Understanding Quantum Physics: An Advanced Guide for the Perplexed

pictures for quantum entanglement and wavefunction collapse.
    5.1.2
Picture of quantum entanglement
    Now let’s analyze
the motion of two particles in quantum entanglement. For the random
discontinuous motion of two particles in an entangled state, the two particles
have a joint propensity to be in any two possible positions, and the
probability density of the two particles appearing in each position pair x 1 and x 2 at a given instant t is determined by the modulus square of
their wave function at the instant, namely ρ(x 1 , x 2 , t) =
|ψ(x 1 , x 2 , t)| 2 .
    Suppose two
particles are in an entangled state ψ u ϕ u + ψ d ϕ d ,
where ψ u and ψ d are two spatially separated states of
particle 1, ϕ u and ϕ d are two spatially separated states
of particle 2, and particle 1 and particle 2 are also separated in space. The
physical picture of this entangled state is as follows. Particles 1 and 2 are
randomly in the state ψ u ϕ u or ψ d ϕ d at an instant, and then they will still stay in this state or jump to the other
state at the next instant. During a very short time interval, the two particles
will discontinuously move throughout the states ψ u ϕ u and
ψ d ϕ d with the same probability 1/2. In this way, the two
particles form an inseparable whole, and they jump in a precisely simultaneous
way. At an arbitrary instant, if particle 1 is in the state ψ u or ψ d ,
then particle 2 must be in the state ϕ u or ϕ d , and vice
versa. Moreover, when particle 1 jumps from ψ u to ψ d or
from ψ d to ψ u , particle 2 must simultaneously jump from ϕ u to ϕ d or from ϕ d to ϕ u , and vice versa. Note
that this kind of random synchronicity between the motion of particle 1 and the
motion of particle 2 is irrelevant to the distance between them, and it can
only be explained by the existence of joint propensity of the two particles as
a whole.
    The above picture
of quantum entanglement is assumed to exist in one inertial frame. It can be
expected that when observed in another inertial frame, this perfect picture
will be distorted in a similar way as for the single particle case. Let’s give
a concrete analysis below. Suppose in an inertial frame S, at instant t a particle 1 is at position x 1a and in state ψ u and
particle 2 at position x 2a and in state ϕ u , and at
instant t b particle 1 is at position x 1b and in state ψ d and particle 2 at position x 2b and in state ϕ d . Then
according to the Lorentz transformation, in another inertial frame S with velocity
v relative to S, where v satisfies:

    the instant at
which particle 1 is at position x 1a and in state ψ u is
the same as the instant at which particle 2 is at position x 2b and
in state ϕ d , namely

    This means that in
S there exists an instant at which particle 1 is in state ψ u but
particle 2 is in state ϕ d . Similarly, in another inertial frame S
with velocity v relative to S, there also exists an instant t at which particle
1 is in state ψ d but particle 2 is in state ϕ u , where v
and t satisfy the following relations:

    Note that since
the two particles are well separated in space, the above two velocities can readily
satisfy the restricting conditions v < c and v < c when the time interval
|t a − t b | is very short.
    In fact, since the
two particles in the above entangled state are separated in space and their
motion is essentially random, in any inertial frame different from S, the
instantaneous correlation between the motion of the two particles in S can only
keep half the time, and the correlation will be reversed for another half of
time, during which the two particles will be in state ψ u ϕ d or ψ d ϕ u at each instant. For a general entangled state
√aψ u ϕ u + √bψ d ϕ d , the proportion of
correlation-reversed time will be 2ab, and the proportion of correlation-kept
time will be a 2 + b 2 . Moreover, the instants at which the
original correlation is kept or reversed are discontinuous and random. This
means that the synchronicity between the jumpings of the two particles is
destroyed too.
    To sum up, the
above analysis indicates that the instantaneous correlation and synchronicity
between the motion of two entangled particles in one inertial frame is
destroyed in other frames due to the Lorentz transformation [103] . As we will see below, however, this
distorted picture of quantum entanglement cannot be measured either.
    5.1.3
Picture of wavefunction collapse
    We have shown that
the picture of the instantaneous