  The inverse problem of extracting the stellar population content of galaxy
spectra is analysed here from a basic standpoint based on information theory.
By interpreting spectra as probability distribution functions, we find that
galaxy spectra have high entropy, thus leading to a rather low effective
information content. The highest variation in entropy is unsurprisingly found
in regions that have been well studied for decades with the conventional
approach. We target a set of six spectral regions that show the highest
variation in entropy - the 4000A break being the most informative one. As a
test case with real data, we measure the entropy of a set of high quality
spectra from the Sloan Digital Sky Survey, and contrast entropy-based results
with the traditional method based on line strengths. The data are classified
into star-forming (SF), quiescent (Q) and AGN galaxies, and show, independently
of any physical model, that AGN spectra can be interpreted as a transition
between SF and Q galaxies, with SF galaxies featuring a more diverse variation
in entropy. The high level of entanglement complicates the determination of
population parameters in a robust, unbiased way, and affect traditional methods
that compare models with observations, as well as machine learning (especially
deep learning) algorithms that rely on the statistical properties of the data
to assess the variations among spectra. Entropy provides a new avenue to
improve population synthesis models so that they give a more faithful
representation of real galaxy spectra.
