In the seminal 1948 paper "a mathematical theory of communication", Shannon introduced the concept of a classical source as a random variable and established its optimal compression rate, given by Shannon entropy. Nearly five decades later, Schumacher rigorously defined the notion of a quantum source and its compressibility. Schumacher's definition involved a quantum system and correlations with a purifying reference system. The exploration of more general models for quantum sources remained challenging for almost two decades. In our work, we build upon Schumacher's quantum source model, extending it to the most general form allowed by quantum mechanics. This extension involves considering the source and the reference in a mixed state, along with the presence of additional systems treated as side information. We address and solve various problems posed by these modifications, determining the optimal compression rates. While our work contributes significant progress in quantum source compression, we point out remaining open questions that require further exploration.