Negative Measurement

When a system such as photon impinges on a measurement device, the measurement postulate requires that the theory that defines quantum state evolution change from unitary evolution to non-unitary evolution for which at least two outcomes are always possible. Generally these outcomes are non-deterministic and cannot be predicted as in unitary evolution. The minimum outcomes are 1) the case for which the system is not absorbed by the measurement device, and 2) the case for which the system is absorbed by the device, which we will either refer to as collapse or positive measurement It is a misconception that the measurement postulate is only invoked when the system undergoes collapsed and the detector changes its readout in a manner that shows that the system has been detected. In fact, the measurement postulate is also invoked even if the system is not absorbed and the detector does not change its readout to indicate that the system has been detected. This latter type of measurement is often referred to as interaction-free measurement (IFM) or negative measurement and is an important aspect of measurement theory to understand.

Interaction-Free Measurement

In IFM or negative measurement, the detector does not react as if it has absorbed or positively measured the system, and there is no collapse associated with the wave function. One might be quick to conclude that therefore the interaction is described by Schrodinger’s equation. Such a quick conclusion is false. Quantum state evolution described by the measurement postulate has the potential to differ from state evolution predicted by Schrodinger’s equation in either positive or negative measurement, and in such cases unitary evolution versus evolution prescribed by the measurement postulate could be experimentally distinguished.

In order to illustrate the differences, below is an example from a computer simulation that computes the state evolution of a wave function that impinges on a detector. In the simulation, the detector is a 3-dimensional sphere and the wave-function also exists in 3-dimensional space. The wave function was chosen to vary in an oscillatory manner where the color is indicative of the wave function amplitude as shown on the scale in the video.

Initially the wave function is bounded by a 3-dimensional cube and enters from the upper left of the screen as it slowly moves toward the sphere. For simplicity it is assumed that the unitary evolution of the wave function is dispersionless. The wave function has oscillating purple and black colors indicative of a function of magnitude .3 and 0 respectively. At 16 seconds into the video, the wave function begins to impinge on the detector. There are several features to take note of. As the wave function continues along trajectory, note that the initial cubic shape of the wave function is modified in a manner that the wave function in front of the spherical detector has been clipped or removed. Hence the wave function is changing its shape from the original 3-dimenionsal cube due to the non-unitary interaction. Also note that as the wave function progresses, the magnitude of the remaining wave function is changing..