Understanding Quantum Physics: An Advanced Guide for the Perplexed

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, which is a Gaussian wave packet of eigenstates of X with width w 0 ,
centered around the eigenvalue x 0 . The time-dependent coupling
strength g(t) is also a smooth function normalized to ∫dtg(t) = 1. But
different from conventional impulse measurements, where the interaction is very
strong and almost instantaneous, protective measurements make use of the
opposite limit where the interaction of the measuring device with the system is
weak and adiabatic, and thus the free Hamiltonians cannot be neglected. Let the
Hamiltonian of the combined system be
    H(t) = H S + H D + g(t)PA, (2.12)
    where H S and H D are the Hamiltonians of the measured system and the measuring
device, respectively. The interaction lasts for a long time T , and g(t) is very
small and constant for the most part, and it goes to zero gradually before and
after the interaction.
    The state of the
combined system after T is given by
    By ignoring the
switching on and switching off processes [7] , the full Hamiltonian (with g(t) = 1/T )
is time-independent

    time. For example,
the kinematic energy term P 2 /2M in the free Hamiltonian of the
pointer will spread the wave packet without shifting the center, and the width
of the wave packet at the end of interaction will be w(T ) =[1/2(w 0 2 + T 2 /M 2 w 0 2 )] 1/2 (Dass
and Qureshi 1999). However, the spreading of the pointer wave packet can be
made as small as possible by increasing the mass M of the pointer, and thus it
will not interfere with resolving the shift of the center of the pointer in
principle [8] .
    2.3.2
Measurements with artificial protection
    Protective
measurements can not only measure the discrete nondegenerate energy eigenstates
of a single quantum system, which are naturally protected by energy
conservation, but also measure the general quantum states by adding an
artificial protection procedure in principle (Aharonov and Vaidman 1993). For
this case, the measured state needs to be known beforehand in order to arrange
a proper protection.
    For degenerate
energy eigenstates, the simplest way is to add a potential (as part of the
measuring procedure) to change the energies of the other states and lift the
degeneracy. Then the measured state remains unchanged, but is now protected by
energy conservation like nondegenerate energy eigenstates. Although this
protection does not change the state, it does change the physical situation.
This change can be brought to a minimum by adding strong protection potential
for a dense set of very short time intervals. Then most of the time the system
has not only the same state, but also the original potential.
    The superposition
of energy eigenstates can be measured by a similar procedure. One can add a
dense set of time-dependent potentials acting for very short periods of time
such that the state at all these times is the nondegenerate eigenstate of the
Hamiltonian together with the additional potential. Then most of the time the
system also evolves under the original Hamiltonian. A stronger protection is needed
in order to measure all details of the time-dependent state. The simplest way
is via the quantum Zeno effect. The frequent impulse measurements can test and
protect the time evolution of the quantum state. For measurement of any desired
accuracy of the state, there is a density of the impulse measurements which can
protect the state from being changed due to the measuring interaction. When the
time scale of intervals between consecutive protections is much smaller than
the time scale of the original state evolution, the system will evolve
according to its original Hamiltonian most of the time, and thus what’s
measured is still the property of the system and not of the protection
procedure (Aharonov and Vaidman 1993).
    Lastly, it is
worth noting that the scheme of protective measurement can also be extended to
a many-particle system (Anandan 1993). If the system is in a product state,
then this is easily done by protectively measuring each state of the individual
systems. But this is impossible when the system is in an entangled state
because neither particle is then in a unique state that can be protected. If a
protective measurement is made only on one of the particles, then this would
also collapse the entangled state into one of the eigenstates of the protecting
Hamiltonian. The right method is by adding appropriate protection procedure to
the whole system so that the entangled state is a nondegenerate eigenstate of
the total