Post-starburst or “E + A” galaxies are rapidly transitioning from star-forming to quiescence. While the current star formation rate (SFR) of post-starbursts is already at the level of early-type galaxies, we recently discovered that many have large CO-traced molecular gas reservoirs consistent with normal star-forming galaxies. These observations raise the question of why these galaxies have such low SFRs. Here we present an ALMA search for the denser gas traced by HCN (1-0) and HCO+ (1-0) in two CO-luminous, quiescent post-starburst galaxies. Intriguingly, we fail to detect either molecule. The upper limits are consistent with the low SFRs and with early-type galaxies. The HCN/CO luminosity ratio upper limits are low compared to star-forming and even many early-type galaxies. This implied low dense gas mass fraction explains the low SFRs relative to the CO-traced molecular gas and suggests that the state of the gas in post-starburst galaxies is unusual, with some mechanism inhibiting its collapse to denser states. We conclude that post-starbursts galaxies are now quiescent because little dense gas is available, in contrast to the significant CO-traced lower density gas reservoirs that still remain.