  This note complements the paper "One-Way Ticket to Las Vegas and the Quantum
Adversary" (arxiv:2301.02003). I develop the ideas behind the adversary bound -
universal algorithm duality therein in a different form, using the same
perspective as Barnum-Saks-Szegedy in which query algorithms are defined as
sequences of feasible reduced density matrices rather than sequences of
unitaries. This form may be faster to understand for a general quantum
information audience: It avoids defining the "unidirectional relative
$\gamma_{2}$-bound" and relating it to query algorithms explicitly. This proof
is also more general because the lower bound (and universal query algorithm)
apply to a class of optimal control problems rather than just query problems.
That is in addition to the advantages to be discussed in Belovs-Yolcu, namely
the more elementary algorithm and correctness proof that avoids phase
estimation and spectral analysis, allows for limited treatment of noise, and
removes another $\Theta(\log(1/\epsilon))$ factor from the runtime compared to
the previous discrete-time algorithm.
