Noncommuting conserved quantities have recently launched a subfield of quantum thermodynamics. In conventional thermodynamics, a system of interest and a bath exchange quantities -- energy, particles, electric charge, etc. -- that are globally conserved and are represented by Hermitian operators. These operators were implicitly assumed to commute with each other, until a few years ago. Freeing the operators to fail to commute has enabled many theoretical discoveries -- about reference frames, entropy production, resource-theory models, etc. Little work has bridged these results from abstract theory to experimental reality. This paper provides a methodology for building this bridge systematically: We present an algorithm for constructing Hamiltonians that conserve noncommuting quantities globally while transporting the quantities locally. The Hamiltonians can couple arbitrarily many subsystems together and can be integrable or nonintegrable. Special cases of our construction have appeared in quantum chromodynamics (QCD). Our Hamiltonians may be realized physically with superconducting qudits, with ultracold atoms, with trapped ions, and in QCD.
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