A new notion of purification has recently been put forwards by Giulio Chiribella, which is phrased entirely in categorical terms.

Definition: “a state is pure iff it only has trivial extensions.”[1]

This is within the context of a causal category [3] (i.e. there is a unique deterministic effect) where additionally, “states separate processes”.

An extension to a state is a bipartite state such that applying the unique deterministic effect to half of the state gives back the original state. *This would be better with some diagrams/equations*. Trivial extensions are states that are separable over the bipartition.

This contrasts with the usual view of purity in quantum theory:

Define: A density matrix ‘$\rho$’ is pure iff $tr(\rho^2)=1$

This means that it cannot be written as a probabilistic mixture of other pure states which leads to a more general description of purity in terms of generalised probabilistic theories[2] (GPTs)

Definition: “A process is pure iff it cannot be obtained from some coarse graining over some outcomes of a given test.”[1]

Open Problem

It is not known whether these views of purification are equivalent or not. If the are the same then there is the question of whether this new viewpoint gives us any insight into purity, if they are different then it would be interesting to know why and whether one notion is in any sense ‘better’ than the other. This problem has implications for categorical reconstructions of quantum theory CQM Reconstruction as it could be taken to be a reasonable axiom in such a reconstruction.


Slides on notions of purity Giulio Chiribella


Giulio Chiribella, `Pure, reversible and sharp, a tale of systems in interaction with their environment'
Bob Coecke, `CauCats the backbone of a quantum relativistic universe of interacting processes'
Ray Lal, `Causal categories'
Giulio Chiribella, `Purity and reversibility as a paradigm for Quantum Information Processing'