No Universal Purification in Quantum Mechanics

We prove that the linearity and positivity of quantum mechanics impose general restrictions on quantum purification, unveiling a new fundamental limitation of quantum information processing. In particular, no quantum operation can transform a finite number of copies of an unknown quantum state or channel into a pure state or channel that depends on the input, thereby ruling out an important form of universal purification in both static and dynamical settings. Relaxing the requirement of exact pure output, we further extend our result to establish quantitative sample complexity bounds for approximate purification, independent of any task details or operational constraints. To illustrate the practical consequences of this principle, we examine the task of approximately preparing pure dilation and, for the first time, prove an exponential lower bound on the required sample complexity.