As far as I understand, the illegal access to classified documents in the US is regulated by the Espionage Act. This old act has since been amended to cover electronic copies of classified information.
However, I still do not understand why spies would prefer the original paper versions of any classified documents versus, say, photocopies or electronic scans of those?
From a naïve perspective, a spy tasked to get access to classified materials would have access to those documents for a brief period of time to make copies as quickly as possible, put the papers back where it belongs and to conceal the very fact of accessing it. The last thing this person would do is collect a stockpile of classified document that literally extends from floor to ceiling, like Trump is accused of:
Image: US Department of Justice, via BBC
This is confirmed by historical evidence:
Atomic spies like Morris Cohen used radio, Klaus Fuchs used sketches and photo copies, etc. I did not find any publicly available evidence suggesting that those spies literally sent original classified documents to the KGB.
Hosts on Russian state propaganda outlet openly claim that "Moscow is already studying" the documents they recently got access to. (I assume they are studying the copies as the originals seem to be with the FBI).
This raises the question:
Why would a spying party need the genuine paper originals (with numbered pages, pen-written signatures, and wet seals)? How would this be better (for the spying party) than the scanned contents plus the confidence that the document is genuine?