This page is mostly a stub right now.
See also Wikipedia’s Stinespring factorization theorem
Let be a completely positive trace-preserving (CPTP) map between states on a finite-dimensional Hilbert space, . Then there exists a Hilbert space and a unitary operation on such that
for all , where is the partial trace on the -system. We refer to as an ancilla system and it is chosen such that dimdim. This representation is unique up to unitary equivalence.
TO DO: Write down the proof.
This dilation construction of channels and states is commonly termed the “Church of the Larger Hilbert Space” (coined by John Smolin) since it incorporates the Hilbert space of both the state under consideration as well the environment, even though the operation is on the system only in most cases (for an exception, see the discussion of open quantum systems below). Note that Stinespring’s theorem can also be used to show that every mixed state can be thought of as arising from some pure state on a larger Hilbert space.