A growing body of work shows that autonomic signals provide a privileged evidence-stream to capture various aspects of subjective and neural states. This work investigates the potential for autonomic markers to track the effects of psychedelics — potent psychoactive drugs with important scientific and clinical value. For this purpose, we introduce a novel Bayesian framework to estimate the entropy of heart rate dynamics under psychedelics. We also calculate Bayesian estimates of mean heart rate and heart rate variability, and investigate how these measures relate to subjective reports and neural effects. Results on datasets covering four drugs — lysergic acid diethylamide (LSD), dimethyltryptamine (DMT), psilocybin, and sub-anaesthetic doses of the dissociative agent ketamine — show consistent increases in mean heart rate, high-frequency heart rate variability, and heart rate entropy during the psychedelic experience. Moreover, these effects have predictive power over various dimensions of the psychedelic experience. Changes in heart rate entropy were found to be correlated with increases in brain entropy, while other autonomic markers were not. Overall, our results show that a cost-efficient autonomic measure has the potential to reveal surprising detail about subjective and brain states, opening up a range of new research avenues to explore in both basic and clinical neuroscience.