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This paper introduces the concept and motivates the use of finite-interval based measures for physically realizable and measurable quantities, which we call -measures. We demonstrate the utility and power of -measures by illustrating their use in an interval-based analysis of a prototypical Bell’s inequality in the measurement of the polarization states of an entangled pair of photons. We show that the use of finite intervals in place of real-numbered values in the Bell inequality leads to reduced violations. We demonstrate that, under some conditions, an interval-based but otherwise classically calculated probability measure can be made to arbitrarily closely approximate its quantal counterpart. More generally, we claim by heuristic arguments and by formal analogy with finite-state machines that -measures can provide a more accurate model of both classical and quantal physical property values than point-like, real numbers—as originally proposed by Tuero Sunaga in 1958.

We present first two heuristic arguments, followed by theoretical and numerical demonstrations, to support motivation of a concept to replace point-like real-numbered physical property values with intervals of value we call D -measures, which may be weighted by some function. The exact nature of the weighting is not crucial, however, to the interval-based representation. In Sect. 1.1, we show how these arguments suggest that the conventionally assumed assignment of real numbers to represent physical property values may not be tenable (see, e.g., [

Measurement of any physical property value and generic manifestation of any physical property value are equivalent processes at some fundamental level. This equivalence is foundational to environmental decoherence theory since certain manifestations of value are physically realized^{1} via “implicit measurement” of objects by the environment in which they exist [

^{1}We use the term “physically realized”, while admittedly not rigidly definable, because it offers a working definition of the notion of a physically realized entity as one that can exist in and have influence on physical reality, while having physical properties that are, in principle, measurable by a physical device. It is to be contrasted with an abstracted physical property, which may be formally useful but may not be measurable by a physical device.

From a communication theoretic point of view, suppose a communication signal could have a producible and detectable parameter represented by a real number. Since real numbers are infinitely precise and can be represented mathematically [

Using these and other similar arguments, we assert that realizable and measurable property values are more accurately modeled by D -measures than by point-like real-numbered values of zero measure. We further assert that D -measures apply to both classical and quantal physically realizable values. At some level, the interval ( D -measure) model is in conflict with the convention of classical physics to assume measurable property values are mathematically represented by real numbers; this conventional representation may be too restrictive.

The conflict is perhaps less pronounced for quantal measurement outcomes due to the intrinsic uncertainties and ambiguities in a quantal description, but there is a critical difference in D -measured quantal superposition and conventional quantal superposition: D -measured outcome values, even when weighted by some appropriate function, are not envisaged to be associated with a probability metric across eigenvalues. Because D -measure intervals apply to each single measurement, or manifestation of value, the eigenvalues within an interval are assumed associated with a non-statistical ontic metric. While the exact definition and meaning of this ontic metric is not yet clear (and the subject of a follow-on paper), the assertion that it is non-statistical means that single measurement outcomes have distributed value, i.e., they are D -measures. This interval-based representation suggests that all realizable quantum states that result from measurement are comprised of simultaneously physically existing eigenstates. Every physically realizable quantum state is a superposition of multiple states in every realizable basis, i.e., a basis with physically measurable eigenvalues. A D -measured state cannot be represented by a single, real-numbered direction in an abstract space of realizable eigenvalues.

D -measured quantum state definitions open the opportunity to form an entropy metric calculated just like Shannon information entropy [

Application of D -measures to physical values is analogous to the application of intervals to the values typical of finite-state machines, which are incapable of specifying or processing real-numbered values. The application of interval analysis herein to all physically realizable property values is likewise suggested for fundamentally similar reasons. Physical objects, systems of objects, and processes, such as classical and quantum measurement, are limited in their ability to manifest real-numbered property values by parameters such as spectral limits, process and time limits, and various other constraints. Both classical and quantal physically realizable objects and systems thus can be viewed in some sense as finite-state machines.

The mathematical formalism of interval analysis was developed and has seen its primary application in computing theory for numerical analysis and mathematical modeling. It is a relatively recent cross-disciplinary field pioneered by M. Warmus [

The need for the concept of an interval was spawned by the need in the above numerical applications to enclose a real number when it can be specified only with limited accuracy, i.e., it cannot be exactly represented on any finite-precision machine. In physical systems, inaccuracy in measurement coupled with known or unknown uncertainty and variability in physical parameters, initial and boundary conditions, etc., formally inhibit the manifestation of measurable quantities as real numbers, to be treated via the machineries of real numbers’ arithmetic and algebra. Special axioms and special interval arithmetic and algebra were clearly needed to endow the new field with rigorous mathematical foundations.

In numerical analysis, finite intervals of one or more dimensions are seen as extensions of real (or complex) numbers. As mathematical objects, intervals in themselves do not form proper vector spaces [

In Sect. 2.1, we demonstrate how, under some conditions, an intervalized but otherwise classically calculated correlation function can be made to arbitrarily come close to its quantal counterpart. The demonstration is essentially a re-casting using intervals and interval analysis of a limiting case derived by Bell [

We demonstrate in this section the validity of the following assertion: Expressed as interval-valued functionals, as opposed to real number-valued functions, the distance between a classically calculated correlation function, of two measured interval quantities, and its quantal counterpart can be shown (under some conditions) to arbitrarily approach zero.^{2}

Let the two measured interval-variables be X = [ X min , X max ] and Y = [ Y min , Y max ] , where we assume that both are one-dimensional intervals (generalization to higher dimensions may not be trivial, see, e.g., [

| C x , y c l ( x ^ , y ^ ; λ ) + ( x ^ ⋅ y ^ ) | ≤ ϵ , (1)

^{2}By “demonstration” we mean here that what follows is neither a rigorous nor a general validation of the above assertion.

where ϵ is a small number but which cannot be made arbitrarily small, i.e., will always be bounded from below due to the finite precision of any physical measurement. Our demonstration of the assertion made above is essentially a recast of Equation (1) in its interval analog for intervals X and Y, but in which the analog of ϵ is shown that it can be made arbitrarily small. The conditions pertain to our assumed low dimensionality of the intervals and of the unit vectors, in addition to the assumed forms of the inner and dot products, our proxy correlation functions.

In lieu of the inner product, we will have an interval-valued integral function, or a functional, and in lieu of the dot product for unit vectors, an assumed interval-valued functional related to the range of the first. The interval analog of the inner product can be written as

C X , Y c l ( x ^ , y ^ ; λ ) = [ ∫ [ Z ] , l o w e r ∂ C X , Y c l , ∫ [ Z ] , u p p e r ∂ C X , Y c l ] , (2)

where lower and upper refer to the lower and upper Darboux integrals [

∫ [ Z ] ∂ C x , y c l ∈ ∫ [ Z ] ∂ C X , Y c l (3)

over the same interval “[Z]”, which follows from our assumed interval-extension of C x , y c l . Although generalization to an extended λ is straightforward and could present interesting cases for further analysis, for purposes of this demonstration, we take the parameter λ to be the same in both the real-numbered valued and interval-valued cases. Since the interval [ Z ] = [ X ] [ Y ] has a range [ − 1, + 1 ] , C X , Y c l can be written as:

C X , Y c l = Z × F ( [ Z ] ) , (4)

where F ( [ Z ] ) is a functional of Z. Note that the above form for C X , Y c l is not unique; as uniqueness is not required for this demonstration. Also, the exact form of the interval-valued function F is not required, but that it is analytic and convergent over an interval Z 0 that includes Z, and over which interval the derivative of F exists and does not contain zero. These general properties [

| f ( x ) − f ( y ) | ≤ const | x − y | for x , y ∈ Z 0 , (5)

we have for F,

d ( F ( [ Z ] ) ) ≤ const d ( Z ) for Z ⊆ Z 0 , (6)

where d ( ⋅ ) denotes the diameter (or width) of its interval argument.

A fundamental property of any extended function, F, of an interval is its “enclosure” property, i.e.,

R ( F ; [ Z ] ) ⊆ F ( [ Z ] ) , (7)

where R ( F ; [ Z ] ) is the range of the function F over the interval [ Z ] , F ( [ Z ] ) is now a “functional” of the interval [ Z ] , and “ ⊆ ” denotes “a subset of”. Almost all derived properties of intervals, including their mapping, differentiation and integration, differential (or integral) equations-based applications are based on the enclosure property [

The extended functional F ( [ Z ] ) is further assumed divisible into smaller subintervals, where it can be regarded as the union of these subintervals,

F ( [ Z ] ; k ) = ∪ l = 1 k F ( [ Z ] l ) , (8)

and where smaller refers to the diameter of each sub-interval Z l being reduced by the factor l = 1 , 2 , ⋯ , k .

For the interval-valued dot product of the unit vectors, being a projection of one unit vector onto the other, and since the range is also [ − 1, + 1 ] , we make the ansatz and connect this with the functional F via its range,

( x ^ ⋅ y ^ ) = const R ( F ; [ Z ] ) , (9)

or with any linear function of R ( F ; [ Z ] ) , where R ( F ; [ Z ] ) is the range of F over the interval Z. Again, this relation is not unique for x ^ ⋅ y ^ .

Next, we take advantage of two basic theorems of interval analysis [

Q ( R ( F ; [ Z ] ) , F ( [ Z ] ) ) ≤ const ‖ d ( [ Z ] ) ‖ ∞ , (10)

where d ( F ( [ Z ] ) ) ≤ const ‖ [ Z ] ‖ ∞ , and the constants ≥ 0 . | | ⋅ | | ∞ denotes the maximum norm. Applied to the subdivided F ( [ Z ] ; k ) , the Hausdorff distance becomes

Q ( R ( F ; [ Z ] ) , F ( [ Z ] ; k ) ) ≤ const ‖ d ( [ Z 0 ] ) ‖ ∞ 2 / k 2 . (11)

What the above theorem suggests is that R ( F ; [ Z ] ) , our proxy for the intervalized dot product, can be arbitrarily close to F ( [ Z ] ; k ) , our proxy for the intervalized inner product, if the subdivision of F ( [ Z ] ) is made sufficiently fine. Clearly, this is only true under the conditions (i.e., low dimensionality of the intervals and the unit vectors) and assumptions made (i.e., the assumed specific forms of the inner and dot products). Applications to different forms and/or any generalization are clearly beyond the scope of the assertion.

Our first application of the D -measure concept is to an interval-based analysis of a prototypical [

p x y ∝ sin 2 ( θ 1 − θ 2 ) or ∝ cos 2 ( θ 1 − θ 2 ) (12)

for each of the four possible combinations that add up to unity. When a third detector is introduced, a Bell’s inequality can constrain the degree of polarization correlation among the angular separations in such a way that

sin 2 ( θ 2 − θ 1 ) + sin 2 ( θ 3 − θ 2 ) ≥ sin 2 ( θ 3 − θ 1 ) . (13)

To intervalize Equation (13), we re-express the measured angles, θ 1 , θ 2 , and θ 3 as angle intervals, Θ 1 = [ θ 1 − δ θ 1 , θ 1 + δ θ 1 ] , Θ 2 = [ θ 2 − δ θ 2 , θ 2 + δ θ 2 ] and Θ 3 = [ θ 3 − δ θ 3 , θ 3 + δ θ 3 ] , where δ θ i is the total uncertainty in measuring θ i , i.e., including all system and random errors in the set-up and the measuring devices. Note that in the limit of δ θ i → 0 , Θ 1 = [ θ i , θ i ] , i.e., it is a degenerate interval. Being a statement about probability measures and their correlations, the form of Equation (2) is retained when expressed as

sin 2 ( Θ 3 − Θ 1 ) ⊆ sin 2 ( Θ 2 − Θ 1 ) + sin 2 ( Θ 3 − Θ 2 ) . (14)

Intervalized, Equation (14) suggests that the interval sin 2 ( Θ 2 − Θ 1 ) + sin 2 ( Θ 3 − Θ 2 ) will always include the interval sin 2 ( Θ 3 − Θ 1 ) . Note that the sine of an interval is also an interval since the sine function will map every point in the interval argument to a point in the interval image of the function.

The enclosure property, Equation (7), can be used, as an example, to show that

sin 2 ( Θ 3 − Θ 1 ) ⊆ sin ( Θ 3 − Θ 1 ) × sin ( Θ 3 − Θ 1 ) . (15)

Another intriguing property of interval functionals is their dependence on the algebraic form or structure of the enclosing function f, or its extended pair F. This dependence stems from the set-theoretic attributes of intervals. For example, another form of Equation (14) that is equivalent for degenerate intervals, i.e., real numbers, but is not for finite intervals is

sin 2 ( Θ 3 − Θ 1 ) − sin 2 ( Θ 2 − Θ 1 ) ≠ ( sin ( Θ 3 − Θ 1 ) + sin ( Θ 2 − Θ 1 ) ) ( sin ( Θ 3 − Θ 1 ) − sin ( Θ 2 − Θ 1 ) ) . (16)

“Inequality violation” of Equation (14) is when the left hand side of the equation minus the right hand side becomes negative. This is indeed seen for the quantum-mechanically calculated probabilities at various angles and over extended domains of none-zero measures (see ^{−4}.

For δ θ ∈ [ 0.01,0.25 ] , we calculate the probability (at θ 2 = 36 deg and θ 3 = 72 deg) of no violation for each δ θ . This is when that difference in the two parts of Equation (14) crosses the zero plane. We assume that both the intervalized difference and the difference that is calculated using error propagation are centered Gaussians. To arrive at the probability of no violation, we simply integrate from the center of the interval to the zero point, after normalizing to unity and subtracting 1/2. Since, for purposes of this demonstration, we do not

ascribe any weighting function to the intervalized angles, the calculated probabilities are more representative of upper limits rather than most likely values.

As mentioned above, the calculated no-violation probabilities using interval-based quantities appear to depend on the algebraic structure of the inequality itself. A critical parameter in the interval estimation for the probabilities is the Hausdorff distance, Equation (11). In

normalized to the diameter of the interval at each k, such that a distance of unity is the smallest possible distance. Here, k = 16 seems to give a rapid (but not necessarily too rapid) of a convergence, almost in an exponential rather than a geometric fashion. This feature may be important in designing Bell tests optimized for error constraints and the algebraic form or structure of the inequality. Rapid convergence (the “right” form of the inequality) can compensate for the size of the measurement error. In this particular illustration, however, given the exponential convergence, the form of the inequality, Equation (14), seems less of a consequence to the calculated no-violation probabilities than the size of the measurement error.

We have introduced and motivated the use of finite intervals to represent physically measurable quantities, we call D -measures, in place of the real-numbered representation, which we consider untenable. We demonstrate the utility of D -measures using theoretical and numerical illustrations. Our theoretical demonstration, an interval-based recasting of Bell’s inequality using proxy correlation functionals, shows that, under some conditions, two measured interval quantities―a classically calculated correlation function and its quantal counterpart―can come arbitrarily close to each other. This is in stark contrast to the Bell theorem claim, which assumes classical property values are real-numbered, that no hidden variable theory [

In our numerical demonstration, we apply interval analysis to a measurement of the polarization states of an entangled pair of photons. We calculate the probabilities of no violation and demonstrate that quantal violation of the Bell inequality is likely less severe under the assumption of D -measured values. This means that Bell tests should be considered less compelling as proof of quantum correlations and non-locality [

These demonstrations, along with our heuristic arguments, motivate the need for and use of D -measures to more accurately model physical property values than the traditionally assumed real-numbered representation. We assert that the interval-based D -measured representation applies to both classical and quantal physical values, and that their desirability and need for broader application to physical theories in general seems apparent. The development of interval analysis for computing theory and its application to finite-state computing machines was predicated by the need to represent numerical values and quantities that are only approximate by necessity in a real world computing machine. It may be at first counterintuitive to think, for example, any microscopic or macroscopic object can have two or more simultaneous values for any one of its physical properties. Upon analysis, however, it becomes evident that distributed values as provided by D -measures are more tenable than real-numbered values, just like in finite-state machines. Thus we suggest that the application of D -measures and interval analysis should see rapid and pervasive growth in applications to many physical and other theories.

More than 60 years ago, mathematician Tuero Sunaga, working in the field of communication theory at the University of Tokyo wrote [

More specifically to NASA, the need for advancements in communication and computing theories and related technologies make broader applications of D -measures to physical systems even more compelling. Our own future work on this effort will include more rigorous interval-based mathematical modeling of Bell-like tests, re-formulation of some well known models of physical systems using interval-based analysis, and a better appreciation of the benefits and limitations of the new analysis when applied to physical theory, with the goal of supporting the advancement of quantum-based analysis, modeling and technologies.

The authors declare no conflicts of interest regarding the publication of this paper.

Eblen, F.P. and Barghouty, A.F. (2019) Interval Based Analysis of Bell’s Theorem. Journal of Modern Physics, 10, 585-600. https://doi.org/10.4236/jmp.2019.106041

In 1964, Irish physicist John S. Bell proposed a revolutionary theorem that could possibly prove the existence of quantal correlations of entangled objects. His theorem showed that violations of a classical probability inequality could be tested so as to prove classical correlations of detected particles cannot be made arbitrarily close to quantal correlations (see, e.g., [

If quantal correlations are as predicted by the theory, Bell test data show a cosine squared-relationship of correlation with respect to the relative detector angle (the red curve in

The classical and quantum correlations are most easily illustrated using photons with the same spin, though twin polarization photons are essentially the same. The angle of Detector 1, designated D1, is used as a reference angle of 0-deg. The angle of D2 relative to D1 is θ . For the classical case, the green arc in

The quantal case is very different, as illustrated by

cos θ . Since quantum probability is the square of the state amplitude, the multiplier becomes cos 2 θ . This means the probability of a D1 detection being the same as a D2 detection, i.e., the probability of agreement, or correlation, is a function of cos 2 θ .

So, if quantum predictions are correct, Bell test data will reproduce the red curve in

But, as we have argued in this paper, if one replaces real-numbered values with “quasi classical” interval values, or D -measures, the differences between the two results may not be as pronounced or as differentiated, at least under some conditions (see