  In this paper we apply ideas from information theory to create a method for
the design of optimal filters for photometric redshift estimation. We show the
method applied to a series of simple example filters in order to motivate an
intuition for how photometric redshift estimators respond to the properties of
photometric passbands. We then design a realistic set of six filters covering
optical wavelengths that optimize photometric redshifts for $z <= 2.3$ and $i <
25.3$. We create a simulated catalog for these optimal filters and use our
filters with a photometric redshift estimation code to show that we can improve
the standard deviation of the photometric redshift error by 7.1% overall and
improve outliers 9.9% over the standard filters proposed for the Large Synoptic
Survey Telescope (LSST). We compare features of our optimal filters to LSST and
find that the LSST filters incorporate key features for optimal photometric
redshift estimation. Finally, we describe how information theory can be applied
to a range of optimization problems in astronomy.
