Entropy, majorization and thermodynamics in general probabilistic theories
In this note we lay some groundwork for the resource theory of thermodynamics in general probabilistic theories (GPTs). We consider theories satisfying a purely convex abstraction of the spectral decomposition of density matrices: that every state has a decomposition, with unique probabilities, into...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Open Publishing Association
2015-11-01
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| Series: | Electronic Proceedings in Theoretical Computer Science |
| Online Access: | http://arxiv.org/pdf/1508.03107v2 |
| _version_ | 1818063835444019200 |
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| author | Howard Barnum Jonathan Barrett Marius Krumm Markus P. Müller |
| author_facet | Howard Barnum Jonathan Barrett Marius Krumm Markus P. Müller |
| author_sort | Howard Barnum |
| collection | DOAJ |
| description | In this note we lay some groundwork for the resource theory of thermodynamics in general probabilistic theories (GPTs). We consider theories satisfying a purely convex abstraction of the spectral decomposition of density matrices: that every state has a decomposition, with unique probabilities, into perfectly distinguishable pure states. The spectral entropy, and analogues using other Schur-concave functions, can be defined as the entropy of these probabilities. We describe additional conditions under which the outcome probabilities of a fine-grained measurement are majorized by those for a spectral measurement, and therefore the "spectral entropy" is the measurement entropy (and therefore concave). These conditions are (1) projectivity, which abstracts aspects of the Lueders-von Neumann projection postulate in quantum theory, in particular that every face of the state space is the positive part of the image of a certain kind of projection operator called a filter; and (2) symmetry of transition probabilities. The conjunction of these, as shown earlier by Araki, is equivalent to a strong geometric property of the unnormalized state cone known as perfection: that there is an inner product according to which every face of the cone, including the cone itself, is self-dual. Using some assumptions about the thermodynamic cost of certain processes that are partially motivated by our postulates, especially projectivity, we extend von Neumann's argument that the thermodynamic entropy of a quantum system is its spectral entropy to generalized probabilistic systems satisfying spectrality. |
| first_indexed | 2024-12-10T14:26:25Z |
| format | Article |
| id | doaj.art-acb050d928d84c4d9fd6d608c2dde9e0 |
| institution | Directory Open Access Journal |
| issn | 2075-2180 |
| language | English |
| last_indexed | 2024-12-10T14:26:25Z |
| publishDate | 2015-11-01 |
| publisher | Open Publishing Association |
| record_format | Article |
| series | Electronic Proceedings in Theoretical Computer Science |
| spelling | doaj.art-acb050d928d84c4d9fd6d608c2dde9e02022-12-22T01:45:04ZengOpen Publishing AssociationElectronic Proceedings in Theoretical Computer Science2075-21802015-11-01195Proc. QPL 2015435810.4204/EPTCS.195.4:68Entropy, majorization and thermodynamics in general probabilistic theoriesHoward Barnum0Jonathan Barrett1Marius Krumm2Markus P. Müller3 University of New Mexico University of Oxford University of Heidelberg, University of Western Ontario University of Heidelberg, University of Western Ontario, Perimeter Institute for Theoretical Physics In this note we lay some groundwork for the resource theory of thermodynamics in general probabilistic theories (GPTs). We consider theories satisfying a purely convex abstraction of the spectral decomposition of density matrices: that every state has a decomposition, with unique probabilities, into perfectly distinguishable pure states. The spectral entropy, and analogues using other Schur-concave functions, can be defined as the entropy of these probabilities. We describe additional conditions under which the outcome probabilities of a fine-grained measurement are majorized by those for a spectral measurement, and therefore the "spectral entropy" is the measurement entropy (and therefore concave). These conditions are (1) projectivity, which abstracts aspects of the Lueders-von Neumann projection postulate in quantum theory, in particular that every face of the state space is the positive part of the image of a certain kind of projection operator called a filter; and (2) symmetry of transition probabilities. The conjunction of these, as shown earlier by Araki, is equivalent to a strong geometric property of the unnormalized state cone known as perfection: that there is an inner product according to which every face of the cone, including the cone itself, is self-dual. Using some assumptions about the thermodynamic cost of certain processes that are partially motivated by our postulates, especially projectivity, we extend von Neumann's argument that the thermodynamic entropy of a quantum system is its spectral entropy to generalized probabilistic systems satisfying spectrality.http://arxiv.org/pdf/1508.03107v2 |
| spellingShingle | Howard Barnum Jonathan Barrett Marius Krumm Markus P. Müller Entropy, majorization and thermodynamics in general probabilistic theories Electronic Proceedings in Theoretical Computer Science |
| title | Entropy, majorization and thermodynamics in general probabilistic theories |
| title_full | Entropy, majorization and thermodynamics in general probabilistic theories |
| title_fullStr | Entropy, majorization and thermodynamics in general probabilistic theories |
| title_full_unstemmed | Entropy, majorization and thermodynamics in general probabilistic theories |
| title_short | Entropy, majorization and thermodynamics in general probabilistic theories |
| title_sort | entropy majorization and thermodynamics in general probabilistic theories |
| url | http://arxiv.org/pdf/1508.03107v2 |
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