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Quantum teleportation understood
though first-order pilot-wave theoryReport Bachelor Project Physics and Astronomy, size 15 EC,
conducted between 09 - 04 - 2018 and 13 - 07 - 2018at the
Descartes Centre for the History and Philosophy of the Sciences and theHumanities
Wouter van SoestSupervisor: Guido Bacciagaluppi
Second examiner: Jeroen van Dongen
Faculty of Science
University of Amsterdam
June 2018
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 geınterpreteerd 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.
Contents
Abstract
1 Introduction 1
2 Chronological overview 3
2.1 The first Pilot-Wave theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Criticism and the abandonment of pilot-wave theory . . . . . . . . . . . . . . 7
2.3 1952: Bohm’s revival of Pilot-Wave theory . . . . . . . . . . . . . . . . . . . 10
2.3.1 Bohm’s theory of measurement . . . . . . . . . . . . . . . . . . . . . 12
2.4 Bell’s treatment of spin measurement . . . . . . . . . . . . . . . . . . . . . . 13
3 Teleportation protocol 15
4 Teleportation through first order pilot wave dynamics 17
5 Conclusion 18
References
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
1This is not undisputed, for example, see [2]
1
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.
2
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
)∇2Ψ + VΨ (1)
midxidt
= ∇iS, (2)
where equation (1) is the ordinary time dependent Schrodinger 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
midxidt
=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,
3
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) |black〉 where ψa(x)
is the positional part of the wave function and suppose there is an observable, ”hardness” of
which the Eigenstates are |hard〉 and |soft〉 with |black〉 = 1√2
(|hard〉 + |soft〉
). 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 |black〉 −→1√2
(ψb |hard〉+ ψc |soft〉
),
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
4
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 |hard〉 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 |hard〉
side, since the input wave function is already in the |hard〉 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.
5
Figure 1: An example of the separation of wave packets: the evolution of the wave functionis ψa |black〉 −→ 1√
2|hard〉ψb + |soft〉ψc. ([9])
6
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
2for a more detailed description, see [5]
7
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 Leon 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.
8
Figure 2: Wave packets scattering off (top) a spatially limited grating → wave packetsseparate, (middle and bottom)a spatially unlimited grating → wave packets don’t separateif they all have the same velocity (middle) but separate for different velocities (bottom) [5]
9
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 Schrodinger equation and the polar form wave function also
used by De Broglie
i~∂Ψ
∂t= −
( ~2
2m
)∇2Ψ + VΨ (1)
Ψ = |Ψ|ei~S. (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
10
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
md2x
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 Schodinger 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,
11
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 |Ψ〉 = |Ψa〉e |Black〉e. This time, we
also prepare the measuring device in a definite state, namely, its ready state |ψr〉m. 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 (|ψ〉e) has travelled all the way though the box, the measuring de-
vice wave function will be either |ψh〉m or |ψs〉m. Hence, the total wave function evolution is
given by
|ψr〉m |ψa〉e |Black〉e −→1√2
(|ψh〉m
∣∣ψ1a
⟩e|Hard〉e + |ψs〉m
∣∣ψ1a
⟩e|Soft〉e . (11)
So even though the particle wave function is not separated in space, the total wave function
has separated in configuration space Now the position of the pointer will show which part
12
Figure 3: An example of the inclusion of the measuring device in the wave function,the evolution of the wave function is |ψr〉m |ψa〉e |Black〉e −→
1√2
(|ψh〉m |ψ1
a〉e |Hard〉e +
|ψs〉m |ψ1a〉e |Soft〉e. ([9])
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 |ψa〉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)〉 =1√2
(ψ0(x) |↑〉+ ψ0(x) |↓〉). (12)
13
When this wave function is fed through the Stern-Gerlach device, the wave function will
evolve according to the Schrodinger 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)〉 =1√2
(ψ↑(x− g~2
) |↑〉+ ψ↓(x− g(−~2
) |↓〉), (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.
3The experiment is set up so that the initial wave function is given in spin components in the directionthe spin will be measured
14
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)〉 =1√2
(a |↑〉1 + b |↓〉1)(|↑〉2 |↓〉3 − |↓〉2 |↑〉3)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
|β1〉 =1√2
(|↑〉1 |↑〉2 + |↓〉1 |↓〉2)
|β2〉 =1√2
(|↑〉1 |↓〉2 + |↓〉1 |↑〉2)
|β3〉 =1√2
(|↑〉1 |↑〉2 − |↓〉1 |↓〉2)
|β4〉 =1√2
(|↑〉1 |↓〉2 − |↓〉1 |↑〉2),
(17)
15
known as the Bell states for particles 1 and 2, (16) can be rewritten as
Ψ(x, t) =1
2
(|β1〉 (−b |↑〉3+a |↓〉3)+|β2〉 (b |↑〉3+a |↓〉3)+|β3〉 (−a |↑〉3+b |↓〉3+|β4〉 (−a |↑〉3−b |↓〉3)
).
(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.
16
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)〉 =1
2
(|β1〉 (−b |↑〉3 + a |↓〉3) + |β2〉 (b |↑〉3 + a |↓〉3)
+ |β3〉 (−a |↑〉3 + b |↓〉3 + |β4〉 (−a |↑〉3 − b |↓〉3))φ(x, t)
(19)
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)〉 =1
2
(|β1〉 (−b |↑〉3 + a |↓〉3)φ(x− gB1t) + |β2〉 (b |↑〉3 + a |↓〉3)φ(x− gB2t)+
|β3〉 (−a |↑〉3 + b |↓〉3)φ(x− gB3t) + |β4〉 (−a |↑〉3 − b |↓〉3)φ(x− gB4t)).
(20)
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 Bn are 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.
17
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 |β2〉 (−b |↑〉3 +a |↓〉3) 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
)=∑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
)=∑n
(Bn)2|φ(x− gBnt)|2.
(21)
Finally, the velocity field is
∂
∂tX(t) =
g
16
∑mBm|φ(x− gBmt)|2∑
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
18
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.
19
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