Whether the carrier wave carries information depends on modulation method.Another thing, do carrier waves ever carry any "Useful information" or is the information only in the side bands?
For example, in the DSB-C AM, the power in the carrier conveys no useful information. Therefore, DSB-SC AM is basically an amplitude modulation wave without the carrier for reducing power waste. In addition, since both sidebands in DSB-SC AM contain identical copies of the information, thus SSB-SC AM can further increase the efficiency by transmitting one sideband only.
DSB-C AM - Double Sideband Large Carrier AM
DSB-SC AM - Double Sideband Suppressed Carrier AM
SSB-SC AM - Single Sideband Suppressed Carrier AM