  State conversion generalizes query complexity to the problem of converting
between two input-dependent quantum states by making queries to the input. We
characterize the complexity of this problem by introducing a natural
information-theoretic norm that extends the Schur product operator norm. The
complexity of converting between two systems of states is given by the distance
between them, as measured by this norm.
  In the special case of function evaluation, the norm is closely related to
the general adversary bound, a semi-definite program that lower-bounds the
number of input queries needed by a quantum algorithm to evaluate a function.
We thus obtain that the general adversary bound characterizes the quantum query
complexity of any function whatsoever. This generalizes and simplifies the
proof of the same result in the case of boolean input and output. Also in the
case of function evaluation, we show that our norm satisfies a remarkable
composition property, implying that the quantum query complexity of the
composition of two functions is at most the product of the query complexities
of the functions, up to a constant. Finally, our result implies that discrete
and continuous-time query models are equivalent in the bounded-error setting,
even for the general state-conversion problem.
