arXiv subjects

Marius Krumm

Publications and source records attributed to Marius Krumm.

21 records · Page 2Linked to original sources

arXiv2017

Thermodynamics and the structure of quantum theory

Despite its enormous empirical success, the formalism of quantum theory still raises fundamental questions: why is nature described in terms of complex Hilbert spaces, and what modifications of it could we reasonably expect to find in some regimes of physics? Here we address these questions by studying how compatibility with thermodynamics constrains the structure of quantum theory. We employ two postulates that any probabilistic theory with reasonable thermodynamic behavior should arguably satisfy. In the framework of generalized probabilistic theories, we show that these postulates already imply important aspects of quantum theory, like self-duality and analogues of projective measurements, subspaces and eigenvalues. However, they may still admit a class of theories beyond quantum mechanics. Using a thought experiment by von Neumann, we show that these theories admit a consistent thermodynamic notion of entropy, and prove that the second law holds for projective measurements and mixing procedures. Furthermore, we study additional entropy-like quantities based on measurement probabilities and convex decomposition probabilities, and uncover a relation between one of these quantities and Sorkin's notion of higher-order interference.

quant-ph↗

arXiv2015

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 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.

quant-ph↗

arXiv2015

Thermodynamics and the Structure of Quantum Theory as a Generalized Probabilistic Theory

This thesis investigates the connection between quantum theory, thermodynamics and information theory. Theories with structure similar to that of quantum theory are considered, mathematically described by the framework of "Generalized Probabilistic Theories". For these theories, a thought experiment by von Neumann is adapted to obtain a natural thermodynamic entropy definition, following a proposal by J. Barrett. Mathematical properties of this entropy are compared to physical consequences of the thought experiment. The validity of the second law of thermodynamics is investigated. In that context, observables and projective measurements are generalized to prove an entropy increase for projective measurements of ensembles. Information-theoretically motivated definitions of the entropy are compared to the entropy from the thermodynamic thought experiment. The conditions for the thermodynamic entropy to be well-defined are considered in greater detail. Several further properties of the theories under consideration (e.g. whether there is higher order interference, Pfister's state discrimination principle) and their relation to entropy are investigated.

quant-ph↗