Understanding Quantum Physics: An Advanced Guide for the Perplexed

definite
measurement result in each world always denotes the result of a conventional
impulse measurement. However, this does not guarantee consistency for all types
of measurements. Indeed, it can be seen that the existence of the many worlds
defined above is inconsistent with the results of protective measurements. The
reason is that the whole superposed wave function of a quantum system including
a measuring device can be measured by a protective measurement [56] . The result of the protective measurement
as predicted by quantum mechanics indicates that all components of the
superposed wave function of the measuring device exist in the same world where
the protective measurement is made. Therefore, according to protective
measurement, the components of the superposed wave function of a measuring
device, each of which represents a definite measurement result, do not
correspond to many worlds, in each of which there is only one such component
and a copy of the measuring device that obtains a definite result; rather, the
whole superposed wave function of the measuring device, if it exists, only
exists in one world, namely our world, and in this world there is only one
measuring device that obtains no definite result. In this way, protective
measurement provides a strong argument against the many-worlds interpretation [57] .
    Four points are
worth stressing. First of all, the above argument does not depend on how the
many worlds are precisely defined in the many-worlds interpretation. For
example, it is irrelevant to whether the many worlds are fundamental or
emergent, and in particular, it also applies to Wallace’s formulation of the
many-worlds interpretation based on a structuralist view on macro-ontology. The
key point is that all components of the superposed wave function of a measuring
device can be detected by protective measurement in one world, namely our
world, and thus they all exist in this world. Therefore, it is impossible that
the superposed wave function of a measuring device corresponds to many worlds,
only one of which is our world [58] .
    Next, the above
argument is not influenced by environment-induced decoherence. On the one hand,
even if the superposition state of a measuring device is entangled with the
states of other systems, the entangled state of the whole system can also be
measured by protective measurement in principle (Anandan 1993). The method is
by adding appropriate protection procedure to the whole system so that its
entangled state is a nondegenerate eigenstate of the total Hamiltonian of the
system together with the added potential. Then the entangled state can be
protectively measured. On the other hand, environment-induced decoherence is
not an essential element of the many-worlds interpretation. Even when a
measuring device is isolated from environment (and the measured particle is
absorbed by the device), the interpretation also requires that each component
of the wave function of the measuring device in which there is a definite
measurement result corresponds to each world among the many worlds; otherwise
the many-worlds interpretation will not give the same predictions of
measurement results as standard quantum mechanics (so long as the latter gives
unambiguous predictions).
    Thirdly, the above
argument does not require protective measurement to be able to distinguish the
superposed wave function of a measuring device (in each component of which
there is a definite measurement result) from one of its components, or whether
the superposed wave function collapses or not during a conventional impulse
measurement. Since the determination demands the distinguishability of two
non-orthogonal states, which is prohibited by quantum mechanics, no
measurements consistent with the theory including protective measurement can do
this. What protective measurement tells us is that such a superposed wave
function, which existence is assumed by the many-worlds interpretation, does
not correspond to the many worlds defined by the many-worlds interpretation. In
other words, protective measurement reveals inconsistency of the many-worlds
interpretation. Lastly, we stress again that the principle of protective
measurement is irrelevant to the controversial process of wavefunction collapse
and only depends on the linear Schrödinger evolution and the Born rule. As a
result, protective measurement can (at least) be used to examine the internal
consistency of the no-collapse solutions to the