Review of A. V. Rau, J. A. Dunningham, K. Burnett, Measurement-Induced Relative-Position Localization Through Entanglement, Science 301, 1081 (2003).

Summary:

Rau, Dunningham and Burnett (RDB) present an analysis of particles localizing in position space under quantum measurement by studying how a robust relative position emerges during the scattering of photons from particles with positions that are initially smeared out over space. As a result, the scattering process transfers information about the relative position and broadens the relative-momentum wave function, initially in center-of-mass momentum eigenstates, so that the conjugate effect is reduced uncertainty in relative-position space.

The lack of information of which particle scattered the photon and subsequent projection of the quantum state results in entanglement of the two particles and this further drives the localization process. With each scattering event, the masses become increasingly entangled and their relative position better defined with each scattering event. In addition to scattering, the situation when the photon does not scatter from a particle must also be considered as the measurement of a non-scattering event also leads to a change in the state. RDB summarize the situation:

One might think that we could localize a single particle to a definite position… However, observing a photon scattered off a lone particle yields no position information: The particle simply recoils, shifting its momentum wave function without any effect on its spatial distribution. Only when there is an extra element in the interaction, such as a lens in the case of Heisenberg’s microscope, can we gain any position information and thus localize the particle … It is then entangled with the lens, and its position is determined relative to the axis of the lens…

Two thought experiments are used to examine this localization process: (1) an interferometer with recoiling mirrors; and (2) photons scattering from a pair of particles and the resulting emergence of a Young’s interference pattern.

Strengths:

The authors show through analysis of their two thought experiments that subsequent scattering events only reinforce localization at existing separations so that localization is therefore robust to further measurements and relative position can be considered a classical property of the particles. The illustrated localization process suggests a prominent role for entanglement and relative observables at the boundary between quantum and classical mechanics.

The analyses presented by RDB constitute quantitatively worked out examples which illustrate a key aspect of measurement: that experiments rarely directly measure the system of interest. Instead, for instance, an atom interacts with surrounding electromagnetic modes, which in turn trigger a photodetector, and on to display systems, then the experimenter’s eye and brain, etc. Although RDB do not discuss their work in this context, such a chain of systems is now known as a “von Neumann chain”. Von Neumann demonstrated that the chain must be cut by applying the projection postulate at any point along the chain of the experimenter’s choosing to indicate a measurement [1]. Due to the entanglement between system and surrounding field, a projective measurement of the field effectively becomes a measurement of the atom, though the resulting measurements are generally not projections but POVM’s instead.

Weaknesses:

RDB do not appear to understand the physical measurement problem as we have defined it [5, Ch.4] which acknowledges that further work is required to address the physical reasons and the conditions under which measurement occurs and that the two von Neumann postulates of quantum mechanics are incomplete. Instead, they view their work as a more tractable implementation of decoherence theory to explain measurement. RDB state:

One of the most successful descriptions of this transition between the microscopic and macroscopic is the theory of decoherence, in which a system is coupled to an environment with many degrees of freedom, causing the decay of its macroscopic coherences…. In practice, however, decoherence theory is often difficult to apply, because of the complex nature of the environment and its influence on the system. We present here a simpler approach, from the perspective that all measurements of position are intrinsically relative measurements, that eliminates the need for environmental dissipation by focusing instead on entanglement between objects.

However, decoherence theory only describes the transfer of first-order coherence into entanglement but does not resolve the measurement problem one iota. Environmental decoherence predicts a pointer basis but does not tell us the conditions for which measurement will occur.

Additional Observations:

Note that the general circumstances of measurement and RDB’s quantitative example of measuring relative position are quite similar to Heisenberg’s early qualitative description of determining an electron’s position via scattering of photons with sufficient resolution to reproduce the electron wave function as proportional to $|\psi|^2$ [2, p.33]:

The orbit is the temporal sequence of the points in space at which the electron is observed. As the dimensions of the atom in its lowest state are of the order of $10^-8$ cm, it will be necessary to use light of wave-length not greater than $10^-9$ cm in order to carry out a position measurement of sufficient accuracy for the purpose. A single photon of such light is, however, sufficient to remove the electron from the atom, because of the Compton recoil. Only a single point of the hypothetical orbit is thus observable. One can, however, repeat this single observation on a large number of atoms, and thus obtain a probability distribution of the electron in the atom. According to Born, this is given mathematically by $\psi\psi^*$ (or, in the case of several electrons, by the average of this expression taken over the co-ordinates of the other electrons in the atom). This is the physical significance of the statement that $\psi\psi^*$ is the probability of observing the electron at a given point.

However, in his initial collision paper [3], where Born proposed the probabilistic interpretation of quantum mechanics for the process of electrons scattering from an atom, Born did not initially connect the wave-function with probability of position in any single iteration. Instead, the $\psi$ controlled the energetic transitions of an atom in the stationary states $\psi_n$ and the energy and direction of motion of colliding electrons. In this case, the probability that the atom is in the nth state is given by the quantity $|c_n|^2=|\int\psi(x,t)\psi_n^*(x)dx|^2$ . The first formal statement of a probability interpretation of position actually appeared in a 1927 paper on degenerate gasses and paramagnetism by Pauli [4, p.83, Footnote 1]:

We shall interpret this function in the spirit of Born’s view of the ”ghost field” … as follows: $|\psi(q_1\cdot\cdot\cdot q_f)|^2dq_1\cdot\cdot\cdot dq_f$ is the probability that, in the named quantum state of the system, these coordinates lie simultaneously in the named volume element $dq_1\cdot\cdot\cdot dq_f$ of position space.

This was the beginning of generally identifying $|\psi(x)|^2$ with quantum mechanical probabilities. As discussed above, this generally requires indirect measurement with a resulting von Neumann chain of physical systems.

Conclusions:

RDB present very clear and tractable examples of how the measurement of relative particle position can emerge progressively via photon scattering which entangles the particles and illustrates the prominent role of entanglement in localization. The authors conclude that relative position becomes well defined but absolute position of the particles and mirrors remain undefined. The work can also be viewed as a detailed illustration of von Neumann chains which generally appear in measurement due to the necessity of accessing observables only indirectly. Absent from the paper are any discussion that further understanding of the physical conditions for which measurement will occur is necessary to complete the measurement problem.

[1] J. von Neumann, Mathematische Grundlagen der Quantenmechanik. Springer, Berlin, 1932; English translation by E. T. Beyer, as Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton, NJ, 1955.

[2] W. Heisenberg, The Physical Principles of the Quantum Theory. Dover, New York, 1949.

[3] M. Born, “Zur Quantenmechanik der Stossvorginge”, Zeitschrift für Physik 37, 863-867 (1926).

[4] W. Pauli, “Uber Gasentartung und Paramagnetismus”, Z. Physik 41, 81-102 (1927).

[5] M. Steiner and R. Rendell, The Quantum Measurement Problem. Inspire Institute, 2018.