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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.