  We present a multi-wavelength study of the brightest cluster galaxies (BCGs)
in a sample of the 95 most massive galaxy clusters selected from South Pole
Telescope (SPT) Sunyaev-Zeldovich (SZ) survey. Our sample spans a redshift
range of 0.3 < z < 1.7, and is complete with optical spectroscopy from various
ground-based observatories, as well as ground and space-based imaging from
optical, X-ray and radio wavebands. At z~0, previous studies have shown a
strong correlation between the presence of a low-entropy cool core and the
presence of star-formation and a radio-loud AGN in the central BCG. We show for
the first time that a central entropy threshold for star formation persists out
to z~1. The central entropy (measured in this work at a radius of 10 kpc) below
which clusters harbor star-forming BCGs is found to be as low as $K_\mathrm{10
~ kpc} = 35 \pm 4$ keV cm$^2$ at z < 0.15 and as high as $K_\mathrm{10 ~ kpc} =
52 \pm 11$ keV cm$^2$ at z~1. We find only marginal (~1$\sigma$) evidence for
evolution in this threshold. In contrast, we do not find a similar high-z
analog for an entropy threshold for feedback, but instead measure a strong
evolution in the fraction of radio-loud BCGs in high-entropy cores as a
function of redshift. This could imply that the cooling-feedback loop was not
as tight in the past, or that some other fuel source like mergers are fueling
the radio sources more often with increasing redshift, making the radio
luminosity an increasingly unreliable proxy for radio jet power. We also find
that our SZ-based sample is missing a small (~4%) population of the most
luminous radio sources ($\nu L_{\nu} > 10^{42}$ erg/s), likely due to radio
contamination suppressing the SZ signal with which these clusters are detected.
