Understanding Quantum Physics: An Advanced Guide for the Perplexed

yields the
value close to one of the eigenvalues. By contrast, the limit of weak
measurement corresponds to w 0 >>a i for all
eigenvalues a i . Then, we can perform the Taylor expansion of the sum
(2.8) around x = 0 up to the first order and rewrite the final probability
distribution of the pointer variable in the following way:

    This is the
initial probability distribution shifted by the value Σ i |c i | 2 a i .
This indicates that the result of the weak measurement is the expectation value
of the measured observable in the measured state:

    Certainly, since
the width of the pointer wavepacket is much greater than the shift of the
center of the pointer, namely w 0 >>A , the above weak
measurement of a single system is very imprecise [5] .
However, by performing the weak measurement on an ensemble of N identical
systems the precision can be improved by a factor √N. This scheme of weak
measurement has been realized and proved useful in quantum optical experiments
(see, e.g. Hosten and Kwiat 2008).
    Although weak
measurements, like conventional impulse measurements, also need to measure an
ensemble of identical quantum systems, they are conceptually different. For
conventional impulse measurements, every identical system in the ensemble
shifts the pointer of measuring device by one of the eigenvalues of the measured
observable, and the expectation value of the observable is then regarded as the
property of the whole ensemble. By contrast, for weak measurements, every
identical system in the ensemble shifts the pointer of measuring device
directly by the expectation value of the measured observable, and thus the
expectation value may be regarded as the property of individual systems.
    2.3 Protective measurements
    Protective
measurements are improved methods based on weak measurements, and they can
measure the expectation values of observables on a single quantum system
without disturbing its state.
    As we have seen
above, although the measured state is not changed appreciably by a weak
measurement, the pointer of the measuring device hardly moves either, and in
particular, its shift due to the measurement is much smaller than its position
uncertainty, and thus little information can be obtained from individual
measurements. A possible way to remedy the weakness of weak measurements is to
increase the time of the coupling between the measured system and the measuring
device. If the state is almost constant during the measurement, the total shift
of the pointer, which is proportional to the duration of the interaction, will
be large enough to be identified. However, under normal circumstances the state
of the system is not constant during the measurement, and the weak coupling
also leads to a small rate of change of the state. As a result, the reading of
the measuring device will correspond not to the state which the system had
prior to the measurement, but to some time average depending on the evolution
of the state influenced by the measuring procedure.
    Therefore, in
order to be able to measure the state of a single system, we need, in addition
to the standard weak and long-duration measuring interaction, a procedure which
can protect the state from changing during the measuring interaction. A general
method is to let the measured system be in a nondegenerate eigenstate of the
whole Hamiltonian using a suitable protective interaction, and then make the
measurement adiabatically so that the state of the system neither collapses nor
becomes entangled with the measuring device appreciably. In this way,
protective measurement can measure the expectation values of observables on a
single quantum system. In the following, we will introduce the principle of
protective measurement in more detail (Aharonov and Vaidman 1993; Aharonov,
Anandan and Vaidman 1993; Aharonov, Anandan and Vaidman 1996) [6] .
    2.3.1
Measurements with natural protection
    As a typical
example, we consider a quantum system in a discrete nondegenerate energy
eigenstate |E n >. In this case, the system itself supplies the
protection of the state due to energy conservation and no artificial protection
is needed.
    The interaction
Hamiltonian for a protective measurement of an observable A in this state
involves the same interaction Hamiltonian as the standard measuring procedure:
    H I =
g(t)PA, (2.11)
    where P is the
momentum conjugate to the pointer variable X of an appropriate measuring
device. Let the initial state of the pointer at t = 0 be |φ(x 0