Related papers: Incomplete information and correlated electrons
Complementarity is a phenomenon explaining several core features of quantum theory, such as the well-known uncertainty principle. Roughly speaking, two objects are said to be complementary if being certain about one of them necessarily…
A detailed and rigorous analysis of G\"odel's proof of his first incompleteness theorem is presented. The purpose of this analysis is two-fold. The first is to reveal what G\"odel actually proved to provide a clear and solid foundation upon…
Although quantum entanglement has already been verified experimentally and applied in quantum computing, quantum sensing and quantum networks, most of the existing measures cannot characterize the entanglement faithfully. In this work, by…
Understanding a complex system entails capturing the non-trivial collective phenomena that arise from interactions between its different parts. Information theory is a flexible and robust framework to study such behaviours, with several…
G\"odel's first and second incompleteness theorems are corner stones of modern mathematics. In this article we present a new proof of these theorems for ZFC and theories containing ZFC, using Chaitin's incompleteness theorem and a very…
Information theory is a statistical theory concerned with the relative state of detectors and physical systems. As a consequence, the classical framework of Shannon needs to be extended to deal with quantum detectors, possibly moving at…
In this paper, we use G\"{o}del's incompleteness theorem as a case study for investigating mathematical depth. We take for granted the widespread judgment by mathematical logicians that G\"{o}del's incompleteness theorem is deep, and focus…
We review with a tutorial scope the information theory foundations of quantum statistical physics. Only a small proportion of the variables that characterize a system at the microscopic scale can be controlled, for both practical and…
The world of mathematics is often considered abstract, with its symbols, concepts, and topics appearing unrelated to physical objects. However, it is important to recognize that the development of mathematics is fundamentally influenced by…
Decoherence processes in quantum electrodynamics due to neglecting either the radiation [L. Landau, Z. Phys. 45, 430 (1927)] or the charged matter [N. Bohr and L. Rosenfeld, K. danske vidensk. Selsk, Math.-Fys. Medd. XII, 8 (1933)] have…
An investigation of Einstein's ``physical'' reality and the concept of quantum reality in terms of information theory suggests a solution to quantum paradoxes such as the Einstein-Podolsky-Rosen (EPR) and the Schroedinger-cat paradoxes.…
State of a $d$-dimensional quantum system can only be inferred by performing an informationally complete measurement with $m\geqslant d^2$ outcomes. However, an experimentally accessible measurement can be informationally incomplete. Here…
In this essay we'll prove G\"odel's incompleteness theorems twice. First, we'll prove them the good old-fashioned way. Then we'll repeat the feat in the setting of computation. In the process we'll discover that G\"odel's work, rightly…
Consideration of the von Neumann measurement process underlying interference experiments shows that the uncertainty in the incoming wave, responsible for its interference, translates during measurement into an uncertainty at the measuring…
We study, in theory and experiment, the quantum properties of correlated light fields measured with click-counting detectors providing incomplete information on the photon statistics. We establish a correlation parameter for the conditional…
Complementarity relations for wave-particle duality are saturated only for pure, single-quanton, quantum states. For a completely incoherent state, it is known that wave and particle quantifiers can reach zero, and hence no information…
We demonstrate that incomplete quantum tomography can give conclusive information in experimental realizations. We divide the state space into a union of multiple disjoint subsets and determine conclusively which of the subsets a system,…
Computing accessible information for an ensemble of quantum states is a basic problem in quantum information theory. We show that the recently obtained optimality criterion (A.S. Holevo, Lobachevskii J. Math., \textbf{43}:7 (2022),…
The overarching theme of the following pages is that mathematical logic -- centered around the incompleteness theorems -- is first and foremost an investigation of $\textit{computation}$, not arithmetic. Guided by this intuition we will…
Irreversible phenomena, such as the production of entropy and heat, arise from fundamental reversible dynamics because the forward dynamics is too complex, in the sense that it becomes impossible to provide the necessary information to keep…