On Feasibility of Quantum Computation and Quantum Communication

1. Introduction

Bell tests, namely, the experimental tests of Bell inequalities [1,2,3,4], are intended to resolve the Einstein-Bohr debate on the conceptual foundations of quantum mechanics [5,6]. According to the opinion of most physicists, Bell’s theorem is the standard interpretation of the test results [1,7,8]. According to Bell’s theorem, the Einstein-Bohr debate seems to have been resolved “in the way which Einstein would have liked least” [9]. But some researchers consider Bell’s theorem questionable [10,11]. Although the Einstein-Bohr debate has not been resolved yet as we shall expound shortly, the Bell tests and Bell’s theorem opened the door to quantum computation and quantum communication [12]. In this study, we investigate the feasibility of quantum computation and quantum communication. The results are formulated as two propositions.

Proposition 1.

Experimentally confirmed statistical predictions of quantum mechanics are not evidence of experimentally realized quantum information processing systems.

Proposition 2.

Physical carriers of quantum information coded by quantum bits (qubits) do not exist in the real world.

Based on a well-established mathematical fact in point-set topology, we can justify the propositions and show that Einstein’s ensemble interpretation of $\psi$-function not only will eliminate inexplicable weirdness in quantum physics but also can help us see clearly none of quantum objects in the real world carries quantum information. By justifying the propositions, we can also clarify confusion caused by a widespread opinion in a popular text [13]. According to this opinion, quantum information processing cannot be precluded unless quantum mechanics fails to be correct. As we shall see, it is not appropriate to connect quantum information processing with quantum mechanics. An inevitable conclusion then follows: Without carriers representing quantum information, physical implementations of quantum information processing systems, such as quantum computers, quantum communication networks, quantum cryptography, quantum key distribution, and quantum teleportation, are merely an unrealizable myth. The above conclusion cannot be found in the literature; the conclusion actually contradicts relevant results in the literature. While all relevant results published nowadays support quantum information processing, the present work shows that quantum information processing systems depend on the assumption that physical carriers representing quantum information exist, but the assumption does not hold as we shall see in the rest of this paper. The purpose of this paper is to question published results that support quantum information processing. Thus, it is not appropriate to include such results in the reference list of the present work without questioning them.

In the rest of this paper, we first expound why the Einstein-Bohr debate remains unsettled and why the unresolved debate suggests that Bell’s theorem is problematic (Section 2). Then we justify the propositions (Section 3). Next, we examine the Bell tests and revisit Einstein’s ensemble interpretation of $\psi$-function (Section 4). After illustrating the reported results with examples (Section 5), we discuss briefly the generalization of the results (Section 6) and conclude the paper (Section 7).

2. Unresolved Debate and Bell’s Theorem

As we can see from [14] and from Ref./23/ in [15], Bell and his followers regarded Einstein as a proponent of hidden variables and tried to reinterpret quantum mechanics while keeping its current form intact by resorting to hidden-variables theories [7,9,16]; thus, the Bell tests presume the legitimacy of quantum superposition. Regarding Einstein as a proponent of hidden variables is based on oral tradition transmitted by Podolsky ([17] p. 254). This oral tradition ascribes to Einstein the belief that future research would reveal significant discrepancies between quantum mechanics and experience. Bell’s understanding of the oral tradition is “Einstein expected quantum mechanics ultimately to come in conflict with experiment.” See [14]. His understanding motivated the Bell tests. However, the oral tradition transmitted by Podolsky may not truthfully reflect the historical fact. The fact is that Einstein did not write the EPR paper. Podolsky wrote the paper. Einstein never endorsed any hidden-variables theory ([17], p.254) and argued against the legitimacy of quantum superposition in his debate with Bohr. Einstein’s argument differs from the EPR argument ([18], p. 178). It seems that Bell and his followers were not aware of this historical fact. As indicated by the above fact, the Einstein-Bohr debate is actually irrelevant to the Bell tests. Therefore, the debate still remains unsettled.

Bell and his followers intended to reproduce statistical predictions of quantum mechanics in the Bell tests but failed. The failure of the Bell tests is an irrefutable experimental fact. Nowadays nobody doubts the correctness of experimentally confirmed quantum-mechanical predictions. However, Bell’s theorem erroneously links Einstein’s argument to the failure of the Bell tests. Misguided by such nonexistent linkage, most physicists consider Einstein’s argument wrong. Also misguided by the nonexistent linkage, experimental physicists and quantum-information theorists have been attempting to realize physically unrealizable quantum information processing systems. As we shall see in the next section, quantum mechanics is a powerful tool for making statistical predictions on empirical results and has nothing to do with quantum information processing. Because Einstein’s argument is irrelevant to the failure of the Bell tests, the unresolved debate suggests that Bell’s theorem is problematic.

For the purpose of this study, we need only consider ideal experiments, focus on identical observables of individual quantum-mechanical systems taken from a pure ensemble, and adhere to the following terminology for ease of exposition.

• The term “repetitions” means “repeating a fixed experiment multiple times under the same experimental conditions”.
• Quantum objects that are described by the same pure state and will be measured in different repetitions of a given experiment are called “identically prepared quantum-mechanical systems”.
• An ensemble consisting of identically prepared quantum-mechanical systems is called “a pure ensemble”.
• An experiment with quantum objects taken from a pure ensemble is called “an ideal experiment”.

Considering pure ensembles allow us to avoid distraction caused by anything that is not essential here. Thus, we can better appreciate the essence of the Einstein-Bohr debate. It is worth emphasizing that any single measurement performed in only one repetition of a fixed experiment can only produce a deterministic result and makes no sense statistically. This is simply a banal fact. This fact can be observed not only in classical physics but also in quantum physics. For example, if we toss a coin only once, then we see only one of two possible outcomes: either a head or a tail. It is necessary for us to toss the coin at least twice if we want to observe two different outcomes, which reminds us that tossing a coin is a random experiment. Similarly, if we use an idealized polarizer to measure the polarization of a single photon in one repetition of an experiment as show in [19], then only one of two possible outcomes can be observed: the photon is either transmitted or not transmitted. Unlike a coin, which can be tossed repeatedly, each photon can only be measured once in one repetition of the experiment. The two different outcomes can only be obtained by performing polarization measurements on at least two photons in at least two repetitions. As shown above, in both classical physics and in quantum physics, a single outcome is not statistically meaningful and must be a deterministic result. This is why we need to explain the meaning of “repetitions” explicitly in the terminology.

3. Quantum Mechanics and Quantum Information

In quantum physics, physicists somehow treat continuous observables and continuous parameters differently. The continuous parameters are involved in specifying experimental conditions. As everybody knows, precise values of all continuous quantities cannot be obtained by measurements; their values are unattainable. However, physicists consider precise values of continuous parameters attainable. To see this, let us quote the late A. Peres ([20], p.39): “Quantum tests may depend on classical parameters which can be varied continuously, and nevertheless these tests have fixed, discrete, outcomes.” Peres also gave two examples of continuous parameters used to specify experimental conditions. One example is the angle of orientation of a calcite crystal used to test the linear polarization of a photon; the other is the angle of orientation of a Stern-Gerlach magnet. The precise values of the angles are elements of $\mathbb{R}$, the set of all real numbers. We shall examine these two examples in Section 5. While experimental conditions specified by precise values of continuous parameters allow us to obtain meaningful results in classical physics, specifying experimental conditions using precise values of continuous parameters is misleading in quantum physics as we shall see below based on a concise point-set topological analysis. As a prelude to justifying the propositions formulated in Section 1, let us first recall a few definitions in point-set topology.

Denote by ${\mathbb{R}}^3$ the three-dimensional Euclidean space, which is the mathematical model of the space in which we live and measure all measurable quantities. The range of a quantity is the set of all values the quantity can have. To avoid overusing mathematical notations, we use the same symbol G to denote both the range of a continuous parameter and the range of a continuous observable. Let X be a topological space. A metric is defined on X. The metric is identical to the usual distance function. If X is the three-dimensional Euclidean space, the distance function is defined on ${\mathbb{R}}^3$. If X is the real line, the distance function is defined on $\mathbb{R}$. Suppose a one-to-one correspondence exists between elements of G and elements of an uncountable set $S\subset X$. Let $s_0$ be an element of S.

Definition 1.

If there is a number $r>0$ such that the distance between $s_0$ and any other point of S is at least r, then $s_0$ is an isolated point of S.

A well-established mathematical fact in point-set topology follows immediately from Definition 1. We state this fact as a lemma.

Lemma 1.

The set S has no isolated points.

An attainable element of G implies the existence of an isolated point of S, thus contradicting Lemma 1. Concerning experiments in quantum physics, the practical implication of Lemma 1 is self-evident; we state it as a theorem. The theorem looks trivial, but it is still worth emphasizing that continuous quantities mentioned in the theorem include both continuous observables and continuous parameters.

Theorem 1.

Precise values of all continuous quantities are unattainable.

In the quoted examples ([20], p.39), the observables are not continuous quantities. In Heisenberg’s position–momentum uncertainty relation, the position and momentum of a particle are continuous observables. Neither the position nor the momentum can be measured perfectly accurately. It is Theorem 1 that prohibits us from obtaining precise values of the position or momentum. Theorem 1 also prohibits us from obtaining precise values of continuous parameters. Ironically, as shown in the quoted examples, physicists and quantum-information theorists only consider precise values of continuous observables unattainable while considering precise values of continuous parameters attainable and capable of specifying experimental conditions in the Bell tests.

To deliberate further the practical implication of Lemma 1, which is stated as Theorem 1, let us first consider continuous observables. Precise values of continuous observables are unattainable by measurements, because the mathematical model of any continuous observable, such as the set of all real numbers or the Euclidean space, does not have isolated points. Were this not true, a precise value of a continuous quantity, which corresponds to an isolated point of the mathematical model of the observable, could be obtained by measurement. Now consider a continuous parameter involved in specifying experimental conditions of a fixed experiment. For the same reason as we have seen in the case of continuous observables, namely, the mathematical model of any continuous quantity does not have isolated points, precise values of continuous parameters are unattainable by measurements. Therefore, the absence of isolated points does imply the unattainability of continuous quantities, including both continuous observables and continuous parameters.

As a common practice in quantum physics, physicists use a fictitious quantum-mechanical system to represent all the members of a pure ensemble, because they believe experimental conditions specified by precise values of continuous parameters are exactly the same in different repetitions of a given experiment [20,21]. The following corollary forbids such practice; the corollary is an immediate consequence of Theorem 1.

Corollary 1.

Neither experimental conditions specified by precise values of continuous parameters nor fictitious quantum-mechanical systems exist in the real world.

Surely, a prerequisite of using a fictitious quantum object is to keep the experimental conditions unchanged in different repetitions of the corresponding experiment. As we have seen, the prerequisite does not hold. We can also justify this result based on Einstein’s ensemble interpretation of $\psi$-function as shown in Section 5. Consequently, it is not legitimate to use the fictitious object. The use of fictitious quantum objects is largely responsible for Bell’s erroneous interpretation of the test results. The above analysis paves the way for us to justify the propositions formulated in Section 1. Note that some kinds of fictitious quantum objects are considered as physical carriers of quantum information coded by qubits in quantum information theory.

In any given experiment with quantum objects taken from a pure ensemble, experimental conditions specified by precise values of any continuous parameter do not exist in the real world, and outcomes of measuring identical observables of the quantum objects correspond to mutually exclusive properties represented by orthogonal vectors spanning a Hilbert space. The Hilbert space is the mathematical setting for quantum mechanics, and the outcomes are obtained in different repetitions of the experiment. Attaching the outcomes measured in different repetitions to a fictitious quantum object, physicists unable to understand the exhibited random phenomena. Thus Bell and many other physicists interpreted the random phenomena exhibited in the Bell tests as inherent randomness. Einstein disliked such interpretation; this is why Einstein questioned current quantum theory by calling it “the fundamental dice-game” ([22], p.149).

In fact, the random phenomena are due to lack of knowledge about precise values of the continuous parameters used to specify the experimental conditions. Therefore, the existence of purported inherent randomness can be excluded. In current quantum theory, the logical relation between the orthonormal vectors is conjunction (“and”). Experimentally confirmed statistical predictions of quantum mechanics are correct, because we can replace conjunction (“and”) with disjunction (“or”). Using disjunction (“or”) to replace conjunction (“and”) can keep quantum-mechanically calculated probabilities unchanged while not modifying the mathematical setting essentially. As a powerful tool for predicting empirical results, quantum mechanics is irrelevant to quantum information processing. In quantum information theory, fictitious quantum objects are supposed to carry quantum information coded by qubits, but none of such objects exists in the real world; it is misleading to use a fictitious quantum object to represent all the members taken from a pure ensemble. Therefore, there is no evidence to support experimentally realized quantum information processing systems. By examining one of the Bell tests in the next section, we shall address the issues further and show that Einstein’s ensemble interpretation of $\psi$-function can remove inexplicable weirdness in quantum physics. As we shall see in Section 5, Einstein’s ensemble interpretation can also exclude the existence of physical carriers of quantum information.

4. Bell Tests and Einstein’s Ensemble Interpretation of $\mathbf{\psi }$-Function

Consider the experimental test of the CHSH inequality [2]. Because this inequality is well-known, we need not give its expression or repeat its derivation here. Like other Bell inequalities, the CHSH inequality is not a result about quantum mechanics [13], but it can be tested by real experiments using technologies of modern optics [3,4]. In experiments with single pairs $({\nu }_1,{\nu }_2)$ of correlated photons, experimental physicists have intensively tested this inequality against quantum mechanics [3,4]. In the test, each pair can be detected once in only one repetition. The photons are analyzed by two linear polarizers. For simplicity, the polarizers are assumed to be perfectly efficient.

The polarization part of the state vector consists of ${\displaystyle \frac{1}{\sqrt{2}}}|x,x\rangle$ and ${\displaystyle \frac{1}{\sqrt{2}}}|y,y\rangle$, which are the superposed components, where $|x\rangle$ and $|y\rangle$ are linear polarizations states. With conjunction (“and”) being the logical relation between the superposed components, a quantum superposition expresses an entangled state, see Figure 1 in [12].

$$|\Psi ({\nu }_1,{\nu }_2)\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left\{\right|x,x\rangle +|y,y\rangle \}.$$

The entangled state $|\Psi ({\nu }_1,{\nu }_2)\rangle$ describes each pair. Physicists also use $|\Psi ({\nu }_1,{\nu }_2)\rangle$ given above to calculate quantum-mechanical probabilities of obtaining the corresponding outcomes by measuring the polarizations of the photons. In the experiment designed to test the CHSH inequality, the parameters specifying the experimental conditions can be varied continuously. The values of the parameters are coordinates of points on a unit sphere $E\subset {\mathbb{R}}^3$.

The unit sphere E is not the “Bloch sphere” in quantum information theory [13]. The Bloch sphere is not contained in ${\mathbb{R}}^3$ and should not be confused with E; they are not isomorphic mathematical objects. An isomorphism is a correspondence between two isomorphic mathematical objects. The correspondence preserves all the structures in question. If two mathematical objects are isomorphic, the objects are essentially two different representations of the same object and can be used interchangeably. The Bloch sphere in quantum information theory presumes the legitimacy of qubits. But E does not have any structure possessed by the Bloch sphere. Clearly, the Bloch sphere has nothing to do with any subset of the Euclidean space. By no means can these two objects be considered the same thing.

The coordinates of some points on the unit sphere E specify the polarizations and propagating directions of different photons detected in different repetitions of the experiment; the coordinates of some other points specify the orientations of the polarizers used to analyze the polarizations of the photons. When interpreting the experimental results, the experimental physicists take precise coordinates for granted and believe they can keep the experimental conditions specified by the coordinates exactly the same in different repetitions of the experiment. Thus, the random phenomenon observed in this Bell test is interpreted as inherent randomness. However, by Theorem 1, the precise coordinates of points on E are unattainable. The observed random phenomenon is actually due to lack of knowledge about the coordinates used to specify the experimental conditions and should not be characterized as inherent randomness. Again, the existence of purported inherent randomness can be excluded.

For a pair of correlated photons described by $|\Psi ({\nu }_1,{\nu }_2)\rangle$ with conjunction (“and”) being the logical relation between ${\displaystyle \frac{1}{\sqrt{2}}}|x,x\rangle$ and ${\displaystyle \frac{1}{\sqrt{2}}}|y,y\rangle$, physicists claim that no well-defined state can be ascribed to each photon, because they do not know how to assign any polarization to either photon [4]. This claim is based on the legitimacy of quantum superposition. As a historical fact, Einstein argued against the legitimacy of quantum superposition in his debate with Bohr. When calculating the quantum-mechanical probabilities, physicists rely on the reduction postulate in current quantum theory and assert that any polarization measurement performed on a photon in a pair will trigger an abrupt collapse of $|\Psi ({\nu }_1,{\nu }_2)\rangle$ [4]. The collapse of $|\Psi ({\nu }_1,{\nu }_2)\rangle$ in such a telepathic way is a typical situation unspeakable in current quantum theory. So long as the logical relation between the superposed components is conjunction (“and”), the entangled state $|\Psi ({\nu }_1,{\nu }_2)\rangle$ cannot describe anything physically meaningful in the real world.

In contrast, according to Einstein’s argument based on his separability principle [18], the correlation between the photons in each pair is determined by the common source that emits the photons, and either of the correlated photons in each pair possesses its autonomous polarization state independent of whatever happened non-locally, and measuring the polarization of either photon cannot affect the other photon in any way. Consequently, corresponding to the autonomous polarization states simultaneously possessed by both correlated photons in each pair to be detected jointly in the real world, the outcome $(+,+)$ or $(-,-)$ is obtained in one repetition of the experiment, even though the precise coordinates used to detect the pair are unattainable by measurements and unknown. However, we can never detect $(+,+)$ and $(-,-)$ for the same pair in the same repetition. The above deliberation is actually based on Einstein’s ensemble interpretation of $\psi$-function. In Einstein’s ensemble interpretation, different pairs of correlated photons constitute an ensemble. The autonomous polarization states of the correlated photons in each pair are detected in different repetitions of the experiment. For each detected pair taken from the ensemble, the logical relation between ${\displaystyle \frac{1}{\sqrt{2}}}|x,x\rangle$ and ${\displaystyle \frac{1}{\sqrt{2}}}|y,y\rangle$ is disjunction (“or”), which will remove inexplicable telepathic collapse of $|\Psi ({\nu }_1,{\nu }_2)\rangle$ triggered by measurements. Such collapse is what Einstein called “spooky actions at a distance” in his debate with Bohr ([22], p.158). We still need probabilities to describe the observed random phenomenon, and the probabilities in Einstein’s ensemble interpretation are exactly the same as those quantum-mechanically calculated probabilities in current quantum theory.

Will Einstein’s ensemble interpretation make the interference pattern in a double-slit experiment disappear? Of course not. In a double-slit experiment, a single spot on a screen produced by a single quantum object cannot form an interference pattern; a large number of spots produced by a large number of quantum objects constitute the interference pattern. The pattern is described by the graph of a probability density function (pdf). In the experimental test of the CHSH inequality, single pairs of correlated photons are not described by the graph of the pdf used to describe the interference patten in the double-slit experiment.

To conclude this section, let us compare Bell’s interpretation of the test result with Einstein’s ensemble interpretation. In Bell’s interpretation, inexplicable telepathic collapse of $|\Psi ({\nu }_1,{\nu }_2)\rangle$ is claimed to be a property of the real world, and “in the way which Einstein would have liked least”, the Einstein-Bohr debate seems to have been resolved [9]. In sharp contrast to Bell’s interpretation, there is no inexplicable weirdness in Einstein’s ensemble interpretation; everything is intuitively understandable and nothing is illusive. As we shall see in the next section, the correlation between the two photons in each single pair is deterministic.

5. Examples

A qubit is a quantum superposition with conjunction (“and”) being the logical relation between the orthonormal vectors or superposed components in question. Some experiments with two-level quantum systems are explained in quantum information theory as evidence for physical carriers of quantum information coded by qubits. Were such explanation physically meaningful, different measurement outcomes might be observed in the corresponding experiment with the experimental condition specified by precise values of a continuous parameter remaining exactly the same in different repetitions of the experiment. The outcomes correspond to mutually exclusive properties, which are possessed by a two-level quantum system and represented by the orthogonal vectors or superposed components. Clearly, the explanation contradicts Lemma 1, the well-established mathematical fact in point-set topology.

With disjunction (“or”) serving as the logical relation between the orthonormal vectors or superposed components, Einstein’s ensemble interpretation of $\psi$-function not only will remove the inexplicable weirdness in the experiments but also can help us see clearly none of quantum objects in the real world carries quantum information. All the purported physical carriers of quantum information are fictitious quantum systems based on the legitimacy of quantum superposition in current quantum theory. Such fictitious quantum systems correspond to nothing whatever in the real world; Einstein’s ensemble interpretation can eliminate all of them. Thus quantum information has no physical carriers. For readers who are unfamiliar with point-set topology, concrete examples given in this section may alleviate difficulty in understanding the above results.

Bell and his followers wanted to reproduce statistical predictions of quantum mechanics in the Bell tests. Because inequalities make sense only for numbers, different measurement outcomes are treated as numbers, namely, $+1$ and $-1$, in the derivation of Bell inequalities and other relevant mathematical expressions. Instead of treating the outcomes as numbers, we use different symbols to represent the outcomes. By doing so, we can easily see the probability of “keeping the experimental condition specified by a precise value of a continuous parameter unchanged in different repetitions of a given experiment” equals zero. Furthermore, treating the outcomes as numbers is not only unnecessary but also misleading. Because of such treatment, deterministic quantum correlation is confused with the so-called quantum entanglement. Without treating the outcomes as numbers, we can avoid the confusion.

Let a and b stand for two different measurement outcomes observed in a given experiment with two-level quantum systems. The systems are all taken from a pure ensemble. The different outcomes correspond to mutually exclusive properties of each system. As explained in the last paragraph, it is not necessary for us to treat the outcomes as numbers. Unlike the systems studied in quantum physics, classical objects such as coins do not possess mutually exclusive properties simultaneously. Thus, different outcomes of tossing a coin can of course be represented as two different numbers. Let different repetitions of an experiment in quantum physics be labeled by $n=1,2,\dots$. In the n-th repetition, $d_n\in \{a,b\}$ is the outcome obtained by measurement, and $c_n$ is the precise value of a continuous parameter used to specify the experimental condition in question. For $n=1,2,\dots$, we cannot obtain $c_n$ by measurement; they are all unknown quantities. In contrast to the unknown quantities $c_n$, the outcomes a and b can still be observed by measurement in the experiment, and the set

$$M=\{(c_n,d_n):n=1,2,\dots \}$$

consists of two disjoint subsets A and B given below.

$$A=\{(c_{n_j},a):j=1,2,\dots \},B=\{(c_{n_k},b):k=1,2,\dots \}.$$

Namely,

$$M=A\cup B,A\cap B=\varnothing .$$

Interpreted in probability theory, A and B are two mutually exclusive events that cannot occur simultaneously. In deed, if $A\cap B\ne \varnothing$, then for some j and k, $(c_{n_j},a)=(c_{n_k},b)$. While $c_{n_j}$ and $c_{n_k}$ are unknown quantities, they might happen to be identical. But $(c_{n_j},a)=(c_{n_k},b)$ implies $\left\{a\right\}=\left\{b\right\}$, which is absurd. Actually, $\left\{a\right\}\ne \left\{b\right\}$ always holds. The meaning of $\left\{a\right\}\ne \left\{b\right\}$ is self-evident: a and b are not identical; they are two different things. Even if an event of probability zero, namely, “$c_{n_j}=c_{n_k}=c$ for all j and k”, somehow occurs, we still have $\left\{a\right\}\ne \left\{b\right\}$. The meaning of “$c_{n_j}=c_{n_k}=c$ for all j and k” is exactly “keeping the experimental condition specified by a precise value of a continuous parameter unchanged in different repetitions of a given experiment.” It is impossible for us to pick c out of a continuous range of possibilities by measurement. We already justified this result based on the point-set topological analysis in Section 3.

Therefore, it is impossible for any system to possess mutually exclusive properties simultaneously as required by physical carriers of quantum information coded by qubits, whether we measure the system or not. The above probabilistic-theoretical analysis is based on Einstein’s ensemble interpretation of $\psi$-function; it also shows why the experimental condition specified by precise values of a continuous parameter cannot remain exactly the same in different repetitions of the experiment. While “$c_{n_j}=c_{n_k}=c$ for all j and k” is an event of probability zero, quantum-information theorists insist that the logical relation between the orthonormal vectors or superposed components should be conjunction (“and”) and claim that $\left\{a\right\}=\left\{b\right\}$ holds if no measurement is performed. They believe this is the way Nature works and do not consider $\left\{a\right\}=\left\{b\right\}$ absurd, even though they have never seen anything like $\left\{a\right\}=\left\{b\right\}$ in the real world.

Example 1.

In quantum information theory, photons are claimed to be highly stable carriers of quantum information coded by qubits [13]. Let us consider an experiment with single photons taken from a pure ensemble. In this experiment, a photon is described by a quantum superposition with conjunction (“and”) being the logical relation between two superposed polarization states. A polarizer (i.e., a crystal) is used to measure the linear polarization of each photon. Precise values of a continuous parameter are supposed to specify the experimental condition. The parameter is the angle of the polarizer axis relative to the polarization plane [20]. For this experiment, $\left\{a\right\}=\{+\}$ with “+” representing “a single photon is transmitted,” and $\left\{b\right\}=\{-\}$ with “−” standing for “the photon is not transmitted.” In the real world, the photons are measured in different repetitions of the experiment; each photon can at most be detected only once. Quantum-information theorists take the experimental condition specified by precise values of the angle for granted and are unaware that the experimental condition cannot be the same in different repetitions. Consequently, they use a fictitious photon to represent all the photons in the experiment. While photons are all real physical objects, there is no such fictitious photon in the real world. Therefore, photons are not physical carriers of quantum information.

Example 2.

Consider the famous Stern–Gerlach experiment with single spin-1/2 particles taken from a pure ensemble. In this experiment, a spin-1/2 particle is described by a quantum superposition with conjunction (“and”) being the logical relation between two eigenvectors spanning a Hilbert space. Similar to Example 1, precise values of a continuous parameter are supposed to specify the experimental condition. The parameter is the angle of orientation of a Stern-Gerlach magnet [20]. Like single photons in Example 1, the particles are measured in different repetitions of the experiment; each particle can at most be measured only once. For this experiment, $\left\{a\right\}=\{\uparrow \}$ with “↑” representing “spin up”, and $\left\{b\right\}=\{\downarrow \}$ with “↓” representing “spin down”. In quantum information theory [13], the experimental condition specified by precise values of the angle is considered exactly the same in different repetitions, and quantum-information theorists use a fictitious particle to represent all the particles in the experiment. Although the particles are all real physical objects, the fictitious particle does not exist in the real world and cannot be a physical carrier of quantum information.

Example 3.

In quantum physics, the energy of a particle is quantized to discrete energy levels. Consider a single particle confined to a rigid one-dimensional box with impenetrable walls. If only two lowest energy levels are considered, the particle is a two-level system described by a time-independent wave-function. The wave-function is a quantum superposition with the logical relation between the superposed components being conjunction (“and”). In current quantum theory, physicists need perform an experiment to decide whether the particle can have different energies at the same time. Instead of considering only one particle, physicists must consider a large number of particles taken from a pure ensemble, and a precise value τ of a continuous time variable is supposed to specify the experimental condition in different repetitions of the experiment. In this experiment, $\left\{a\right\}=\left\{e_0\right\}$ and $\left\{b\right\}=\left\{e_1\right\}$ representing two lowest energy levels. By measuring the particles at the purported same time τ in different repetitions of the experiment, physicists assert that the particles possess different energies simultaneously. Based on this assertion, quantum-information theorists then claim that each particle taken from the ensemble is a physical carrier of quantum information coded by a qubit [13]. Both the assertion and the claim are wrong. The particle can indeed have different energies. But in no sense can the same particle have different energies at the same time. Different energies observed in the experiment actually belong to different particles measured in different repetitions. The particle is a real physical object, possesses different energies, but does not carry quantum information.

The above three examples do not touch upon the so-called quantum entanglements. Actually, quantum entanglements should be called quantum correlations; the two notions differ essentially. Just like correlations in classical physics, quantum correlations are deterministic. Bell used two examples to illustrate his understanding of quantum correlations. One example goes as follows [14]: “Suppose I take from my pocket a coin and, without looking at it, split it somehow down the middle so that the head and tail are separated. Suppose then that, still without anyone looking, the two different pieces are pocketed by two different people who go on different journeys. The first to look, finding that he has head or tail, will know immediately what the other will subsequently find. Are the quantum mechanical correlations any different? Indeed they are not, according to Einstein, if I have understood him correctly. In the example of the coin, the head and the tail were head and tail all along, even while hidden. The person who first looked was just the first to know. But in fact everything was determined from the handing over the pieces (and even before, in fully deterministic classical theory). It is by not explicitly containing the ‘hidden variables’ reading already head or tail, (or ‘up’ or ‘down’), before observation, that quantum mechanics makes a mystery of a perfectly simple situation.” Another example, “Bertlmann’s socks” [15], is similar to the coin example. Bell’s examples actually show that he was quite close to realizing the deterministic nature of quantum correlations, as we shall see below based on a slightly different version of his own coin example.

Example 4.

Suppose the procedure used by Bell is repeated for a large number of coins. Without looking, the side of head and the side of tail of each coin is separated. Still without anyone looking, the two different pieces, which are correlated deterministically, are pocketed by two different messengers. The messengers then go on different journeys, delivering the pockets to Alice and Bob, respectively. When finding she has the piece of head or tail, Alice will know immediately what Bob will subsequently find. Similarly, if Bob finds the piece of head or tail, he will immediately know what Alice will find. According to Einstein, quantum correlations are not different from such correlations in classical physics. But where does the randomness come from? Caused by lack of knowledge about the procedure used here to separate the coins and to pocket the pieces, the randomness can be explained classically, just like explaining the randomness in an experiment of tossing a coin repeatedly. Unlike Einstein, Bell eventually did not awake to the deterministic nature of quantum correlations, because he seemed not to know the randomness cannot manifest itself in only one repetition. Similar to the randomness caused by lack of knowledge about Bell’s procedure in this slightly different version of his coin example, the randomness exhibited in a Bell test is due to lack of knowledge about precise values of a continuous parameter used to specify the experimental condition. It is impossible for the experimental condition to remain the same in different repetitions of the Bell test.

6. Discussion

The analysis of experiments with two-level quantum systems suggests a way to generalize the results illustrated by the corresponding examples in the previous section. For any quantum-mechanical system described by a quantum superposition with conjunction (“and”) being the logical relation between the orthonormal vectors or superposed components in question, if physicists want to decide whether the system can possess mutually exclusive properties at the same time as illustrated in Example 3, they must perform an experiment and rely on a continuous time variable to specify the experimental conditions in different repetitions of the experiment. We distinguish two situations. At the purported same time, the exclusive properties observed in different repetitions of the experiment either belong to different microscopic systems that constitute a pure ensemble, or belong to the same macroscopic system that can be repeatedly measured in different repetitions of a time ensemble [19]. In both situations, the microscopic systems and the macroscopic system are indeed real physical objects, possess mutually exclusive properties at different times, but cannot have such properties at the same time as required by physical carriers of quantum information. While the macroscopic system can be measured repeatedly, the measurement outcomes obtained in different repetitions of the time ensemble correspond to the exclusive properties and will only be observed at different, unknown times. Of course, physicists can imagine a fictitious macroscopic system that might possess exclusive properties simultaneously; the system is analogous to Schrödinger’s cat described by a quantum superposition in current quantum theory [23].

$$|\mathrm{Schr}ö{\mathrm{dinger}}^{’}\mathrm{s}\mathrm{cat}\rangle =p|\mathrm{Schr}ö{\mathrm{dinger}}^{’}\mathrm{s}\mathrm{cat}\mathrm{is}\mathrm{alive}\rangle +q|\mathrm{Schr}ö{\mathrm{dinger}}^{’}\mathrm{s}\mathrm{cat}\mathrm{is}\mathrm{dead}\rangle .$$

But such a fictitious system is not a real physical object; it only exists in the imagination of physicists rather than in the real world. The above argument, which is based on Einstein’s ensemble interpretation, can be readily generalized to eliminate any purported physical carrier of quantum information.

As shown in the previous sections, inexplicable weirdness in quantum physics is due to the notion of quantum superposition with conjunction (“and”) being the logical relation between the orthogonal vectors or superposed components. Based on a recent experiment, some experimental physicists claim that a macroscopic quantum state exists in the real world; the state is analogous to Schrödinger’s cat: “a macroscopic object that defies intuition because it involves a superposition of classically distinct trajectories” [24]. Quantum-information theorists might interpret the purported macroscopic object as evidence for the physical existence of qubits. However, if we replace conjunction (“and”) with disjunction (“or”), we can easily get rid of anything that defies intuition, meanwhile, everything else in quantum mechanics will remain unchanged. By doing so, we can make quantum mechanics intuitively understandable. While physicists believe quantum mechanics is counterintuitive, an intuitively understandable quantum theory is not impossible. A famous philosopher once said: “Everything that can be thought at all can be thought clearly.” To conclude our discussion, let us say something similar: “Everything that can be understood at all can be understood intuitively.”

7. Conclusion

Bell’s theorem interpreted the results of the Bell tests erroneously, which opened door to quantum information processing, such as quantum computation and quantum communication. In this study, we investigated the feasibility of quantum computation and quantum communication. The investigation is based on the well-established mathematical fact in point-set topology, see Lemma 1 in Section 3. The findings are as follows. (a) Experimentally confirmed statistical predictions of quantum mechanics are not evidence of experimentally realized quantum information processing systems. (b) Physical carriers of quantum information coded by qubits do not exist in the real world. (c) Einstein’s ensemble interpretation of $\psi$-function will eliminate not only inexplicable weirdness in quantum physics but also all the purported physical carriers of quantum information. A regrettable conclusion then follows inevitably: Without carriers representing quantum information, physical implementations of quantum information processing systems are merely an unrealizable myth.