We give a characterization of trace-preserving and positive linear maps preserving trace distance partially, that is, preservers of trace distance of quantum states or pure states rather than all matrices. Applying such results, we give a characterization of quantum channels leaving Helstrom's measure of distinguishability of quantum states or pure states invariant and show that such quantum channels are fully reversible, which are unitary transformations.
"On Partially Trace Distance Preserving Maps and Reversible Quantum Channels." J. Appl. Math. 2013 1 - 5, 2013. https://doi.org/10.1155/2013/474291