  Analogue Hamiltonian simulation is a promising near-term application of
quantum computing and has recently been put on a theoretical footing. In
Hamiltonian simulation, a physical Hamiltonian is engineered to have identical
physics to another - often very different - Hamiltonian. This is qualitatively
similar to the notion of duality in physics, whereby two superficially
different theories are mathematically equivalent in some precise sense.
However, existing characterisations of Hamiltonian simulations are not
sufficiently general to extend to all dualities in physics. In particular, they
cannot encompass the important cases of strong/weak and
high-temperature/low-temperature dualities. In this work, we give three
physically motivated axiomatisations of duality, formulated respectively in
terms of observables, partition functions and entropies. We prove that these
axiomatisations are equivalent, and characterise the mathematical form that any
duality satisfying these axioms must take. A building block in one of our
results is a strengthening of earlier results on entropy-preserving maps to
maps that are entropy-preserving up to an additive constant, which we prove
decompose as a direct sum of unitary and anti-unitary components, which may be
of independent mathematical interest.
