Experiments in Wave Function Collapse

Consider the classic spin-EPR experiment where a singlet decays into a pair of correlated spin-1/2 particles. Measurements show that the spins are correlated in a non-local way. But measurements by definition also involve state-function collapse, which is commonly believed to be a thermodynamic effect, a many-body interaction with a chaotic sea of photons and electrons in the detectors. If this is indeed true, then can non-local correlations be used to achieve non-local thermodynamic equilibrium between pairs of detectors?

Possibly. This page discusses several possible (gedanken-)experiments, involving fermions and photons, and tries to make deductions about entropy and the nature of measurement.

Experiment One

Say we have created a pair of correlated fermions, and we are using a Stern-Gerlach magnet to measure their spin along a common axis. Well, the Stern-Gerlach device works by immersing the fermion in a sea of low-energy photons (in the form of a non-uniform magnetic field). Whenever one fermion interacts with the photon sea to experience a net force in one direction, the other fermion must (sooner or later, somehow or another) experience a similar force in the opposite direction. To maintain correlation between spins, the fermions must necessarily connect the two (spacelike-separated) magnetic fields in some way. One might suppose the connection to be thermodynamic in some way, exchanging a miniscule number of low-energy photons in one location for those in another location. If it were possible to somehow make one device cold, and the other hot, might heat be exchanged from one detector to the other?

Probably not. One reason is that if this happened, then we would have a faster-than light signalling device. How is that? If the two devices were space-like separated by a large distance, then experimenter L could send messages to experimenter R by alternately switching in a hot or cold detector. Experimenter R would only have to watch which direction the temperature of their device went.

Another problem with this experiment is that there is some question of where the wave-function collapse actually took place. Did it occur while the fermion was in the magnetic field? Or did it only occur later, when the fermion collided with the film plate (or other particle detector)?

A better way to understand this type experiment might be by thinking about entropy. Entropy is usually defined in a classical sense, as the logarithm of the number of (thermodynamically accessible) states. In the following, it will be convenient to use the logarithm base-two, and talk entropy as being a number of bits.

One way to tie entropy to a quantum experiment is through the 'quantum teleportation' effect. There are two variants of this effect. In one variant, a quantum state in one location is destroyed, and reproduced in another location by conveying two bits of information and an entangled state to the second location. The other variant, the one of interest to us, chews up two bits in one location, and communicates an entangled states to the second location, at which the two bits are extracted. (Need diagrams here).

Experiment Two

In this experiment, we use the second variant of the 'quantum teleportation' effect. However, instead of manually feeding in two bits of data to be communicated, we let these two bits be chosen at random from a thermodynamic sea. However, we've arranged the thermodynamic sea so that certain bit combinations occur far more frequently than others. In the traditional wording of the quantum teleportation experiment, the two bits represent one of four states: the singlet l=0, and one of the triplet l=1, lz=-1, 0, +1. Say we could arrange things so that the four states were associated with different energies: e.g. the lz=-1 was low energy, and the lz=0,1 where higher energy. Then, given a thermodynamic distribution of states e-E/kT, the data being sent from location A to B would be bit-patterns that followed this distribution. Imagine now that the sender A was cooler than the detector at B. As the data gets reconstructed at B, shouldn't we find that the entropy at B decreases? We conclude this because at B, we are reconstructing data that doesn't equally occupy all four possible states, but rather tends to prefer one outcome.

Leap: By considering the quantum teleportation experiment, it is tempting to split macroscopic state into two pieces. One is a set of classically-behaving 'bits', which contribute to the entropy and general thermodynamic behavior. The other piece is one big quantum-entangled state. The teleportation concept tells us that we can take bits and convert them into entangled states, and then pull them out again. A 'measurement' especially of the EPR type, is then really the statement that the two detectors have become (are) quantum correlated, and that we have also extracted bits from the state function. Experiments such as the 'quantum eraser' are 'just' a yanking of bits out, and then pushing them back into the quantum state.

What I find curious is that this view seems to indicate that detectors, e.g. a pair of stern-Gerlach detectors, become quantum-entangled with each other, and remain irrevocably so, even as we pull 'bits' out of the system that represent the actual measurements made. (Similarly, different grains of silver on a photographic film behind a two-slit arrangement become, and remain, quantum-entangled, even as bits are yanked out and mark the grains as 'exposed' and 'unexposed' grains).

We now come to the meta-questions: is the universe a pure state, or is it a mixture of disconnected states? If it is a mixture of disconnected states, how do we find them? If we find them, can we use quantum teleportation to 'shrink' one and 'grow' another? And what would the utility of that be?

Bibliography

August 2000
March 2001
Linas Vepstas [email protected]