The first law of general quantum resource theories

Carlo Sparaciari1, Lídia del Rio2, Carlo Maria Scandolo3, Philippe Faist4,5, and Jonathan Oppenheim1

1Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom
2Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland
3Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK
4Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
5Institute for Quantum Information and Matter, Caltech, Pasadena CA, 91125 USA

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We extend the tools of quantum resource theories to scenarios in which multiple quantities (or resources) are present, and their interplay governs the evolution of physical systems. We derive conditions for the interconversion of these resources, which generalise the first law of thermodynamics. We study reversibility conditions for multi-resource theories, and find that the relative entropy distances from the invariant sets of the theory play a fundamental role in the quantification of the resources. The first law for general multi-resource theories is a single relation which links the change in the properties of the system during a state transformation and the weighted sum of the resources exchanged. In fact, this law can be seen as relating the change in the relative entropy from different sets of states. In contrast to typical single-resource theories, the notion of free states and invariant sets of states become distinct in light of multiple constraints. Additionally, generalisations of the Helmholtz free energy, and of adiabatic and isothermal transformations, emerge. We thus have a set of laws for general quantum resource theories, which generalise the laws of thermodynamics. We first test this approach on thermodynamics with multiple conservation laws, and then apply it to the theory of local operations under energetic restrictions.

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