Her father and guru Alauddin Khan, founder of the "Senia Maihar gharana" or "Senia Maihar School" of Hindustani classical music, was a noted musician and guru of Indian classical music. Her uncles, Fakir Aftabuddin Khan and Ayet Ali Khan, were noted musicians at their native place Shibpur, in the present-day Bangladesh. Her brother Ali Akbar Khan was a legendary Sarod maestro and was considered a "national living treasure" in India and the USA. Annapurna was initially instructed by her father in the sitar and vocals. She then switched to the surbahar which she played in the traditional style of a rudra veena. One of her earliest concerts of note was with the surbahar when she played in honour of the Raja of Maihar. She was rewarded with a large tract of land for her performance.