Quantum Horn’s lemma, finite heat baths, and the third law of thermodynamics

Jakob Scharlau1 and Markus P. Mueller2,3,4,5,1

1Department of Theoretical Physics, University of Heidelberg, Heidelberg, Germany
2Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria
3Department of Applied Mathematics, University of Western Ontario, London, ON N6A 5BY, Canada
4Department of Philosophy, University of Western Ontario, London, ON N6A 5BY, Canada
5Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada

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Interactions of quantum systems with their environment play a crucial role in resource-theoretic approaches to thermodynamics in the microscopic regime. Here, we analyze the possible state transitions in the presence of "small" heat baths of bounded dimension and energy. We show that for operations on quantum systems with fully degenerate Hamiltonian (noisy operations), all possible state transitions can be realized exactly with a bath that is of the same size as the system or smaller, which proves a quantum version of Horn's lemma as conjectured by Bengtsson and Zyczkowski. On the other hand, if the system's Hamiltonian is not fully degenerate (thermal operations), we show that some possible transitions can only be performed with a heat bath that is unbounded in size and energy, which is an instance of the third law of thermodynamics. In both cases, we prove that quantum operations yield an advantage over classical ones for any given finite heat bath, by allowing a larger and more physically realistic set of state transitions.

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