Would apartment complex be responsible for someone entering my apartment as I slept. Stole my purse, my phone, keys and style my car?

That depends on whether your loss was caused by the apartment complex's negligence.  If, for example, the robber stole a pass key from the apartment complex which it had failed to secure properly, or if a security guard fell asleep and allowed the robber to enter, the complex might be deemed liable.  Even if you have some evidence of such negligence, however, you should check your lease to see if there are any limitations of liability clauses.