Separation of Church and state is incompatible with state
religion
Yes, the title is tautological. However, there is some important insight in the defense of this obvious claim. Separation of Church and state is typically thought of as a check on religious tyranny via division of power. Proponents of the concept are often very strong advocates of state religious affiliations.
This presents a contradiction when carefully evaluated. All mainline religions make claims of primacy and exclusivity. A state which authentically espouses such a religion reserves no right to qualify its adherence to its chosen religious tenets. In so doing, it fully submits to the religious authority.
Conversely, a state which maintains separation of civic and religious authority must be coequal in primacy to any religion it considers. Since religions make exclusive claims to primacy, they are not compatible with such a state. The only possible outcome is the secularization of the political space.
The key takeaway is that anything less that total submission to a religious authority is a secular state of being.