  Von Neumann projections are the main operations by which information can be
extracted from the quantum to the classical realm. They are however static
processes that do not adapt to the states they measure. Advances in the field
of adaptive measurement have shown that this limitation can be overcome by
"wrapping" the von Neumann projectors in a higher-dimensional circuit which
exploits the interplay between measurement outcomes and measurement settings.
Unfortunately, the design of adaptive measurement has often been ad hoc and
setup-specific. We shall here develop a unified framework for designing
optimized measurements. Our approach is two-fold: The first is algebraic and
formulates the problem of measurement as a simple matrix diagonalization
problem. The second is algorithmic and models the optimal interaction between
measurement outcomes and measurement settings as a cascaded network of
conditional probabilities. Finally, we demonstrate that several figures of
merit, such as Bell factors, can be improved by optimized measurements. This
leads us to the promising observation that measurement detectors which---taken
individually---have a low quantum efficiency can be be arranged into circuits
where, collectively, the limitations of inefficiency are compensated for.
