I understand the statement that 'X QFT is unitary' is shorthand for saying 'the S-matrix of X QFT is unitary', cf. e.g. this Phys.SE post.
Is there some definition of unitarity that is stronger than this, which is a property of the theory and not just its S-matrix? The S-matrix is a very special time evolution operator which takes asymptotically free states at minus infinity in time to asymptotically free states at positive infinity. Is there a is there a definition of unitarity in terms of a finite time evolution operator which implies the S-matrix definition?
I have also heard people discussing whether a certain conformal field theory is unitary, which suggests to me that it could be a property of the theory itself. To my understanding, the S-matrix is defined in a CFT only by taking a limit in dimensional regularisation. Is it that when people say '$d$-dimensional CFT X is unitary', they mean 'the S-matrix of CFT X in dimensional regularisation where $d' = d + \epsilon$ for $\epsilon >0$ is unitary'?
This would seem strange to me, I believe that unitarity is a central property of a CFT, but the S-matrix is not always the most natural quantity to compute.