Hamiltonian Purification
Davide Orsucci, Daniel Burgarth, Paolo Facchi, Hiromichi Nakazato,, Saverio Pascazio, Kazuya Yuasa, and Vittorio Giovannetti
TL;DR
This paper formalizes the Hamiltonian purification problem, exploring how to embed non-commuting Hamiltonians into a larger space as commuting ones, with solutions for specific algebraic structures.
Contribution
It introduces formal definitions of Hamiltonian purification, spanning-set, and generator purification, providing optimal solutions for the u(d) algebra.
Findings
Formalization of Hamiltonian purification problem
Introduction of spanning-set and generator purification concepts
Optimal solutions for u(d) algebra
Abstract
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.
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