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arXiv:quant-ph/0202109

The No Cloning Theorem versus the Second Law of Thermodynamics

Gavriel Segre

Published Feb 19, 2002• 1 citation
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Abstract

Asher Peres' proof that a violation of No Cloning Theorem would imply a violation of the Second Law of Thermodynamics is shown not to take into account the algorithmic-information's contribution to the Thermodynamical Entropy of the semi-permeable membranes of Peres' engine.
DOI
10.48550/arxiv.quant-ph/0202109

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Two statements by von Neumann and a thought-experiment by Peres prompts a discussion on the notions of one-shot distinguishability, orthogonality, semi-permeable diaphragm, and their thermodynamic implications. In the first part of the paper, these concepts are defined and discussed, and it is explained that one-shot distinguishability and orthogonality are contradictory assumptions, from which one cannot rigorously draw any conclusion, concerning e.g. violations of the second law of thermodynamics. In the second part, we analyse what happens when these contradictory assumptions comes, instead, from _two_ different observers, having different pieces of knowledge about a given physical situation, and using incompatible density matrices to describe it.
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It is known that the stronger no-cloning theorem and the no-deleting theorem taken together provide the permanence property of quantum information. Also, it is known that the violation of the no-deletion theorem would imply signalling. Here, we show that the violation of the stronger no-cloning theorem could lead to signalling. Furthermore, we prove the stronger no-cloning theorem from the conservation of quantum information. These observations imply that the permanence property of quantum information is connected to the no-signalling and the conservation of quantum information.
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 for the most part, von Neumann's argument 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.