Understanding Quantum Physics: An Advanced Guide for the Perplexed

Hamiltonian of the system together with the added potential. Then the
entangled state can be protectively measured. Note that the additional
protection usually contains a nonlocal interaction for separated particles.
However, this measurement may be performed without violating Einstein causality
by having the entangled particles sufficiently close to each other so that they
have this protective interaction. Then when the particles are separated they
would still be in the same entangled state which has been protectively
measured.
    2.3.3
Further discussions
    According to the
standard view, the expectation values of observables are not the physical
properties of a single system, but the statistical properties of an ensemble of
identical systems. This seems reasonable if there exist only conventional
impulse measurements. An impulse measurement can only obtain one of the
eigenvalues of the measured observable, and thus the expectation value can only
be defined as a statistical average of the eigenvalues for an ensemble of
identical systems. However, as we have seen, there exist other kinds of quantum
measurements, and in particular, protective measurements can measure the
expectation values of observables for a single system, using an adiabatic
measuring procedure. Therefore, the expectation values of observables should be
considered as the physical properties of a single quantum system, not those of
an ensemble (Aharonov and Vaidman 1993; Aharonov, Anandan and Vaidman 1993;
Aharonov, Anandan and Vaidman 1996) [9] .
    It is worth
pointing out that a realistic protective measurement (where the measuring time
T is finite) can never be performed on a single quantum system with absolute
certainty because of the tiny unavoidable entanglement in the final state (e.g.
Eq.(2.17)) [10] . For example, we can only obtain the exact
expectation value A with a probability very close to one, and the measurement
result may also be the expectation value A ⊥ with a probability proportional to 1/T 2 ,
where ⊥ refers to the normalized state in the
subspace normal to the initial state as picked out by the first-order
perturbation theory (Dass and Qureshi 1999). Therefore, a small ensemble is
still needed for a realistic protective measurement, and the size of the
ensemble is in inverse proportion to the duration of measurement. However, the
limitation of a realistic protective measurement does not influence the above
conclusion. The key point is that a protective measurement can measure the
expectation values of observables on a single quantum system with certainty in
principle, using an adiabatic measuring procedure, and thus they should be
regarded as the physical properties of the system.
    In addition, we
can also provide an argument against the standard view, independent of our
analysis of protective measurement. First of all, although the expectation
values of observables can only be obtained by measuring an ensemble of
identical systems in the context of conventional impulse measurements, this
fact does not necessarily entails that they can only be the statistical
properties of the ensemble. Next, if each system in the ensemble is indeed
identical as the standard view holds (this means that the quantum state is a
complete description of a single system), then obviously the expectation values
of observables will be also the properties of each individual system in the
ensemble. Thirdly, even if the quantum state is not a complete description of a
single system and hidden variables are added as in the de Broglie-Bohm theory
(de Broglie 1928; Bohm 1952), the quantum state of each system in an ensemble
of identical systems is still the same, and thus the expectation values of
observables, which are calculated in terms of the quantum state, are also the
same for every system in the ensemble. As a result, the expectation values of
observables can still be regarded as the properties of individual systems.
    Lastly, we stress
that the expectation values of observables are instantaneous properties of a
quantum system (Aharonov, Anandan and Vaidman 1996). Although the measured
state may be unchanged during a protective measurement and the duration of
measurement may be very long, for an arbitrarily short period of time the
measuring device always shifts by an amount proportional to the expectation
value of the measured observable in the state. Therefore, the expectation
values of observables are not time-averaged properties of a quantum