Von Neumann's information engine without the spectral theorem

Abstract

Von Neumann obtained the formula for the entropy of a quantum state by assuming the validity of the second law of thermodynamics in a thought experiment involving semipermeable membranes and an ideal gas of quantum-labeled particles. Despite being operational in the most part, von Neumann's argument crucially departs from an operational narrative in its use of the spectral theorem. In this work we show that the role of the spectral theorem in von Neumann's argument can be taken over by the operational assumptions of repeatability and reversibility, and using these we are able to explore the consequences of the second law also in theories that do not possess a unique spectral decomposition. As a byproduct, we obtain the Groenewold--Ozawa information gain as a natural monotone for a suitable ordering of instruments, providing it with an operational interpretation valid in quantum theory and beyond.

10 pages, 4 figures. [v2]: Accepted version for the publication on PRR

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• A derivation of quantum theory from physical requirements
• Higher-order interference and single-system postulates characterizing quantum theory
• Clean Positive Operator Valued Measures
• Thermodynamic Constraints on Quantum Information Gain and Error Correction: A Triple Trade-Off
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Cited by in corpus (3)

• Universal validity of the second law of information thermodynamics
• Pseudo standard entanglement structure cannot be distinguished from standard entanglement structure
• One-shot and asymptotic classical capacity in general physical theories