The Stern-Gerlach experiment
This is a schematic representation of the classic Stern-Gerlach
experiment with spin-half particles. A particle emerges from the source
and travels towards the screen. The spin of the particle, at time t=0,
is in a superposition state of pointing "up" (|+>) and pointing "down"
(|->). It can be represented by
As it passes through the magnets, the "up" component of the spin pulls the particle up, and the "down" component pulls it down. As a result, the particle becomes "fuzzy" and splits into a superposition of two wave-packets travelling in different directions. This is a bizarre state of the particle - it is simultaneously in two positions!
When this superpostion of wave-packets, on reaching the other end, interacts with the detectors, the quantum coherence is destroyed, and the particle is detected at one of the two positions. It will be detected up with a probability |a|2 , and down with a probablity |b|2 . We are back to the familiar classical world.
This last stage where the superposition of the two wave-packets is destroyed, and one ends up with a particle either "here" or "there", is loosely referred to as the wave function collapse.
For those who are not shocked by the above experiment,
look at the case of the
What is unknown, cannot be known!
As one might have guessed, after a measurement like the one made in
the example described above, one doesn't acquire any information
about the original state |t=0>. In a single measurement one either gets
an "up" spin or a "down" spin - one doesn't know what original state
led to the result obtained. And after one measurement, the state is
irretrievably destroyed. So, any subsequent measurements too do not
tell us anything about the original state. In effect, if the state is
initially unknown, there is absolutely no way of knowing it! This is the
generally accepted credo of quantum mechanics.
This scenario seems to indicate that the state, or the wavefunction, is not real - only if we possess sufficient knowledge about a system, we construct a wave-function for it. The fact that the only wave-functions we can measure are the ones already known to us, leads one to conclude that the wave-function is an entity which is formed out of our knowledge of the system - it doesn't have an independent existence. To put it in the philosophers jargon, the status of wave-function is "epistemological" (based on knowledge), and not "ontological" (real).
Can an unknown wave-function be "measured"?
Yakir Aharonov, Jeeva Anandan and Lev Vaidman
came up with a scheme which they called "protective measurement", which is claimed to yield
information about unknown wave-functions, although of a very special
class. Let us try to understand what these protective measurements
Let us first see what happens in a conventional measurement, like the
one in the SG experiment described above. Assume a particle of mass m,
travelling along the x axis, a magnetic field along the z-axis, and
that the field is inhomgeneous along the y-axis. Let the particle be in a
wave-packet state centered around y=0, and can be respresented by the
wavefunction f(0). Let the state of the spin be written as a|+>+b|->.
The full wave-function of the particle at time t=0, when the particle
just enters the region of the magnetic field, is thus written as
Protective measurements are similar to conventional measurements
except that they are weak and
adiabatic (slow). But this makes the crucial difference as we
will see now. Because the interaction is weak and acts for a long
time, we cannot neglect the "free" Hamiltonian of the system and the
apparatus. So the total Hamiltonian of the system-apparatus can be
We can divide the time T into N tiny intervals so that DelT=T/N. So the full wavefunction at the time T can be written as
Are there any problems?
This problem will be unavoidable when one tries to practically implement protective measurements. One might argue that the apparatus has to be of a classical nature, but here during the adiabatic interaction of the system and the apparatus the evolution of the apparatus has to be unitary. This whole analysis hinges on that. So, classical nature has to emerge later. It might emerge through interaction with some kind of an environment, by the mechanism of decoherence.
On the other hand one can treat the apparatus like a quantum system, as in the Stern-Gerlach experiment, and "detect" the apparatus using a classical apparatus. This will be a conventional measurement leading to a collapse of the apparatus wae-packet. In this scenario, we suggest two ways out of this problem.
Repeated measurement of a single state
Quantum Nondemolition Measurement of the Apparatus
There is another interesting way out of this problem. For this we make use of certain special kind of measurements, called Quantum Non-Demolition (QND) measurements. This scheme is based on repeated weak quantum nondemolition measurements performed on the apparatus. Recently Alter and Yamamoto analyzed the problem of a series of repeated weak QND measurement on a quantum system, to address the question of getting information about the unknown wave function of a single quantum system from such measurements. They concluded that it is possible to obtain the mean value of an observable in an unknown state, but no information can be obtained about the uncertainty of the observable. While the conclusions of Alter and Yamamoto were negative as far as using repeated weak QND measurements to determine the unknown wave function of a single system, it appears tailor-made to solve the problem of "reading the pointer position" in protective measurements. Thus we apply their scheme not to the system part of the protective measurement, but to the apparatus part instead. Then we can get information about the center of the wave packet, which in the protective measurement scheme carries information on the expectation values of observables in the system state.
Thus one may proceed with a protective measurement by first allowing an adiabatic interaction of the system with an apparatus which can be treated quantum mechanically. This would result in a shifted wave packet of the pointer. One can then do a series of weak QND measurements on this wave packet to get the position of the center. This seems the most promising possibility for experimentally realizing protective measurements.
Does it really work for single system?
In a realistic situation, the time of interaction can only be very large, but not infinite. As a result, there is always a very tiny entanglement left, which is of the order 1/T. This entanglement does change the wavefunction to a tiny extent. In the unlucky situation the detection of the apparatus wave-packet might lead to a collapse to this infinitesimal, but non-zero, branch of the entangled wave-function.
Thus, if you are just given one single system in an unknown energy eigenstate, you cannot be hundred percent sure that you have measured the right state.
For a detailed analysis of protective measurements, see ( N.D. Haridass and T. Qureshi, quant-ph/9805012).