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S4d. How much information is in a particle?
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Knowing a particular electron intimately is infinitely precious.
A pure state of an electron is defined by its wave function
(up to a phase). Thus knowing all about an electron requires in the
traditional interpretation to know all about this wave function -
an infinite amount of information.
The information humans are interested in is however always finite,
since they can hardly remember even 20 decimal digits seen only once.
And the amount of information humans are capable of retrieving
by experiment is still limited, since each experiment has only a finite
accuracy.
Thus they simplify things to the point that all they want to know about
an electron is its mass, charge and its state to a small number
of decimal places.
This is only a few bits. But if you want to tell someone else exactly
where the electron is that you are referring to, you have an
infinitely more difficult task. Of course, any human 'else' will not
be patient enough to hear the whole (infinite) story but will be
satisfied with a crude position and momentum
estimate consistent with the uncertainty relation. But this is not the
best possible statement about the electron, which would be telling
its complete wave function. You can do it only if you force the
electron into a prison where it has to behave in a dull (and hence
completely describable) way, being
restricted in its freedom to at most a few bits of change.
This is indeed done when studying qubits for quantum information
processing.
For an N-state system, one needs N^2-1 independent pieces of
information to reconstruct (by quantum tomography) the density matrix
of a finite mixed quantum system, and a fortiori the wave function of
a finite pure quantum system. Most natural systems, unlike those
systems carefully prepared by modern technology, have infinitely many
states, and therefore need an infinite amount of information for their
reconstruction to full accuracy.