determines the frequencies of realizations of the concrete values, where ν N ( α j ) is the number of the 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 theoryterm·mth of frequency probability was developed by von Mises [ 30 ] (see also [ 22 ] for an introduction). A sequence x generated by observations is called a collective . Von Mises’ theoryterm·mth is a theoryterm·mth of collectives. Instead of operations on sets, as done in the Kolmogorov measure-theoretic theoryterm·mth, von Mises constructed a probability theoryterm·mth based on operations with collectives. We remark that P C ( A = α j ) ≡ P C A ( α j ) can be considered as the probability generated by the collective x ≡ x C , A , i.e. P C A ( α j ) = P x C , A ( α j ) . It is important to note that not all collectives are compatible or combinable in von Mises’ terminology. Two collectives x and y are combinable if their combination is also a collective, and the probability distributions P x and P y are marginal for the probability distribution P z , i.e.

The frequency probability theoryterm·mth contains the notion of conditional probabilityterm·key that is similar to CMMterm·mth’s conditioning, and the post-measurement contextterm·dom C A = a corresponds to the post-measurement collective x C , A = a (see [ 22 , 30 ] for details).

In contrast to the Kolmogorov measure-theoretic probability theoryterm·mth, in the von Mises frequency probability theoryterm·mth, classical FTPterm·mth is violated [ 22 , 29 ], and the probabilities can interfere and generate the additive perturbation of FTPterm·mth 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 a collective was the seed for the future growing theoryterm·mth of random sequences. Besides the existence of the limits for frequencies, equation (2.42), a collective is characterized by the stability of these limits with respect to 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 theoryterm·mth of randomness (e.g. [ 69 ]). In particular, the monograph presents a ‘light version’ of the von Mises theoryterm·mth. In physics (at least in quantum physics), one does not analyse in the random structure of the sequences of the measurement’s outcomes. In principle, in QMterm·dom, one can proceed with ‘light-collectives’ determined solely by the existence of limits (2.42). The calculus of such ‘light-collectives’ can be explored for the description of the probabilistic structure of QMterm·dom [ 69 ]. The first version of the contextual probabilityterm·key was presented in such a ‘light-frequency’ framework.

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

Contextual measurement modelterm·mth for Kolmogorov theoryterm·mth Let K = ( Ω , F , P ) be a Kolmogorov probability space [ 36 ]. Here, Ω is a set of any origin, F is a collection of its subsets forming σ -algebra, i.e. F is closed with respect to countable unions and intersections and the operation of complement. (If Ω is finite, then F is a 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 contextsterm·dom. For each contextterm·dom C , the Bayes formula defines the conditional probabilityterm·key measure We highlight that the statistical mixtures of contextsterm·dom 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 subset C of Ω that can be identified with the weighted sum p 1 C 1 + p 2 C 2 .

As an illustrative example, consider some agricultural region Ω , and as contextsterm·dom, 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 the poorness of the set of possible experimental contextsterm·dom.

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 } . The contextual probabilityterm·key coincides with the conditional probabilityterm·key given by the Bayes formula: Thus, the set of probability distributions P = { P C a : C ∈ C , a ∈ O } .

For any set D ∈ F and random variable a ∈ R d , x ∈ X a , we define that set and the map For any contextterm·dom C ∈ C and random variable a ∈ R d , x ∈ X a , we define the family of contextsterm·dom Each random variable a and its outcome x determine a map with the domain of definition C a ( x ) .

Thus, classical CMMterm·mth M c l consists of measurement contextsterm·dom composed of pre-measurement contextsterm·dom—elements of F with non-zero probabilities, observables—(discrete) random variables and contextterm·dom update maps, T = { T a ( x ) } . Here, each observable a , random variable, determines uniquely the contextterm·dom update maps T a ( x ) , and, hence, the contextual instrumentterm·mth.

The conditional probabilityterm·key 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 FTPterm·mth (§2.3): In this measurement model, all observables are compatible, conditional JPD coincides with JPD (again by Bayes formula); no pair of observables show OEterm·dom, since All observables show repeatability, since P C ( a = x , a = x ) = P C ( a = x ) , and RREterm·dom, since The Bell inequalitiesterm·key are not violated, since their derivation is based on the existence of JPD.

We can summarize the properties of classical CMMterm·mth by referring to the aforementioned list of possibilities: – violation of FTPterm·mth—no – OEterm·dom—no – violation of replicability—no – RREterm·dom—yes – OEterm·dom+RREterm·dom—yes – violation of Bell inequalitiesterm·key—no One of the problems of the above contextual representation of the classical probability is that the uniqueness conditions (2.2) and (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 For D 1 , D 2 ∈ F , we set D 1 ∼ D 2 if P ( D 1 Δ D 2 ) = 0. This is an equivalence relation on F ; it splits F into disjoint equivalence classes. We denote the set of equivalence classes by the symbol F ~ and the equivalence class of zero probability sets by the symbol Z ~ . The set of pre-measurement contextsterm·dom 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, and we denote the set of these classes by the symbol R ~ d and set O ~ = R ~ d .

For a pre-measurement contextterm·dom C ~ ∈ C ~ and observable a ~ ∈ R ~ d , we define the probability distribution P C ~ a ~ ( x ) = P C ( a = x ) for some representatives C ∈ C ~ and a ∈ R ~ d (the correctness of this definition is proved below), and set P ~ = { P C ~ a ~ } . The modified classical contextual probability spaceterm·mth is the triple Σ ~ = ( C ~ , R ~ d , P ~ ) .

The map T a ( x ) , see equation (3.3), generates a map of F ~ into itself Set C ~ a ( x ) = { C ~ ∈ C ~ : T ~ a ( x ) C ~ ∈ C ~ } . Then, T ~ a ( x ) : C ~ a ( x ) → C ~ . A measurement contextterm·dom is a triple ( C ~ , a ~ , T ~ a ) . Modified classical CMMterm·mth M ~ c l is given by the set of such measurement contextsterm·dom.

Now, we demonstrate that in M ~ c l , the uniqueness conditions (2.2) and (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 P ( D 1 ) = P ( ( D 1 ∩ D 2 ) ∪ ( D 1 ∖ D 2 ) ) = P ( D 1 ∩ D 2 ) = P ( D 2 ) . Let now C 1 , C 2 ∈ C ~ , then, as we have seen, P ( C 1 ) = P ( C 2 ) ; also, for any G ∈ F , P ( G ∩ C 1 ) = P ( G ∩ C 1 ∩ C 2 ) = P ( G ∩ C 2 ) . Hence, P C 1 ( G ) = P C 2 ( G ) . So, conditional probabilityterm·key measure P C does not depend on the choice of a representative C ∈ C ~ , and it can be denoted as P C ~ .

We show that implication (2.2) holds. Let for random variables a 1 , a 2 , P C a 1 = P C a 2 for any contextterm·dom C . Let, for some x , P ( Ω a 1 = x ) > 0. Take C = Ω a 1 = x , then i.e.

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 (2.3). 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 ∩ C 2 ) / P ( C 2 ) , i.e. P ( C 1 ∩ C 2 ) = P ( C 2 ) = P ( ( C 1 ∩ C 2 ) ∪ ( C 2 ∖ C 1 ) , i.e. P ( C 2 ∖ C 1 ) = 0 and symmetrically P ( C 1 ∖ C 2 ) = 0 , i.e. contextsterm·dom belong to the same equivalence class.

Contextual measurement modelterm·mth for von Neumann observables We restrict the consideration to finite-dimensional Hilbert state spaces. The space of pre-measurement contextsterm·dom is mathematically represented as the space of density operators D , i.e. C = D , and observables are Hermitian operators (von Neumann observables [ 1 , 37 ]). We 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 ⟨ A ^ | B ^ ⟩ = T r A ^ B ^ .

The operator A ^ ∈ L H has the spectral decomposition: A ^ = ∑ x ∈ X A x E ^ A ( x ) , where E ^ A ( 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 C A ( x ) = { ρ ^ ∈ D : P ρ ( A = x ) > 0 } .

Thus, P = { P ρ A : ρ ∈ D , A ∈ L H } and T = { T A ( x ) : A ∈ L H , x ∈ X A } . We remark that observable A uniquely determines the family of maps T A ( x ) , x ∈ X A by equality (4.2). So, measurement contextsterm·dom can be represented by pairs ( ρ ^ , A ^ ) , ρ ^ ∈ D , A ^ ∈ L H . We denote this CMMterm·mth by the symbol M Q V N .

In this CMMterm·mth, one need not define separately the contextterm·dom update maps, they are automatically encoded in observables. On the one hand, this simplifies theoryterm·mth. On the other hand, this is the misleading path in measurement theoryterm·mth, cf. with quantum instrumentterm·key theoryterm·mth.

In CMMterm·mth M Q V N , the conditional probabilityterm·key is given by the formula It can be rewritten as In this LSR-based CMMterm·mth, it is convenient to introduce the maps: Then, the above formulas can be rewritten as and the conditional probabilityterm·key is written in the form and conditional JPD as These formulas lead to the quantum instrumentterm·key theoryterm·mth (§5): ( A , I A ( x ) ) is a special quantum instrumentterm·key, and ( A , T A ( x ) ) is the corresponding contextual instrumentterm·mth.

We note that in CMMterm·mth M Q V N probabilities determine contextsterm·dom (states) and observables (operators), i.e. equations (2.2) and (2.3) hold (Mackey’s axiom 2).

Let P ρ A 1 ( x ) = P ρ A 1 ( x ) for all ρ ^ ∈ D and real x . Then, T r ( E A 1 ( x ) − E A 2 ( x ) ) ρ ^ = ⟨ E A 1 ( x ) − E A 2 ( x ) | ρ ^ ⟩ = 0. Hence, E A 1 ( x ) = E A 2 ( x ) for any x ; so A ^ 1 = A ^ 2 . Thus, an observable can be identified with the set of probability distributions P A = { P ρ A : ρ ^ ∈ D } .

Now let P ρ 1 A ( x ) = P ρ 2 A ( x ) for all A ^ ∈ L H and real x . Then, T r E A ( x ) ( ρ ^ 1 − ρ ^ 2 ) = ⟨ E A ( x ) | ρ ^ 1 − ρ ^ 2 ⟩ = 0. Hence, ρ ^ 1 = ρ ^ 2 . Thus, a quantum state can be identified with the set of probability distributions P ρ = { P ρ A : A ∈ L H } .

In the von Neumann measurement theoryterm·mth, two observables are compatible if they are represented by commuting operators A ^ , B ^ : [ A ^ , B ^ ] = 0. Compatibility is interpreted as guaranteeing 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 In particular, for compatible observables, there is no OEterm·dom for any state ρ . If operators A ^ 1 , A ^ 2 do not commute, then there exists a state ρ ^ showing OEterm·dom for these observables, i.e. P ρ ( A = x , B = y ) ≠ P ρ ( B = y , A = x ) .

Each observable A shows replicability, e.g.

If observables are compatible, then for any ρ ^ , they show RREterm·dom, e.g.

We highlight that it is impossible to combine OEterm·dom and RREterm·dom within CMMterm·mth M Q V N [ 60 ] (in the finite-dimensional case).

FTPterm·mth can be violated; classical FTPterm·mth is additively perturbed by the interference term; for instance, consider a pure state | ψ ⟩ , and then where On the r.h.s., the first summand corresponds to classical FTPterm·mth, and the second one is the interference term; it quantifies the degree of non-classicality for this CMMterm·mth; see §2.3 for FTPterm·mth 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 where the angle θ = θ ( B = y | A ; ψ ) .

We can summarize the properties of von Neumann CMMterm·mth with the list of possibilities presented above: – iolation of FTPterm·mth—yes (equations (4.14) and (4.15)).

– OEterm·dom—yes. This is a straightforward consequence of the existence of incompatible observables represented by non-commuting operators.

– iolation of replicability—no (equation (4.13)).

– RREterm·dom—no. This is again a consequence of the incompatibility of some observables. Consider non-commuting operators A ^ and B ^ . For simplicity, assume that they have non-degenerate spectra; consider the orthonormal bases ( e j A ) and ( e j B ) consisting of operators’ eigenvectors with eigenvalues ( a j ) and ( b j ) . At least for one pair of indexes k , m , 0 < | ⟨ e k A | e m B ⟩ | < 1. Let the initial state be a pure state | ψ ⟩ . Suppose that the measurement of observable A gives the outcome a k and the sequential measurement observable B gives the outcome b m . Let observable A be measured again. Then, the probability to get again the outcome a k , given by | ⟨ e k A | e m B ⟩ | 2 , does not equal one. The case of operators with degenerate spectra is studied in the article [ 60 ]. It is important to note that we consider only the finite-dimensional case.

– OEterm·dom+RREterm·dom—no. Observables demonstrating OEterm·dom should be incompatible. But, for incompatible observables, RREterm·dom is violated.

– Violation of Bell inequalitiesterm·key—yes [ 43 , 44 ].

Contextual measurement modelterm·mth for quantum instrumentsterm·key In this section, we present CMMterm·mth M Q I , where a measurement process is mathematically described by quantum instrumentterm·key theoryterm·mth [ 7 , 8 , 11 , 12 , 46 – 52 ]. This CMMterm·mth extends CMMterm·mth M Q V N , where a measurement process is mathematically described by a Hermitian operator (von Neumann observable).

The space of linear Hermitian operators 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 ≥ 0 , T ( g ) ≥ 0.

Consider an observable A with a 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 → I ( x ) , where for each x ∈ X , the map I ( x ) is a positive superoperator, and is called a quantum instrumentterm·key . It determines some observable; we denote it by the symbol A ; see equation (5.9).

The probability of the output A = x is given by the generalized Born rule in the form We note that the measurement with the output A = x generates the state update through the transformation with the domain of definition C A ( x ) = { ρ ^ ∈ D : P ρ ( A = x ) > 0 } .

Let where ( 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 ( V ^ ( x ) ) be a family of linear operators constrained by the normalization condition: An atomic quantum instrumentterm·key is a superoperator of the form Applications of the quantum instrumentterm·key theoryterm·mth 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 be written as where I is the unit operator and I ⋆ ( x ) is the superoperator that is adjoint to I ( x ) in Hilbert space L H . Hence, the generalized Born rule has the form where Operators A ^ ( x ) , x ∈ X are called effects ; they are positive semi-definite Hermitian and sum up to the unit operator: The family of operators A = ( A ^ ( x ) , x ∈ X ) is called a POVM; for a subset Δ of X , we set this is an additive operator-valued measure, i.e. for Δ 1 , Δ 2 ⊂ X , Δ 1 ∩ Δ 2 = ∅ , Instruments of the projection type, equation (5.4), determine the special class of POVMs, projection-valued measures (PVMs).

Two POVMs A = ( A ^ ( x ) , x ∈ X ) and B = ( B ^ ( y ) , y ∈ Y ) are called compatible , if there exists a POVM C = ( C ^ ( x , y ) , ( x , y ) ∈ X × Y ) such that The compatibility is interpreted as guarantying the possibility of the joint measurement of these observables, and their JPD is given by generalization of the Born rule for compatible von Neumann observables: In the contextual probability spaceterm·mth, contextsterm·dom are mathematically represented by density operators (quantum states), C = D , and observables by POVMs (also known as generalized quantum observables), and the probability distributions are determined by the generalized Born rule, equation (5.2). CMMterm·mth M Q I is endowed by quantum instrumentterm·key maps updating quantum states (contextsterm·dom) due to the measurement feedback, equation (5.3).

Consider POVM A ^ = ( A ^ ( x ) ) and all quantum instrumentsterm·key generating it via (5.9). Then, the corresponding state (contextterm·dom) update maps are defined by equality (5.3). The same POVM, a 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 byproducts generated by quantum instrumentsterm·key.

Quantum instrumentsterm·key considered earlier were invented in the article [ 7 ] (see also monograph [ 8 ]), and these are Davies–Levis instruments , so M Q I = M Q I ; D L . In quantum information theoryterm·mth, one uses the special class of quantum instrumentsterm·key given by completely positive maps I ( x ) ; we denote the corresponding CMMterm·mth M Q I ; O , where I use the index ‘O’ to mention Masanao Ozawa who contributed so much to the theoryterm·mth of such quantum instrumentsterm·key [ 11 , 12 , 46 – 49 ]. It is commonly assumed that the instruments belonging to M Q I ; D L ∖ M Q I ; 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 through the indirect measurement scheme [ 11 ]. This scheme is adequate for quantum measurement processes, and any deviation from this scheme is non-physical. Nevertheless, it might be that in quantum-like modellingterm·key, even instruments that are not completely positive can find applications. Such applications would lead to the modification of the indirect measurement scheme, may be through the consideration of non-unitary interactions.

We remark that instrument maps I ( x ) are linear in the Hilbert space L H . In terms of contextterm·dom (state) update, these maps can be written as Hence, in a quantum instrumentterm·key, CMMterm·mth scaling of the update map T A ( x ) by probability P ρ ( A = x ) is a linear map. Generally, LSR-CMM with the contextterm·dom (state) space C = D can be endowed with contextterm·dom (state) update maps such that scaling (5.12) can be nonlinear. CMMterm·mth M Q I ; N L with nonlinear contextterm·dom update maps might be useful for quantum-like modellingterm·key. It is interesting to find concrete applications of M Q I ; N L . Of course, such applications would lead to the modification of the indirect measurement scheme.

We can summarize properties of quantum instrumentterm·key CMMterm·mth, both models M Q I ; D L and M Q I ; O . Since the quantum instrumentterm·key model extends the von Neumann model, the majority of properties follow automatically from its properties.

– violation of FTPterm·mth—yes – OEterm·dom—yes – violation of replicability—yes – RREterm·dom—no – OEterm·dom+RREterm·dom—generally no, but can be realized by special instruments – violation of Bell inequalitiesterm·key—yes So, M Q I differs from M Q V N with respect to replicability and OEterm·dom+RREterm·dom combination. Replicability in M Q V N is a consequence of projection state update. Such an update is idempotent, and thus the value of an observable is replicable. Model M Q I permits more general updates; generally, they are not idempotent. The possibility to reproduce OEterm·dom+RREterm·dom combination was demonstrated in the article [ 61 ]. This is a technically complicated construction within the theoryterm·mth of quantum instrumentsterm·key, and it is impossible to present this construction in the present paper.

It is interesting to find a property distinguishing M Q I ; D L and M Q I ; O through 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 a 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 the PVM type, i.e. A ^ = ( E ^ A ( x ) ) and B ^ = ( E ^ B ( y ) ) . They are jointly measurable and the JPD is given by equation (4.11). The conditional JPD is given by The r.h.s. of equations (4.11) and (5.13) coincide only if the instrument superoperators are of the projection type, i.e. I ( x ) ρ = E ^ ( x ) ρ E ^ ( x ) .

Moreover, in this case, two projection-type observables, PVMs, can have a variety of conditional probabilityterm·key distributions corresponding to different instruments generating them by the rule E ^ ( x ) = I ⋆ ( x ) I .

Ordered space measurement model with probability measure states In this section, we connect the generalized probability theoryterm·mth (the Davies–Lewis approach [ 7 ]) for probability measures with CMMterm·mth. 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 ; we denote this set by the symbol S ; this is the set of states and S is a convex subset of M . The latter is endowed with the variation norm, | | μ | | = v a r ( μ ) , and it is a Banach space. Consider its dual space M ′ , the space of continuous linear functionals f : M → R . We denote by A the subset of M ′ , consisting of functionals mapping S into [ 0 , 1 ] . Elements of A are called effects, and these are basic observables. They can be described solely in terms of the state space S as affine functionals valued in [ 0 , 1 ] , i.e. A ≡ A ( S ) .

Consider the functional u ∈ M ′ defined as μ → ⟨ u | μ ⟩ = μ ( Ω ) . Its characteristic property is that ⟨ u | μ ⟩ = 1 for any state μ ∈ 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 analogues of POVMs; we call them M -POVMs. These are observables of the contextual probability spaceterm·mth Σ m e a s u r e with contextsterm·dom C = S , and the set of probability distributions P defined as As we learn from the quantum instrumentterm·key theoryterm·mth, 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 I : X → L ( M ) such that I ( x ) M + ⊂ M + and 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 Set Then, for each x , A ( x ) is an effect, i.e. A ( x ) ∈ A ( S ) . So, each M -instrument determines a M -POVM.

The M -CMM consists of contextterm·dom (states) given by probability measures and POVM-observables with state updates given by instruments.

Linear space representationterm·mth for contextual probability spaceterm·mth 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 To each outcome x ∈ X and state s ∈ S , this function is a probability of the outcome x for the system that is in the state s . The generalized probability model is a triple ( S , p ( ⋅ , ⋅ ) , X ) . We 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 (7.1) is injection. Thus, each state s can be mapped to a function belonging to space Φ [ 0 , 1 ] , and 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 .

We denote by A ( S ¯ ) the space of all affine functionals In particular, for any x ∈ X , f x ∈ A ( S ¯ ) . Any functional f ∈ A ( S ¯ ) describes an outcome of some observable, and thus f ( s ) is the probability for that outcome in state s .

In QMterm·dom, S ¯ is the set of density operators, and elements of A ( S ¯ ) are called effects—components of POVMs, f ( ρ ) = T r E ^ f ρ ^ , where E ^ f 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 analogue of quantum observable given by the unit operator I . Consider the point-wise order structure on A ( S ¯ ) , f ≤ g if 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 LSRterm·mth to 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 ] denote the space of functions from X to [ 0 , 1 ] . We map S into Φ [ 0 , 1 ] , s = ( C , A ) → 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 ∈ 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 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 contextterm·dom C can be represented as a vector belonging to Φ [ 0 , 1 ] , C ( x ) = C ( A , x ) = P C A ( x ) . Due to equation (2.3), embedding of the set of contextsterm·dom C into Φ [ 0 , 1 ] is injection. Again, we 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 C A ( x ) ∈ [ 0 , 1 ] . Now fix A ∈ O and consider the family of functionals F = ( F A ( x ) = f A , x : x ∈ X ) . This is the representation of observable A .

So, any contextual probabilityterm·key model can be realized like COM—an observational COM.

Concluding remarks As was emphasized in §1, CMMterm·mth can be considered as the most general probabilistic model for measurement. It can also be considered as a minimalist restructuring of Mackey’s project [ 6 ]. Mackey proceeded to quantum logic, and this made the mathematical construction more complicated. One may even say that mathematics shadowed measurement theoryterm·mth. Surprisingly, even this minimalist model (CMMterm·mth) has a complex structure and represents the basic elements of quantum probabilityterm·dom and measurement theoryterm·mth, e.g. interference of probability, OEterm·dom, entanglementterm·key and the violation of the Bell inequalitiesterm·key.

CMMterm·mth can be employed not only in quantum foundations but also in quantum-like modellingterm·key that can employ contextual probabilityterm·key calculi and CMMsterm·mth that are not based on the complex Hilbert space formalism.

Finally, we reproduce a list of the basic properties that can be used to classify CMMsterm·mth: – violation of FTPterm·mth – OEterm·dom – violation of replicability – RREterm·dom – OEterm·dom+RREterm·dom – violation of Bell inequalitiesterm·key.

Although two basic quantum measurement models, von Neumann’s model M Q V N and the quantum instrumentsterm·key model M Q I , can be distinguished with respect to violation of replicability and OEterm·dom+RREterm·dom combination, they both violate FTPterm·mth, demonstrate OEterm·dom and violate Bell inequalitiesterm·key. It would be interesting to find other CMMsterm·mth that are distinguished, for example, with respect to the violation of FTPterm·mth and Bell inequalitiesterm·key.