If we achieve sentience using mutable hardware, will it be possible to make a copy of that "brain" and its active state? Hardware comes in two forms, basically: immutable, such as RAM, and mutable, such as FPGAs.

In animals, neurological connections gain in strength by changing the physical structure of the brain. This is analogous to FPGAs whereby signal strength is increased by changing the pathways themselves.

If we achieve sentience using mutable hardware (e.g., neuromemristive systems), will it be possible to make a copy of that "brain" and its active state?

For this question, assume that the brain is how the hardware has "reconfigured" [or etched, if you will] its pathways to strengthen them and the brain's state is captured by how electrons are physically flowing throughout those pathways.
Theoretically, there shouldn't be a problem copying either of the artificial brains in any state. 

Difficulty in measuring a state doesn't seem to really be a problem until you get down to the quantum level, where the means of measurement affect the state.

The configuration of the artificial brains, including pathway structures and states, should be reducible to a single string, which could then be used to reconfigure the artificial brain the information is being copied to.



Definitely look into the concept of a Turing Machine and Universal Turing Machine.