A Physicist’s Perspective on How One Converts Observation into Information

Andréa Oldofredi, Michaël Esfeld

Paul Dirac has been undoubtedly one of the central figures of the last century physics, contributing in several and remarkable ways to the development of quantum mechanics; he was also at the centre of an active community of physicists, with whom he had extensive interactions and correspondence. In particular, Dirac was in close contact with Bohr, Heisenberg and Pauli. For this reason, among others, Dirac is generally considered a supporter of the Copenhagen interpretation of quantum mechanics. Similarly, he was considered a physicist sympathetic with the positivistic attitude which shaped the development of quantum theory in the 1920s. Against this background, the aim of the present essay is twofold: on the one hand, we will argue that, analyzing specific examples taken from Dirac's published works, he can neither be considered a positivist nor a physicist methodologically guided by the observability doctrine. On the other hand, we will try to disentangle Dirac's figure from the mentioned Copenhagen interpretation, since in his long career he employed remarkably different—and often contradicting—methodological principles and philosophical perspectives with respect to those followed by the supporters of that interpretation. Quanta 2019; 8: 68–87.

Antonio Saggion H, Rossella Faraldo, Matteo Pierno

Antoine Browaeys

T. A. Fritz

Resources and their use and consumption form a central part of our life. Many branches of science and engineering are concerned with the question of which given resource objects can be converted into which target resource objects. For example, information theory studies the conversion of a noisy communication channel instance into an exchange of information. Inspired by work in quantum information theory, we develop a general framework for this type of question. The convertibility of resources into other ones and the possibility of combining resources is accurately captured by the mathematics of ordered commutative monoids. As an intuitive example, we consider chemistry, where chemical reaction equations such as $\mathrm{2H_2 + O_2} \longrightarrow \mathrm{2H_2O}$ are concerned both with a convertibility relation $\longrightarrow$ and a combination operation $+$. We study ordered commutative monoids from an algebraic and functional-analytic perspective and derive a wealth of results which should have applications to concrete resource theories, such as a formula for rates of conversion. As a running example showing that ordered commutative monoids are also of purely mathematical interest, we exemplify our results with the ordered commutative monoid of graphs. While closely related to both Girard's linear logic and Deutsch's constructor theory, our framework also produces results very reminiscent of the utility theorem of von Neumann and Morgenstern in decision theory and of a theorem of Lieb and Yngvason on thermodynamics. Concerning pure algebra, our observation is that some pieces of algebra can be developed in a context in which equality is not necessarily symmetric, i.e. in which the equality relation is replaced by an ordering relation. For example, notions like cancellativity or torsion-freeness are still sensible and very natural concepts in our ordered setting.

T. A. Fritz

Resources and their use and consumption form a central part of our life. Many branches of science and engineering are concerned with the question of which given resource objects can be converted into which target resource objects. For example, information theory studies the conversion of a noisy communication channel instance into an exchange of information. Inspired by work in quantum information theory, we develop a general mathematical toolbox for this type of question. The convertibility of resources into other ones and the possibility of combining resources is accurately captured by the mathematics of ordered commutative monoids. As an intuitive example, we consider chemistry, where chemical reaction equations such as \mathrm{2H_2 + O_2} \lra \mathrm{2H_2O,} are concerned both with a convertibility relation ‘→’ and a combination operation ‘+.’ We study ordered commutative monoids from an algebraic and functional-analytic perspective and derive a wealth of results which should have applications to concrete resource theories, such as a formula for rates of conversion. As a running example showing that ordered commutative monoids are also of purely mathematical interest without the resource-theoretic interpretation, we exemplify our results with the ordered commutative monoid of graphs. While closely related to both Girard's linear logic and to Deutsch's constructor theory, our framework also produces results very reminiscent of the utility theorem of von Neumann and Morgenstern in decision theory and of a theorem of Lieb and Yngvason on the foundations of thermodynamics. Concerning pure algebra, our observation is that some pieces of algebra can be developed in a context in which equality is not necessarily symmetric, i.e. in which the equality relation is replaced by an ordering relation. For example, notions like cancellativity or torsion-freeness are still sensible and very natural concepts in our ordered setting.

Dirk Bouwmeester, ARTUR K. EKERT, Anton Zeilinger