It is likely soil borne fungus called Thielaviopsis basicola or black root rot. It can be problematic to many holly species but it seems to effect Japanese holly and inkberry holly most. 

There isn't a cure for it and doesn't warrant a fungicide application as it will not fix the current infection and applications will have to be ongoing . You can just prune out the dead branches and monitor its health. It may continue to decline and then you can remove the plant and would recommend either choosing a less susceptible plant or just making sure the replacement holly is cared for to mitigate stress. 

Here is an additional source from the University of Kentucky.

Emily