This article is from the Star Trek Tech FAQ, by Joshua Bell inexorabletash@hotmail.com with numerous contributions by others.
No. It is not possible (with 24th century technology, at least) to
replicate something at the quantum level. First, the amount of
information needed to define a living, thinking being at that level of
detail is incredibly large, far surpassing the computer capacity of
any 24th century database. (TNG TM)
Presumably, Data and other Soong-type androids which use positronic
brains have components which function at a quantum, or sub-molecular
level which cannot be easily replicated.
Secondly, there is no way to scan at quantum resolution without
destroying the subject. The transporter ACB need not know the precise
details of every particle being transported - where they are and what
they are doing is enough. Further, attempting to retrieve such
information from the ACB would destroy it.
To duplicate a living being, a hypothetical effect, which I call an
Annular Confinement Beam-Splitter, would be needed. As the ACB was
passed through it, along with a supply of raw phased matter, it would
duplicate the ACB's contents in the raw stream.
 
Continue to: