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              <p>Abstract The large deviations at various levels that are explicit for Markov jump processes satisfying detailed balance are revisited in terms of the supersymmetric quantum Hamiltonian H that can be obtained from the Markov generator via a similarity transformation. We first focus on the large deviations at level 2 for the empirical density p ^ ( C ) of the configurations C seen during a trajectory over the large time window [ 0 , T ] , and rewrite the explicit Donsker–Varadhan rate function as the matrix element I [ 2 ] [ p ^ ( . ) ] = ⟨ p ^ | H | p ^ ⟩ involving the square-root ket | p ^ ⟩ . (The analog formula is also discussed for reversible diffusion processes as a comparison.) We then consider the explicit rate functions at higher levels, in particular for the joint probability of the empirical density p ^ ( C ) and the empirical local activities a ^ ( C , C ′ ) characterizing the density of jumps between two configurations ( C , C ′ ) . Finally, the explicit rate function for the joint probability of the empirical density p ^ ( C ) and of the empirical total activity A ^ that represents the total density of jumps of a long trajectory is written in terms of the two matrix elements ⟨ p ^ | H | p ^ ⟩ and ⟨ p ^ | H off | p ^ ⟩ , where H off represents the off-diagonal part of the supersymmetric Hamiltonian H . This general framework is then applied to pure or random spin chains with single-spin-flip or two-spin-flip transition rates, where the supersymmetric Hamiltonian H corresponds to quantum spin chains with local interactions involving Pauli matrices of two or three neighboring sites. It is then useful to introduce the quantum density matrix ρ ^ = | p ^ ⟩ ⟨ p ^ | associated with the empirical density p ^ ( . ) in order to rewrite the various rate functions in terms of reduced density matrices involving only two or three neighboring sites.</p>
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