I wonder if we'll be needing to rebuy our PSP games (but with trophies) or we don't need to rebuy them again (but with no trophies). Although the ideal solution is we don't need to rebuy and they have trophies.
Frankly, we shouldn't even have to
ask whether we would need to rebuy PSP games on PS4 or not. After Sony had cross-buy for such a long time, and Nintendo and Microsoft followed suit, and the digital media industry at large supports transfer of content across multiple platforms, it should be
expected. The fact that we are now in a position where people question whether PSP games (or PS1 games) would transfer over or not is a sad state of affairs.
Sony needs to sorts this out as soon as possible. They need to figure out how they are going to honor cross-buy with PS2 classics between those purchased on PS3 and those purchased on PS4. They have now fragmented their own digital PS2 game library by implementing them differently on different platforms within their own ecosystem.
I would go as far as to say that Sony should step back and not release any more PS2 classics onto PS4, until they can figure a customer-friendly solution for cross-buy or migrating content as their competitors have done. If they continue to sell PS2 classics on multiple platforms as separate purchases without cross-buy, then this just creates a bigger mess that they'll ultimately have to clean up later.