Destructive Interference as a Path to Resolving the Quantum Measurement Problem

Abstract

Over the past several decades, there has been an accelerating trend to ever more accurate quantum sensors: sensors of time intervals (i.e., atomic clocks), sensors of magnetic fields (i.e., quantum magnetometers), and sensors of inertial motions (i.e., atom interferometers), to name just a few. With this trend has come a renewed interest in the problem of quantum mechanical measurement (i.e., collapse of the wavefunction), and though there have been many attempts to resolve the problem, there is still no wholly accepted resolution. Here, we discuss a little-explored path for resolving the issue that exploits wavefunction phase. To illustrate this path’s potential, we consider the notion of “eigenphase” sets that are disjoint among orthogonal eigenvectors. Wavefunction collapse then occurs because of constructive/destructive interference when a classical measuring device “phase-locks” to an incoming wavefunction. While the present work examines one method for exploiting wavefunction phase, its primary purpose is to more generally re-focus attention on wavefunction phase as a means for resolving the measurement problem that avoids many other solutions’ problematic aspects.

1. Introduction

Over the past several decades, there has been an accelerating trend towards the creation of quantum sensors: atomic clocks for sensing time intervals [1,2], atom interferometers for sensing non-inertial motions [3], quantum magnetometers [4], Rydberg-atom [5] and atomic candle [6,7] sensors of microwave fields, and many others. In a very real sense, “quantum engineering” is emerging as a vibrant cousin to electrical and mechanical engineering, and with this trend comes a closer scrutiny of quantum mechanical measurement as technologists question the ultimate limits and limitations of quantum sensors. Of these questions, there is no more important issue than the “measurement problem” (i.e., collapse of the wavefunction), which is receiving renewed attention in the wider physics community [8,9].

To illustrate the significance of the issue, one need only consider NASA’s Deep Space Atomic Clock [10], where an atomic superposition state created by a microwave field collapses to a hyperfine eigenstate of ¹⁹⁹Hg⁺ as the ion absorbs a UV photon from an rf-discharge lamp. This device can assess the frequency of that microwave field out to the 15th significant figure and is already easily capable of sensing relativistic planetary alterations of the metric. As space-qualified atomic clocks continue to improve [11], researchers will likely need to better understand how quantum measurement plays out on a changing metric background during fly-by missions of massive astronomical objects.

The measurement problem in quantum mechanics consists principally of two parts: (1) No matter the initial wavefunction of an object, the result of a “strong” measurement [12] on that object always results in the collapse of the wavefunction into one of the eigenstates associated with the measuring apparatus. (2) Once the wavefunction collapses, it remains collapsed regardless of the object’s initial wavefunction [13]. To clarify the first issue by way of example, consider the Stern–Gerlach experiment where spin-1/2 atoms are passed through an inhomogeneous magnetic field [14,15] and assume that the apparatus is arranged to measure the atom’s spin along the z-axis of the experimental coordinate frame (i.e., the eigenstates of the measuring apparatus correspond to ∣+〉 and ∣−〉 relative to the z-axis). Regardless of the initial wavefunction ∣Φ〉 of the atoms entering the apparatus (e.g., ∣Φ〉 = a₊∣+〉 + a₋∣−〉), the result of the strong measurement will always be a manifestation of either ∣+〉 or ∣−〉. The measurement will never correspond to some superposition of measuring-apparatus eigenstates, even though that may have been the initial state of the object’s wavefunction.

The second part of the measurement problem states that the collapse is irreversible. In a sense, the complexity of the object is reduced by the strong measurement. Prior to the collapse, while in a superposition state, there was complexity to the object’s z-axis nature: it was partly spin-up and partly spin-down. After the collapse that complexity is lost; the atom is either all z-axis spin-up or all z-axis spin-down.

The reason that quantum-state measurement is often viewed as problematic has to do with the “schizophrenic” split (to quote from DeWitt [16]) that it creates in the foundational nature of change in quantum mechanics. On the one hand, in the absence of measurement, a wavefunction evolves in time under the action of a Hamiltonian in a deterministic manner: von Neumann’s T2 process [17]. However, when a strong measurement takes place, the wavefunction evolves discontinuously; it evolves in an irreversible manner, and its evolution is probabilistic as opposed to deterministic with probabilities given by the Born rule [18]: von Neumann’s T1 process [17]. How one resolves these two quite distinct natures of change into a coherent whole has been a foundational problem in quantum mechanics since its beginning [19], and the success or failure of various attempts to solve the problem is still largely a matter of opinion.

In this paper, we want to revisit a solution pathway to the measurement problem first touched on by Bohm [20] and then more carefully pointed out by Pearle [21]. Bohm noted that every measurement involves an interaction, and that through the interaction the combined quantum-entity/measuring-device wavefunction develops a random phase. On average, this random phase keeps measuring-device “pointer states” solely associated with their corresponding quantum-particle eigenstates. Unfortunately, in Bohm’s analysis, these random phases do nothing to resolve the problem of wavefunction collapse. Pearle took the phase idea a bit further, arguing that if the phase of a wavefunction were a random variable, then along with a nonlinear Schrödinger equation this random phase would lead to wavefunction collapse. Here, we also focus on wavefunction phase as central to the measurement problem. However, different from Pearle, we take quantum mechanics as linear.

Prior to discussing our wavefunction-phase pathway, we present in the next section a very brief review of some of the present leading candidates for solving the measurement problem, considering their advantages as well as their problematic aspects. We then turn to some preliminary issues: the notion of global phase in textbook quantum mechanics and the very general concept of measurement itself. Following these preliminaries, we discuss one wavefunction-phase pathway as a concrete illustration of the pathway’s broader potential for resolution of the measurement problem and its ability to circumvent other solutions’ problematic aspects.

2. Some Attempts to Solve the Measurement Problem

2.1. Copenhagen Interpretation

The standard solution to the measurement problem routinely found in textbooks follows from the Copenhagen Interpretation of quantum mechanics. Briefly, the wavefunction is interpreted solely as a computational device that provides probabilities for the results of experimental outcomes; it is not indicative of a quantum entity’s actual nature [22]. When a measurement occurs, new information regarding the quantum entity is obtained. In the Copenhagen Interpretation, the wavefunction does not really collapse during a strong measurement. Rather, measurement provides new information regarding the quantum entity, and consequently the tool for predicting further experimental outcomes (i.e., the wavefunction) must be updated. Thus, in the Copenhagen Interpretation, there is no measurement problem. What we call collapse of the wavefunction is nothing more than the manifestation of one of the entity’s stochastic potentialities and an updating of the tool for future computation. (To be clear, the Copenhagen Interpretation does not imply that the quantum object has some well-defined eigenstate prior to measurement; rather, in the Copenhagen Interpretation, it is meaningless to talk about the state of a quantum object prior to measurement, since, by definition, no measurement has taken place to determine the object’s state.)

Though pragmatic in the sense that the Copenhagen Interpretation allows the practicing physicist to proceed with their use of quantum mechanics without worry of foundational issues, it does nonetheless have detractors. Many practicing physicists would likely rebel against the view of the wavefunction as nothing more than a computational tool. They would likely see the wavefunction as an actual reflection of a quantum entity’s nature. Some might even argue that they do in fact “touch” the wavefunction through (for example) atom interferometry experiments [23,24] and weak measurements [25].

2.2. Many Worlds Interpretation

Another solution to the measurement problem originated with Everett as the Many Worlds Interpretation of quantum mechanics [26]. In this interpretation, the total wavefunction of the object/observer (i.e., object/measuring-device) system is viewed as an entangled superposition state of object eigenvectors and their corresponding measurement-device eigenvectors. It is not that the object’s wavefunction collapses, but rather that each of the object/observer potentialities is realized, with each realization manifesting itself as a splitting of the universe into a set of disconnected branches. There is one observer for each branch of the splitting, and that observer measures one specific manifestation of the quantum object’s multitude of potentialities.

As with the Copenhagen solution, the Many Worlds solution to the measurement problem has its supporters and detractors. On the one hand, it neatly avoids the dualism of von Neumann’s T1 and T2 processes: there are only T2 processes, and the appearance of T1 processes is an artifact of an observer confined to one branch of a multiverse. However, for many, this requires the uncomfortable acceptance of a multiverse, with branches rapidly multiplying and forever isolated from one another.

2.3. Decoherence Theory

In recent years, attention has been directed towards the Decoherence Theory of measurement [27]. In any measurement of a quantum entity by a laboratory measuring apparatus, we are dealing with a macroscopic device that is, at heart, an open quantum system coupled to the environment. Thus, as the quantum entity interacts with the measuring device, the near-infinite degrees of freedom the environment brings into the problem result in a very rapid loss of coherence among linearly superposed eigenvectors in the entity’s wavefunction. Consequently, the density matrix describing a sequence of measurements on the quantum entity is always diagonal; there is zero probability of having a measurement that represents a superposition of eigenvectors. Thus, in Decoherence Theory, collapse of the wavefunction is an observational consequence of measuring quantum entities with macroscopic devices coupled to the environment.

Again, as with the other solutions to the measurement problem, Decoherence Theory has its adherents and detractors. On the one hand, it recognizes that ideal measurements, isolated from the world around them, exist primarily in the physicist’s imagination. Actual measurements with real measuring devices always have some coupling to the world at large, and, in Decoherence Theory, that coupling plays a central role in what can and cannot be observed and measured. Critics, of course, could argue that Decoherence Theory only “sweeps the measurement problem under the rug,” since it relies on a general very rapid loss of coherence that by rights should be clearly articulated and defined in every measurement situation. Further, critics could reasonably argue that if one could decouple a macroscopic measuring device from the environment at large, then one could find the measuring device in a superposition of measuring-apparatus eigenstates. One might also argue that Decoherence Theory creates an epistemological problem for quantum mechanics, since it is only the density matrix that reflects experimental outcomes. Consequently, it can be argued that it forces the physicist to accept the density matrix’s Liouville equation as more foundational than the Schrödinger equation.

2.4. Spontaneous Collapse Models

Spontaneous Collapse Models start from the assumption (even if not specifically articulated) that there is some random field permeating the universe, which cannot be a standard quantum field. This field couples to “quantum matter through an anti-Hermitian and nonlinear coupling” [28]. At random times, this coupling causes the wavefunction to localize and thereby results in the collapse of the wavefunction into one of its eigenstates.

Though logically consistent, and amenable to experimental investigation, many would argue that the requirement for a nonlinear Schrödinger equation is at a minimum distasteful. Furthermore, requiring a non-quantum field that permeates all of space, likely with a cosmological origin, opens a host of new problems related to the structure and evolution of the universe.

2.5. General Character of Solutions to the Measurement Problem

In each of these potential measurement problem solution cases, the solution should be seen as having a dyadic nature. For one part of the solution, there is a statement regarding textbook quantum mechanics: either textbook quantum mechanics requires modification, or it does not. For the second part, there is a measuring “protocol” statement. For example, in the Many Worlds interpretation, there is no specific change to textbook quantum mechanics, but there is a measuring protocol statement regarding the introduction of an unknowable multiverse. In Spontaneous Collapse Models, nonlinearity is introduced into textbook quantum mechanics, while the measurement protocol hypothesizes the operation of a quantum-entity/non-quantum-field interaction. In Decoherence Theory, textbook quantum mechanics is taken without modification, and the measuring protocol is the interaction of a heat-bath with the coupled quantum-entity/measuring-device system. Even in the Copenhagen Interpretation, there is what amounts to an epistemological protocol: “one cannot know without measurement; therefore, it is meaningless to talk about a quantum entity existing in a superposition or entangled state.”

2.6. Overview of Wavefunction Phase as a Path to Resolution of the Measurement Problem

There are, of course, many other suggested solutions to the measurement problem [29,30] with some of these distinct from those mentioned above and some novel modifications [31,32,33,34,35]. Here, we want to re-open discussion of a path to resolution of the measurement problem that avoids many of the issues listed in the preceding and follows from ideas touched on by Bohm [20] and introduced by Pearle [21]: a wavefunction-phase path.

To demonstrate the potential of this pathway, we focus on the global phase invariance of operator eigenvectors. Rather than taking global phase invariance as continuous (modulo two pi), we allow each eigenvector to have a set of unique phases (of extremely large cardinality) with the sets disjoint between eigenvectors: no phase appearing in one set equals the phase appearing in an orthogonal eigenvector’s phase set. This does not destroy the U(1) symmetry of the eigenvectors since all measurements remain invariant with respect to a U(1) transformation of the wavefunction.

Similar to other solutions to the measurement problem, we introduce a measurement protocol: during the measurement process, the classical measuring device “phase-locks” to the incoming quantum particle. With this slight change to textbook quantum mechanics (eigenphase sets) and the measurement protocol (quantum-particle/measuring-device phase-locking), wavefunction collapse occurs as a consequence of constructive/destructive interference between the macroscopic measuring device and the quantum particle.

Advantages to this solution pathway are several. Contrary to the Copenhagen Interpretation, the wavefunction is seen as an object that describes the actual nature of a quantum object, and in addition the wavefunction-phase solution does not require postulating a rapidly expanding multiverse. The wavefunction-phase pathway also gives primacy to the Schrödinger equation, which is (arguably) different from Decoherence Theory. Importantly, the wavefunction-phase pathway to resolution of the measurement problem takes quantum mechanics as linear, which all experiments to date support. There is no nonlinear aspect to the fundamental equations of Nature, nor is there a need for a cosmological random field that couples to matter as in Spontaneous Collapse Models.

We also note that the wavefunction-phase pathway is not a Hidden Variables Theory [36] as will be partially demonstrated subsequently. A wavefunction’s global phase is a fundamental element of standard quantum theory. It is in no way an extra parameter added into quantum mechanics and manifests itself in countless atomic interference experiments (e.g., atomic clocks and atom interferometers). An excellent experimental/technological example of this is provided by atomic clocks based on Coherent Population Trapping (CPT) [37]. With CPT, two routes of optical excitation within an atom are simultaneously excited. However, since the two routes (i.e., matrix elements) have different phases, they destructively interfere, which results in no optical absorption though both optical excitation routes are resonantly excited.

3. Some Preliminary Considerations

3.1. Wavefunction Phase in Standard Quantum Mechanics

It is well known that the solution of the Schrödinger equation is not unique. There are an infinite number of solutions, distinguished from one another by a global phase. Specifically, if Φe^iα is a solution to the Schrödinger equation (with α a constant), then so is Φe^iβ with α ≠ β. This is to be distinguished from the fact that the evolution of every quantum entity plays out in spacetime, which gives the wavefunction a spacetime dependence. Specifically, with every wavefunction, there is typically a phase factor of the form ${\mathrm{e}}^{-\mathrm{i}{\mathrm{k}}_{\mathsf{\mu }}\left({\mathrm{x}}^{\mathsf{\mu }}-{\mathrm{x}}_{\mathrm{o}}^{\mathsf{\mu }}\right)}$, where the x^μ are spacetime coordinates. It is worth noting ${\mathrm{e}}^{-\mathrm{i}{\mathrm{k}}_{\mathsf{\mu }}\left({\mathrm{x}}^{\mathsf{\mu }}-{\mathrm{x}}_{\mathrm{o}}^{\mathsf{\mu }}\right)}={\mathrm{e}}^{-\mathrm{i}{\mathrm{k}}_{\mathsf{\mu }}{\mathrm{x}}^{\mathsf{\mu }}}{\mathrm{e}}^{\mathrm{i}\mathsf{\phi }}$, where x_o^μ is routinely taken as the origin of an arbitrarily defined coordinate system. As such, x_o^μ is one manifestation of a wavefunction’s global phase.

All of this is routinely illustrated in quantum textbooks through the example of a free particle’s wavefunction:

$$\mathsf{\Phi }\left(\mathrm{r},\mathrm{t}\right)={\mathsf{\Phi }}_{\mathrm{o}}{\mathrm{e}}^{\mathrm{i}\mathsf{\alpha }}{\mathrm{e}}^{-\mathrm{i}{\mathrm{k}}_{\mathsf{\mu }}{\mathrm{x}}^{\mathsf{\mu }}}={\mathsf{\Phi }}_{\mathrm{o}}{\mathrm{e}}^{\mathrm{i}\mathsf{\alpha }}{\mathrm{e}}^{-\mathrm{i}\left(\overrightarrow{\mathrm{k}}\cdot \overrightarrow{\mathrm{r}}-\mathsf{\omega }\mathrm{t}\right)},$$

(1)

where we have explicitly included the global phase α. In many experiments and quantum devices, the k_μx^μ phase factor plays an important role, such as in atomic clocks where it leads to Ramsey’s separated oscillatory fields method for atomic signal generation [38,39]. However, in these devices and experiments, the global phase factor e^iα plays no role.

As a textbook illustration of the usual irrelevance of the global phase factor, consider a perturbation V that causes transitions between the eigenstates of a Hamiltonian. The transition rate between the initial state ∣i〉 and the final state ∣f〉, Γ_if, is given by Fermi’s Golden Rule:

$${\mathsf{\Gamma }}_{\mathrm{i}\mathrm{f}}={\displaystyle \frac{2\mathsf{\pi }}{\hslash }}{\left|\left.〈\mathrm{f}\right|\mathrm{V}\left|\mathrm{i}\right.〉\right|}^2\mathsf{\rho }\left({\mathrm{E}}_{\mathrm{f}}\right),$$

(2)

where ρ(E_f) is the density of final states. If we allow these two orthogonal eigenvectors to have different phases: α_i and β_f, Equation (2) becomes

$${\mathsf{\Gamma }}_{\mathrm{i}\mathrm{f}}={\displaystyle \frac{2\mathsf{\pi }}{\hslash }}{\left|\left.〈\mathrm{f}\right|\mathrm{V}\left|\mathrm{i}\right.〉{\mathrm{e}}^{\mathrm{i}\left({\mathsf{\alpha }}_{\mathrm{i}}-{\mathsf{\beta }}_{\mathrm{f}}\right)}\right|}^2\mathsf{\rho }\left({\mathrm{E}}_{\mathrm{f}}\right)={\displaystyle \frac{2\mathsf{\pi }}{\hslash }}{\left|\left.〈\mathrm{f}\right|\mathrm{V}\left|\mathrm{i}\right.〉\right|}^2\mathsf{\rho }\left({\mathrm{E}}_{\mathrm{f}}\right).$$

(3)

The global phase factor has dropped out of the transition probability.

The same will be true for collision cross-sections or any other observable. Consequently, in computations, it is routine to set the global phase factor of an eigenvector to zero. With regard to present considerations, all of this illustrates the tremendous freedom within standard quantum mechanics regarding an eigenvector’s global phase. No matter how one chooses to define a wavefunction’s global phase, the results of textbook quantum mechanics are unaltered.

3.2. Measurement

As discussed more fully in Appendix A, a measurement is very generally a mapping from the entity to be measured (i.e., the measurand) to a categorical manifestation of the measured entity (often quantifiable). In other words, we can define the measuring process through a mapping operator that projects the measurand’s state onto a measuring-device “response state.” Of course, for a strong quantum measurement [25], which is the only type we consider here, the mapping irreversibly couples the measurand and measuring device. In other words, the wavefunction of the measurand collapses.

It is also important to note that physical measurement processes are one-to-one: one quantum entity gives rise to one measuring-device response. For example, in the detection of a single photon, conservation of energy requires the generation of one photodiode current-pulse equivalent to the photon’s energy (independent of any measuring-device gain). In the destructive detection of a single charge-carrying particle, conservation of charge requires the measuring device to create one charge-equivalent response (again prior to any measuring-device gain).

Here, we capture this notion of measurement through a mapping operator M that projects the combined initial state of the quantum wavefunction and measuring device to some final state. To keep the discussion focused, we restrict our considerations to a two-state quantum entity interacting with a three-state measuring device. The quantum measurand can therefore be in the “down” state ∣−〉 or the “up” state ∣+〉, or it can be in some down/up superposition state. (We can also think in terms of a photon’s polarization, with the down state corresponding to horizontal polarization and the up state to vertical polarization.) The measuring device can be in the neutral state ∣m₀〉 (e.g., producing no measurement response), the down measurement response state ∣m₋₁〉, or the up measurement response state ∣m₊₁〉; for each of these measurement response states, there is a unit-valued, measurement pointer: A_q, which is the measuring device’s output for a single quantum-measurand detection. Thus, we write M as

$$\mathbf{M}=\left[\begin{array}{cc}\sqrt{{\mathrm{A}}_{+1}}\left|{\mathrm{m}}_{+1}\right.〉\left|+\right.〉\left.〈+\right|\left.〈{\mathrm{m}}_0\right|& 0\\ 0& \sqrt{{\mathrm{A}}_{-1}}\left|{\mathrm{m}}_{-1}\right.〉\left|-\right.〉\left.〈-\right|\left.〈{\mathrm{m}}_0\right|\end{array}\right].$$

(4)

Note that similar to Decoherence Theory [27] and Bohm’s analysis of measurement [20], there is no phase coherence among the measuring-device’s response states (i.e., the off-diagonal elements are zero). This is consistent with the coupling of a classical, macroscopic measuring device to an environmental heat-bath. Of course, loss of phase coherence among the measuring-device’s response states can also occur via incoherent coupling of the quantum entities making up the measuring device to one another (e.g., the individual atoms or domains making up a classical magnet).

As an example of this measuring operator’s application, consider the incoming wavefunction in a superposition state:

$$\left|\mathsf{\Phi }\right.〉=\left[{\mathrm{a}}_{+}\left|+\right.〉{+\mathrm{a}}_{-}\left|-\right.〉\right].$$

(5)

The initial state of the measurand/measuring device is ∣Φ〉∣m₀〉, so that the mapping operator acting on this wavefunction yields

$$\mathbf{M}\left|\mathsf{\Phi }\right.〉\left|{\mathrm{m}}_0\right.〉{=\mathrm{a}}_{+}\sqrt{{\mathrm{A}}_{+1}}\left|{\mathrm{m}}_{+1}\right.〉\left|+\right.〉{+\mathrm{a}}_{-}\sqrt{{\mathrm{A}}_{-1}}\left|{\mathrm{m}}_{-1}\right.〉\left|-\right.〉$$

(6)

The measurement process has entangled measuring-device response states and quantum-entity eigenstates. Thus, taking the norm of this final measurand/measuring-device state (i.e., ∣Ψ〉 = M∣Φ〉∣m₀〉) as the output of the measuring-device, we have

$$〈\mathsf{\Psi }\left|\mathsf{\Psi }\right.〉={\left|{\mathrm{a}}_{+}\right|}^2{\mathrm{A}}_{+1}+{\left|{\mathrm{a}}_{-}\right|}^2{\mathrm{A}}_{-1}$$

(7)

The output of the measuring device can be either A₊₁ or A₋₁ with probabilities ∣a₊∣² and ∣a₋∣², respectively.

Given this result, it would appear that we have done nothing with the measurement operator of Equation (4) but arrive back at the measurement problem (i.e., no collapse of the wavefunction within standard quantum mechanics). However, that is precisely the point. There is nothing novel in writing Equation (4); it is fully consistent with standard quantum mechanics [20]. As we will show in the following sections, by positing disjoint eigenvector phase sets, the entanglement of Equation (7) is eliminated, with the result that the measuring device produces either A₊₁ or A₋₁ but not both.

4. Eigenphase Sets and Measurement

4.1. Overview

As previously stated, there is considerable freedom within standard quantum mechanics in setting an eigenvector’s global phase. We can take advantage of this freedom by postulating an addition to textbook quantum mechanics, which in no way violates the tenants of standard quantum mechanics. To each eigenvector of an Hermitian operator (e.g., angular momentum), we assign a set of unique “eigenphases” (which we take as distributed uniformly between −π and π radians for convenience), and define these sets as disjoint for orthogonal eigenvectors. Consequently, no matter where or when an eigenvector manifests itself, the global phase of that eigenvector will always be drawn from its unique set of eigenphases.

In the present case, for an operator with two eigenvectors, ∣+〉 and ∣−〉, there will be two eigenphase sets: {θ} and {ϕ}. Specifically, ∣+〉e^iθ1, ∣+〉e^iθ2,… ∣+〉e^iθN are all equivalent ∣+〉 eigenvectors, while ∣−〉e^iϕ1, ∣−〉e^iϕ2,… ∣−〉e^iϕN are all equivalent ∣−〉 eigenvectors; for all i and j, θ_i ≠ ϕ_j. Note that while the cardinality of the eigenphase sets will be very large (e.g., 10¹⁷), every manifestation of an eigenvector is nonetheless a single ray in Hilbert space.

Clearly, a significant caveat to this assumption is that the phase sets contain a finite number N_φ of elements. Though we speculate on possible origins for the eigenphase sets’ finite size in Appendix D, here we simply note that the value of N_φ will be exceedingly large. Consequently, Δφ ~ 2π/N_φ is so small that, in any experiment sensitive to an eigenvector’s phase (e.g., atom interferometry), the global phases within a set would have the appearance of continuity and no measurable influence on experimental outcomes.

With this modification to textbook quantum mechanics comes the issue of a solution to the Schrödinger equation when the Hamiltonian and operator of interest (e.g., S_z) do not commute, such that in expressing the Schrödinger equation’s solution ∣Φ〉 in terms of the operator-of-interest’s eigenvectors, we have essentially two choices regarding the global phase, namely

$$\left|\mathsf{\Phi }\right.〉=\left[{\mathrm{a}}_{+}\left|+\right.〉{\mathrm{e}}^{\mathrm{i}\mathsf{\theta }}{+\mathrm{a}}_{-}\left|-\right.〉{\mathrm{e}}^{\mathrm{i}\mathsf{\varphi }}\right],$$

(8a)

or

$$\left|\mathsf{\Phi }\right.〉=\left[{\mathrm{a}}_{+}\left|+\right.〉{+\mathrm{a}}_{-}\left|-\right.〉\right]{\mathrm{e}}^{\mathrm{i}\mathsf{\eta }}$$

(8b)

The problem with Equation (8a) is readily apparent: for any general Hermitian operator O, the expectation value of the operator would become a random variable depending on how the disjoint global eigenphases θ and ϕ manifest:

$$\left.〈\mathsf{\Phi }\right|\mathbf{O}\left|\mathsf{\Phi }\right.〉={\left|{\mathrm{a}}_{+}\right|}^2\left.〈+\right|\mathbf{O}\left|+\right.〉+{\left|{\mathrm{a}}_{-}\right|}^2\left.〈-\right|\mathbf{O}\left|-\right.〉+2\mathrm{Re}\left[{\mathrm{a}}_{+}^{*}{\mathrm{a}}_{-}\left.〈+\right|\mathbf{O}\left|-\right.〉{\mathrm{e}}^{\mathrm{i}\left(\mathsf{\varphi }-\mathsf{\theta }\right)}\right].$$

(9)

In other words, with Equation (8a), the matrix elements of a Hermitian operator become random variables, which is clearly not borne out by experiment. Thus, the only way to write a superposition state consistent with experience is Equation (8b).

To relate the phase η to the sets of eigenphases {θ} and {ϕ}, we note from Born’s rule [18] that ∣a₊∣² and ∣a₋∣² are the probabilities for finding the state ∣Φ〉 in either ∣+〉 or ∣−〉, respectively. Thus, it should be that

$$\mathrm{P}\left[\mathsf{\eta }=\left\{\mathsf{\theta }\right\}\right]={\left|\mathrm{a}_{+}\right|}^2\mathrm{and}\mathrm{P}\left[\mathsf{\eta }=\left\{\mathsf{\varphi }\right\}\right]={\left|\mathrm{a}_{+}\right|}^2,$$

(10)

where P[η = {X}] is the probability that η takes on one of the values in the set {X}. Consequently, from the perspective of eigenphases, Born’s rule is not so much an “added on” statement regarding experimental outcomes, but rather a quantum rule assigning the probability for choosing a superposition wavefunction’s phase from one of the possible eigenphase sets.

It is also important to note that this modification of textbook quantum mechanics has no effect on expectation values. Given any Hermitian operator O and considering the wavefunction of Equation (8b),

$$\left.〈\mathsf{\Phi }\right|\mathbf{O}\left|\mathsf{\Phi }\right.〉={\left|{\mathrm{a}}_{+}\right|}^2\left.〈+\right|\mathbf{O}\left|+\right.〉+{\left|{\mathrm{a}}_{-}\right|}^2\left.〈-\right|\mathbf{O}\left|-\right.〉+2\mathrm{Re}\left[{\mathrm{a}}_{+}^{*}{\mathrm{a}}_{-}\left.〈+\right|\mathbf{O}\left|-\right.〉\right],$$

(11)

which is independent of any stochastic nature to the wavefunction’s phase. Moreover, Equation (8b) maintains the U(1) transformation invariance of inner products involving eigenvectors. Recognizing that the U(1) transformation (i.e., e^iα) is independent of the global phase choice, U(1)∣Φ〉 = [a₊∣+〉 + a₋∣−〉]e^i(η+α) ⟹ U(1) [〈Φ∣O∣Φ〉] = 〈Φ∣O∣Φ〉.

4.2. The Measurement Operator and Eigenphase Sets

To incorporate the concept of eigenphase sets into the measurement operator, we now rewrite Equation (4) as

$$\begin{array}{l}\mathbf{M}\left(\mathsf{\eta },\mathsf{\theta },\mathsf{\varphi }\right)=\\ \left[\begin{array}{cc}{\displaystyle \frac{{\mathrm{g}}_{+1}\sqrt{{\mathrm{A}}_{+1}}}{2}}{\displaystyle \sum _{\mathrm{J}=1}^{{\mathrm{N}}_{\mathsf{\theta }}}{\mathrm{e}}^{\mathrm{i}{\mathsf{\theta }}_{\mathrm{J}}}\left(1+{\mathrm{e}}^{\mathrm{i}\left(\mathsf{\eta }-{\mathsf{\theta }}_{\mathrm{J}}\right)}\right)}\left|{\mathrm{m}}_{+1}\right.〉\left|+\right.〉\left.〈+\right|\left.〈{\mathrm{m}}_0\right|& 0\\ 0& {\displaystyle \frac{{\mathrm{g}}_{-1}\sqrt{{\mathrm{A}}_{-1}}}{2}}{\displaystyle \sum _{\mathrm{J}=1}^{{\mathrm{N}}_{\mathsf{\varphi }}}{\mathrm{e}}^{\mathrm{i}{\mathsf{\varphi }}_{\mathrm{J}}}\left(1+{\mathrm{e}}^{\mathrm{i}\left(\mathsf{\eta }-{\mathsf{\varphi }}_{\mathrm{J}}\right)}\right)}\left|-\right.〉\left.〈-\right|\left.〈{\mathrm{m}}_0\right|\end{array}\right]\end{array}$$

(12)

In this expression, the global phase terms in the two summations capture interference between the incoming quantum measurand and the multitude of quantum objects making up the classical, macroscopic measuring device, each with its own global eigenphase. Further, since the macroscopic measuring device is obviously constructed from an Avogadro’s number of quantum entities, all interacting with the quantum measurand, the classical measuring device presents the complete set of eigenphase values to the incoming quantum measurand. This is appropriate so long as the number of quantum entities in the measuring device, N_m, is significantly greater than N_φ. (Thoughts on the N_m/N_φ ratio are discussed in Appendix D.) Finally, g_±1 captures the fact that for a single measurement with N_m measuring-device quantum objects, the result of the measurement must nonetheless be one-to-one: one quantum measurand produces one measuring-device response. Note that if all the eigenphases were zero, and the g_±1 were equal to unity, then Equation (12) would reduce to Equation (4).

4.3. Quantum-Measurand and Measuring-Device Phase-Locking

As already noted, in all present solutions to the measurement problem, there is a protocol that leads to wavefunction collapse. To re-iterate,

• In the case of the Copenhagen interpretation of quantum mechanics the protocol is an appeal to epistemology. Specifically, since one cannot know without measurement; it is pointless to consider the state of the wavefunction prior to measurement. While theory may allow one to write the state of the wavefunction as a superposition of eigenvectors prior to measurement, this mathematical statement is solely a convenient way of capturing how an experiment might turn out. It is in no way a description of reality, since reality cannot be known without measurement. Clearly, there is no possibility of experimentally testing this protocol since the appeal to epistemology effectively redefines the measurement problem away.
• In the Many Worlds hypothesis, the protocol is an appeal to a multiverse. There is no collapse of the wavefunction, each possible result of a measurement is in fact realized. Since observers are confined to one particular universe within the multiverse, an observer experiences only one of the myriad results that are obtained from measurement. In fact, however, all possible results are obtained. Again, there is no way of experimentally testing this protocol, since each universe in the multiverse is forever cut-off from all others. One must take it “on faith” that the multiverse exists.
• The protocol of Decoherence Theory clearly has an intuitive appeal, since it focuses on the coupling of a real measuring device to its laboratory environment. Through this coupling, off-diagonal elements of the density matrix rapidly decay to zero, so that experimental outcomes must devolve to the diagonal elements of the density matrix, which describes the probability of finding the wavefunction in one eigenvector or another. Nevertheless, the protocol does lead to a provocative (and some might say disturbing) thought experiment. If one could manage to decouple a classical measuring device from the environment, would (for example) a Stern–Gerlach apparatus lead to the detection of a spin-up/spin-down superposition state?
• Finally, the protocol of Spontaneous Collapse Models is an appeal to a random (non-quantum) field pervading the universe and, while controversial, does have the advantage of possible experimental verification [40]. Nevertheless, we note again (as with all other solutions to the measurement problem) that Spontaneous Collapse Models require a measurement protocol to achieve wavefunction collapse.

Here, our measurement protocol involves a classical measuring device “phase-locking” to an incoming quantum measurand. Prior to obtaining the measurement output, the measuring operator transitions via this phase-locking: M ⟶ T_PL[M], where T_PL indicates the phase-locking transformation of the measuring device.

As discussed in Appendix B, depending on the phase of the measurand’s wavefunction, two possibilities present themselves for phase-locking:

(1)

If η is an element of the set {θ}, then all terms in the top left sum of Equation (12), except for η = θ_K (i.e., the actual phase of the incoming wavefunction) have (η − θ_J) = π, and all terms in the lower right sum have (η − ϕ_J) = π.

(2)

If η is an element of the set {ϕ}, then all terms in the lower right sum of Equation (12), except η = ϕ_K (again the actual phase of the incoming wavefunction), have (η − ϕ_J) = π, and all terms in the top left sum have (η − θ_J) = π.

Consequently, after phase-locking, the sums in Equation (12) reduce to a series of Kronecker deltas:

$${\mathrm{T}}_{\mathrm{P}\mathrm{L}}\left[\mathbf{M}\right]=\left[\begin{array}{cc}{\mathrm{g}}_{+1}\sqrt{{\mathrm{A}}_{+1}}{\mathrm{e}}^{\mathrm{i}\mathsf{\eta }}\left({\displaystyle \sum _{\mathrm{J}=1}^{{\mathrm{N}}_{\mathsf{\theta }}}{\mathsf{\delta }}_{\mathsf{\eta },{\mathsf{\theta }}_{\mathrm{J}}}}\right)\left|{\mathrm{m}}_{+1}\right.〉\left|+\right.〉\left.〈+\right|\left.〈{\mathrm{m}}_0\right|& 0\\ 0& {\mathrm{g}}_{-1}\sqrt{{\mathrm{A}}_{-1}}{\mathrm{e}}^{\mathrm{i}\mathsf{\eta }}\left({\displaystyle \sum _{\mathrm{J}=1}^{{\mathrm{N}}_{\mathsf{\theta }}}{\mathsf{\delta }}_{\mathsf{\eta },{\mathsf{\varphi }}_{\mathrm{J}}}}\right)\left|-\right.〉\left.〈-\right|\left.〈{\mathrm{m}}_0\right|\end{array}\right].$$

(13)

In effect, the incoming quantum measurand and phase-locked measuring device have destructively interfered for all eigenphases other than η.

To examine this measurement process concretely, we apply T_PL[M] to the superposition state of Equation (8b), considering (for illustrative purposes) the specific case of η = θ_K: the phase of the incoming quantum-state measurand for this one particular realization of the quantum entity comes from the set of unique eigenphases associated with the ∣+〉 eigenvector. Applying T_PL[M] to this incoming quantum measurand then yields the measurand/measuring-device combined state after measurement, ∣Ψ〉:

$$\left|\mathsf{\Psi }\right.〉={\mathrm{T}}_{\mathrm{P}\mathrm{L}}\left[\mathbf{M}\right]\left|\mathsf{\Phi }\right.〉\left|{\mathrm{m}}_0\right.〉=\left[\begin{array}{c}{\mathrm{a}}_{+}{\mathrm{g}}_{+1}\sqrt{{\mathrm{A}}_{+1}}{\mathrm{e}}^{\mathrm{i}{\mathsf{\theta }}_{\mathrm{K}}}\left|{\mathrm{m}}_{+1}\right.〉\left\{{\displaystyle \sum _{\mathrm{J}=1}^{{\mathrm{N}}_{\mathsf{\theta }}}{\mathsf{\delta }}_{{\mathsf{\theta }}_{\mathrm{K}},{\mathsf{\theta }}_{\mathrm{J}}}}\right\}\left|+\right.〉\\ {\mathrm{a}}_{-}{\mathrm{g}}_{-1}\sqrt{{\mathrm{A}}_{-1}}{\mathrm{e}}^{\mathrm{i}{\mathsf{\theta }}_{\mathrm{K}}}\left|{\mathrm{m}}_{-1}\right.〉\left\{{\displaystyle \sum _{\mathrm{J}=1}^{{\mathrm{N}}_{\mathsf{\theta }}}{\mathsf{\delta }}_{{\mathsf{\theta }}_{\mathrm{K}},{\mathsf{\theta }}_{\mathrm{J}}}}\right\}\left|-\right.〉\end{array}\right]=\left[\begin{array}{c}{\mathrm{a}}_{+}{\mathrm{g}}_{+1}\sqrt{{\mathrm{A}}_{+1}}{\mathrm{e}}^{\mathrm{i}{\mathsf{\theta }}_{\mathrm{K}}}\left|{\mathrm{m}}_{+}\right.〉\left|+\right.〉\\ 0\end{array}\right].$$

(14)

It should be noted that ∣Ψ〉 is not a state in the usual quantum sense, since it is a mixture of a quantum state and a classical measuring-device response-state (i.e., ∣m₊〉) as it must be. Consequently, the norm of ∣Ψ〉 is not unity but rather the magnitude of the classical measuring device’s output A_±1.

The value of g_±1 in Equation (14) is fixed by the requirement that the measurement be one-to-one:

$${\mathrm{A}}_{+1}=〈\mathsf{\Psi }\left|\mathsf{\Psi }\right.〉={\left|{\mathrm{a}}_{+}\right|}^2{\mathrm{g}}_{+1}^2{\mathrm{A}}_{+1},$$

(15)

yielding

$${\mathrm{g}}_{+1}={\left|{\mathrm{a}}_{+}\right|}^{-1}.$$

(16)

Interestingly, for the classical measuring device to always produce a one-to-one response, regardless of the value of a₊, the classical measuring device must in effect increase its sensitivity to an eigenvector as that eigenvector’s contribution to the overall measurand wavefunction decreases. Given Equation (16), we arrive at

$$\left|\mathsf{\Psi }\right.〉=\sqrt{{\mathrm{A}}_{+1}}\left|{\mathrm{m}}_{+}\right.〉\left|+\right.〉{\mathrm{e}}^{\mathrm{i}{\mathsf{\theta }}_{\mathrm{K}}}.$$

(17)

The measuring device is in the single response-state ∣m₊〉 with norm A₊₁, and the quantum-measurand wavefunction has collapsed to ∣+〉e^iθK with unit norm.

If the measurement process leading to Equation (17) is repeated many times, then different results will be obtained depending on the values of the quantum-measurand’s phase. The probability of finding the macroscopic measuring device in the state ∣m₊〉 and obtaining the quantity A₊₁ for the measurement’s manifestation will be equal to the probability of having the quantum-measurand’s phase η equal to a member of the eigenphase set {θ}, which is just ∣a₊∣². Similarly, the probability of finding the macroscopic measuring device in the state ∣m₋〉 and obtaining the measurement manifestation A₋₁ will be equal to the probability of η = {ϕ}, which is just ∣a₋∣². Consequently, the statistics of measurement outcomes is consistent with the Born rule. Moreover, as seen here, the Born rule is not an after-the-fact addition to quantum mechanics for the prediction of experimental outcomes. Rather, the Born rule follows from the probability that a wavefunction’s phase is drawn from one or another eigenphase set.

5. Hidden Variables and Eigenphase Wavefunctions

5.1. Eigenphase Sets and Bell’s Inequality

As is well known, modifications to quantum mechanics involving hidden variables are inconsistent with experiment, and so it is reasonable to ask if the present use of eigenphase sets does not somehow correspond to a hidden-variables solution to the measurement problem. To demonstrate that this is not the case (at least partially), it will be useful to show that quantum mechanics with eigenphase sets yields the exact same results for a Bell’s inequality experiment as standard quantum mechanics without eigenphase sets. Saying this differently, if the inclusion of eigenphase sets to standard quantum mechanics (QM) was a Local Hidden Variables Theory (LHVT), it must necessarily lead to different results for a Bell’s inequality experiment.

To investigate the question, we first consider the typical Bell’s inequality experiment as shown in Figure 1, from the viewpoint of standard quantum mechanics. In this case, two spin-1/2 particles are in a singlet entangled state ∣Φ〉 given by

$$\left|\mathsf{\Phi }\right.〉={\displaystyle \frac{1}{\sqrt{2}}}\left[{\left|\uparrow \right.〉}_1{\left|\downarrow \right.〉}_2-{\left|\downarrow \right.〉}_1{\left|\uparrow \right.〉}_2\right].$$

(18)

One of the particles in the entangled state is sent towards the a-detector system, while the other is sent towards the b-detector system. Further, we will assume that the a-detector is set to measure particles in a rotated basis set:

$$\left|{\mathrm{U}}_{\mathrm{a}}\right.〉=\cos \left[\mathsf{\alpha }\right]\left|\uparrow \right.〉-\sin \left[\mathsf{\alpha }\right]\left|\downarrow \right.〉$$

(19a)

$$\left|{\mathrm{D}}_{\mathrm{a}}\right.〉=\sin \left[\mathsf{\alpha }\right]\left|\uparrow \right.〉+\cos \left[\mathsf{\alpha }\right]\left|\downarrow \right.〉$$

(19b)

Similarly, the b-detector is set to measure particles in an oppositely rotated basis set with a different rotation angle:

$$\left|{\mathrm{U}}_{\mathrm{b}}\right.〉=\cos \left[\mathsf{\beta }\right]\left|\uparrow \right.〉+\sin \left[\mathsf{\beta }\right]\left|\downarrow \right.〉$$

(20a)

$$\left|{\mathrm{D}}_{\mathrm{b}}\right.〉=-\sin \left[\mathsf{\beta }\right]\left|\uparrow \right.〉+\cos \left[\mathsf{\beta }\right]\left|\downarrow \right.〉$$

(20b)

The probability of detecting a spin-up signal in the a-detector system, while simultaneously detecting a spin-down signal in the b-detector system (i.e., P_a↑,b↓) is

$${\mathrm{P}}_{\mathrm{a}\uparrow ,\mathrm{b}\downarrow }=|\left.〈{\mathrm{U}}_{\mathrm{a}}\right|\left.〈{\mathrm{D}}_{\mathrm{b}}\right|\left|\mathsf{\Phi }\right.〉{|}^2.$$

(21)

Substituting from Equations (18), (19a) and (20b), we then have

$${\mathrm{P}}_{\mathrm{a}\uparrow ,\mathrm{b}\downarrow }={\displaystyle \frac{1}{2}}|\left\{\cos \left[\mathsf{\alpha }\right]\left.〈\uparrow \right|-\sin \left[\mathsf{\alpha }\right]\left.〈\downarrow \right|\right\}\left\{-\sin \left[\mathsf{\beta }\right]\left.〈\uparrow \right|+\cos \left[\mathsf{\beta }\right]\left.〈\downarrow \right|\right\}\left\{{\left|\uparrow \right.〉}_1{\left|\downarrow \right.〉}_2-{\left|\downarrow \right.〉}_1{\left|\uparrow \right.〉}_2\right\}{|}^2,$$

(22)

and working through the multiplications and trigonometry we find that

$${\mathrm{P}}_{\mathrm{a}\uparrow ,\mathrm{b}\downarrow }={\displaystyle \frac{1}{2}}{\cos }^2\left[\mathsf{\alpha }-\mathsf{\beta }\right].$$

(23)

This is the Bell’s inequality experimental result that is predicted from standard quantum mechanics, and which is inconsistent with any Local Hidden Variables Theory. (See the discussion in Ref. [41].)

Considering now quantum mechanics with eigenphase sets, let {θ} be that set of eigenphases associated with ∣↑〉 and {ϕ} be the set of eigenphases associated with ∣↓〉; Equation (18) then becomes

$$\left|\mathsf{\Phi }\right.〉={\displaystyle \frac{1}{\sqrt{2}}}\left[{\left|\uparrow \right.〉}_1{\left|\downarrow \right.〉}_2-{\left|\downarrow \right.〉}_1{\left|\uparrow \right.〉}_2\right]{\mathrm{e}}^{\mathrm{i}\left({\mathsf{\theta }}_1+{\mathsf{\varphi }}_2\right)}.$$

(24)

Here, θ₁ and ϕ₂ are particular random choices out of the sets {θ} and {ϕ} for particle-1 and particle-2, respectively, and we have written the phase term consistent with Equation (8b), recognizing that the entangled wavefunction is a combination of product states with each term in the product state bringing with it a global phase. Clearly, for some other realization of the entangled wavefunction, we could have ϕ₁ and θ₂, namely particle-1’s phase drawn from the set {ϕ} and particle-2’s phase drawn from the set {θ}. Note also that because Equation (24) refers to a singlet state, the phase factor consists of one phase from the set {θ} and one phase from the set {ϕ}. If we had considered a triplet state, then the phase factor could have consisted of two phases from the set {θ} or two phases from the set {ϕ}.

Using Equation (24) along with Equations (19a) and (20b), we obtain for P_a↑,b↓

$${\mathrm{P}}_{\mathrm{a}\uparrow ,\mathrm{b}\downarrow }={\displaystyle \frac{1}{2}}|\left\{\cos \left[\mathsf{\alpha }\right]\left.〈\uparrow \right|-\sin \left[\mathsf{\alpha }\right]\left.〈\downarrow \right|\right\}\left\{-\sin \left[\mathsf{\beta }\right]\left.〈\uparrow \right|+\cos \left[\mathsf{\beta }\right]\left.〈\downarrow \right|\right\}\left\{{\left|\uparrow \right.〉}_1{\left|\downarrow \right.〉}_2-{\left|\downarrow \right.〉}_1{\left|\uparrow \right.〉}_2\right\}{\mathrm{e}}^{\mathrm{i}\left({\mathsf{\theta }}_1+{\mathsf{\varphi }}_2\right)}{|}^2$$

(25a)

or taking the e^i(θ1+ϕ2) phase factor out of the absolute value

$${\mathrm{P}}_{\mathrm{a}\uparrow ,\mathrm{b}\downarrow }={\displaystyle \frac{1}{2}}|\left\{\cos \left[\mathsf{\alpha }\right]\left.〈\uparrow \right|-\sin \left[\mathsf{\alpha }\right]\left.〈\downarrow \right|\right\}\left\{-\sin \left[\mathsf{\beta }\right]\left.〈\uparrow \right|+\cos \left[\mathsf{\beta }\right]\left.〈\downarrow \right|\right\}\left\{{\left|\uparrow \right.〉}_1{\left|\downarrow \right.〉}_2-{\left|\downarrow \right.〉}_1{\left|\uparrow \right.〉}_2\right\}{|}^2.$$

(25b)

But this is just Equation (22), yielding

$${\mathrm{P}}_{\mathrm{a}\uparrow ,\mathrm{b}\downarrow }={\displaystyle \frac{1}{2}}{\cos }^2\left[\mathsf{\alpha }-\mathsf{\beta }\right].$$

(26)

The identity of Equation (26) with Equation (23) shows that including eigenphase sets in standard quantum mechanics leads to the same result as textbook quantum mechanics for a Bell’s inequality experiment. Therefore, including eigenphase sets into quantum mechanics cannot be a Local Hidden Variables Theory. Moreover, to try and state that wavefunction phase is somehow a hidden variable would be inconsistent with standard quantum mechanics where eigenvectors exist in a complex Hilbert space: ∣Φ〉 = abs[∣Φ〉]e^iψ. In the next subsection, we consider how wavefunction collapse occurs in the Bell’s inequality experiment with the inclusion of eigenphases.

5.2. Collapse of the Wavefunction in a Bell’s Inequality Experiment

To elucidate the nature of wavefunction collapse in a Bell’s inequality experiment using eigenphase sets, we consider the simplified experimental situation where α = β = 0 in Equations (19) and (20). Further, we should recognize that both the ‘a’ and ‘b’ measuring devices of Figure 1 will have their own measurement operators, M^(a) and M^(b), and that these will only respond to those components of the entangled wavefunction reaching them. For conceptual concreteness, we will imagine that Equation (24) describes an HD molecule in its singlet ground state (i.e., a hydrogen-deuterium molecule), and that for the Bell’s inequality experiment we are “gently” stretching the molecule, adding kinetic energy to the molecular bond.

$$\left|\mathsf{\Phi }\right.〉={\displaystyle \frac{1}{\sqrt{2}}}\left[{\left|\uparrow \right.〉}_{\mathrm{H}}{\left|\downarrow \right.〉}_{\mathrm{D}}-{\left|\downarrow \right.〉}_{\mathrm{H}}{\left|\uparrow \right.〉}_{\mathrm{D}}\right]{\mathrm{e}}^{\mathrm{i}\left({\mathsf{\theta }}_{\mathrm{H}}+{\mathsf{\varphi }}_{\mathrm{D}}\right)}.$$

(27)

Again, this is only one realization of the molecular wavefunction, and in some other realization, we could have θ_H ⟶ ϕ_H and ϕ_D ⟶ θ_D.

For clarity in this gedanken experiment, we will also imagine that the system in Figure 1 is modified by adding a “mass-filter” prior to each Stern–Gerlach magnet. If we consider that the HD stretching energy is equally partitioned between the H and D atoms, then the separation velocity of these two must necessarily be different by the ratio (m_H/m_D)^1/2. Knowing this, we could place the deuterium detector (e.g., the b-detector) closer to the HD source and restrict our measurement results to coincidental counts from the two detector systems. If a deuterium atom is emitted towards the b-detector, then the spin measurements of the ‘a’ and ‘b’ detectors will be coincident. If a hydrogen atom is emitted towards the b-detector, then the b-detector’s spin measurement will precede the ‘a’ detector’s measurement. Note that this mass filter does not alter the spin entanglement. Rather, it simply reduces the number of spin measurements in the Bell’s inequality experiment. Consequently, we will consider the b-detector system as examining only the spin of the deuterium atom and the a-detector only measuring the spin of the hydrogen atom. Note that this does nothing to the entangled nature of the spins; it simply reduces the Bell measurement results by one half, since any time a hydrogen atom is stretched towards the b-detector, that measurement will not be included in the Bell’s inequality results.

To proceed, with the H-atom traveling towards the a-detector and the D-atom traveling towards the b-detector, Equation (13) becomes

$${\mathrm{T}}_{\mathrm{P}\mathrm{L}}\left[{\mathbf{M}}^{\left(\mathrm{a}\right)}\right]=\left[\begin{array}{cc}{\mathrm{g}}_{\uparrow }\sqrt{{\mathrm{A}}_{\uparrow }}{\mathrm{e}}^{\mathrm{i}{\mathsf{\theta }}_{\mathrm{H}}}\left({\displaystyle \sum _{\mathrm{J}=1}^{{\mathrm{N}}_{\mathsf{\theta }}}{\mathsf{\delta }}_{{\mathsf{\theta }}_{\mathrm{H}},{\mathsf{\theta }}_{\mathrm{J}}}}\right)\left|{\mathrm{m}}_{\uparrow }\right.〉\left|\uparrow \right.〉\left.〈\uparrow \right|\left.〈{\mathrm{m}}_0\right|& 0\\ 0& {\mathrm{g}}_{\downarrow }\sqrt{{\mathrm{A}}_{\downarrow }}{\mathrm{e}}^{\mathrm{i}{\mathsf{\theta }}_{\mathrm{H}}}\left({\displaystyle \sum _{\mathrm{J}=1}^{{\mathrm{N}}_{\mathsf{\theta }}}{\mathsf{\delta }}_{{\mathsf{\theta }}_{\mathrm{H}},{\mathsf{\varphi }}_{\mathrm{J}}}}\right)\left|\downarrow \right.〉\left.〈\downarrow \right|\left.〈{\mathrm{m}}_0\right|\end{array}\right].$$

(28a)

$${\mathrm{T}}_{\mathrm{P}\mathrm{L}}\left[{\mathrm{M}}^{\left(\mathrm{b}\right)}\right]=\left[\begin{array}{cc}{\mathrm{g}}_{\uparrow }\sqrt{{\mathrm{A}}_{\uparrow }}{\mathrm{e}}^{\mathrm{i}{\mathsf{\varphi }}_{\mathrm{D}}}\left({\displaystyle \sum _{\mathrm{J}=1}^{{\mathrm{N}}_{\mathsf{\theta }}}{\mathsf{\delta }}_{{\mathsf{\varphi }}_{\mathrm{D}},{\mathsf{\theta }}_{\mathrm{J}}}}\right)\left|{\mathrm{m}}_{\uparrow }\right.〉\left|\uparrow \right.〉\left.〈\uparrow \right|\left.〈{\mathrm{m}}_0\right|& 0\\ 0& {\mathrm{g}}_{\downarrow }\sqrt{{\mathrm{A}}_{\downarrow }}{\mathrm{e}}^{\mathrm{i}{\mathsf{\varphi }}_{\mathrm{D}}}\left({\displaystyle \sum _{\mathrm{J}=1}^{{\mathrm{N}}_{\mathsf{\theta }}}{\mathsf{\delta }}_{{\mathsf{\varphi }}_{\mathrm{D}},{\mathsf{\varphi }}_{\mathrm{J}}}}\right)\left|\downarrow \right.〉\left.〈\downarrow \right|\left.〈{\mathrm{m}}_0\right|\end{array}\right].$$

(28b)

As a result, each measuring device phase-locks to either the H-atom’s eigenphase or the D-atom’s eigenphase, and after phase-locking, the result of this one particular ‘a’ and ‘b’ detector measurement yields

$$\left|{\mathsf{\Psi }}^{\left(\mathrm{a}\right)}\right.〉=\sqrt{{\mathrm{A}}_{\uparrow }}\left|{\mathrm{m}}_{\uparrow }\right.〉\left|\uparrow \right.〉{\mathrm{e}}^{\mathrm{i}{\mathsf{\theta }}_{\mathrm{H}}}.$$

(29a)

$$\left|{\mathsf{\Psi }}^{\left(\mathrm{b}\right)}\right.〉=\sqrt{{\mathrm{A}}_{\downarrow }}\left|{\mathrm{m}}_{\downarrow }\right.〉\left|\downarrow \right.〉{\mathrm{e}}^{\mathrm{i}{\mathsf{\varphi }}_{\mathrm{D}}}.$$

(29b)

Detector ‘a’ collapses the entangled spins into a spin-up state for the H-atom, while the b-detector collapses the entangled spins into a spin-down state for the D-atom. Though we only considered the simplified case of α = β = 0, nothing precludes the analysis from recreating the situation of α ≠ β ≠ 0 discussed previously.

6. The Uncertainty Principle and Eigenphase Sets

To be fully consistent with quantum mechanics, the use of a wavefunction’s global phase to resolve the measurement problem must be consistent with the uncertainty principle, which is arguably the foundation upon which all of quantum mechanics is built. As discussed in a number of textbooks [42,43], for two Hermitian operators A and B, the uncertainty principle follows from their commutator:

$$\Delta \mathrm{A}\cdot \Delta \mathrm{B}\ge {\displaystyle \frac{1}{2}}\left|\left.〈\mathsf{\Phi }\right|\left[\mathbf{A},\mathbf{B}\right]\left|\mathsf{\Phi }\right.〉\right|,$$

(30)

where ∣Φ〉 is the state of the particle in which the eigenvalues of A and B are obtained. Here, ΔA is the uncertainty in the measured eigenvalue of A, which is defined as ∣(A − 〈A〉I)Φ∣ with a similar definition for ΔB.

To demonstrate that global phase is consistent with the uncertainty principle, we again consider the uncertainty for A and B measurements of a two-state particle in a superposition state as given by Equation (8b):

$$\left|\mathsf{\Phi }\right.〉=\left[{\mathrm{a}}_{+}\left|+\right.〉{+\mathrm{a}}_{-}\left|-\right.〉\right]{\mathrm{e}}^{\mathrm{i}\mathsf{\eta }}.$$

(31)

Using Equation (31) to determine ΔA (and similarly ΔB), we have

$$\Delta \mathrm{A}=\left|\left(\mathbf{A}-〈\mathrm{A}〉\mathbf{I}\right)\left[{\mathrm{a}}_{+}\left|+\right.〉{+\mathrm{a}}_{-}\left|-\right.〉\right]{\mathrm{e}}^{\mathrm{i}\mathsf{\eta }}\right|=\left|\left(\mathbf{A}-〈\mathrm{A}〉\mathbf{I}\right)\left[{\mathrm{a}}_{+}\left|+\right.〉{+\mathrm{a}}_{-}\left|-\right.〉\right]\right|,$$

(32a)

and we see that the global phase has dropped out as a consequence of the absolute value in the definition of ΔA. For the right-hand-side of Equation (30), we obtain

$$\begin{array}{c}\left|\left.〈\mathsf{\Phi }\right|\left[\mathbf{A},\mathbf{B}\right]\left|\mathsf{\Phi }\right.〉\right|=\left|\left.〈\left[{\mathrm{a}}_{+}^{*}\left.〈+\right|{+\mathrm{a}}_{-}^{*}\left.〈-\right|\right]{\mathrm{e}}^{-\mathrm{i}\mathsf{\eta }}\right|\left[\mathbf{A},\mathbf{B}\right]\left|\left[{\mathrm{a}}_{+}\left|+\right.〉{+\mathrm{a}}_{-}\left|-\right.〉\right]{\mathrm{e}}^{\mathrm{i}\mathsf{\eta }}\right.〉\right|\\ =\left|\left.〈\left[{\mathrm{a}}_{+}^{*}\left.〈+\right|{+\mathrm{a}}_{-}^{*}\left.〈-\right|\right]\right|\left[\mathbf{A},\mathbf{B}\right]\left|\left[{\mathrm{a}}_{+}\left|+\right.〉{+\mathrm{a}}_{-}\left|-\right.〉\right]\right.〉\right|\end{array}$$

(32b)

and again the global phase has dropped out. Thus, the uncertainty principle is transparent to eigenphase sets, because (1) each particle’s phase is global and unaffected by the action of any Hermitian operator (i.e., Equation (32b)) and (2) because the definition of uncertainty is in terms of an eigenvalue’s magnitude (i.e., Equation (32a)).

7. Summary

In this work, we focused on one possible pathway for a wavefunction-phase resolution of the measurement problem with the goal of highlighting the broader potential of such a pathway. Here, we considered a modification to textbook quantum mechanics and a protocol regarding the interaction between a single quantum entity and a classical, macroscopic measuring device:

• Modification—To each eigenvector of an Hermitian operator, there is a set of discrete eigenphases, with orthogonal eigenvectors having disjoint eigenphase sets.
• Protocol—A classical, macroscopic measuring device phase-locks to the global phase of an incoming wavefunction.

Though a number of ideas regarding resolution of the measurement problem are under active debate, there are several advantages to the wavefunction-phase pathway that should be noted:

(1)

There is no need to define the wavefunction as solely a computational device, which is not indicative of a quantum entity’s actual nature—Copenhagen Interpretation.

(2)

There is no need to postulate a rapidly expanding multiverse—Many Worlds Theory. Here, wavefunction collapse is a consequence of destructive interference between a phase-locked macroscopic object and a single quantum-entity measurand.

(3)

There is no need to hypothesize the existence of a non-quantum random field pervading the Universe—Spontaneous Collapse Models. Stochasticity arises from a wavefunction’s random “choice” of global phase.

(4)

There is no epistemological problem for quantum mechanics. The Schrödinger equation is more foundational than the density matrix’s Liouville equation—Decoherence Theory.

(5)

There is no need to hypothesize a fundamental nonlinear theory of nature to which standard quantum mechanics is an approximation. Quantum mechanics is fundamental, and if nonlinearity enters the measurement problem at all, it is through the interface between the quantum and classical regimes (e.g., a classical measuring device’s phase-locking process).

(6)

There is no schizophrenic split between von Neumann’s T1 and T2 processes. Quantum mechanics solely involves T2 processes. Apparent T1 processes arise from destructive interference between a measuring device and a quantum measurand.

All of this is not to say that the use of a wavefunction’s global phase to resolve the measurement problem has no outstanding questions requiring further study.

• Is it indeed true that a macroscopic object can phase-lock to a single quantum entity, and are there experiments that might test this? Though this is a question future work will hopefully address, we present some preliminary thoughts on how phase-locking might occur and be tested in Appendix C.
• How might discretization of a wavefunction’s global phase arise? We present some conjectures in Appendix D arguing that the concept of eigenphase sets with very large cardinality is viable.

Regardless of the questions noted above, what we have really demonstrated in this work is the potential of a little-explored path for resolving the measurement problem. It is our contention that this pathway has much to recommend it and should be studied further and critically discussed.