  The observing strategy of a galaxy survey influences the degree to which its
resulting data can be used to accomplish any science goal. LSST is thus seeking
metrics of observing strategies for multiple science cases in order to
optimally choose a cadence. Photometric redshifts are essential for many
extragalactic science applications of LSST's data, including but not limited to
cosmology, but there are few metrics available, and they are not
straightforwardly integrated with metrics of other cadence-dependent quantities
that may influence any given use case. We propose a metric for observing
strategy optimization based on the potentially recoverable mutual information
about redshift from a photometric sample under the constraints of a realistic
observing strategy. We demonstrate a tractable estimation of a variational
lower bound of this mutual information implemented in a public code using
conditional normalizing flows. By comparing the recoverable redshift
information across observing strategies, we can distinguish between those that
preclude robust redshift constraints and those whose data will preserve more
redshift information, to be generically utilized in a downstream analysis. We
recommend the use of this versatile metric to observing strategy optimization
for redshift-dependent extragalactic use cases, including but not limited to
cosmology, as well as any other science applications for which photometry may
be modeled from true parameter values beyond redshift.
