  The problem of Hamiltonian purification introduced by Burgarth et al. [D. K.
Burgarth et al., Nat. Commun. 5, 5173 (2014)] is formalized and discussed.
Specifically, given a set of non-commuting Hamiltonians {h1, . . ., hm}
operating on a d-dimensional quantum system Hd, the problem consists in
identifying a set of commuting Hamiltonians {H1,...,Hm} operating on a larger
dE-dimensional system H_{dE} which embeds H_d as a proper subspace, such that
hj = PHjP with P being the projection which allows one to recover Hd from HdE .
The notions of spanning-set purification and generator purification of an
algebra are also introduced and optimal solutions for u(d) are provided.
