Plasmonic sensors are important detectors of biochemical trace compounds, but those that utilize optical readout are approaching their absolute limits of detection as defined by the Heisenberg uncertainty principle in both differential intensity and phase readout. However, the use of more general minimum uncertainty states in the form of squeezed light can push the noise floor in these sensors below the shot noise limit (SNL) in one analysis variable at the expense of another. Here, we demonstrate a quantum plasmonic sensor whose noise floor is reduced below the SNL in order to perform index of refraction measurements with sensitivities unobtainable with classical plasmonic sensors. The increased signal-to-noise ratio can result in faster detection of analyte concentrations that were previously lost in the noise. These benefits are the hallmarks of a sensor exploiting quantum readout fields in order to manipulate the limits of the Heisenberg uncertainty principle.