Quantum State Teleportation understood through the Bohm Interpretation

Abstract

1  Introduction

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

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

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 = -[(ѲR)/ 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

Y(x₁,x₂,¼,x_N) = R(x₁,x₂,¼,x_N)e^{iS(x₁,x₂,¼,x_N)} produces a quantum potential, Q_i, for each particle given by:

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,x_N) = f(x₁)x(x₂,¼,x_N). In this case

R(x₁,x₂,¼,x_N) = R₁(x₁)R_2¼N(x₂,¼,x_N), and therefore:

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

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

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) = f_a(x₁)x_a(x₂)+f_b(x₁)x_b(x₂), where the active information depends upon the simultaneous position of both particle 1 and particle 2. If the f_a and f_b are overlapping wave functions, but the x_a and x_b are non-overlapping, and the actual position of particle 2 is contained in just one wave-packet, say x_a, the active information will be contained only in f_a(x₁)x_a(x₂), the information in the other branch will be passive. Therefore only the f_a(x₁) wave-packet will have an active effect upon the trajectory of particle 1. In other words although f_a and f_b are both non-zero in the vicinity of particle 1, the fact that particle 2 is in x_a(x₂) will mean that only f_a(x₁)x_a(x₂) is active, and thus particle 1 will only be affected by f_a(x₁).

If f_a(x₁) and f_b(x₁) are separated, particle 1 will always be found within the location of f_a(x₁). The position of particle 2 may therefore be regarded as providing an accurate measurement of the position of particle 1. Should the f_a and f_b 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 f_a(x₁)x_a(x₂) will be active. To restore activity to the passive branches of the superposition requires that both f_a(x₁) and f_b(x₁) and x_a(x₂) and x_b(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

Q_i = [1/ 2m](-Ñ_i²R+Ñ_iF^fÑ_iF+(F^fÑ_iF)²). B is the magnetic field and m_i 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 s_i = ¹/₂Y^fs_iY, where s_i 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 Y^fs_i leads the equation of motion for the particle i spin vector:

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:

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

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 O_l = b_lb_l^f. This creates the state

As the position variable x₀ of the measuring device enters one of the non- overlapping wave-packets h_i(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

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 Y_f. This will produce a quantum potential that can be written in the form

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) = - åp_alog₂p_a is regarded as the entropy of the source. Here p_a 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

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

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