The most fundamental theories in physics, provide a meaning for information or even a way of measuring it. And it has a counterfactual character: a message cannot carry information unless a different message is also possible. Statements about information were therefore long regarded in physics as second-class, non-fundamental approximations. Information itself was considered an a priori abstraction like Euclid's perfect triangles and circles, whose physical instantiations are inevitably approximate. Here, Deutsch and Marletto talk about the current theories of physics
The most fundamental theories in physics, provide a meaning for information or even a way of measuring it. And it has a counterfactual character: a message cannot carry information unless a different message is also possible. Statements about information were therefore long regarded in physics as second-class, non-fundamental approximations. Information itself was considered an a priori abstraction like Euclid's perfect triangles and circles, whose physical instantiations are inevitably approximate. Here, Deutsch and Marletto talk about the current theories of physics
Information has a well-defined meaning in physics. The most fundamental analysis of the nature of information carried out so far originates from and in physics. In physics, the sub-discipline that comes closest to the derived meaning is communication theory. Originating traditionally as it does, communication theory concerns itself primarily with the challenge of information transmission rather than with the problems of information content or meaning. It is abstract in its approach to information, though concerned with the technical characteristics of communication channels independently of the nature of message sources, senders, receivers, or message destinations and coding strategies.
What is information? A short answer could be: Anything that changes our minds and states of thinking! Information is virtual. It can be carried by speech, an image, scratches on a stone, patterns in a photon field, the connections of neurons in our brains or even by the wavefunction of an electron. Information does not depend on the actual means of transmission nor does its identity change when changing the carrier. Information is physical. It needs to be sustained by a physical substrate and rearranging that substrate to contain the wanted information requires to spend energy or work. Without a physical world, information can not be stored, processed, or transmitted. Our world is permeated with different representations of information. Actually Information about the physical laws governing our Universe can be found everywhere and in everything. Could it then be that the real fundamental elements of this world are tiny bits of information? Could we get it from bit?1 A number of researcher investigate this possibility, and several of their attempts to recover physical laws from the principles governing information are reported about in this special issue on physics of information. The articles by Gregg Jaeger,2 by Ariel Caticha,3 and by John Skilling and Kevin Knuth4 address the reconstruction of quantum mechanics from information theory. Furthermore, Kevin Knuth and James Walsh5 show how to recover space-time from particles sending each others signals in an initially geometry-free setting. A central element of most works on information is the concept of a probability. Probabilities can be defined via various routes. Most familiar might be their definitions by the frequentists school as relative frequencies of events (in the limit of infinite amounts of such). A more general definition of probabilities is the Bayesians' one, which regards probabilities as quantifiers of statements, expressing how much those should be expected to be true. Although this definition looks subjective on the first glance, it rests on a solid mathematical theorem by Richard Cox6 that any extension of binary logic to a continuum of single valued truthfulness quantifier that respects a minimum of requirements must be isomorphic to probability theory. And this definition turns out to embrace the frequentists' definition in case that infinite numbers of comparable events are actually available and can be counted. The majority of authors of this special issues seem to favor the Bayesian perspective, which is probably a result of our own, the guest editors', preference and not necessarily a reflection of the prevailing position of contemporary scientists. The different views on probabilities of frequentists and Bayesians are addressed by the article of Allen Caldwell.7 He builds a bridge between these antagonistic camps by explaining frequentists' constructions in a Bayesian language. In particular, his argumentation why the omnipresent p-value in statistical and experimental literature might often capture a relevant signal of model misfit is reassuring, given that a large fraction of our medical system is based on decisions backed up by p-values. A Bayesian might add that it would be even more reassuring, if a system rigorously build on logic and its continuous extension would be used for decisions affecting live-or-death questions. The laws governing probabilities themselves seem to reflect that of more elementary mathematical structures. Kevin Knuth shows how these emerge from lattice theory, which deals with relations in partially ordered sets like that of collections of possibilities.8 The sum rule of probability theory is a direct consequence of the underlying lattice structure. Interestingly, a number of other theories are resting also on such lattices and therefore exhibit their own versions of sum rules: measures of sets, mutual information, the Euler characteristics of topological spaces, as well as quantum mechanical interactions that are quantified by pairs of numbers, the real and imaginary parts of complex probability amplitudes (see also4). The theory of information has very practical application in physics and elsewhere. Manfred Opper discusses how the solutions of stochastic differential equations can be inferred via variational methods, which have their origin in thermodynamics.9 Torsten Enßlin introduces information field theory (IFT), the information theory for fields, as a means to infer the configuration of a physical field from incomplete and noisy data.10 The mathematical apparatus for this is that of quantum field theory (QFT), just with ordinary probabilities instead of complex probability amplitudes as in the latter. Could it be that the difference between IFT and QFT is just be the same as between classical information theory and quantum mechanics, the transition from real valued to complex valued quantifier for statements? Anyhow, any practical inference machinery, be it for dynamical systems or for fields, requires sophisticated numerical methods. Michael Betancourt reviews Markov Chain Monte Carlo Methods to calculate expectation values over high dimensional probabilities.11 Theo Steininger and coworkers present the NIFTy3 software package for numerical information field theory calculations, which can handle very high dimensional field inference problems.12 Finally, in an article focused on the subtleties of the Monty Hall problem and mostly aiming at information theoretical entertainment, Torsten Enßlin and Margret Westercamp investigate the possibility that human intuition might be more rational than contemporary psychology thinks.13 A number of possible topics for an issue on physics of information are not addressed here. We would have loved to have articles on Landauer's principle, which states that erasing information has to cost energy, on QBism, a Bayesian version of quantum mechanics, on artificial and biological neural networks, as well as any other aspect of the information-soaked biological world. What has a reader of this special issue to be equipped with in order to follow the lines of thoughts? In addition to the usual physicist's education, the minimal additional equipment are the basics of probability theory and the concept of entropy. Most of the other concepts used in the articles rest on those and are introduced there. Let A, B, and C be propositions (or possibilities). The probability P ( A | B ) of A being true assuming that B is true is a number between 0 and 1, with P ( A | B ) = 1 expressing certainty of A under B ( B ⇒ A ) and P ( A | B ) = 0 expressing certainty of not A being the case under B ( B ⇒ ¬ A ).
Abstract. At this point in time, two major areas of physics, statistical mechanics and quantum mechanics, rest on the foundations of probability and entropy. The last century saw several significant fundamental advances in our understanding of the process of inference, which make it clear that these are inferential theories. That is, rather than being a description of the behavior of the universe, these theories describe how observers can make optimal predictions about the universe. In such a picture, information plays a critical role. What is more is that little clues, such as the fact that black holes have entropy, continue to suggest that information is fundamental to physics in general. In the last decade, our fundamental understanding of probability theory has led to a Bayesian revolution. In addition, we have come to recognize that the foundations go far deeper and that Cox’s approach of generalizing a Boolean algebra to a probability calculus is the first specific example of the more fundamental idea of assigning valuations to partially-ordered sets. By considering this as a natural way to introduce quantification to the more fundamental notion of ordering, one obtains an entirely new way of deriving physical laws. I will introduce this new way of thinking by demonstrating how one can quantify partially-ordered sets and, in the process, derive physical laws. The implication is that physical law does not reflect the order in the universe, instead it is derived from the order imposed by our description of the universe. Information physics, which is based on understanding the ways in which we both quantify and process information about the world around us, is a fundamentally new approach to science.
In last years, we are witnessing a trend, of trying to describe the laws of nature using informational tools instead of by functional tools. This change towards more abstract description of the nature, is a natural development, as from the time of the Greeks, through the time of Newton, followed by Analytical Mechanics, all the way up to General Relativity and String Theory, there was always a preference to the more abstract and general description of nature. The reason we prefer the more general and abstract descriptions, is that they give us a more objective point of view at the laws of Nature, which allow us more effectively to separate between Phenomena that are the manifestation of the deep laws of Nature, and Phenomena that are just being resulted by our very specific point of view at the universe, and the specific description that we have decided to give it. The more abstract descriptions give us a better frame of reference in our attempted to understand the origins of the laws of Physics. Informational description of the Universe is suspected to be better than the Functional description, as it is a more abstract and general point of view of the Universe, and yet, there is no complete and simple way to translate every Physical Theory to informational language. In this article I have attempted to do just that, by taking two tied Philosophical arguments, and giving them a Formal and Mathematical meaning, by so upgrading them into Physical Principals. The first Principle is the idea that the only objective thing in Nature, that does not depend at the observer point of view, is the different options that realty can exist in. This set of options of reality is meaningless and lacks structure in any way besides their combined number (cardinality), and I name these options of reality “cosmoses”. The second Principle is that there are observers, that are bound to one option of reality, but unable to separate between their cosmos to other cosmoses. This inability to completely know in what cosmos they do exist, creates a local structure. This structure, and the way it changes over time is what creates the structure of the universe that the observers (including you dear reader) measure. By using this simple language, I was able to suggest some new tools and predictions.
We take it for granted that our physical environment communicates information, making things observable and measurable. However, there are very strong constraints on the fundamental physics of any universe that can do this. Measuring or communicating any kind of information always requires an appropriate interactive context, and these contexts are necessarily complex, involving other kinds of information determined in different contexts. This makes measurement hard to grasp theoretically, since every measurement depends on other kinds of measurements. Even so, we can identify some basic functional requirements for a physics that determines and communicates facts. These are sufficient to explain the peculiar features of quantum mechanics, combining the unitary evolution of superpositions with the mysterious "collapse" that occurs whenever the context allows new facts to be defined. Moreover, the precise determinism of classical physics can be understood on the same basis. It seems likely, in fact, that most of the complexity and fine-tuning we see in our most fundamental theories is needed to make any kind of information measurable.
The field of quantum mechanics has revolutionised physics as a subject. Areas such as information theory, computer science and physical sensing have all been affected by the tremendous successes of various quantum protocols. In this thesis I present my contribution to the development of such non-classical protocols. In classical communication theory a message is always carried by physical particles that interact with a transmitter, after which they travel to a receiver. In this thesis I outline a quantum protocol which allows a receiver to obtain a message without receiving any physical object or particles that have interacted with the transmitter—that is, counterfactually. I build my protocol for counterfactual communication on the principles of interaction-free measurements, ensuring that information always propagates in the opposite direction to the protocol particles. The protocol shows how quantum mechanics breaks the previous premise of communication theory. From the perspective of local observers, it is a beautiful manifestation of the non-locality of interaction-free measurements. Furthermore, it is highly robust against experimental errors and external disturbances. The majority of this part of the thesis is based on my published article ‘Quantum counterfactual communication without a weak trace’ [Phys. Rev. A 94, 062303 (2016)]. Previous to my work, Salih et al. attempted to design a counterfactual communication protocol [Phys. Rev. Lett. 110, 170502 (2013)]. This protocol has been highly controversial. As counterfactual phenomena impose restrictions on the inter-measurement paths of quantum particles, and the physical reality of such paths lacks description in the Copenhagen interpretation of quantum mechanics, an extension of current quantum theory is required to facilitate a discussion. In this thesis I present an operational and interpretation-independent methodology, enabling the discussion of inter-measurement paths of quantum particles. I start by considering the interferometers of counterfactual protocols, making the basic assumption that any quantum evolution naturally involves uncontrolled weak interactions. I then show how the Fisher information of these weak interactions, available at the output of counterfactual experiments, can be used to discuss the pre-measurement past of the particles. Based on this analysis, the protocol developed by Salih et al. is found to strongly violate counterfactuality. However, my protocol is more flexible in that it allows particles to propagate in the opposite direction to the message. This leads to counterfactuality being satisfied—even in the presence of large experimental errors. These results are observed both analytically and numerically. This part of the thesis is based on my article ‘Evaluation of counterfactuality in counterfactual communication protocols’ [Phys. Rev. A 96, 062316 (2017)]. The numerical methods are inspired by another of my publications: ‘Protocol for fermionic positive-operator-valued measures’ [Phys. Rev. A 96, 052305 (2017)]. Finally, as the Fisher information measure is found to be useful in evaluating counterfactual protocols, I extend my work by investigating the quantum Fisher information in experiments with general discrete quantum circuits. I prove that the quantum Fisher information of a two-level interaction in a quantum circuit can be expressed by a simple formula. Under certain phase-relations, the formula provides a straightforward connection between the abstract concept of the inter-measurement wavefunction and the quantum Fisher information at the output. With regard to how the information obtained from a certain volume of space influences our perception of classical objects, I argue that the quantum Fisher information measure is highly useful in describing quantum objects. If this measure is applied to observers with a limited set of the experimental measurement outcomes, a quantum object can appear to follow non-classical discontinuous paths. This supports the remarkable conclusion that our perception of the past of a quantum object is subjectively dependent on the measurement we conduct on it.