'''[[Quantum mechanics]] and [[Theory of relativity|relativity]] theory''' are two of the foundational stones of [[theoretical physics]], while [[information theory]] is one of the most widely applied of all theories in [[mathematics]].
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 solutions in general relativity|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 <math>I(X,Y)</math> which measures the information about <math>X</math> which is supplied when one learns <math>Y</math>, ''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 [[entropy (information theory)|communication entropy]]
:<math>H(X/\sim) = -\sum_{j=1} \mu(A_j) \log \mu(A_j)</math>
had earlier appeared in statistical mechanics as an approximation to Boltzmann's notion of a statistical entropy.
(''Note'': in the formula, the <math>A_j</math> 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:
*Fundamentally, what is information in physics?
*How can information be obtained physically?
*By what means can information be transmitted?
*Can information be complete?
The review article [Asher Peres and Daniel Terno 2004] states:
<blockquote>
[[Quantum theory]] and [[relativity theory]] emerged at the beginning of the twentieth century to give answers to unexplained issues in [[physics]]: the [[black body]] [[spectrum]], the structure of [[atom]]s and [[Atomic nucleus|nuclei]], the [[electrodynamics]] of moving bodies. Many years later, [[information theory]] was developed by [[Claude Shannon]] [1948] for analyzing the efficiency of communication methods. How do these seemingly disparate disciplines affect each other? In this review, we shall show that they are inseparably related.
</blockquote>
==Unsolved problems==
In [Asher Peres and Daniel Terno 2004] the following unsolved problems (among others) are pointed out:
*There is a trade-off between the reliability of detectors and their localization. This is an important practical problem.
*Progressing from special to general relativity, what is the meaning of parallel transport of a spin?
*We still need a method for detection of relativistic entanglement that involves the spacetime properties of the quantum system,
*After the above problems have been solved, we’ll still have to find a theory of the quantum dynamics for the spacetime structure.
==References==
*A. Einstein, B. Podolsky, and N. Rosen,''Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?'' Phys. Rev. 47, 777–780 (1935).
* Claude Shannon: ''A mathematical theory of communication.'' Bell System Technical Journal, vol. 27, pp. 379-423 and 623-656, July and October, 1948.
*W. Pauli, letter to M. Fierz dated 10 August 1954, reprinted and translated in K. V. Laurikainen, Beyond the Atom: The Philosophical Thought of Wolfgang Pauli, (Springer-Verlag, Berlin, 1988), p. 226.
* Werner Heisenberg. ''Physics and Beyond: Encounters and Conversations'' translated by A. J. Pomerans (Harper & Row, New York, 1971), pp. 63–64.
*M. Jammer, ''The EPR Problem in Its Historical Development'' in Symposium on the Foundations of Modern Physics: 50 years of the Einstein-Podolsky-Rosen Gedankenexperiment, edited by P. Lahti and P. Mittelstaedt (World Scientific, Singapore, 1985), pp. 129–149.
*A. Fine, ''The Shaky Game: Einstein Realism and the Quantum Theory,'' (University of Chicago Press, Chicago, 1986)
*A. Peres, ''Quantum Theory: Concepts and Methods,'' (Kluwer, Dordrecht, 1993).
*C. M. Caves and C. A. Fuchs,''Quantum Information: How Much Information in a State Vector?'' in The Dilemma of Einstein, Podolsky and Rosen – 60 Years Later, edited by A. Mann and M. Revzen, Ann. Israel Phys. Soc. 12, 226–257 (1996).
* Christopher Fuchs, ''Quantum mechanics as quantum information (and only a little more)'' in A. Khrenikov (ed.) Quantum Theory: Reconstruction of Foundations (Växjo: Växjo University Press, 2002).
*Asher Peres and Daniel Terno. ''Quantum Information and Relativity Theory'' Rev.Mod.Phys. 76 (2004) 93.
{{relativity-stub}}
This site is not affiliated with or endorsed in any way by the Wikimedia Foundation or any of its affiliates. In fact, we fucking despise them.
|
