Understanding Quantum Physics: An Advanced Guide for the Perplexed

partial collapse of the measured wave function. For a helpful discussion see
Miller (2010).
    [6] For a review of earlier objections to the validity and meaning of
protective measurements and the answers to them see Aharonov, Anandan and
Vaidman (1996), Dass and Qureshi (1999) and Vaidman (2009).
    [7] The change in the total Hamiltonian during these processes is smaller
than PA=T, and thus the adiabaticity of the interaction will not be violated
and the approximate treatment given below is valid. For a more strict analysis
see Dass and Qureshi (1999).
    [8] As in conventional impulse measurements, there is also an issue of
retrieving the information about the center of the wave packet of the pointer (Dass
and Qureshi 1999). One strategy is to consider adiabatic coupling of a single
quantum system to an ensemble of measuring devices and make impulse position
measurements on the ensemble of devices to determine the pointer position. For
example, the ensemble of devices could be a beam of atoms interacting
adiabatically with the spin of the system. Although such an ensemble approach
inevitably carries with it uncertainty in the knowledge of the position of the
device, the pointer position, which is the average of the result of these
position measurements, can be determined with arbitrary accuracy. Another
approach is to make repeated measurements (e.g. weak quantum nondemolition
measurements) on the single measuring device. This issue does not affect the
principle of protective measurements. In particular, retrieving the information
about the position of the pointer only depends on the Born rule and is
irrelevant to whether the wave function collapses or not during a conventional
impulse measurement.
    [9] Anandan (1993) and Dickson (1995) gave some initial analyses of the
implications of this result for quantum realism. According to Anandan (1993),
protective measurement refutes an argument of Einstein in favor of the ensemble
interpretation of quantum mechanics. Dickson’s (1995) analysis was more philosophical.
He argued that protective measurement provides a reply to scientific empiricism
about quantum mechanics, but it can neither refute that position nor confirm
scientific realism, and the aim of his argument is to place realism and
empiricism on an even score in regards to quantum mechanics.
    [10] This point was discussed and stressed by Dass and Qureshi (1999).
    [11] Quoted
in Moore (1994), p.148.
    [12] This important point was also stressed by Aharonov, Anandan and Vaidman
(1993).
    [13] It is worth stressing that the added protection procedure depends on the
measured state, and different states need different protection procedures in
general.
    [14] Whether the charge is real or effective will be investigated in the next
section.
    [15] Any physical measurement is necessarily based on some interaction
between the measured system and the measuring system. One basic form of
interaction is the electrostatic interaction between two electric charges as in
our example, and the existence of this interaction during a measurement, which
is indicated by the deviation of the trajectory of the charged measuring system
such as an electron, means that the measured system also has the charge
responsible for the interaction. If one denies this point, then it seems that
one cannot obtain any information about the measured system by the measurement.
Note that the arguments against the naive realism about operators and the
eigenvalue realism in the quantum context are irrelevant here (Daumer et al
1997; Valentini 2010).
    [16] Strictly speaking, the mass density is m|ψ(x)| 2 +ψ ∗ Hψ/c 2 in the non-relativistic
domain, but the second term is very small compared with the first term and can
be omitted.
    [17] Alternatively one might simply insist that even if the mass and charge
distributions of a charged quantum system are real, they still have no
gravitational and electrostatic self-interactions. One may further argue that
this is because the system is of quantum nature (for a classical charged system
these self-interactions do exist), and the superposition principle of quantum
mechanics prohibits the existence of these self-interactions. However, this
view is untenable. On the one hand, even if the superposition principle may be
used to explain the absence of self-interactions for a charged quantum system,
it does not tell us whether the mass and charge distributions of the quantum
system are real or not. One cannot simply