Can I sue my apartment complex for loss of a stolen vehicle

It depends heavily on the language of your lease agreement and the particular facts and circumstances of your case.  In particular, does your complex say that it is providing security or security cameras, or does your lease say that the complex is not liable for lost or stolen items?  Who stole your vehicle?  Why?  Was the open gate a material factor in the theif's decision to steal your car?  How many othe vehicles have been stolen from the complex recently?  Was an employee of the complex involved in the theft?  There are far too many specific facts which impact whether your complex owed you a duty, whether it breached any duty owed, and whether any such breach was a proximate cause of the theft of your vehicle.