Completing Quantum Mechanics within the Framework of Local Realism

1. Introduction

Consider two spatially separated, individual microscopic objects. A correlation between them always exists. If the condition demanded by Einstein’s separability principle is satisfied [1], namely, the objects possess their autonomous, real states independent of each other, then they constitute a separable system, and the possibility of describing the objects by non-locality will be precluded. For ease of exposition, call such objects distant components of the separable system, or simply call them distant components when no confusion will arise if the corresponding system is not mentioned. Non-locality is also referred to as “nonlocal-interaction” ([2], pp. 1886-1887). Were there nonlocal-interaction in the real world, a measurement performed on one of the distant components would change the real state of the other component telepathically; all such telepathic changes are excluded by Einstein’s separability principle. Einstein illustrated his separability principle with a simple, non-quantum-theoretical example [1]. The following is a slightly different version of Einstein’s original illustration (see Table 1 below).

Two colored balls are spatially separated in two boxes. A correlation exists between the separated balls. For example, if the color of one ball is blue, the color of the other ball must be red. Before a person lifts the lid of a box and looks inside, which box contains the blue (or red) ball is unknown. Will the color of the ball in the box change when its lid is lifted? Of course not. The colors of the separated balls will remain unchanged regardless of whatever happens to either box. Clearly, the separated balls are deterministically correlated. There is no need of using probability here because no randomness appears in the illustration. In Section 8, Einstein’s original illustration will be discussed to show a connection between the purported probabilistic nature attached to current quantum theory and the role played by observers as an integral part rather than a consequence of the axioms of quantum mechanics.

In his debate with Bohr [3,4], Einstein grounded both locality and realism on his separability principle. Thus, in Einstein’s local-realist description of the world, measurements performed on either distant component cannot affect the other component according to locality, and values of all variables describing a distant component exist objectively according to realism. The essence of the Einstein-Bohr debate is the legitimacy of quantum superpositions. Einstein disagreed with Born’s probabilistic interpretation of wave-functions, which are expressed as quantum superpositions. Born’s probabilistic interpretation cannot account for indeterminism in quantum physics. Once a measurement is performed on a quantum-mechanically described object, the corresponding wave-function collapses abruptly. Because the quantum-mechanical description “cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance” ([5], p.158), Einstein called it “the fundamental dice-game” ([5], p.149). Besides Born’s probabilistic interpretation, observers in the axioms of quantum mechanics also disturbed Einstein.

Inspired by Einstein, Bell and his followers intended to complete quantum mechanics within the framework of local realism [6]. Bell derived the first of Bell inequalities to be tested by experiments [7,8,9,10] and proved Bell’s theorem to interpret the experimental results [7]. To his surprise, his theorem shows Einstein to be wrong in the Einstein-Bohr debate [2,7], which is the opposite of what Bell intended [6,11]. Why does Bell’s approach lead him to prove the opposite of what he intended? Can quantum mechanics be completed in a way different from Bell’s approach? The questions are still open and concern different understandings of Bell’s theorem [1,12,13,14,15,16,17,18], including those disproving or questioning it [17,18]. This study aims to answer the above questions by introducing a new principle, the general principle of measurements, which is proved as a mathematical theorem. The main findings are as follows.

Bell’s approach is problematic; the deterministic correlation between distant components of a separable system in Einstein’s local-realist description of the world is mistaken for non-locality in the world described by Bell’s theorem [2]. Quantum mechanics can be completed within the framework of local realism in a way consistent with the formal axiomatic definition of a general Hilbert space, so the completed quantum theory can provide the same probabilistic predictions of empirical results as those provided by current quantum theory. Using disjunction (“or”) as the logical relation between superposed orthonormal vectors, the completed theory precludes observers in the axioms of the current theory and inexplicable collapses of wave-functions by explaining indeterminism in quantum physics; it is intuitively comprehensible and alleviates much difficulty in understanding quantum mechanics. Among various world views, Einstein’s local-realist world view is correct. Relinquishing Einstein’s separability principle as suggested in [1] is unnecessary.

2. Local Realism and EPR Argument

Locality and realism are not mathematical assumptions. Some people attribute the conflict between quantum mechanics and Bell inequalities to some mathematical assumptions needed to prove the inequalities. How they justify their opinion is irrelevant to this study and will not be considered here. According to [1], Einstein himself did not write the EPR paper (i.e., [3]). He neither mentioned the “elements of physical reality” in his standard argument for the incompleteness of quantum mechanics nor considered the well-known EPR argument (as given in [3]) satisfactory; Podolsky wrote the EPR paper. Thus the EPR argument cannot fully reflect Einstein’s own views and deeper philosophical assumptions. Nevertheless, because the EPR argument is widely known, it is still necessary to analyze the role played by the assumptions of locality and realism (local realism) underlying the EPR argument in the context of its logical structure.

The purpose of the EPR argument [3] is to question the completeness of quantum mechanics based on local realism and some less important assumption. As stated by EPR [3]: “In a complete theory there is an element corresponding to each element of reality. A sufficient condition for the reality of a physical quantity is the possibility of predicting it with certainty, without disturbing the system.” In their argument [3], EPR then continued to reveal a contradiction in the conceptual foundations of current quantum theory: if quantum mechanics is assumed to be a complete theory, then this assumption, together with the criterion of reality, leads to a contradiction. But what is the contradiction they revealed?

To answer the above question, let us consider two well-known statements in the EPR argument [3]: “either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) two physical quantities described by two non-commuting operators cannot have simultaneous reality.” One of the two statements must be wrong. According EPR [3]: “Consideration of the problem of making predictions concerning a system on the basis of measurements made on another system that had previously interacted with it leads to the result that if (1) is false then (2) is also false.”

In other words, the negation of (1) implies the negation of (2). By contra-position, the statement “the negation of (1) implies the negation of (2)” is equivalent to the statement “(2) implies (1)”, i.e., “two physical quantities described by two non-commuting operators cannot have simultaneous reality” implies “the description of reality given by the wave function in quantum mechanics is not complete.” If the two non-commuting operators are the position operator and the momentum operator, then (2) is a consequence of Heisenberg’s uncertainty relation. The negation of (2), i.e., the statement “(2) is false”, actually implies the statement “Heisenberg’s uncertainty relation is false.” Thus the negation of (2) further implies the falsity of Heisenberg’s uncertainty relation. The negation of (1) is the statement “quantum mechanics is complete.”

As shown above, the contradiction in the conceptual foundations of current quantum theory is this: the completeness of quantum mechanics implies the falsity of Heisenberg’s uncertainty relation, or equivalently, Heisenberg’s uncertainty relation, if it holds, implies the incompleteness of current quantum theory. Because the negation of (2) implies the falsity of Heisenberg’s uncertainty relation and is the only other alternative if the uncertainty relation holds, EPR “are thus forced to conclude that the quantum-mechanical description of physical reality given by wave functions is not complete.” However, is it possible for EPR to endorse Heisenberg’s uncertainty relation? As can be seen shortly, the answer to this question must be no.

But what forces EPR to reach their conclusion? In other words, what exactly in the EPR argument imply the incompleteness of current quantum theory? Can locality or realism imply the incompleteness of quantum mechanics? A positive answer to the above question would lead EPR to reject local realism. The EPR argument aims to question the completeness of current quantum theory based on local realism. It is the belief of EPR that a complete quantum theory within the framework of local realism is possible [3]. Considering locality or realism responsible for the incompleteness of quantum mechanics contradicts EPR’s belief. Clearly, rejecting either of these assumptions would appear entirely unacceptable to EPR.

According to current quantum theory, before any measurement is performed on a particle, its state is described by a wave-function, which is a quantum superposition, such that the commutation relation for the position operator and the momentum operator holds for this wave-function; after a measurement is performed on the particle, the wave-function collapses immediately. The EPR argument rests on the existence of a correlation between two spatially separated particles ([19] p.225). According to Einstein’s separability principle in his own incompleteness argument [1], the particles are distant components of a separable system. As distant components of a separable system, the particles always possess their autonomous, real states independent of each other. The collapses of wave-functions triggered by measurements are exactly due to what Einstein called “spooky actions at a distance”. Therefore, it is impossible for EPR to endorse Heisenberg’s uncertainty relation.

Bohr defended quantum mechanics. He raised issues concerning practical measurements related to the uncertainty relation but did not mention the correlation between the spatially separated particles [4]. Thus the contradiction in the conceptual foundations of current quantum theory cannot be explained away.

3. Bell’s Approach Revisited

Consider the following opinions: 1) regarding hidden-variables theories, what Bell’s theorem tell us is that for such theories to reproduce the statistical predictions given by current quantum theory, they must be either nonlocal or super-deterministic; the essence of Bell’s investigation was whether the perfect correlations predicted by current quantum theory could be explained locally by introducing “hidden variables”, and 2) questioning Bell’s theorem amounts to suggesting that quantum mechanics is incorrect.

Such opinions reflect a fact: there are different understandings of Bell’s theorem as shown in the literature [1,12,13,14,15,16,17,18]. But whatever Bell’s theorem tell us may have nothing to do with Einstein’s local-realist description of the world, and questioning Bell’s theorem should not be considered as saying “quantum mechanics is incorrect.” Bell’s theorem is questionable, because Bell mistook the correlation between distant components of a separable system in Einstein’s local-realist description of the world for non-locality in the world described by his theorem [2]. To address the above issues, let us begin with revisiting Bell’s approach. The issues will be further discussed in Section 8.

When deriving Bell inequalities [7,8,9,10] by resorting to a hidden-variables theory [20,21,22], Bell and his followers merely tried to reinterpret quantum mechanics while keeping the theory in its current form intact [11]. Thus Bell’s approach presumes the legitimacy of current quantum theory. Purportedly obtained from the assumptions underlying the EPR argument, Bell inequalities cannot capture the essence of the Einstein-Bohr debate, namely, the legitimacy of quantum superpositions. Einstein never endorsed any hidden-variables theory ([19], p.254). Nevertheless, Bell regarded Einstein as a proponent of hidden variables [7] and maintained his views (see Ref. /23/ in [23] and [24]). Although neither Bell nor Einstein ever mentioned the “elements of physical reality” [1], some physicists believe that there exists a linkage between hidden variables and the “elements of physical reality” [2,10]. Such a linkage is nonexistent.

Bell inequalities are not results about quantum mechanics. But the world described by Bell’s theorem is the world described by quantum mechanics, which differs substantially from Einstein’s local-realist world view. In Einstein’s local-realist description of the world, the correlation between distant components of a separable system is due to some interaction occurred before the components spatially separated; after the separation, there is no longer any interaction between the distant components. But the correlation still exists, even though the components have spatially separated [3]. Regrettably, the correlation between distant components of a separable system in Einstein’s local-realist description of the world is mistaken for nonlocal-interaction in the world described by Bell’s theorem [2], leading to serious consequences as shown below.

One of Bell inequalities, the CHSH inequality [8], has been intensively tested by actual experiments with single pairs of correlated photons using technologies of modern optics [9,10]. According to Bell’s theorem, the Einstein-Bohr debate seems to have been resolved “in the way which Einstein would have liked least” [22]. Nowadays people believe that Einstein’s local-realist world view conflicts with the experimental results of testing the CHSH inequality [2,10]. However, derived based on the nonexistent linkage between hidden variables and the “elements of physical reality”, the CHSH inequality merely represents an unsuccessful attempt to reinterpret quantum mechanics by reproducing its statistical predictions while presuming the legitimacy of quantum superpositions. Thus the CHSH inequality cannot capture the essence of the Einstein-Bohr debate, and the assumptions underlying the EPR argument as well as Einstein’s separability principle are all irrelevant to Bell’s theorem.

But why is the correlation between distant components of a separable system in Einstein’s local-realist description of the world mistaken for non-locality in the world descried by Bell’s theorem? Before answering this question, it is necessary to introduce a new principle, the general principle of measurements, which can be proved as a mathematical theorem.

4. General Principle of Measurements

Physical quantities are all measured in the real world based on mathematical models of space and time. Thus the corresponding models must be identified first before the new principle is introduced and proved mathematically. The mathematical model of space is the three-dimensional Euclidean space ${\mathbb{R}}^3$ endowed with the metric given by the usual distance function between two points in space; a point $z\in {\mathbb{R}}^3$ represents a precise space coordinate. The mathematical model of time is the set of nonnegative real numbers $R_0$ equipped with the metric given by the usual distance function between two nonnegative real numbers; an element $t\in R_0$ is a precise time coordinate.

To prove the general principle of measurements as a mathematical theorem, let us recall a few definitions in “metric space” and “point-set topology”. A metric space is denoted by $(X,d)$, where X is a set, and d is a metric on X. Let $r>0$ be a real number. For $x\in X$, the open ball with center x and radius r is

$$B(x;r)=\{y\in X:d(x,y)<r\}.$$

(1)

Any open subset of X is a union of open balls. All open subsets of X constitute a metric topology ${\mathcal{T}}_X$ for X. The metric topology ${\mathcal{T}}_X$ and X form a metric topological space. Consider $x\in S$ where $S\in {\mathcal{T}}_X$. If there exists $r>0$ such that

$$B(x;r)\cap S=\left\{x\right\},$$

(2)

then x is an isolated point of S. Denote by ${\mathcal{T}}_{{\mathbb{R}}^3}$ and ${\mathcal{T}}_{R_0}$ the metric topologies for ${\mathbb{R}}^3$ and $R_0$ associated with the metrics given by the corresponding distance functions. Apparently, measuring a point z in space perfectly precisely requires z to be an isolated point of ${\mathbb{R}}^3$. Similarly, unless time t is an isolated point of $R_0$, it is impossible to measure t perfectly precisely.

Theorem 1.

(The General Principle of Measurements): Precise space and time coordinates are practically unattainable by measurements, or equivalently, neither ${\mathbb{R}}^3$ nor $R_0$ have isolated points.

Proof. 

Consider first an arbitrarily given $z\in S$, where $S\in {\mathcal{T}}_{{\mathbb{R}}^3}$ is arbitrary. Evidently, there is no $r>0$ such that

$$B(z;r)\cap S=\left\{z\right\}.$$

(3)

Thus ${\mathbb{R}}^3$ has no isolated point. Now consider $t\in S$, where $S\in {\mathcal{T}}_{R_0}$ is arbitrary. An open “ball” now is an open interval

$$B(t;r)=(t-r,t+r).$$

(4)

There are two cases: $t=0$, and $t>0$. If $t=0$, then $B(0;r)\notin {\mathcal{T}}_{R_0}$ for any $r>0$, and there is no $S\in {\mathcal{T}}_{R_0}$ such that

$$S\cap B(0;r)=\left\{0\right\}.$$

(5)

Thus 0 is not an isolated point of $R_0$. If $t>0$, there is no $r>0$ such that

$$S\cap B(t;r)=\left\{t\right\}.$$

(6)

Thus t is not an isolated point of $R_0$ either. Consequently, $R_0$ has no isolated point. □

When physical events are considered in special relativity, the theorem proved above is also valid. Mathematically, the space-time of events in special relativity is a four-dimensional differential manifold called the Minkowski manifold, which is endowed with a topology, such that any subset of this manifold is either open or not in the usual sense. Thus the definition of an isolated point also applies to the Minkowski space-time manifold. Clearly, the Minkowski manifold has no isolated point.

Of course, real numbers and other mathematical objects constructed based on real numbers are all precisely defined. But the general principle of measurements is not about precise definitions of such mathematical objects; its significance is this: specified by precise space and time coordinates, “the same experimental conditions” are not physically meaningful. Any random phenomenon in physics can only be observed in different repetitions of a given experiment under the experimental conditions that can only be approximately the same. According to the general principle of measurements, if “the same experimental conditions” are specified by precise space and time coordinates, then such experimental conditions are physically meaningless, because precisely defined coordinates are unattainable by practical measurements. Proved as a mathematical theorem, the general principle of measurements does not involve issues concerning practical measurements raised by Bohr [4] or the accuracy of results obtained by measuring the values of space or time variables in various wave-functions.

Before the advent of quantum mechanics, physicists held a commonsense: “the same experimental conditions” always produce the same results. In other words, results produced by “the same experimental conditions” are deterministic. This commonsense is approximately true, because the experimental conditions in classical physics can be considered approximately the same, and random phenomena observed in classical physics are mainly due to lack of knowledge needed to describe physical situations that typically involve a large number of single microscopic objects; their behaviors are usually assumed to be independent. Randomness in such situations is explainable using statistical mechanics. The general principle of measurements is ignorable. Quantum mechanics changed this commonsense. Nowadays physicists hold a new commonsense: “the same experimental conditions” do not produce the same results in quantum physics, or equivalently, the results produced by “the same experimental conditions” in quantum physics are indeterministic [25]. However, this new commonsense is misleading and largely responsible for erroneously interpreted experimental results in quantum physics.

In an experiment with quantum objects of a given kind, mutually exclusive properties are actually observed in different repetitions of the experiment and correspond to different measurement outcomes associated with different objects of the same kind. But such properties are attached to an imaginary object, which does not exist in the real world as a consequence of violating the general principle of measurements by taking precise space and time coordinates for granted to specify “the same experimental conditions”. Thus the origin of indeterminism in quantum physics is concealed by “the same experimental conditions”. In various experiments involving quantum superpositions, indeterminism actually stems from violating the general principle of measurements and is not explainable using statistical mechanics.

Consider, again, the optical experiment designed to test the CHSH inequality [10]. Expressed as a quantum superposition, an “entangled state” describes the single pairs of correlated photons and is used to calculate the probabilities of obtaining the corresponding outcomes by measuring the polarizations of the correlated photons in the pairs, which implies the legitimacy of Born’s probabilistic interpretation of wave-functions. Thus the failure of the CHSH inequality is inevitable when tested by actual experiments against quantum mechanics. The “entangled state” and the optical experiment depend on “the same experimental conditions” specified by precise space coordinates, which are the corresponding points on a unit sphere $U\subset {\mathbb{R}}^3$. This sphere U should not be confused with the “Bloch sphere”, which is not contained in ${\mathbb{R}}^3$.

The points on U correspond to a) the polarizations as well as propagating directions of different photons detected in different repetitions of the experiment, and b) the orientations of the polarizers for measuring the polarizations of the photons. Taking the precise space coordinates for granted [10], the “entangled state” violates the general principle of measurements and is illegitimate. The indeterminism observed in the measurement outcomes is exactly due to lack of knowledge about the precise space coordinates used to specify “the same experimental conditions” for measuring the polarizations of different photons detected in different repetitions. Using statistical mechanics cannot explain this indeterminism.

Taking precise time coordinates for granted to specify “the same experimental conditions” can also conceal the origin of indeterminism in quantum physics and has caused much difficulty in understanding the quantum measurement problem concerning not only microscopic objects but also “macroscopic” objects [13,14]. Experiments with such “macroscopic” objects typically involve a “time ensemble” (see Figure 3 in [14]). Measurements are performed on a “macroscopic” object in different repetitions of the corresponding experiment. In each repetition, the “macroscopic” object is measured at some fixed times used to specify “the same experimental conditions”. The indeterminism due to lack of knowledge about precise time coordinates in the above situation can be analyzed similarly based on the general principle of measurements.

5. Hilbert Space in Quantum Mechanics

The mathematical setting for quantum mechanics is Hilbert space. In 1927, John von Neumann provided the first formal axiomatic definition of a general Hilbert space. Based on the axiomatically defined general Hilbert space, von Neumann further provided an axiomatic formulation of quantum mechanics as its formal foundation. The name of “Hilbert space” is in honor of David Hilbert. At the beginning of the last century, Hilbert studied the classical prototype of what is known today as a Hilbert space in his work on the theory of integral equations. In functional analysis, mathematicians now denote this space by ${\mathsf{\ell }}^2$. As shown below, the general principle of measurements allows quantum mechanics to be completed within the framework of local realism in a way consistent with the formal axiomatic definition of the general Hilbert space.

Concepts used by von Neumann to define axiomatically the general Hilbert space are all highly abstract notions and have no practical meanings. Specified by an inner product, orthogonality is a purely mathematical concept. Assigning practical meanings to orthogonality is unnecessary. Moreover, the logical relation between orthogonal vectors is not needed in the formal axiomatic definition of the general Hilbert space. Elements of ${\mathsf{\ell }}^2$, the prototypical Hilbert space, are infinite sequences of complex numbers. The logical relation between orthogonal vectors spanning ${\mathsf{\ell }}^2$ is neither conjunction (“and”) nor disjunction (“or”); it is not necessary to assign any practical meaning to the logical relation. Only for a given application, practically meaningful concepts are necessary to define a specific Hilbert space used to describe practically meaningful objects.

However, if conjunction (“and”) is the logical relation between orthogonal vectors spanning a Hilbert space, the orthogonal vectors must not correspond to mutually exclusive properties simultaneously belonging to the same object; such an imaginary object is a consequence of violating the general principle of measurements by taking precise space and time coordinates for granted to specify “the same experimental conditions”. As shown in the last section, the “the same experimental conditions” are not physically meaningful, and the imaginary object does not exist in the real world. With the inner product defined for Euclidean vectors, ${\mathbb{R}}^3$ is a Hilbert space. Orthogonal vectors spanning ${\mathbb{R}}^3$ are orthogonal only in the sense of Euclidean geometry but do not represent mutually exclusive properties simultaneously belonging to any geometric object. Thus conjunction (“and”) can serve as the logical relation between the orthogonal vectors spanning ${\mathbb{R}}^3$. The components of such vectors can be measured simultaneously. The measurements will not cause anything in ${\mathbb{R}}^3$ to collapse.

The general Hilbert space axiomatically defined by von Neumann differs from any Hilbert space in quantum mechanics. The difference between the former and the latter is that the concept of orthogonality in the latter has a specific meaning: conjunction (“and”) is the logical relation between orthonormal vectors, which purportedly represent mutually exclusive properties simultaneously belonging to the same physical object before measurements. This specific meaning assigned to orthogonality makes the axiomatic formulation of quantum mechanics questionable. But von Neumann’s formal axiomatic definition of the general Hilbert space is still valid and allows disjunction (“or”) to serve as the logical relation between orthogonal vectors spanning a Hilbert space for practical applications. Some people deny the existence of the above mentioned difference. The reader may judge whether the difference exists.

For a Hilbert space in quantum mechanics completed based on the general principle of measurements, the logical relation is disjunction (“or”). Thus, represented by orthonormal vectors spanning the Hilbert space in quantum mechanics with disjunction (“or”) serving as the logical relation, different outcomes corresponding to mutually exclusive properties of a physical object of a given kind are associated with different objects of the same kind; the objects are measured in different repetitions of a given experiment. Obtained by measuring the corresponding object, each single outcome reveals an “element of the physical reality” considered in the EPR argument under the assumptions of locality and realism [3]. Consequently, a value corresponding to the single outcome can be assigned to the object, even though the precise space and time coordinates used to measure it are unknown; the value can even be taken from a continuum and cannot be obtained by measurements, such as the position or momentum of a particle moving in space.

As shown above, based on the general principle of measurements, quantum mechanics can indeed be completed within the framework of local realism, such that the completed quantum theory is consistent with the formal axiomatic definition of the general Hilbert space without changing the mathematical setting. In von Neumann’s formal axiomatic definition of the general Hilbert space, the axioms concerning various calculations required by quantum mechanics, including the calculations of probabilities and expectation values, will all remain unchanged. However, in von Neumann’s axiomatic formulation of quantum mechanics, some axioms are questionable and should be removed. These axioms are irrelevant to the calculations and can only make quantum mechanics difficult to understand, such as those implying the purported completeness of current quantum theory, the so-called inherently probabilistic nature of observations on quantum systems, and inexplicable collapses of wave-functions triggered by measurements. The questionable axioms are responsible for bringing the existence of observers into the axiomatically formulated foundation of current quantum theory. Removing such axioms will significantly simplify the axiomatic formulation of quantum mechanics. See Section 8 for further discussion.

6. Implications of Completed Quantum Theory

With disjunction (“or”) serving as the logical relation between superposed orthonormal vectors, the notion of “quantum superposition” in the completed quantum theory will be denoted by “superposition (disjunction)”, which differs essentially from its counterpart in current quantum theory. To avoid confusion, the notion of “quantum superposition” in current quantum theory will be referred to as “superposition (conjunction)”. Violating the general principle of measurements can result in using an imaginary object described by a superpositions (conjunction) to characterize different objects measured in different repetitions. No outcome is obtained by measuring the imaginary object, which is nonexistent in the real world.

The EPR argument reveals a contradiction in the conceptual foundations of quantum mechanics, namely, the truth of Heisenberg’s uncertainty relation implies the incompleteness of current quantum theory, or equivalently, the completeness of quantum mechanics in its current form implies the falsity of Heisenberg’s uncertainty relation [3]. After quantum mechanics is completed within the framework of local realism, does the completed quantum theory imply the falsity of Heisenberg’s uncertainty relation? The answer is yes. In the real world, there is no particle described by a wave-function expressed as a superposition (conjunction), which violates the general principle of measurements and implies an inexplicable collapse of the wave-function triggered by a measurement performed on an imaginary particle. In contrast, using superposition (disjunction) to describe different particles of the same kind measured in different repetitions of the corresponding experiment, the completed quantum theory precludes inexplicable collapses of wave-functions and does imply the falsity of Heisenberg’s uncertainty relation.

In the optical experiment with single pairs of correlated photons [10], the pairs are described by the “entangled state”, which is a superposition (conjunction). For a pair of correlated photons so described, no polarization can be assigned to either photon if no polarization measurement is performed, and a measurement triggers an abrupt collapse of the “entangled state” [10]. The “entangled state” cannot describe anything physically meaningful in the real world. According to Einstein’s separability principle, either of the correlated photons in each pair possesses its autonomous polarization state independent of the other photon, and measuring the polarization of either photon cannot affect the other photon. Consequently, corresponding to the autonomous polarization states simultaneously possessed by both correlated photons (i.e., distant components) in each single pair (i.e., a separable system) to be detected jointly in the real world, each single outcome is obtained in one repetition of the experiment, even though the precise space coordinates used to detect the pair are unattainable by measurements and unknown.

This is actually Einstein’s ensemble interpretation of a wave-function [1], if the wave-function is expressed as a superposition (disjunction). Different pairs of correlated photons with their autonomous polarization states are detected in different repetitions of the experiment; they form an ensemble described by the superposition (disjunction). According to the general principle of measurements, the indeterminism exhibited in the outcomes obtained by measuring the polarization states of the photons is due to lack of knowledge about precise space coordinates and cannot be explained by statistical mechanics. Violating the general principle of measurements brings about using an imaginary pair to characterize different pairs detected in different repetitions of the experiment.

The way to complete quantum mechanics suggested in this article should not be considered as questioning the correctness of quantum-mechanically predicted probabilities of obtaining different outcomes by measurements. After quantum mechanics is completed within the framework of local realism, the axioms relevant to calculating probabilities and expectation values needed by quantum mechanics are all unchanged. This is an important difference between the completed quantum theory and any hidden-variables theory. According to Bell’s theorem [2], “a theory can achieve complete agreement with quantum mechanics only if it is non-local.” This claim is incorrect. By providing the same probabilistic predictions of empirical results as those provided by current quantum theory, the completed quantum theory can indeed achieve perfect agreement with current quantum theory. In contrast, the predictions given by Bell inequalities derived by resorting to a hidden-variables theory differ from the quantum-mechanical predictions. More importantly, the completed quantum theory is local rather than non-local.

In addition to the above theoretical implications concerning the conceptual foundations of quantum mechanics, the completed theory also has important practical implications concerning experiments in quantum physics. Such experiments usually require to construct very expensive equipments. For an experiment with quantum objects described by superpositions (conjunction), the current theory and the completed theory can both provide correct probabilistic predictions of empirical results; however, the current theory cannot correctly explain the empirical results, which may lead to wasting effort, time, and money. The completed theory can explain the the empirical results correctly and is helpful to avoid the waste. The implications will be further discussed in Section 8.

7. Correlation Versus Non-Locality

Consider the experimental tests of Bell inequalities [26] with single pairs (i.e., separable systems of the same kind) of correlated photons (i.e., the corresponding distant components), where each separable system is measured in only one repetition of a given experiment. In the interpretation of the experimental results given by Bell’s theorem, the correlation between distant components (i.e., correlated photons) of a separable system (i.e., a single pair) in Einstein’s local-realist world view is confused with nonlocal-interaction in the world described by Bell’s theorem. According to Bell’s theorem, not only local realism but also Einstein’s separability principle have to be rejected. The confusion must be clarified. The correlation between distant components of a separable system in Einstein’s local-realist description of the world differs essentially from non-locality in the world described by Bell’s theorem. The general principle of measurements can reveal the difference.

In the experimental tests of Bell inequalities [26], the “locality loophole” has been closed, which is relevant to the Einstein-Bohr debate. Other loopholes, such as those concerning detections of photons and various far-fetched interpretations of Bell’s theorem, are not fundamental. Detection loopholes cannot disprove Bell’s theorem or change the world it describes. Far-fetched interpretations of Bell’s theorem are not the experiments themselves and will not be considered here.

As implied by a condition necessary to observe any random phenomenon, the correlation between distant components of a separable system in Einstein’s local-realist description of the world is deterministic. The necessary condition is a banal fact: any single measurement makes no sense statistically. This condition is ignored in the existing literature regarding the present status of the problem. Clearly, jointly detecting the correlated photons in a single pair will produce one and only one outcome in one repetition of the corresponding experiment; indeterminism cannot manifest itself in only one repetition. Thus the polarizations simultaneously possessed by both correlated photons in each pair can be detected jointly, and the correlation between the photons in each single pair must be deterministic.

Therefore, using superpositions (disjunction) rather than superpositions (conjunction) to describe the single pairs precludes collapses of “entangled states” triggered by measurements. Indeed, if there are no superpositions (conjunction), there will be no inexplicable collapses of wave-functions triggered by measurements. Compared to the deterministic correlation, non-locality implies the legitimacy of “entangled states”, which amounts to presuming the legitimacy of current quantum theory. Without non-locality, Bell’s theorem cannot interpret the experimental results of testing Bell inequalities, which are expressed in terms of statistical correlations. By considering an example (Bertlmann’s-socks), Bell almost realized the deterministic nature of quantum correlation (see Figure 1 in [23]). Bell’s example is similar to the slightly different version of Einstein’s non-quantum-theoretical illustration of the separability principle (Table 1). But why is the deterministic correlation between distant components of a separable system in Einstein’s local-realist description of the world mistaken for non-locality in the world described by Bell’s theorem? There are three reasons.

First, Bell and his followers did not attempt to explain indeterminism in outcomes obtained by experiments involving superpositions (conjunction); they merely tried to “reinterpret quantum mechanics in terms of a statistical account of an underlying hidden-variables theory” [2], keeping quantum mechanics in its current form intact. To reproduce the statistical predictions of quantum mechanics, Bell inequalities have to be expressed by statistical correlations. Secondly, the condition necessary to observe random phenomena is ignored. According to this condition, indeterminism cannot manifest itself in only one repetition of the corresponding experiment. Finally, “entangled states” used in the experimental tests of Bell inequalities violate the general principle of measurements. Consequently, an imaginary pair is used to characterize different pairs detected in different repetitions, and mutually exclusive properties corresponding to different outcomes are attached to the imaginary pair. Eventually, the deterministic correlation is mistaken for non-locality.

8. Discussion

Quantum mechanics is one of the greatest discovers in the history of science. But its conceptual foundations have been controversial since its inception and are still debatable even today. In the Einstein-Bohr debate, two issues disturbed Einstein very much: (i) the purported probabilistic nature of quantum mechanics, and (ii) the role of observers treated as an integral part rather than a consequence of the axiomatically formulated formal foundation of quantum mechanics. Because quantum-mechanically predicted probabilities are always in good agreement with statistical data obtained by experiments, nowadays most physicists are no longer worried about the so-called probabilistic nature of current quantum theory. However, the role played by observers in the axioms of quantum mechanics is still worrisome [27].

In fact, the issues (i) and (ii) are closely related, as can be readily seen from Einstein’s illustration of his separability principle mentioned in Section 1. Einstein’s illustration is non-quantum-theoretical (see Table 2 below). There are two boxes, I and II, which may serve as two spatially separated regions. One of the boxes (say box I) contains a single ball, and box II is empty. Before an observer lifts the lid of a box and looks inside, which box contains the ball is unknown. According to the quantum-mechanical way of thinking, if the observer does not choose to lift the lid of a box and look at its content, the ball is not really in either box, and the state of each box is completely described by a probability equal to 1/2 [1]. In contrast, Einstein’s way of thinking based on his separability principle is this: the contents of the boxes are independent of one another; one of the boxes, along with everything having to do with its contents, is independent of whatever happens to the other box.

Probability appeared in Table 2 is not meaningful. If probability used here makes sense, a large number of box pairs will be needed; the boxes must be prepared by putting a single ball randomly into one of the two boxes to form an ensemble. Clearly, the contents in the two boxes are deterministically correlated: if one box contains the ball, the other box must be empty. For each single pair of boxes in the ensemble, the deterministic correlation always exists. Because of this deterministic correlation, when lifting the lid of a box and looking inside, what the observer finds in the box must be deterministic: the box is either empty or contains the ball, and from the content of the box, the observer knows immediately the content of the other box.

Table 2 shows clearly what causes the randomness and why probability is not an intrinsic property of the world. Although this illustration is non-quantum-theoretical, Einstein’s separability principle as illustrated in Table 2 implies an important hint for us to see a connection between the so-called probabilistic nature of quantum mechanics and the role played by observers in the axioms of current quantum theory. The non-quantum-theoretical illustration of the separability principle can also be readily extended and used to analyze the quantum-mechanical interpretation of the perfect correlations observed in the experimental tests of Bell inequalities with technologies of modern optics (Table 3).

In the optical experiment with single pairs of correlated photons denoted by (${\nu }_1,{\nu }_2$) [10], two parallel polarizers are spatially separated for measuring the polarizations of (${\nu }_1,{\nu }_2$) described by the “entangled state”, which is a superposition (conjunction). There are only two categories of polarizations measured for (${\nu }_1,{\nu }_2$), which are represented by (+, +) and (-, -). If no observer is involved here to perform a polarization measurement, then according to the quantum-mechanical interpretation, no polarization can be assigned to either of correlated photons in any single pair, and what can be assigned to each pair is the quantum-mechanically calculated probability. Moreover, polarization measurements performed by observers will abruptly trigger inexplicable collapses of the “entangled state” [10]. Such collapses of the “entangled state” triggered by measurements are nothing but what Einstein called “spooky actions at a distance”. All such “spooky actions at a distance” are excluded irrefutably by Einstein’s separability principle. Although the statistical correlations obtained from the quantum-mechanically calculated probabilities are in good agreement with the experimental data, it is the quantum-mechanical interpretation that not only attaches “the probabilistic nature” to quantum mechanics but also brings observers into the axioms of current quantum theory. As shown in Section 4, the quantum-mechanical interpretation violates the general principle of measurements and cannot account for indeterminism in quantum physics due to the violation. The indeterminism can be explained by the general principle of measurements.

Table 4. Polarizations of (${\nu }_1,{\nu }_2$) measured in different repetitions.

Polarizations of (${\mathit{\nu }}_1,{\mathit{\nu }}_2$) Measured in a Repetition	Polarizations of (${\mathit{\nu }}_1,{\mathit{\nu }}_2$) Measured in Another Repetition
(+, +)	(-, -)

Table 4. Polarizations of (${\nu }_1,{\nu }_2$) measured in different repetitions.

Polarizations of (${\mathit{\nu }}_1,{\mathit{\nu }}_2$) Measured in a Repetition	Polarizations of (${\mathit{\nu }}_1,{\mathit{\nu }}_2$) Measured in Another Repetition
(+, +)	(-, -)

According to Einstein’s separability principle, either ${\nu }_1$ or ${\nu }_2$ possesses its autonomous polarization state, and measuring the polarization of either photon cannot affect the other photon. Corresponding to the autonomous polarization states simultaneously possessed by both ${\nu }_1$ and ${\nu }_2$ in each single pair (${\nu }_1,{\nu }_2$) to be detected jointly in the real world, each single outcome, (+, +) or (-, -), is obtained in one repetition of the experiment as illustrated in Table 4. But (+, +) and (-, -) can never be detected for the same single pair in the same repetition. This shows again what the “entangled state” can describe is only an imaginary pair rather than any single pair in the real world.

Table 5. Einstein’s ensemble interpretation of perfect correlations.

Measured Polarizations of (${\mathit{\nu }}_1,{\mathit{\nu }}_2$) in Ensemble	Probability Assigned to (${\mathit{\nu }}_1,{\mathit{\nu }}_2$) in Ensemble
(+, +)	1/2
(-, -)	1/2

Table 5. Einstein’s ensemble interpretation of perfect correlations.

Measured Polarizations of (${\mathit{\nu }}_1,{\mathit{\nu }}_2$) in Ensemble	Probability Assigned to (${\mathit{\nu }}_1,{\mathit{\nu }}_2$) in Ensemble
(+, +)	1/2
(-, -)	1/2

Table 5 illustrates Einstein’s ensemble interpretation of the perfect correlation. Different pairs of perfectly correlated photons form an ensemble described by a superposition (disjunction). Compare Table 5 with Table 3. The comparison shows clearly the difference between the quantum-mechanical interpretation and Einstein’s ensemble interpretation; the purported probabilistic nature of the current theory is attached by the the quantum-mechanical interpretation and precluded by Einstein’s ensemble interpretation. It also shows the difference between the completed theory and the current theory; the completed theory is local-realist but can achieve complete agreement with the current theory by providing the same probabilistic predictions of empirical results as those provided by the current theory.

9. Conclusion

Inspired by Einstein, Bell and his followers intended to complete quantum mechanics within the framework of local realism [6,11]. However, they adopted a problematic approach, which cannot reveal the essential difference between the deterministic correlation in Einstein’s local-realist description of the world and non-locality in the world described by Bell’s theorem. The deterministic correlation is between distant components of a separable system that satisfies the condition demanded by Einstein’s separability principle [1]. This regrettable situation might have been avoided, had they focused on explaining indeterminism in quantum physics rather than reinterpreting current quantum theory. Described himself as a follower of Einstein, Bell hoped for better theories than our current quantum theory, insisting that the current theory was no more than a temporary expedient [6]; he would have been happy to see quantum mechanics completed within the framework of local realism without relinquishing Einstein’s separability principle.

By explaining indeterminism in quantum physics, the general principle of measurements allows quantum mechanics to be completed within the framework of local realism while keeping the formal axiomatic definition of a general Hilbert space unchanged, so the completed theory can provide the same probabilistic predictions of empirical results as those provided by the current theory. The general principle of measurements can also reveal the essential difference between the deterministic correlation and non-locality. In addition, using disjunction (“or”) as the logical relation between superposed orthonormal vectors, the completed theory precludes observers in the axioms of the current theory and inexplicable collapses of wave-functions; it is intuitively comprehensible and alleviates much difficulty in understanding quantum mechanics experienced by many people, including Bell [11]. Among various world views, Einstein’s local-realist world view is correct.