Finite-resource cooling of qudits beyond thermodynamic length

Abstract

Cooling a quantum system with $N$ thermal machines costs more than Landauer's bound. For swap-based collision models the excess is set by thermodynamic length, which is optimal for qubits and was suggested by numerics to be optimal in general. We show that for qudits, permutations of energy eigenstates that use several resonant transitions at once do better for every spectrum, reducing the excess by up to 83% near the ground state. To second order, coherent collisions do no better, and at infinite temperature $d_M$-level machines reduce the excess by at most a factor $d_M-1$.

The paper asks whether thermodynamic length determines the minimum excess Landauer cost of cooling with $N$ fresh thermal machines. Optimal swap protocols dissipate $(L^*)^2/(2N)+O(N^{-2})$. This is optimal for qubit systems with qubit machines; previous numerics suggested broader optimality. For qudits, resonant permutations of joint system–machine eigenstates distribute population transfer across several channels and beat swaps. A second-order expansion defines a local cost whose convexification gives a generally Finsler, rather than Fisher–Rao, geometry. A theorem using Kubo–Mori metric monotonicity and Birkhoff decomposition shows that arbitrary coherent collisions near dissipationless ones are dominated by mixtures of resonant permutations for first-order spectral changes. Constructions beat swaps near infinite temperature for every nondegenerate spectrum with $d\geq3$, using machines of the same dimension. Near the ground state, with a nondegenerate first excited level, the dominant-transfer cost approaches $1/(d-1)$ of the swap cost. Exact 41-machine protocols reduce excess dissipation by approximately 50% for qutrits and 83% for eight-level systems. Enumeration and primal–dual cone programs characterize the qutrit local geometry. A second theorem bounds the improvement at a maximally mixed system by a factor $d_M-1$; two-level transfers attain this bound. The optimality statements are local and perturbative, not unrestricted finite-$N$ results. Protocol bounds assume uniformly quasistatic collisions and populations bounded away from zero. Whether the $d_M-1$ bound holds at all temperatures remains conjectural for $d_M\geq3$. Near-resonant offsets may provide additional gains, intermediate-temperature improvements can depend on commensurability, and the general higher-dimensional geometry remains unresolved.