Quantum teleportation understood though first-order pilot-wave theory

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Quantum teleportation understood

though first-order pilot-wave theory

Report Bachelor Project Physics and Astronomy, size 15 EC,

conducted between 09 - 04 - 2018 and 13 - 07 - 2018

at the

Descartes Centre for the History and Philosophy of the Sciences and the

Humanities

Wouter van Soest

Supervisor: Guido Bacciagaluppi Second examiner: Jeroen van Dongen

Faculty of Science University of Amsterdam

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Abstract

Pilot wave theory is an alternative interpretation of quantum mechanics that might solve the measurement problem. In this paper, the development of pilot wave theory by Louis de Broglie, David Bohm and John Stewart Bell will be discussed. De Broglie first presented the idea in 1927, but it was met with criticism and forgotten, I will argue that disappearance of pilot wave theory was not justified. In 1952, Bohm revived the theory by including the measuring device in the wave function when describing measurements. After reviewing this, I will develop the mathematics to describe the puzzling teleportation protocol in pilot wave terms. I will show that the expected result will arise by following the dynamics of the theory rather than by the collapse of the wave function. Also, I will argue that nothing is actually teleported and thus that teleportation is not the right term to describe the phenomenon, at least, not in the pilot wave interpretation. There is already one treatment in the litera-ture, which uses Bohm’s second-order formalism, but I will be using the simpler first-order formalism developed by de Broglie and Bell.

Popular summary

Quantumtheorie beschrijft de wereld op de allerkleinste schaal, waar de natuur niet meer blijkt te werken zoals beschreven door bijvoorbeeld Newton’s wetten. De wiskunde die in quantumtheorie gebruikt wordt kan op verschillende manieren ge¨ınterpreteerd worden. De conventionele interpretatie gaat ervanuit dat de toestand waarin een systeen zich bevind compleet beschreven kan worden door een golf waaruit afgeleid kan worden wat de kansen zijn op uitkomsten van experimenten, de uitkomsten van deze experimenten zouden dus willekeurig zijn, en de natuur niet deterministisch. Deze interpretatie heeft echter een prob-leem waar nog geen antwoord voor is. In een andere, veel minder populaire interpretatie van quantumtheorie, de zogenaamde pilot-wave theorie, is dit probleem niet relevant. Voor dit project heb ik gekeken naar hoe pilot-wave theorie ontwikkeld is om te proberen te begrijpen waarom deze niet zo populair als de conventionele interpretatie is geworden. Ook bekijk ik hoe het quantumteleportatiephenomeen beschreven kan worden met als uitgangspunt de pilot-wave interpretatie.

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1 Introduction

Pilot-wave theory is an alternative interpretation of quantum mechanics. It is a non-local hidden variable theory. It is worth mentioning that this type of theory is not disproved by Bell’s inequality [1], on the contrary, Bell argues that any hidden variable theory has to have this feature of non-locality1 _{and even developed pilot-wave theory further himself [3].}

Besides being non-local, pilot-wave theory is a deterministic theory in which particles have well defined positions and momenta and thus trajectories can be calculated.

But why would we want an alternative for the conventional Copenhagen interpretation? A few points on the basis of which one might want to choose the pilot-wave interpretation over the conventional one are the following. First of all, the measurement problem, in the Copenhagen interpretation, when a measurement is made, the wave function collapses into an Eigenstate of the measured observable. However, there is no good definition of what a measurement exactly is and, perhaps more cumbersome, there are no dynamics that properly describe the collapse of the wave function. Moreover, to do a measurement, one needs an observer. This causes a problem when talking about quantum cosmology, since if the system to be measured is the entire Universe, what (or who) is left outside the system to make a measurement? This question remains unanswered in the conventional interpretation of quantum mechanics. In pilot-wave theory however, as will be explained in section 2.3, measurements are perfectly well described by the dynamics of the theory, and therefore these problems are not present. A more conceptual question in quantum theory is the question of nonlocality. Quantum teleportation was described by the EPRB paradox, in which Einstein called whatever happened a “spooky action at a distance” [4]. However, in a nonlocal theory like the pilot-wave theory, this might not be as spooky as it looks.

An interesting phenomenon that arises in the conventional interpretation is the phenomenon of quantum teleportation, the apparent instantaneous transportation of a quantum state from one place to an other (or the instantaneous recreation of that state at another place). The different views on nonlocality make teleportation a really interesting phenomenon to look at through both interpretations, especially since the intuitive pilot wave picture might

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shed light on weather something spooky is really happening.

Pilot wave theory was first formulated in first-order dynamics by Louis de Broglie [5] and later by David Bohm in second order dynamics. More about the difference will be explained in section 2. Owen Maroney and Basil J. Hiley treated quantum teleportation in Bohm’s second order dynamics [6] and arrive to similar conclusions as myself, but no accurate description of the phenomenon is given in the original first order formulation. This is interesting because the first order dynamics are even more intuitive than Bohm’s variation, and they allow for an easy treatment of quantum spin. Also, recently the first order theory is becoming more popular, Samuel Colin and Antony Valentini even found possible grounds to reject the second order formulation [7].

In the next section, I will give a chronological overview of the development of pilot-wave theory, with a focus on the understanding of measurements. Subsequently, Bell’s treatment of quantum spin measurements will be explained by giving the example of a simple Stern-Gerlach measurement. I will then explain the quantum teleportation protocol and finally, quantum teleportation will be treated through the Bell spin measurement formulation.

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2 Chronological overview

In this section I will start by explaining De Broglie’s first pilot-wave theory and how simple measurements can be done accordingly. Next, I will discuss some comments that were made regarding De Broglie’s ideas. I will show that the theory was not complete, but argue that there was no clear reason as to abandon the theory. I will then elaborate on Bohm’s later theory (which answers the criticism De Broglie initially received).

2.1 The first Pilot-Wave theory

As mentioned in section 1, Louis de Broglie was the first to come up with the idea of a pilot-wave. This idea came from the particle-wave duality that Einstein discovered with respect to light. De Broglie took this duality as more fundamental and assumed that particles had to have wave characteristics as well [8]. From this idea he kept on working through the analogy with optics. He described his first pilot-wave dynamics with

i~∂Ψ ∂t = − ~2 2m ∇2_{Ψ + V Ψ} ₍₁₎ mi dxi dt = ∇iS, (2)

where equation (1) is the ordinary time dependent Schr¨odinger equation that governs the evolution of the wave function and (2) gives the velocities of particles with wave functions Ψi = |Ψ|e

i

~S, with S the phase of the wave. The particles thus move perpendicular to the

wave front, they are ”riding” the wave. Equation (2) can also be written in the form

mi dxi dt = Ψ∗∇iΨ |Ψ|2 = ji X(t) ρ X(t) (3)

where J X(t) is the probability current and ρ X(t) is the probability density. We will use this is section 2.4. From (3) it can be seen that the particles move along with the probability current as expected. Note that X(t) is the list (x1, x2, ..., xn) of the positions of all particles,

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thus the trajectory of the particle xi depends on the positions of all particles, no matter

their distance to xi. This demonstrates the nonlocality that is fundamentally embedded in

the theory.

Measurement in De Broglie’s formalism

Powered by these dynamics, I will now give an example of a simple measurement in the pilot-wave theory. Suppose we prepare some particle in the state Ψ = ψa(x) |blacki where ψa(x)

is the positional part of the wave function and suppose there is an observable, ”hardness” of which the Eigenstates are |hardi and |sof ti with |blacki = √1

2 |hardi + |sof ti . In

pilot-wave theory, we can predict outcomes of all measurements on the particle if we know its initial position. Say we want to measure the hardness. To do this, we feed the wave function though a hardness box. This process is shown in figure 1. The box works as follows, wave functions fed though the box separate due to hardness dependent forces into Eigenstate packets, the hard parts of wave functions get deflected up, and the soft parts get deflected down. Thus, the evolution of the wave function we prepared is

ψa|blacki −→

1 √

2 ψb|hardi + ψc|sof ti,

where eventually, ψb will only be nonzero in a region above the box and ψc will only be

nonzero in a region below the box. Now, if we assume the wave function and the hardness box are prepared symmetrically (with respect to the soft and hard parts), and remember that the particle moves along with the probability current as described by (3), then, a particle that starts in the top half of ψa will move up, along with the hard wave packet and a particle that

starts in the bottom half of ψa will move down along with the soft wave packet. If the wave

packets were brought together again, the measurement would be ”undone”. However, the interaction the packets have with their environments once they are fully separated will cause correlations between the environment and packets. This means that for the measurements to be undone, the wave functions of the degrees of freedom that interacted with the packets should overlap too. The chances of this happening once the macroscopic degrees of freedom

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are involved are practically zero and thus the packets are separated for good. Because the wave packets don’t overlap, one of the two will always be zero anywhere in configuration space, and will never cease to be so, thus the system will be trapped in one of the packets. If we now know the initial position, we know which packet will become the new effective wave function and the outcome of the measurement.

In the case of figure 1, where the particle starts in the top half of ψa, note that the wave

function has not collapsed, but is reduced, obeying to the dynamics of the theory, to the new effective wave function ψb|hardi in the sense that the other component has become

irrelevant to guiding the particle. This way, if we consequently feed the particle and its wave though an other hardness box like in the figure, it will definitely come out at the |hardi side, since the input wave function is already in the |hardi Eigenstate. In this way, not only consecutive measurements of the same observable will yield the expected result, but any quantum measurements that are recorded in the measured particles positions can be modelled.

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Figure 1: An example of the separation of wave packets: the evolution of the wave function is ψa|blacki −→ √1₂|hardi ψb+ |sof ti ψc. ([9])

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2.2 Criticism and the abandonment of pilot-wave theory

When De Broglie first presented his theory to his colleagues at the 1927 Solvay conference, it was met with some criticism. I will now point out two of the comments made by Wolf-gang Pauli and Hans Kramers, both concerning scattering problems. The first comment, raised by Pauli concerned the inelastic scattering of a particle off a rigid quantum rotator (a body with one rotational degree of freedom and discrete energy levels). Before treating this comment, it is necessary to note that the wave functions in De Broglie’s theory don’t always propagate through regular 3-space, but actually live in configuration space. In terms of the experiment in section 2.1 this means that the total wave function of the particle would live in a configuration space defined by two continuous position coordinates and one discrete hardness coordinate which can only have two values.

Without going into to much detail, Pauli’s objection comes down to the following2. Pauli uses an optical analogy derived by Fermi in which he uses a 3-dimensional light wave scattering off a grating instead of a 2-dimensional wave scattering off a single body with a rotational degree of freedom. The grating separations here account for the discrete energy levels, where the rotational degree of freedom is accounted for by the extra spatial dimension, this way the configuration space in which the wave function lives has the same number of dimensions. To make calculations easier, in optics waves are treated as if they are infinite in space, but this is not the physical reality, since actually they are limited in space. However, the rotator degree of freedom is not limited in energy, thus the wave in the analogy should be unlimited in this dimension. This, according to Pauli causes the final wave packets to overlap (see figure 2), and without the separation of wave packets, there is no definite out-come, which is in contrast with experimental data.

This problem posed by Pauli can be answered easily if one can disregard the optical analogy[5], or by including the measuring apparatus in the wave function like Bohm would have done 30 years later (more about this in the next section). At the time, however, De Broglie failed to give a satisfactory answer. The answer to Pauli’s problem lies in the fact that the velocities of all the outgoing wave packets in the analogy will be the same (the speed

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of light, c), however, in the real case these velocities actually depend on the electron energy, hence even on an infinite grating with an infinite incoming wave, the packets will separate and a definite outcome will be present. Neither Pauli or De Broglie, nor any of the other great minds present at the Solvay conference noticed this misconception in Pauli’s objection. By trying to answer Pauli’s objection in terms of the optical analogy, De Broglie had put himself before an impossible task.

The second comment on De Broglie’s theory I will discuss was posed by Kramers with regard to a single photon being reflected by a mirror. The treatment of a cloud or ensemble of photons was given by De Broglie but the question how his theory would account for the change in momentum of the mirror for a single photon remained unanswered. One remark made by L´eon Brillouin, a supporter of the pilot-wave theory, was that this question would pose a problem in any interpretation of quantum mechanics [5]. This is true because at the time, quantum mechanics was only applied to microsystems without looking at the macroscopic consequences like the movement of the mirror, and that approach can’t answer the problem posed by Kramer’s. So what was missing? The answer, in the case of pilot-wave theory at least, came in 1952 when Bohm published a series of two articles in which he presented his renewed version of De Broglie’s theory[10]. The most important feature of Bohm’s theory is that he included the measuring device in the wave function when describing measurements of any kind. It is this feature that solved Kramer’s problem.

This being the two main criticisms of De Broglie’s theory, was it justified that it was so soon forgotten? I think not, yes there were some unanswered questions, but isn’t that normal for a completely new theory? Moreover, the Copenhagen interpretation is still facing problems, as mentioned in section 1 and one of the main problems posed did in fact apply to other interpretations as well.

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Figure 2: Wave packets scattering off (top) a spatially limited grating → wave packets separate, (middle and bottom)a spatially unlimited grating → wave packets don’t separate if they all have the same velocity (middle) but separate for different velocities (bottom) [5]

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2.3 1952: Bohm’s revival of Pilot-Wave theory

As mentioned in the previous section, Bohm revived pilot-wave theory in 1952. The key novelty Bohm introduced is the inclusion of the measuring device in the wave function, but there are more differences between the two theories. I will point out one essential difference that arises from the fact that Bohm used second order dynamics instead of first order like De Broglie. Since this paper is mostly about De Broglie’s theory, I won’t elaborate too much on Bohm’s theory. I will summarise his derivation up to a point where it can be seen that the two theories are equivalent and the aforementioned essential difference can be pointed out. I will briefly discuss Colin and Valentini’s approach ([7]) to why Bohm’s second order theory should be discarded because of this difference. Also, I will give an example analogous to the one in the previous section where it is necessary to include the measuring device in the wave function.

Derivation of Bohm’s pilot-wave theory

Here I will discuss the derivation of Bohm’s dynamics as described by Bohm and Hiley ([11]). The derivation starts with the Schr¨odinger equation and the polar form wave function also used by De Broglie i~∂Ψ ∂t = − ~2 2m ∇2_{Ψ + V Ψ} ₍₁₎ Ψ = |Ψ|e~iS. (4)

By inserting (4) into (1), and separating the real and imaginary parts, two equations arise; ∂S ∂t + (∇S)2 2m + V − ~ 2m ∇2_|Ψ| |Ψ| = 0 (5) ∂|Ψ| ∂t + ∇ · |Ψ| 2∇S m = 0. (6)

Equation (6) is the polar form of the continuity equation, which assures that the probability P = |Ψ|2 _{is a conserved quantity. Hence, if the total probability equals one initially, it will}

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do so at all times. Equation (5) can be approximated, in the classical regime, as a Hamilton-Jacobi equation describing a particle with momentum p = ∇S. This approximation, known as the WKB approximation, comes down to discarding the term

Q = − ~ 2m

∇2_|Ψ|

|Ψ| . (7)

Bohm and Hiley then argue that equation (5) can be viewed as the quantum Hamilton-Jacobi equation describing a particle with the same momentum. The extra term (7) can then be seen as an extra potential which they call the quantum potential. From the form of the quantum potential, it can be seen that the amplitude of the wave function |Ψ| has no influence on the momentum since it appears in the numerator and the denominator. This illustrates the nonlocality of the theory, the shape of the wave function is all that matters and for large distances, even though the amplitude diminishes, the overall shape doesn’t change. If (5) is viewed as a Hamilton-Jacobi equation, it follows that the equation of motion is

md

2_x

dt2 = −∇(V + Q). (8)

Now it can be seen that the dynamics of De Broglie and Bohm are equivalent. That is, if we take the time derivative of (2) and plug in the Sch¨odinger equation, we arrive precisely at (8).

A difference between De Broglie and Bohm

In both De Broglie’s and Bohm’s mechanics, one crucial assumption has to be made; that initially, the Born rule holds, that is

P = |Ψ(t = 0)|2 (9)

In De Broglie’s case, there are arguments and simulations to the effect that if this constraint is dropped, and in the early Universe there was no quantum equilibrium, nowadays systems would have relaxed to quantum equilibrium anyway and thus we would observe quantum equilibrium anyway (which we do)([12]). Bohm however, has to make an other assumption,

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namely that the initial momenta are given by

p(t = 0) = ∇S(t = 0). (10)

In De Broglie’s theory, this is not a constraint but simply the equation of motion aplied to t = 0, but in Bohm’s case it is not, and if one wants to view Bohm’s theory as different from De Broglie’s, one should be able to drop this constraint without contradictions arising. Colin and Valentini ([7]) used this as a possible argument to reject Bohm’s theory, since they show that if both assumptions (9) and (10) are dropped, quantum equilibrium would not arise and our world wouldn’t look like we observe it.

2.3.1 Bohm’s theory of measurement

As mentioned before, Bohm found a way to treat quantummechanical measurements through pilot-wave theory. The key point: one has to include the degrees of freedom of the measuring apparatus in the wave function. By doing this one can make sure the wave packets corre-sponding to different outcomes will never overlap and thus a definite outcome will always be present. This way, the remark made by Kramers, mentioned in section 2.2, can be solved. To illustrate how this works, I’ll treat an example analogous to the one in section 2.1. Again, suppose we prepare some particle in the state |Ψi = |Ψai_e|Blacki_e. This time, we

also prepare the measuring device in a definite state, namely, its ready state |ψrim. The

forces in the box don’t result in the particle’s position and momentum changing, but in the position and momentum of the device’s pointer changing, see figure 3. Once the particle’s positional wave function (|ψi_e) has travelled all the way though the box, the measuring de-vice wave function will be either |ψhim or |ψsim. Hence, the total wave function evolution is

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Figure 3: An example of the inclusion of the measuring device in the wave function, the evolution of the wave function is |ψrim|ψaie|Blackie −→

of the right hand side of 11 will be nonzero, and thus what the effective wave function will be(which determines the further trajectory of the particle). Note that this will of course depend ultimately on the initial position of the particle in |ψai_e and the way the box works.

2.4 Bell’s treatment of spin measurement

To do spin measurements in first-order pilot-wave theory, I will use the formalism described by J.S. Bell [3]. Before applying this to the teleportation case, I will use it to describe a simple Stern-Gerlach mesurement. In this example there is no need to include the wave function of a measuring device, since the outcome of the measurement can be determined from the particle positions of the measured particles.

We start with the initial wave function

|Ψ(t = 0)i = √1

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When this wave function is fed through the Stern-Gerlach device, the wave function will evolve according to the Schr¨odinger equation with Hamiltonian

H = gσ~ i

∂

∂x (13)

where g is a coupling constant that depends on the setup of the experiment (the magnetic field strength among other parameters) and σ is one of the Pauli matrices, depending on the orientation of the measuring device3. This means the wave packets will get deflected according to their spin parts; the spin-up packet will get deflected up and the spin-down packet will get deflected down. The wave function will evolve to become

|Ψ(t)i = √1 2(ψ↑(x − g ~ 2) |↑i + ψ↓(x − g(− ~ 2) |↓i), (14)

here it can be seen that the wave packets deflect in opposite directions because of their opposite Eigenvalues with respect to the spin operator S = σ~

i. Once the wave packets are

fully separated, if the particle is observed in one of them, the other part will be zero and the system becomes trapped in its new effective wave function consisting of one of the terms in (14).

Finally, we can plug this wave function into equation 3 to find the velocity field for the particle d dtxi(t) = g ~ 2 |ψ↑|2− |ψ↓|2 |ψ↑|2+ |ψ↓|2 . (15)

This is simply the average velocity of the wave packets, however only one of the terms will be nonzero and thus the particle trajectory corresponds to one of them.

3_{The experiment is set up so that the initial wave function is given in spin components in the direction}

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3 Teleportation protocol

Quantum teleportation is the apparent instantaneous teleportation of information from lo-cation A to B. The standard method to demonstrate this is by using three spin−1₂ particles, two of which are entangled. In this section, I will treat the standard quantum teleportation protocol, the protocol is the same for any interpretation.

The initial state of the three particles is

|Ψ(t)i = √1

2(a |↑i1+ b |↓i1)(|↑i2|↓i3− |↓i2|↑i3)ψ1(x1)ψ2(x2)ψ3(x3) (16) Where particles 2 and 3 are maximally entangled in a so called EPRB (Einstein-Podolsky-Rosen-Bohm) state and the state of particle 1 is specified by parameters a and b. Note that in the rest of this section, and in section 4 I will omit ψ1(x1)ψ2(x2)ψ3(x3) (the spatial

part of the wave function), by doing this, I assume that any measurement will be recorded in the position of some pointer (like in section 2.3.1) rather than in the position of the measured particles (like in section 2.1). If this is the case, the particles don’t move during the teleportation protocol. Thus, the spatial parts of the wave function are static, and nonzero only in a small area around the particles, they area localised wave-packets.

The information in parameters a and b is the information that will appear to be teleported. Suppose that particles 2 and 3 are brought to the locations A and B respectively, and that particle 1 is already at location A. By introducing the entangled states

|β1i =

1 √

2(|↑i1|↑i2 + |↓i1|↓i2) |β2i =

1 √

2(|↑i1|↓i2 + |↓i1|↑i2) |β3i =

1 √

2(|↑i1|↑i2 − |↓i1|↓i2) |β4i =

1 √

2(|↑i1|↓i2 − |↓i1|↑i2),

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known as the Bell states for particles 1 and 2, (16) can be rewritten as

Ψ(x, t) = 1 2

|β1i (−b |↑i3+a |↓i3)+|β2i (b |↑i3+a |↓i3)+|β3i (−a |↑i3+b |↓i3+|β4i (−a |↑i3−b |↓i3)

. (18)

All we have done is rewrite the wave function in a different basis, but it already appears that the information in a and b is now attached to particle 3 instead of particle 1. It can’t be the case that the information was already at location B, linked to particle 3, since this might be any other particle at any other location.

To complete the teleportation, we do a Bell state measurement of the particles 1 and 2 at location A, that is, we measure in which of the states (17) particles 1 and 2 are. At this point, for an observer of this measurement, the information will be instantaneously teleported to location B, since the observer now knows exactly how the information is embedded in the state of particle 3. To complete the teleportation of the entire state of particle 1, the observer at A now sends the result of the measurement, which consists of two bits of classical information, to B, according to which he applies one of the unitary operations

U1 =   0 1 −1 0   U2 =   0 1 −1 0   U3 =   0 1 −1 0   U4 =   0 1 −1 0  

on particle 3, after which the particle will be in the exact same state as particle 1 started. Since the parameters a and b are continuous, the information they hold is an infinite amount of classical bits, so by only sending two bits, infinite bits are teleported from A to B.

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4 Teleportation through first order pilot wave

dynam-ics

Now we have all the perquisites to go through the teleportation protocol once more, this time using first-order pilot-wave theory and Bell’s treatment of spin.

Starting with (18), we now preform the Bell state measurement. There is no need to write down the explicit operator for this measurement, but all we need is that the outcome is registered by some position wave function and is macroscopic. Assume the measuring device works analogous to figure 3, but now there are four possible outcomes (and a ready state). The initial wave function is the same as in the previous section, but now includes the measuring device.

|Ψ(t = 0)i = 1 2

|β1i (−b |↑i₃+ a |↓i₃) + |β2i (b |↑i₃+ a |↓i₃)

+ |β3i (−a |↑i₃+ b |↓i₃+ |β4i (−a |↑i₃ − b |↓i₃)

φ(x, t)

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where φ(x, t) is the ready state of the measurement apparatus. Now, as we do the measurement, the wave function of the apparatus splits in four wave-packets, turning the wave function into

|Ψ(t)i = 1 2

|β1i (−b |↑i₃+ a |↓i₃)φ(x − gB1t) + |β2i (b |↑i₃+ a |↓i₃)φ(x − gB2t)+

|β3i (−a |↑i₃ + b |↓i₃)φ(x − gB3t) + |β4i (−a |↑i₃− b |↓i₃)φ(x − gB4t)

.

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Here, Bn are the eigenvalues of the Bell state measurement and g is the coupling constant,

these constants depend on how exactly the Bell state measurement is done and determine how the pointer moves, as long as the Bnare sufficiently different (which they are in a proper

Bell state measurement device), the wave packets will separate. Again, the position of the pointer now shows which of the wave packets is nonzero, and thus what the effective wave function of particle 3 is. Now one of the unitary operations from section 3 can be carried out as to change the effective wave function of particle 3 to the exact state particle 1 was at the start.

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To complete the pilot-wave picture, we can go on to calculate the velocity field, just like in section 2.4. The inner products of the Eigenstates (for example |β2i (−b |↑i3+ a |↓i3) with

themselves will all be 1. So the probability density and probability flow respectively are

ρ(x, t) = 1 2 |φ(x − gB1t)|2+ |φ(x − gB2t)|2+ |φ(x − gB3t)|2+ |φ(x − gB4t)|2 =X n |φ(x − gBnt)|2 and j(x, t) = g 4 (B1)2|φ(x − gB1t)|2+ (B2)2|φ(x − gB2t)|2 + (B3)2|φ(x − gB3t)|2+ (B4)2|φ(x − gB4t)|2 =X n (Bn)2|φ(x − gBnt)|2. (21)

Finally, the velocity field is ∂ ∂tX(t) = g 16 P mBm|φ(x − gBmt)| 2 P n|φ(x − gBnt)|2 (22)

Again, this is an average velocity of the four wave packets, only one of which will be nonzero, hence the pointer will end up indicating one of the four possible outcomes.

5 Conclusion

In the first half of this paper, I have argued that the criticism De Broglie received in 1927 was not enough to objectively reject pilot wave theory. Even though in the literature one can find multiple non-physical reasons for this rejection, an overview with a clear summary is not to be found, this would improve the understanding as to why the theory was forgotten. The second aim of this paper was to explore the phenomenon of quantum teleportation in the pilot-wave interpretation and see what intuitions and insights arise. We have seen that quantum wave functions live in configuration space rather than in regular 3-space and hence wave packets will always separate. Consequently, the system will get trapped in one

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of those packets, this packet has then become the new effective wave function. In the case of teleportation, measuring a property of one part of an entangled system traps the entire system, and thus determines the effective wave function of the other (distant) part as well. Can we really call this teleportation? And is it really as “spooky” as Einstein called it in the conventional interpretation? In popular literature (say science fiction) teleportation is something being either transported instantaneously or being recreated instantaneously (with the original ceasing to exist) at a different location. In the conventional interpretation of quantum mechanics, a system is fully described by its wave function, and it is also its wave function that is recreated at a distant location when following the teleportation protocol. So we can say it is justified to call this phenomenon teleportation. In pilot-wave theory, the wave function is not the full description of a system, namely, it also has a well defined position. By following the protocol, what happens is that a particle at a distant location obtains the spin part of the wave function of some other particle. But it is not the same particle, since it has an entirely different position and the original stays in its place (although its spin part wave function is changed). Thus there isn’t any real teleportation happening, at least not in the sci-fi sense of the word. Einstein though entanglement was spooky because he was not used to actions at distances, however, in a nonlocal theory like the pilot-wave theory, action at a distance is inherent, and thus not spooky at all.

Whether the term teleportation does justice to the phenomenon it describes is definitely dependent on the interpretation of quantum theory, and by using the pilot wave theory, some spookiness is eliminated from physics, which could mean it is another step in the right direction.

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References

[1] J. S. Bell, “On the einstein podolsky rosen paradox,” in John S Bell On The Foundations Of Quantum Mechanics, pp. 7–12, World Scientific, 2001.

[2] T. M. Nieuwenhuizen, “Is the contextuality loophole fatal for the derivation of bell inequalities?,” Foundations of Physics, vol. 41, no. 3, pp. 580–591, 2011.

[3] J. S. Bell, “On the impossible pilot wave,” Foundations of Physics, vol. 12, no. 10, pp. 989–999, 1982.

[4] A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of phys-ical reality be considered complete?,” Physphys-ical review, vol. 47, no. 10, p. 777, 1935. [5] G. Bacciagaluppi and A. Valentini, Quantum theory at the crossroads: reconsidering the

1927 Solvay conference. Cambridge University Press, 2009.

[6] O. Maroney and B. J. Hiley, “Quantum state teleportation understood through the bohm interpretation,” Foundations of Physics, vol. 29, no. 9, pp. 1403–1415, 1999. [7] S. Colin and A. Valentini, “Instability of quantum equilibrium in bohm’s dynamics,” in

Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 470, p. 20140288, The Royal Society, 2014.

[8] L. de Broglie, “Interpretation of quantum mechanics by the double solution theory,” in Annales de la Fondation Louis de Broglie, vol. 12, pp. 1–23, 1987.

[9] D. Z. Albert, Quantum mechanics and experience. Harvard University Press, 2009. [10] D. Bohm, “A suggested interpretation of the quantum theory in terms of” hidden”

variables. i ii,” Physical Review, vol. 85, no. 2, p. 166, 1952.

[11] D. Bohm and B. J. Hiley, The undivided universe: An ontological interpretation of quantum theory. Routledge, 2006.

[12] M. Towler, N. Russell, and A. Valentini, “Time scales for dynamical relaxation to the born rule,” Proc. R. Soc. A, p. rspa20110598, 2011.

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[13] T. Bonk, “Why has de broglie’s theory been rejected?,” Studies in History and Philos-ophy of Science Part A, vol. 25, no. 3, pp. 375–396, 1994.

[14] J. Berkovitz, “Action at a distance in quantum mechanics,” in The Stanford Encyclo-pedia of Philosophy (E. N. Zalta, ed.), Metaphysics Research Lab, Stanford University, spring 2016 ed., 2016.

Quantum teleportation understood though first-order pilot-wave theory