Abstract
Understanding when one physical state can be transformed into another is a central problem in quantum information science and thermodynamics. Majorization provides a mathematical tool for describing such transformations. Yet many transitions that are forbidden by majorization can become possible in the presence of a catalyst, an auxiliary system that enables the process without being consumed or altered. Determining the feasibility of such catalytic transformation typically involves checking an infinite set of inequalities involving generalized entropic quantities. Here, we derive a finite sufficient set of inequalities that imply catalysis. Extending this framework to thermodynamics, we also establish a finite set of sufficient conditions for catalytic state transformations under thermal operations. For further examples, we provide a software toolbox implementing these conditions. Our results rely on the connection between a polynomial representation of ℓp norm with Rényi p entropies for any real value of p.
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Acknowledgements
We thank Nelly Ng for useful discussions with an earlier version of this project. A.N. thanks Vjosa Blakaj for discussion of her work on level sets of entropy. S.S. was supported by the Royal Society University Research Fellowship. A.N. acknowledges support from MEXT Quantum Leap Flagship Program (MEXT QLEAP) Grant No. JPMXS0120319794 and ERC Grant Agreement No. 948139. A.N. also acknowledges support by the European Research Council (ERC Grant Agreement No. 948139) and the Excellence Cluster - Matter and Light for Quantum Computing (ML4Q).
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A.M., A.N. worked out the technical derivations and wrote the first draft. D.E., S.S. contributed equally to the supervision of the project. All authors revised the text and prepared the final manuscript.
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Elkouss, D., Maity, A.G., Nema, A. et al. A finite sufficient set of conditions for catalytic majorization. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02583-x
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DOI: https://doi.org/10.1038/s42005-026-02583-x
