Decoherence as an Inherent Characteristic of Quantum Mechanics as

In this study, we show that it is possible to explain the quantum measurement process within the framework of quantum mechanics without any additional postulates. We do not delve into a deep discussion regarding what the measurement problem actually is, and only examine the problems that seem to exist between classical and quantum physics. Relations between quantum and classical equations of motion are briefly reviewed to show that the transition from a superposition of quantum states to an eigenstate, namely, decoherence, is necessary to ensure that the expectation values in quantum mechanics obey the classical equations of motion. Several Bell-type inequalities and the Kochen-Specker theorem are also reviewed to clarify the concepts of nonseparability and counterfactual definiteness in quantum mechanics. The main objective of this study is to show that decoherence is an inherent characteristic of quantum states caused by the quantum uncertainty relation. We conclude that the quantum measurement process can indeed be explained within the framework of pure quantum mechanics. We also show that our conclusion is consistent with the counterfactual indefiniteness of quantum mechanics.


Introduction
The measurement problem in quantum mechanics is an unresolved problem in modern physics, and a subject of considerable debate [1] [2]. The microscopic world in conventional quantum mechanics is treated differently from the macroscopic world in classical mechanics. However, these two worlds are, in fact, linked to each other. Therefore, it may seem that the present formulation of quantum mechanics is insufficient to describe nature. Nevertheless, for many For a better understanding of the discussion in subsequent sections, we first provide some background on the problems that exist between classical physics and quantum mechanics, and the theories suggested to solve them. In the present work, we examine a Stern-Gerlach-like experiment, where the spin in the z-direction of an electron S is measured, to clarify these problems.
In classical mechanics, the relation between the state of a system under investigation and the physical quantities associated with it is trivial. However, certain assumptions are needed to connect these in quantum mechanics. In its standard formulation, the eigenstate-eigenvalue link [3] [4] states that eigenvalues and eigenstates have a one-to-one correspondence except in the case of degeneracy.
Thus, in our Stern-Gerlach-like experiment, we assume that where + and − represent the eigenstates of S with eigenvalues 2 +  and 2 −  , respectively.
However, some studies have suggested that this condition is unnecessary [5].
Nevertheless, if we do not agree with the eigenstate-eigenvalue link, then it follows that standard quantum mechanics is insufficient to describe nature. To better understand this, we examine the above assumption in more detail. Note that because the assumption eigenstate → eigenvalue is a part of Born's rule [6], and discarding this assumption implies discarding Born's rule. Conversely, discarding the assumption eigenvalue → eigenstate implies that there may be states other than the eigenstate that correspond to the measured value of the concerned physical quantity. For instance, let i a be the value obtained after the measurement of a quantity. If the state obtained directly after this measurement is not the eigenstate corresponding to i a , then it is possible to obtain a measured value that is different from i a even after the first measurement. Thus, if we discard the assumption eigenvalue → eigenstate, then the standard theory of quantum mechanics appears to be insufficient.
We introduce an ideal measurement device M and assume that S and M interact by obeying the Schrödinger equation. Let n be the neutral state of M before measurement, and p and m be the states of M after measurement with measured values 2 +  and 2 −  , respectively. Subsequently, taking into account the eigenstate-eigenvalue link, an ideal measurement process Û is defined as ˆ, n U n p Let the initial state of S be ( )( ) 1 2 + + − . Then, the state obtained after ( ) ( ) 1 1 . 2 2 U n p m + + − = + + − (5) Note that either the measured value 2 +  or 2 −  is expected as a result of the measurement in Equation (5). However, the measurement process and the eigenstate-eigenvalue link contradict each other because the initial state of S is not the eigenstate corresponding to the eigenvalues 2 +  or 2 −  .
Thus, the above example shows that it is necessary to adopt a modification of standard quantum mechanics. The most traditional approach is to adopt the projection postulate [7]. As will be discussed in Section 2, the transition from a superposition of states to an eigenstate is necessary so that the expectation values in quantum mechanics obey the classical equations of motion. However, this transition cannot be described by the Schrödinger equation because it is not a unitary process. By adopting the projection postulate, the quantum states not only develop unitarily by obeying the Schrödinger equation but also non-unitarily owing to the collapse of the wave packets during the measurement process. For example, a state ψ changes to the corresponding eigenstates i a after the measurement with corresponding measured values i a . Thus, in the Stern-Gerlach-like experiment, a non-unitary change such as ( ) is permitted in addition to the unitary development represented by Equation (5). Although the projection postulate is a type of nonlocal requirement, there are several studies that demonstrate its consistency with special relativity [8]. Note that classical information cannot be transferred during the collapse of a wave packet.
In contrast to the above description, the collapse of a wave packet is not assumed in the many-worlds interpretation of quantum mechanics [9] [10]. In this interpretation, even macroscopic states maintain coherent superpositions, and therefore, the assumption that only one outcome can be obtained from one appropriate measurement process, which is usually regarded as a matter of course, is discarded. Decoherence [11] [12] may be regarded as one of the most successful theories to explain the quantum measurement process without any additional postulates. Moreover, it is frequently applied to the many-worlds interpretation [13]. Decoherence was first proposed by Zeh et al. [14] [15], which was followed by the important work of Zurek [16]. The efforts of these authors ensured that the decoherence theory was actively studied. In this theory, the observed decay of interference is explained by the interactions between the system and the environmental degrees of freedom. For example, let the state Z that represents a unified system consisting of an observed system and a measurement device be described by the superposition of the two states a and b , such that: The density matrix ˆZ ρ of the interacting system is then Thus, we observe the decay of interference in the system. Moreover, the preferred basis problem has also been studied in this framework by Zurek [17], who also presented a solution for this problem.
However, certain problems remain with the decoherence theory. First, the unitary interaction between the system and the environment does not lead to a complete removal of the interference terms. The reduced density matrix given by Equation (11) does not indicate whether the system is in state a or b because reduced density matrices represent improper mixtures [18] that only provide a probability distribution. This implies that the coherence has been delocalised into a larger system including the environment [2]. The next problem may be more severe. Note that because the coherent terms disappear after in the interaction between the system and the environment, obtaining an outcome for a superposition of states implies direct violation of the eigenstate-eigenvalue link.
Considering the above arguments, it seems necessary to remove certain natural conditions from or add certain new unnatural conditions to standard quantum mechanics to explain the measurement process. In other words, the quantum measurement problem is reduced to the problem of determining these conditions.
Note that in the present study, we do not delve into a discussion regarding what the measurement problem actually is. Instead, we define our problem in the form of the following question: is it possible to explain the measurement process within the framework of pure quantum mechanics? In this study, pure quantum mechanics refers to the theory of quantum mechanics without any additional postulates, such as the projection postulate. We demonstrate that the answer to this question is, in fact, yes; thus, we can indeed explain the measurement process within the framework of pure quantum mechanics.
The remaining paper is organised as follows. In Section 2, the relations between the quantum mechanical and classical equations of motion are briefly reviewed. It is shown that the transition from a superposition of states to an eigenstate is necessary to ensure that the expectation values in quantum mechanics obey the classical equations of motion; specifically, decoherence is necessary to connect the classical and quantum worlds.
In Section 3, several Bell-type inequalities are reviewed. The fact that these inequalities do not hold true for quantum mechanics indicates the presence of some sort of nonlocality. Although it is unlikely that the universe is nonlocal, the measurement problem of quantum mechanics is not a problem of explaining the nonlocality itself. The question we should rather ask is whether such theories of quantum mechanics are reasonable enough to explain the measurement process and experiments that demonstrate the reasonableness of its own. The Kochen-Specker theorem is also discussed in Section 3. It is well known that this theorem prohibits counterfactual definiteness of quantum mechanics. Nevertheless, it is important to consider the meaning of counterfactual definiteness in relation to the nonlocality.
Section 4 presents the main findings of this study. We show that decoherence is an inherent characteristic of quantum states caused by the quantum uncertainty relation, and it can explain the measurement process within the framework of pure quantum mechanics. It is also shown that there is no inconsistency between decoherence and the counterfactual definiteness discussed in Section 3.
Section 5 concludes the paper.

Relation between Quantum Mechanical and Classical Equations of Motion
In this section, we show that the transition from a superposition of states to an eigenstate is necessary to ensure that the quantum mechanical expectation values obey the classical equations of motion.

Schrödinger Equation, Path Integral Formulation, and Hamilton-Jacobi Equation
The time evolution of physical quantities characterising macroscopic objects is However, it can be visualised easily by adopting the path integral formulation.

Relation between Schrödinger Equation and Hamilton-Jacobi Equation
It has been shown that the phase of a wave function obeys the classical Hamilton-Jacobi equation under certain assumptions [19] [20]. Let then Equation (12) can be separated into its real and imaginary parts. The real part is given by and the imaginary part is given by Let us first consider the imaginary part in Equation (15). The right-hand side of Equation (15) multiplied by A yields Assuming that the momentum p of the particle is i i p S = ∇ , then Equation where v p m = is the velocity of the particle and This is the classical Hamilton-Jacobi equation, assuming that S is the action and S ∇ is the momentum of the particle.
Reasonableness of these assumptions based on the similarity between quantum mechanics and optics has been discussed in several textbooks [21] [22].

Path Integral Quantisation
Note that in this section, we omit the subscript i in i q for simplicity. Adopting the path integral quantisation method [23] makes the discussion in the previous section clearer. In this theory, the wave function can be described as evolving from the wave function where ( ) , L q q  is the Lagrangian and On the other hand, applying the Hamiltonian operator where the terms of ( ) or higher are ignored. We can see that and the Schrödinger equation Substituting Equations (21) and (23) into Equation (25), we obtain the Hamilton-Jacobi equation Then, classical mechanics states that F q and F Hamilton's equations of motion: As shown above, the classical equations of motion can be derived from the ψ has been lost in the above formalism; specifically, the discussion here is formal and

Ehrenfest Theorem
The Ehrenfest theorem [24] demonstrates how quantum mechanical expectation values obey the classical equations of motion. Although Ehrenfest [25] originally proposed the theorem in the Schrödinger picture, in this section we employ the Heisenberg picture. Note that because the operators corresponding to the physical quantities depend explicitly on time in the Heisenberg picture, the relation is rather straightforward. Let q and p be the expectation values of the coordinate q and momentum p , respectively; then, the equations obeyed by q and p in quantum mechanics should be the same as the classical Hamilton's equations of motion under certain conditions. Let us suppose that a quantum mechanical system is described by coordinates and their respective conjugate momenta where Ĥ is the Hamiltonian operator and [ ] , represents a commutator. Because the commutator between ˆi q and ˆj p is , .
Next, substituting Â with ˆi q and ˆi p in Equation (29) and using Equations (31) and (32), we obtain the evolution equations for the respective expectation values as dˆ, We note that the above equations are in the form of Hamilton's equations of motion in classical mechanics.
Note that the assumptions in Equations (35) and (36)

Bell-Type Inequalities and the Kochen-Specker Theorem
In this section, we review Bell-type inequalities and the Kochen-Specker theorem.
These theorems indicate that the world described by quantum mechanics is considerably different from the macroscopic world we are familiar with.
It has been experimentally confirmed [

EPR-Bohm Experiment
First, we introduce the Einstein-Podolsky-Rosen (EPR)-Bohm experiment [33] [ 34], in which the spins of two spin 1/2 particles, labelled as 1 and 2 in Figure 1  ( ) are the spin eigenstates in the z direction.
These satisfy where ˆz σ is the spin operator in the z direction. The spin operator in the r direction, which is perpendicular to the direction of the particles' motion (i.e. along x direction) and makes an angle 2θ with the z direction in the yz-plane, is given by cos sin cos sinˆ.
sin cos sin cos The eigenstates of ˆr which satisfy the following equations: By using Equation (43), C can be rewritten as ( ) Let us suppose that two observers simultaneously perform measurements on the spins of particles 1 and 2 along the directions z and r, respectively. Rewriting respectively.
It should be noted that these equations hold true for the measurements in which the angle between the spins of the two particles is 2θ .

Derivation
We examine the EPR-Bohm experiment in which the spin of particle 1 along directions a or a' and the spin of particle 2 along directions b or b' are measured. Let A, A', B, and B' be the measured values of the spin divided by 2  , respectively. In this section, we assume separability: the measurement of the spin of particle 1 never affects the measurement of the spin of particle 2 and vice versa. We also assume counterfactual definiteness: the spin value is determined separately before the measurement. In other words, each spin is assumed to have a definite value even if the measurement has not actually been performed. Thus, A, A', B, and B' assume a definite value +1 or −1 each.
Based on these assumptions, we define the quantity M as ( ) ( ).

M AB AB A B A B A B B
Note that because one of the terms on the right-hand side of Equation (52) is always 0 and the other term is always +2 or −2, M is always +2 or −2. Therefore, the average value M obtained after several measurements of M satisfies This is the CHSH inequality [35].
From inequality (57), it follows that the absolute value of Equation (58) satis- Note that the fact that the sum of the spins of particles 1 and 2 is zero when a b = has not been used in the derivation of inequality (60). If we take this into account and put b a ′ ′ = , then we can substitute 1 A B ′ ′ = − into inequality (60) to obtain This was the first inequality proposed by Bell in 1964 [37].

Contradiction between CHSH Inequality and Quantum Mechanics
In this section, we show that quantum-mechanical expectation values violate the CHSH inequality (53).
We calculate the quantum-mechanical expectation value M of M defined in Equation (52) using Equations (48) to (51). Let a z = and , α β , and γ be the angles that directions , a b ′ , and b′ make with the z direction in the yz-plane, respectively. The expectation value of the first term of M then becomes Therefore, we can conclude that quantum-mechanical expectation values violate the CHSH inequality (53).

Wigner's Inequality and Its Variations
Note that we again assume counterfactual definiteness and nonseparability in the derivation of the inequalities in this section.
Using these equations, we obtain Recalling that probability cannot be less than zero, we obtain Wigner's inequality [38], such that ( ) ( ) ( ) In case 1), it can be shown trivially that the probability of the spins of the two particles being different is 1. In case 2), the corresponding probability is 1/3. Therefore, the probability that the spins of particles 1 and 2 measured in different directions have different signs exists between 1 and 1/3. This can be expressed as ( ) ( ) However, when all the angles between a, a′ and a′′ are 2 3 π , then ( ) ( ) where at least one term on the right-hand side must be 1/6 or more. However, when all the angles between a, a′ and a′′ are 2 3 π , every term on the right-hand side of condition (88) is calculated to be 1/8 using Equations (49)

Kochen-Specker Theorem
In the previous section, we derived a series of Bell-type inequalities assuming not only separability but also counterfactual definiteness. Because quantum mechanical expectation values violate these inequalities, it implies that quantum mechanics must be non-separable or counterfactually indefinite or both. Interestingly, the Kochen-Specker theorem [42] solely requires the assumption of counterfactual definiteness.
Using FUNC, we derive a sum rule for the commuting operators Â and B in an N-dimensional Hilbert space. We also define an operator Ĉ , whose eigenvectors include all the vectors that form an orthonormal basis by simultaneously diagonalising Â and B . Ĉ can be expanded as 1ˆ, where i c represents the eigenvalues of Ĉ and ˆC Note that the second and fourth steps in Equation (95) where Î is the identity operator. By applying the sum rule (given by Equation P must be 1 and the remaining 0 to satisfy Equation (99). In summary, if it is possible to assign every physical quantity a definite value satisfying FUNC, then one of the projection operators onto its eigenstate must be assigned a definite value 1 and the remaining must be assigned a definite value 0. Kochen and Specker [42] proved that it is not possible to assign such definite values to all physical quantities.
We reconsider the assumption FUNC. In FUNC, it is not possible to know which physical quantities are being observed simultaneously. Although it may seem natural that the definite values satisfy FUNC, Bell [43] insisted that contextual definite values do not need to satisfy FUNC. This implies that the Kochen-Specker theorem does not deny the existence of contextual definite values.
We can say that the spin in each direction has a contextual definite value in the EPR-Bohm experiment introduced in Section 3.1, provided the definite value of the spin in the r direction of particle 2 varies in accordance with the direction of the spin of particle 1, which was measured simultaneously. Furthermore, it is known that quantum mechanics must be nonlocal under such an assumption [44] [45].

Nonlocality and Weak Values
In view of the discussion in the previous sections, we need to check whether the counterfactual definiteness assumed during the derivations of the Bell-type inequalities is local. We consider the case in which the observed values are determined by local hidden variables, as shown in Equation (54). Note that Equation (58) is necessary for the derivation of CHSH inequality, and the density matrices for AB and AB' must be the same to satisfy Equation (58). Therefore, we can conclude that the counterfactual definiteness assumed in this case is local. Thus, nonlocal counterfactual definiteness (i.e. existence of contextual definite values) is not forbidden in the above discussion. Conversely, if quantum mechanics is counterfactually indefinite, then we need to introduce nonlocality to explain the correlation between EPR pairs. Therefore, quantum mechanics must be nonlocal irrespective of whether it is counterfactually definite or indefinite, although it may seem difficult to appreciate this fact.
Weak measurements [46] clearly demonstrate the nonlocality of quantum

Decoherence as an Inherent Characteristic of Quantum Mechanics
As stated in the Introduction, most studies in this field have yielded a negative answer to the question regarding whether the measurement process can be explained within the framework of pure quantum mechanics. However, in this section, we demonstrate that the answer is affirmative, that is, we can indeed explain the measurement process within the framework of pure quantum mechanics. We propose a new decoherence theory, in which the uncertainty of microscopic objects gives rise to decoherence as an inherent characteristic of pure quantum mechanics. Note that because this decoherence exists prior to a measurement, the eigenstate-eigenvalue link can be maintained. Moreover, we do not intend to explain the nonlocality or counterfactual indefiniteness of quantum mechanics by means of other concepts, as it is beyond the scope of this work. However, we do attempt to illustrate how the measurement process can be understood within the scope of pure quantum mechanics.
We examine three experiments in this section. First, in Section 4.1, we examine a Stern-Gerlach-like experiment with an electron to illustrate our idea of decoherence. In Section 4.2, we apply our theory to an EPR pair of electrons and show that the correlation between spatially separated particles is not a result of wave packet collapse or similar processes. In Section 4.3, the double-slit experiment with electrons is examined to demonstrate the effectiveness of our theory for cases with continuous eigenvalues. This experiment also illuminates how pure quantum mechanics is able to demonstrate that electrons behave as interfering particles, that is, particles whose detection rate is consistent with interference.

Stern-Gerlach-Like Experiment
We examine a Stern-Gerlach-like experiment with an electron S whose spin is measured in the z direction, as shown in Figure 2. Note that + and − are the eigenstates of the electron with eigenvalues 2 +  and 2 −  , respectively. S initially travels along the x-axis and enters an inhomogeneous magnetic field along the z-axis, where a magnetic force acts on it. When S exits the magnetic field, its momentum in the z direction is −p if its spin is 2 +  and +p if the spin is 2 −  . We define the momentum eigenstates p − and p + with eigenvalues −p and +p, respectively. Furthermore, 0 is defined as the momentum eigenstate having 0 momentum. Note that because the spin and momentum operators along the same direction commute with each other, the state of S can be described as a simultaneous eigenstate of these two operators. We also define a unitary operator ˆS U that represents the interaction between S and the magnetic field, such that ˆ0 , ˆ0 . Next, we consider the uncertainty relation. Because we want to measure S's momentum in the z direction, we must allow some uncertainty in its position in the same direction. To account for this uncertainty, we introduce the density matrix ( ) S ρ ζ of S translated to a distance ζ in the z direction, which is defined as ∆ is the uncertainty in S's position along the z direction and from Equation (105) We want to know what is observed by the macroscopic detector; hence, we set which leads to Therefore, the average density matrix, which describes the state of S to be detected, loses its interference terms and becomes ( )  (103) but is either p + − or p − + . In this calculation, we have not used any additional postulates such as the projection postulate. Thus, it is worth noting that this decoherence is not a result of the interaction between S and the detector or other environmental factors, but is rather due to the uncertainty relation.
Note also that the density matrix of S is not always written as shown in Equation (112). If we want to measure another observable of S, then we need to take an average the density matrix over its conjugate observables. Consequently, we can obtain the average density matrix that has a diagonal form in this observable, as illustrated in the remainder of this section.
Let us suppose that an ideal macroscopic measuring device M that measures S's energy is employed instead of the aforementioned detector. Furthermore, S is prepared in its neutral state r and is at either of the two energy levels E + and E − , in accordance with its spin, after it exits the magnetic field. V is a unitary operator that transforms r to E + or E − , which are the eigenstates whose eigenvalues are E + and E − , respectively. Thus, we have The initial state of the unified system is represented by | J 〉 , which is defined as ( ) Then, the state of S just prior to detection by M can be written as and its density matrix Note that because we want to measure the energy of S, we require a time interval. Therefore, we define the density matrix ˆJ av ρ averaged over the mea- To obtain a macroscopic result, we set ( ) , which leads to ( ) Therefore, the average density matrix in this case becomes ( ) Thus, we observe that equation (126) is diagonal in S's energy.

EPR-Bohm Experiment
In this section, we re-examine the EPR-Bohm experiment, which was briefly reviewed in Section 3.1. We adopt the same setup as in the previous section, where particles 1 and 2 enter inhomogeneous magnetic fields along the z and r directions, respectively. When the particles exit the magnetic fields, each particle gains a momentum in its respective direction of the magnetic field. Initially, neither particle has a momentum in the z or r direction. If we define the zero momentum state of the particles as 0 i , then the state C I before entering the magnetic field can be written as 1 2 0 0 , where C is defined in Equation (39) where the unitary operator ˆC U represents the interaction, and p and q are the momenta in the z and r directions, respectively. As discussed in the previous section, we must allow some uncertainty in the position of the particles along the corresponding directions. Therefore, the density matrix ˆC av ρ that describes the state to be measured is defined as ( ) where ( ) , C ρ ζ ξ is the density matrix of particle 1 translated to a distance ζ in the z direction and particle 2 translated to a distance ξ in the r direction. By  Conversely, we need to consider the relation between the discussions in this and the previous section. Note that because we can obtain only one average density matrix for an observed pair of electrons, as discussed earlier, we do not need to assume that the state prior to the measurement possesses a definite value of the spin in each direction. In other words, the probable counterfactual definiteness is not local but contextual. As discussed in the previous section, contextual counterfactual definiteness is not forbidden by Bell-type inequalities or the Kochen-Specker theorem. The spin in either direction is not fixed in the initial state

Conclusions
In this study, we demonstrated that decoherence is one of the inherent characteristics of pure quantum mechanics. Quantum states describe both the wave as well as particle behaviour of microscopic objects, specifically, the microscopic objects propagate as waves and are observed as particles. In this context, it is worth paying careful attention to the meaning of the phrase observing the wave nature of a state. It is equivalent to saying identifying the state propagating as a wave by observing many particles. As shown in Section 4.3, there is essentially no difference between observing the electrons on the screen and near the slit. Irrespective of where the electrons are observed, they are observed as particles, which helps us determine the amplitude of the corresponding state. Therefore, we conclude that the quantum measurement process can be explained within the framework of pure quantum mechanics. Moreover, we believe that our study can be applied to a more general discussion about the quantum-to-classical transition in nature.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.