On 11/3/2019 10:43 AM, Philip Thrift wrote:
In order for this statement to be useful, we need to solve the
preferred basis problem. For example, consider the mathematical identity
|u a w> + |e d c> = |n+ s+ f+> + |n+ s- f-> + |n- s+ f-> + |n- s- f+> (2)
where
|n+> = |u> + |e>, |n-> = |u> - |e>
|s+> = |a> + |d>, |s-> = |a> - |d>,
|f+> = |w> + |c>, |f-> = |w> - |c>.
Unless we introduce a further piece of interpretive apparatus, we are
in danger of supposing that the system described by |phi> is also
described by |n+ s+ f+> or each of the other components in (2), which
would mean we have not solved the original problem. However, this is
easily solved, as follows:
Postulate 1. A quantum system and environment described by the state
|phi> is also described by one of the states of the basis in which the
reduced density matrix of |phi> is diagonal after the environment has
been traced over.
The `collapse' from |phi> to one of |u a w> and |e d c> is now no
different from the abrupt change in a classical probability
distribution when more information becomes available.
At this point one is then tempted to ask why not just say the wf
collapsed? Tracing over the environment is already something the
experimenter does on paper. It's not part of the Schroedinger equation
unitary evolution. Sure it makes the off-diagonal terms small...but it
doesn't make them zero. It's still just FAPP.
Brent
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