I don't agree with this. Quantum measurements are projective and they are very much one way. People seem to want to dismiss this as being just "epistemic" like the way entropy (thermodynamics) is one-way, but not fundamentally one way. Entropy increases only because we can't see all the details. Quantum measurements are not like that.
The only confirmed T-symmetry violation is that of K and B mesons, I think. But that does not follow from quantum theory I think. It's experimental result.
Edit: And actually, causal relationships aren't directional, they just look that way because we usually care about "What can we infer about the more complicated [thermodynamically later] system state given knowledge about the less complicated [thermodynamically earlier] one?" rather than vice versa. (Well, also because most physical processes exchange information about about particle velocities (heat differentials) for information about the state and layout of macroscopic-ish objects (knowledge), and the laws of thermodynamics result in those exchanges all going in the same direction.)