The question is Does a philosophical zombie have mental states?
One can answer that if one can clarify what it means to be a zombie and what mental states are. Although these terms may be fluid, let's use how Wikipedia defines them as a baseline. My emphasis is in bold.
Wikipedia describes a mental state as follows:
A mental state is a state of mind that an agent is in. Most simplistically, a mental state is a mental condition. It is a relation that connects the agent with a proposition. Several of these states are a combination of mental representations and propositional attitudes. There are several paradigmatic states of mind that an agent has: love, hate, pleasure and pain, and attitudes toward propositions such as: believing that, conceiving that, hoping and fearing that, etc.
Wikipedia describes a zombie:
A similar argument holds that it is conceivable (or not inconceivable) that there could be physical duplicates of people, called "philosophical zombies", without any qualia at all. These "zombies" would demonstrate outward behavior precisely similar to that of a normal human, but would not have a subjective phenomenology. It is worth noting that a necessary condition for the possibility of philosophical zombies is that there be no specific part or parts of the brain that directly give rise to qualia—the zombie can only exist if subjective consciousness is causally separate from the physical brain.
If a zombie is conceivable then there is something about us, who are not zombies, that is not explainable by materialism. This understanding of zombie requires understanding qualia.
Wikipedia describes qualia as follows:
In philosophy and certain models of psychology, qualia ... are defined as individual instances of subjective, conscious experience....
Examples of qualia include the perceived sensation of pain of a headache, the taste of wine, as well as the redness of an evening sky. As qualitative characters of sensation, qualia stand in contrast to "propositional attitudes", where the focus is on beliefs about experience rather than what it is directly like to be experiencing.
Given these as definitions one can distinguish qualia as the "subjective, conscious experience" of having a mental state from the mental state itself.
This allows one to answer the question. A zombie can have all the physical characteristics of being in a mental state. In that sense the zombie has the mental state. However, the zombie would lack the subjective experience associated with having that mental state.
Wikipedia contributors. (2019, July 20). Mental state. In Wikipedia, The Free Encyclopedia. Retrieved 13:47, October 8, 2019, from https://en.wikipedia.org/w/index.php?title=Mental_state&oldid=907154141
Wikipedia contributors. (2019, October 1). Qualia. In Wikipedia, The Free Encyclopedia. Retrieved 13:48, October 8, 2019, from https://en.wikipedia.org/w/index.php?title=Qualia&oldid=918997985