  In the context of two-party interactive quantum communication protocols, we
study a recently defined notion of quantum information cost (QIC), which
possesses most of the important properties of its classical analogue. Although
this definition has the advantage to be valid for fully quantum inputs and
tasks, its interpretation for classical tasks remained rather obscure. Also,
the link between this new notion and other notions of information cost for
quantum protocols that had previously appeared in the literature was not clear,
if existent at all.
  We settle both these issues: for quantum communication with classical inputs,
we provide an alternate characterization of QIC in terms of information about
the input registers, avoiding any reference to the notion of a purification of
the classical input state. We provide an exact operational interpretation of
this alternative characterization as the sum of the cost of transmitting
information about the classical inputs and the cost of forgetting information
about these inputs. To obtain this characterization, we prove a general lemma,
the Information Flow Lemma, assessing exactly the transfer of information in
general interactive quantum processes. Furthermore, we clarify the link between
QIC and IC of classical protocols by simulating quantumly classical protocols.
  Finally, we apply these concepts to argue that any quantum protocol that does
not forget information solves Disjointness on n-bits in Omega (n)
communication, completely losing the quadratic quantum speedup. This provides a
specific sense in which forgetting information is a necessary feature of
interactive quantum protocols. We also apply these concepts to prove that QIC
at zero-error is exactly n for the Inner Product function, and n (1 - o(1)) for
a random Boolean function on n+n bits.
