Post-starburst or "E+A" galaxies are rapidly transitioning from star-forming to quiescence. While the current star formation rate 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 star formation rates. Here we present an ALMA search for the denser gas traced by HCN (1--0) and HCO+

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... --0) in two CO-luminous, quiescent post-starburst galaxies. Intriguingly, we fail to detect either molecule. The upper limits are consistent with the low star formation rates 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 star formation rates relative to the CO-traced molecular gas and suggests 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.