"EA have confirmed that there are issues with the digital build of Battlefield 3 (on the PS Store) which will be affecting people trying sign into Battlelog," Playstation Community Team member "RabidWalker" said on the EU forum. "They are currently working on a patch for this, although no ETA has been given at the moment."
Eurogamer mentions the theory that the issue is caused by incompatibility with Online Pass. The PSN version can't be bought used, the theory suggests, and the requirement was insufficiently patched out of the game. Online Pass issues interacting with stat tracking service issues -- if that's the case, it would be two features that didn't exist just a couple years ago, teaming up to make a game unplayable. To the future!