For two computational problems $A$ and $B$ in complexity class C (let say $NP$), the existence of a reduction $A <_m^L B$ computable in class (say $L$) implies that $A$ is not computationally harder than $B$ and $B$ is at least as hard as $A$.
I am interested in information theoretic characterization and implications for the existence and non-existence of $<_m^L$-reductions between computational problems. I'm aware of concepts such as computational entropy and hardness amplification but my limited understanding of these notions, which I guess originated from cryptographic applications, does not seem to shed light on the information theoretic characterization of $<_m^L$-reductions.
What is known about information theoretic characterization and consequences of the existence and non-existence of such reductions? Can notions such as computational entropy and hardness amplification help in answering my previous question?