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In quantum mechanics, one is concerned with what one can hope to learn about a physical system (for example, according to the Uncertainty Principle , one cannot hope to learn both the position and momentum of an electron to arbitrary accuracy). In relativity theory, one of the basic principles is that signals cannot be propagated faster than light, and that all observers measure the same value for this maximal speed (in a vacuum). Put another way, given a particular Event A in a given Spacetime model (such as an Exact Solution In General Relativity , there is a definite region, called the Absolute Future of A, such that no events outside the absolute future can be causally affected by event A. Information theory is on the other hand a ''strictly statistical theory''. While this theory does have a clear (statistical) notion of causal relationship, ''it has no arrow of time''. Specifically, the quantity which measures the information about which is supplied when one learns , ''or vice versa'', does not allow us to conclude that one event ''causally influenced'' the other, only that the two are ''statistically correlated'' and thus ''causally related''. (''Terminological warning:'' 'event' is used in information theory in the sense of Probability Theory as formulated in terms of Measure Theory by Andrei Kolmogorov , or more properly Ergodic Theory ; this is quite different conceptually and mathematically from the meaning of ''event'' in a Lorentzian Manifold .) Nonetheless, information theory is intimately concerned with signals, and the reception of a signal can certainly result in physically measurable effects. For example, consider a signal sent from Earth to the Mars Rover sitting on the surface of Mars; if upon reception of the signal, a robot arm extends from the Rover, we would naturally say that the transmission event back on Earth influenced the extension event on Mars. And of course, quantum theory is also founded upon notions of probability theory, and the early development of ergodic theory heavily influenced the early development of quantum theory. Moreover, ergodic theory arose in an attempt to rigorously resolve murky early ideas about Statistical Mechanics , and as one of the founders of both quantum theory and of ergodic theory, John Von Neumann , pointed out to Claude Shannon , the founder of information theory, Shannon's fundamental quantity, the Communication Entropy : had earlier appeared in statistical mechanics as an approximation to Boltzmann's notion of a statistical entropy. (''Note'': in the formula, the are the ''blocks'' of an ''equivalence relation'' on a Probability Measure Space X, where the relation has finitely many classes. This is the modern way of capturing Shannon's notion of information having been conveyed when one can choose one alternative from a finite list of choices, perhaps not all having equal a priori likelihood.) Given these considerations, it is natural to speculate that these three theories might enjoy some interesting relationships. Indeed, since the discovery of Hawking Radiation , which is an application of the Semiclassical Approximation for Quantum Field Theory to the region outside the Event Horizon of a Black Hole , and the proof of the laws of Black Hole Thermodynamics , it has become increasingly clear that there are interesting and surprising connections between event horizons in Lorentzian manifolds, quantum field theories, and classical thermodynamics. (A common misconception is these notions are specific to General Relativity ; in fact, Hawking radiation should occur in various physical situations ''completely unrelated to gravitation''-- except formally-- but where something closely analogous to an event horizon occurs; this leads to the idea of Analog Gravity , which includes the notions of Optical Black Hole s and Acoustic Black Hole s.) In addition, in the last decade, the new concept of the Qubit has been intensively developed in the new field sometimes called Quantum Information Theory . This work really does involve both information theory and quantum theory in essential ways. The following questions naturally arise:
The review article Peres and Daniel Terno 2004 states:
UNSOLVED PROBLEMS In Peres and Daniel Terno 2004 the following unsolved problems (among others) are pointed out:
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