Quantum State Teleportation understood through the Bohm Interpretation
O. J. E. Maroney and B. J. Hiley
Theoretical Physics Research Unit
Birkbeck College, University of London
Malet Street, London, WC1E 7HX, United Kingdom
Foundations of Physics, vol. 29, No. 9, 1999
Abstract
Quantum state teleportation has focused attention on the role of quantum
information. Here we examine quantum teleportation through the Bohm interpretation.
This interpretation introduced the notion of active information and we
show that it is this information that is exchanged during teleportation.
We discuss the relation between our notion of active information and the
notion of quantum information introduced by Schumacher.
1 Introduction
The recent discovery of quantum state teleportation [1]
has re-focused attention on the nature of quantum information and the role
of quantum non-locality in the transfer of information. Developments in
this area have involved state interchange teleportation [2],
as well as multi-particle entanglement swapping [3],
and position/momentum state teleportation [4]. Although
these effects arise from a straight forward application of the formalism,
the nature of the quantum information and its transfer still presents difficulties.
Attempts to address the issue from the perspective of information theory
[5],[6]
and without invoking wave function collapse [7] have
clarified certain aspects of this process but problems still remain.
In order to obtain a different perspective on these phenomena we first
review the salient features of the Bohm interpretation that are of direct
relevance to these situations [8], [9],
[10],
[11], before applying its techniques to the specific
example of spin teleportation. One of the advantages of using this approach
in the present context is that to account for quantum processes it is necessary
to introduce of the notion of `active' information. This notion was introduced
by Bohm & Hiley [9] to account for the properties
of the quantum potential which cannot be consistently regarded as a mechanical
potential for reasons explained in Bohm & Hiley [9].
There is also the added advantage that the approach gives a clear physical
picture of the process at all times, and, therefore provides an unambiguous
description of where and how the `quantum information' is manifested. In
this paper we will discuss how the three notions of active, passive and
inactive information are of relevance to the teleportation problem.
2 Quantum Teleportation
The basic structure of quantum teleportation can be expressed using three
spin-
1/2 particles, with particles 2 and 3 initially
in a maximally entangled EPRB state, and particle 1, in an unknown superposition:
Y1 = (a|ñ1+b|¯ñ1)(|ñ2|¯ñ3-|¯ñ2|ñ3)/Ö2 |
|
By introducing the `Bell states'
|
b(ij)1
= (|ñi|ñj+|¯ñi|¯ñj)/Ö2 |
|
b(ij)2
= (|ñi|ñj-|¯ñi|¯ñj)/Ö2 |
|
b(ij)3
= (|ñi|¯ñj+|¯ñi|ñj)/Ö2 |
|
b(ij)4
= (|ñi|¯ñj-|¯ñi|ñj)/Ö2 |
|
|
|
|
we can re-write Y1 as
If we now measure the Bell state of particles 1 and 2, and communicate
the result to the recipient of particle 3 who will,using that information,
then perform one of the local unitary operations on particle 3 given below
In this way we have disentangled particle 3 from particle 2 and produced
the state (a|ñ3+b|¯ñ3)
on particle 3. Thus the information represented by [a,b] has been perfectly
`teleported' from particle 1 to particle 3, without our having measured
a or b directly. Furthermore, during the transfer process we have only
passed 2 classical bits of information (corresponding only to the choice
of U) between the remote particles. Note that as 'a' and 'b' are continuous
parameters, it would require an infinite number of classical bits to perfectly
specify the [a,b] state. This ability to teleport accurately has been shown
to be critically dependant upon the degree of entanglement of particles
2 and 3 [5], [12].
We may note that in the Bell state expansion, the information signified
by the coefficients [a,b] appears on the particle 3 spin states before
any actual measurement has taken place (although this information is encoded
in a different way for each Bell state). What are we to make of this?
It would seem absurd to assume that the information described by a and
b was already attached to particle 3 as, at this stage, particle 1 could
be any other particle in the universe. Indeed all that has happened is
that Y1 has been the re-written in
a different basis to give Y2. Clearly
this cannot be regarded as an actual physical effect.
Following Heisenberg [13] and Bohm [14],
we can regard the wave function as describing potentialities. At this stage
Y2
describes the potentiality that particle 3 could carry the [a,b] information
that would be actualised during the measurement. However, here we have
a problem as Braunstein[7] has shown that a collapse
of the wavefunction (the usual mechanism by which such potentialities become
actualised) is unnecessary to the description of quantum teleportation,
by including the Bell state measuring device within the quantum formalism.
Using this descripton, we find that the attachment of the [a,b] information
to particle 3, after the Bell state interaction, is the same as
in the Y2 expansion
prior
to the interaction. While this is clearly necessary to maintain the no-signalling
theorem, it leaves ambiguous the question of whether the [a,b] information
has been transferred to particle 3, at this stage, or not.
To resolve these issues, we need to give a clearer meaning to the nature
of the information contained in [a, b] and to understand how and when this
information becomes manifested at particle 3. We now turn to the Bohm interpretation
to provide some new insights into these questions.
3 The Quantum Potential as an Information
Potential
The Bohm interpretation of quantum mechanics can be derived from the polar
decomposition of the wave function of the system, Y
= ReiS, which is inserted into the Schrödinger equation1
i |
¶Y
¶t |
= |
æ
ç
è |
- |
Ñ2
2m |
+V |
ö
÷
ø |
Y |
|
yielding two equations, one that corresponds to the conservation of probability,
and the other, a modified Hamilton-Jacobi equation:
- |
¶S
¶t |
= |
(ÑS)2
2m |
+V- |
Ñ2R
2mR |
|
|
This equation can be interpreted in the same manner as a classical Hamilton-
Jacobi, describing an ensemble of particle trajectories, with momentum
p = ÑS, subject to the classical potential
V and a new quantum potential Q = -[(Ñ2R)/
2mR]. The quantum potential, Q, is responsible for all the non-classical
features of the particle. It can be shown that, provided the particle trajectories
are distributed with weight R2 over a set of initial conditions,
the weighted distribution of these trajectories will match the statistical
results obtained from the usual quantum formalism. It should be noted that
although the quantum Hamilton-Jacobi equation can be regarded as a return
to a classical deterministic theory, the quantum potential has a number
of the non-classical features that make the theory very different from
any classical theory. We should regard Q as being a new quality of global
energy that augments the kinetic and classical potential energy to ensure
the conservation of energy at the quantum level. These non-classical features
are of direct relevance to the problem of quantum teleportation. Of particular
importance are the properties of non-locality and form-dependence.
3.1 Non-locality
Perhaps the most surprising feature of the Bohm approach is the appearance
of non-locality. This feature can be clearly seen when the above equations
are generalised to describe more than one particle. In this case the polar
decomposition of
Y(x1,x2,¼,xN)
= R(x1,x2,¼,xN)eiS(x1,x2,¼,xN)
produces a quantum potential, Qi, for each particle given by:
Qi = - |
Ñ2iR(x1,x2,¼,xN)
2mR(x1,x2,¼,xN) |
|
|
This means that the quantum potential on a given particle i will, in
general, depend on the instantaneous positions of all the other particle
in the system. Thus an external interaction with one particle may have
a non-local effect upon the trajectories of all the other particles in
the system. In other words groups of particles in an entangled state are,
in this sense, non-separable. In separable states, the overall wave function
is a product of individual wave functions
For example, when one of the particles, say particle 1, is separable
from the rest, we can write
Y(x1,x2,¼,xN)
= f(x1)x(x2,¼,xN).
In this case
R(x1,x2,¼,xN)
= R1(x1)R2¼N(x2,¼,xN),
and therefore:
Q1 = - |
Ñ21R1(x1)R2¼N(x2,¼,xN)
2mR1(x1)R2¼N(x2,¼,xN) |
= - |
Ñ21R1(x1)
2mR1(x1) |
|
|
In a separable state, the quantum potential does not depend on the position
of the other particles in the system. Thus the quantum potential only has
non-local effects for entangled states.
3.2 Form dependence
We now want to focus on one feature that led Bohm & Hiley [9]
to propose that the quantum potential can be interpreted as an `information
potential'. As we have seen above the quantum potential is derived from
the R-field of the solution to the appropriate Schrödinger equation.
The R-field is essentially the amplitude of the quantum field Y
. However, the quantum potential is not dependant upon the amplitude of
this field (i.e., the intensity of the R-field), but only upon its form.
This means that multiplication of R by a constant has no effect upon the
value of Q. Thus the quantum potential may have a significant effect upon
the motion of a particle even where the value of R is close to zero. One
implication of this is that the quantum potential can produce strong effects
even for particles separated by a large distance. It is this feature that
accounts for the long- range EPRB-type correlation upon which teleportation
relies.
It is this form-dependence (amongst others things) that led Bohm &
Hiley [9] [16] to suggest that the
quantum potential should be interpreted as an information potential, rather
than a classical mechanical potential2.
Here the word `information' takes its meaning from its etymological roots
informo
(or informare) which signifies the action of forming or bringing
order into something. Thus the proposal is that the quantum potential captures
a dynamic, self- organising feature that is at the heart of a quantum process.
For many-body systems, this organisation involves a non-local correlation
of the motion of all the bodies in the entangled state, which are all being
simultaneously organised by the collective R-field. In this situation they
can be said to be drawing upon a common pool of information encoded in
the entangled wave function. The informational, rather than mechanical,
nature of this potential begins to explain why the quantum potential is
not definable in the 3-dimensional physical space of classical potentials
but needs a 3N-dimensional configuration space. When one of the particles
is in a separable state, that particle will no longer have access to this
common pool of information, and will therefore act independently of all
the other particles in the group (and vice versa). In this case, the configuration
space of the independent particle will be isomorphic to physical space,
and its activity will be localised in space-time.
4 Active, Passive and Inactive Information
In order to discuss how and what information is teleported, we must first
distinguish between the notions of active, passive and inactive information.
All three play a central role in our discussion of teleportation. Where
a system is described by a superposition
Y(x)
= Ya(x)+Yb(x),
and Ya(x) and Yb(x)
are non- overlapping wave-packets, the actual particle position will be
located within either one or the other of the wave-packets. The effect
of the quantum potential upon the particle trajectory will then depend
only upon the form of the wave-packet that contains the particle. We say
that the information associated with this wave-packet is active, while
it is passive for the other packet. If we bring these wave-packets together,
so that they overlap, the previously passive information will become active
again, and the recombination will induce complex interference effects on
the particle trajectory.
Now let us see how the notion of information accounts for measurement
in the Bohm interpretation. Consider a two-body entangled state, such as
Y(x1,x2)
= fa(x1)xa(x2)+fb(x1)xb(x2),
where the active information depends upon the simultaneous position of
both particle 1 and particle 2. If the fa
and fb are overlapping wave functions,
but the xa and xb
are non-overlapping, and the actual position of particle 2 is contained
in just one wave-packet, say xa,
the active information will be contained only in fa(x1)xa(x2),
the information in the other branch will be passive. Therefore only the
fa(x1)
wave-packet will have an active effect upon the trajectory of particle
1. In other words although
fa and
fb
are both non-zero in the vicinity of particle 1, the fact that particle
2 is in xa(x2) will mean
that only
fa(x1)xa(x2)
is active, and thus particle 1 will only be affected by fa(x1).
If fa(x1) and fb(x1)
are separated, particle 1 will always be found within the location of fa(x1).
The position of particle 2 may therefore be regarded as providing an accurate
measurement of the position of particle 1. Should the fa
and fb now be brought back to overlap
each other, the separation of the wave-packets of particle 2 will continue
to ensure that only the information described by fa(x1)xa(x2)
will be active. To restore activity to the passive branches of the superposition
requires that both fa(x1)
and fb(x1) and xa(x2)
and xb(x2) be simultaneously
brought back into overlapping positions. If the x(x2)
represents a thermodynamic, macroscopic device, with many degrees of freedom,
and/or interactions with the environment, this will not be realistically
possible. If it is never possible to reverse all the processes then the
information in the other branch may be said to be inactive (or perhaps
better still `deactivated'), as there is no feasible mechanism by which
it may become active again. This process replaces the collapse of the wave
function in the usual approach.
5 Quantum State Teleportation and Active Information
In order to examine how these ideas of information can be used in quantum
teleportation, we must explain how spin is discussed in the Bohm interpretation.
There have been several different approaches to spin [9],
[19],
[20], but this ambiguity need not concern us here as
we are trying to clarify the principles involved. Thus for the purpose
of this article we will adopt the simplest model that was introduced by
Bohm, Schiller and Tiomno [17], [18].
We start by rewriting the polar decomposition of the wave function as Y
= ReiSF where F
is a spinor with unit magnitude and zero average phase. If we write:
F = |
æ
ç
ç
ç
ç
ç
è |
|
|
|
ö
÷
÷
÷
÷
÷
ø |
|
|
where n is the dimension of the spinor space, then åisi
= 0 and åi(ri)2
= 1. The many-body Pauli equation then leads to a modified quantum Hamilton-Jacobi
equation given by:
|
¶S
¶t |
-iFf |
¶F
¶t |
= - |
å
i |
|
æ
ç
è |
|
pi2
2m |
+Qi+ 2miB.si |
ö
÷
ø |
|
|
with a momentum pi = ÑiS+FfÑiF,
a quantum potential
Qi = [1/ 2m](-Ñi2R+ÑiFfÑiF+(FfÑiF)2).
B
is the magnetic field and mi is the
magnetic dipole moment associated with particle i. We can, in addition,
attribute a real physical angular momentum to each particle i given by
si
= 1/2YfsiY,
where si are the Pauli matrices operating
solely in the spinor subspace of particle i.
The information contained in the spinor wave function is again encoded
in the quantum potential, so that the trajectory of the particle is guided
by the evolution of the spinor states, in addition to the classical interaction
of the B field with the magnetic dipole moment of the particle.
Contracting the Pauli equation with Yfsi
leads the equation of motion for the particle i spin vector:
where Ti is a quantum torque. The k components of the
torque are given by
[Ti]k = |
å
j |
|
1
2rmj |
eklm{[si]l[Ñj]n(r[Ñj]n[si]m)+
slr[Ñj]n(r[Ñj]nsmr)} |
|
where r = R2 and sij is
the non-local spin correlation tensor formed from
YfsisjY.
Equations of motion for these tensors can be derived by contracting the
Pauli equation with Yfsisj,
and similarly for higher dimension correlation tensors. Detailed application
of these ideas to the entangled spin state problem has been demonstrated
in Dewdney et al. [18].
To complete the description of the particles, we must attach position
wave functions to each of the particles. We do this by assuming that each
particle can be represented by a localised wave-packet. Thus, for the teleportation
problem:
|
|
|
(a|ñ1+b|¯ñ1)(|ñ2|¯ñ3-|¯ñ2|ñ3)r(x1)f(x2)x(x3)/Ö2 |
|
|
|
{b(12)1[-b|ñ3+a|¯ñ3]+b(12)2[+b|ñ3+a|¯ñ3]+ |
|
|
|
b(12)3[-a|ñ3+b|¯ñ3]+b(12)4[-a|ñ3-b|¯ñ3]}
r(x1)f(x2)x(x3)/2 |
|
|
|
|
Initially, the three position wave packets are separable, and the particle
trajectories will be determined by separate information potentials although
the spin properties of particles 2 and 3 will be linked via the spin quantum
potential. The particle spins can be shown to be
|
s1 = |
1
2 |
(a*b+b*a,ia*b-ib*a,a*a-b*b) |
|
|
|
|
|
|
Note that each of the particles 2 and 3 in a maximally entangled anti-symmetric
state have zero spin angular momentum, a surprising point that has already
been noted and discussed by Dewdney et al. [18] and
by Bohm & Hiley [9]. More significantly for our problem
is that at this stage, the information described by a and b acts only through
the quantum potential, Q1, which organises the spin of particle
1, but not the spin of particles 2 and 3.
Before discussing the measurement involved in the actual teleportation
experiment, let us first recall what happens when a simple spin measurement
is made on particle 2 alone. The wave-packet f(x2)
would divide into two, and the particle would enter one of these packets
with equal probability. Thus the wave function becomes
Y = (a|ñ1+b|¯ñ1)r(x1)(|ñ2|¯ñ3f1(x2)-|¯ñ2|ñ3f0(x2))x(x3)/2 |
|
Particle 2 will enter one of the packets, say f1(x2).
As f1(x2) and f0(x2)
separate, particles 2 and 3 will develop non-zero spins, with opposite
senses, and will be described by |¯ñ2|ñ3
. Any subsequent measurement of the spin of particle 3, would divide x(x3)
into two, but particle 3 would always enter the wave-packet on the same
branch of the superposition as particle 2 had entered earlier, as only
the information in that branch is active. This has been beautifully illustrated
by Dewdney et al. [18]
As the particle 1 is in a separable state for both spin and position,
no local interactions on particle 2 or 3 will have any effect on the trajectory
and spin of particle 1. Neither will any measurement on particle 1 produce
any effect on particles 2 and 3. The behaviour of the spins of particles
2 and 3 will be determined by the pool of information common to them both,
while only the behaviour of particle 1 is determined by the [a,b] information,
regardless of the basis in which the spin states are expanded.
Now let us return to the main theme of this paper and consider the measurement
that produces teleportation. Here we need to introduce a Bell state measurement.
Let the instrument needed for this measurement be described by the wave-packet
h(x0)
where x0 is a variable (or a set of variables) characterising
the state of this apparatus. The measurement is achieved via an interaction
Hamiltonian that can be written in the form H = O(12)Ñ0.
The interaction operator O(12) = lO(12)l
couples the x0 co-ordinate to the Bell state of particles 1
and 2 through the Bell state projection operators Ol
= blblf.
This creates the state
|
|
|
{h1(x0)b(12)1[-b|ñ3+a|¯ñ3]+h2(x0)b(12)2[+b|ñ3+a|¯ñ3]+ |
|
|
|
h3(x0)b(12)3[-a|ñ3+b|¯ñ3]+h4(x0)b(12)4[-a|ñ3-b|¯ñ3]} |
|
|
|
|
|
|
|
where h1(x0), h2(x0),
h3(x0)
and h4(x0) are the wave-packets
of the four non-overlapping position states corresponding to the four outcomes
of the Bell state measuring instrument. Initially all four systems become
entangled and their behaviour will be determined by the new common pool
of information. This includes the [a,b] information that was initially
associated only with particle 1.
As the position variable x0 of the measuring device enters
one of the non- overlapping wave-packets hi(x0),
only one of the branches of the superposition remains active, and the information
in the other branches will become passive. As this happens, particle 3
will develop a non-zero particle spin s3, through the
action of the quantum torque. The explicit non-locality of this allows
the affects of the Bell state measurement to instantaneously have an effect
upon the behaviour of particle 3. The significance of the Y2
Bell state expansion is now revealed as simply the appropriate basis for
which the [a,b] information will be transferred entirely onto the behaviour
of particle 3, if only a single branch of the superposition were to remain
active. The interaction with the Bell state measuring device is required
to bring about this change from active to passive information in the other
branches (and thereby actualising the potentiality of the remaining branch).
However, no meaningful information on [a,b] may yet be uncovered at
particle 3 until it is known which branch is active, as the average over
all branches, occuring in an ensemble, will be statistically indistinguisable
from no Bell state measurement having taken place. Simply by noting the
actual position (x0) of the measuring device, the observer,
near particles 1 and 2, immediately knows which wavepacket x0
has entered, and therefore which state is active for particle 3. The observer
then sends this classical information to the observer at 3 who will then
apply the appropriate unitary transformation U1¼U4
so that the initial spin state of particle 1 can be recovered at particle
3.
6 Conclusion
In the approach we have adopted here, the notion of active information
introduced by Bohm and Hiley [9] has been applied to
the phenomenon of state teleportation. This gives rise to a different perspective
on this phenomenon and provides further insight into the notion of quantum
information. To see more clearly how teleportation arises in this approach
let us re-examine the above spin example in more general terms. The essential
features can be seen by examining the general structure of the quantum
potential. Using the initial wave function, Yi
given above, the quantum potential takes the form
Q(x1, x2, x3)
= Q1(x1, a, b)Q23(x2, x3) |
|
Here the coefficients a and b characterise the quantum potential acting
only on particle 1. This means that initially the information carried by
the pair [a, b] actively operates on particle 1 alone. At this stage the
behaviour of particle 3 is independent of a and b, as we would expect.
To perform a Bell State measurement we must couple particle 1 to particle
2 by introducing the interaction Hamiltonian given above. During this process,
a quantum potential will be generated that will couple all three particles
with the measuring apparatus. When the interaction is over, the final wave
function becomes Yf. This will produce
a quantum potential that can be written in the form
Q(x1, x2, x3,
x0) = Q12(x1, x2, x0)Q3(x3,
x0,a, b) |
|
Thus after the measurement has been completed, the information contained
in a and b has now been encoded in Q3 which provides the active
information for particle 3. Thus we see that the information that was active
on particle 1 has been transferred to particle 3. In turn this particle
has been decoupled from particle 2. Thus the subsequent spin behaviour
of particle 3 will be different after the measurement.
What we see clearly emerging here is that it is active information
that has been transferred from particle 1 to particle 3 and that this transfer
has been mediated by the non-local quantum potential. Let us stress once
again that this information is in-formation for the particle and, at this
stage has nothing to do with `information for us'.
Previous discussions involving quantum information have been in terms
of its relation to Shannon information theory [23].
In classical information theory, the expression H(A) = - åpalog2pa
is regarded as the entropy of the source. Here pa is the probability
that the message source produces the message a. This can be understood
to provide a measure of the mean number of bits, per signal, necessary
to encode the output of a source. It can also be thought of as a capacity
of the source to carry potential information. The interest here is in the
transfer of `information for us'.
Schumacher [23] extended Shannon's ideas to the quantum
domain by introducing the notion of a `qubit' (the number of qubits per
quantum system is log2(H), where H is the dimension of the system
Hilbert space). A spin state with two eigenvalues, say 0 and 1, can be
used to encode 1 bit of information. To relate this to Shannon's source
entropy, Schumacher represents the signal source by a source density operator
where pa = |aiñáai|
is the set of orthogonal operators relevant to the measurements that will
be performed and p(a) is the probability of a given eigenvalue being found.
The von Neumann information S(r) = Tr(rlog2r)
corresponds to the mean number of qubits, per signal, necessary for efficient
transmission. The `information' in a quantum system, under this definition,
is therefore defined only in terms of its belonging to a particular ensemble
r.
It is not possible to speak of the information of the individual system
since the von Neumann information of the individual pure state is zero
(regardless of the actual values of a and b).
In contrast, in the Bohm interpretation, the information given by [a,b]
has an objective significance for each quantum system, it determines the
trajectories of the individual particles. The standard interpretation attributes
significance only to the quantum state, leaving the particle's position
as somewhat ambiguous and, in spite of the appearance of co- ordinate labels
in the wave function, there may be a temptation to think that it is the
particles themselves that are interchanged under teleportation. This of
course is not what happens and the Bohm approach confirms this conclusion,
making it quite clear that no particle is teleported. What it also shows
is that it is the objective active information contained in the wave function
that is transferred from particle 1 to particle 3.
7 Acknowledgment
The authors wish to thank Dr. O. Cohen for many stimulating discussions
on the subject of quantum information.
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Footnotes:
1 We set
(h/2p) = 1 throughout
this paper.
2
It is essential to note that the Bohm intepretation we use in this paper
is conceptually different from the "Bohmian mechanics" discussed by Goldstein
et
al.[21] [For further details see Hiley
[22].]
Bohm [14,15] himself rejected the
notion of "mechanics" as underlying quantum processes. It was for
this reason that Bohm and Hiley [9] were led to introduce
the notion of an information potential.
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