The observation that every mixed state can be purified, that is, can be thought of as a subsystem of a larger system that is in a pure state, is a fundamental fact of quantum mechanics. It also extends to quantum channels, or processes: every channel can be thought of as arising from a global unitary evolution, called a Stinespring dilation, on a larger system. Importantly, purifications are not unique, as they can differ by unitaries on the purifying system. This way of thinking, sometimes referred to as “the Church of the larger Hilbert space”, is mathematically very powerful, yet it is subject to a key operational limitation: the purifying system cannot in general be reconstructed from the mixed state alone, as doing so would violate linearity. In other words, there is no physical map that, given a single copy of an unknown mixed state, produces a purification of it.
Recently, however, a new fundamental primitive has been identified in quantum information processing: the random purification channel. Given n copies of an unknown quantum state, the random purification channel produces n identical copies of one of its purifications, selected at random. The original construction, due to Tang, Wright, and Zhandry, relied on the complicated machinery of Schur-Weyl duality. We will present instead a fully elementary construction that uses no representation theory at all, and requires only basic familiarity with quantum theory. We will then extend the construction to channels, obtaining a random Stinespring superchannel, i.e. a physical transformation—realised by an efficient circuit—that, given access to n parallel uses of a quantum channel, grants access to n parallel uses of one of its Stinespring dilations, selected at random.
We will then discuss some of the many applications of the random purification (super)channel, mainly to quantum learning, where it yields efficient quantum tomography algorithms with optimal query complexity, and to quantum Shannon theory, where it yields versions of Uhlmann's theorem for broad classes of quantum divergences, including at the channel level.
Based on arXiv:2511.23451 and arXiv:2512.20599.