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'''[[Quantum mechanics]] and [[Theory of relativity|relativity]] theory''' comprise two of the foundation stones of [[theoretical physics]], and [[information theory]] is one of the most successful of all theories in [[applied mathematics]].

In quantum mechanics, one is obsessed 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 learns 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 [[qauntum information theory]].  This work really does involve both information theory and quantum theory in essential ways.

Perhaps motivated by these developments, Carl Hewitt (Electrical Engineering, University of Michigan, Emeritus) speculates there should be a new kind of 'information theory' which addresses questions such as these:
*Fundamentally, what is information in physics?
*How can information be obtained physically?
*By what means can information be transmitted?
*Can information be complete?
Hewitt speculates further that his own [[actor model]] in the theory of [[concurrent computing]] might shed some light on these questions, although apparently few if any physics have yet taken an interest in this proposal, which does not appear to have been formulated more precisely than the vague suggestion just stated.  Nor has Hewitt clearly explained just how his speculations are related to work on analog gravity; rather he offers the following quotation from tThe 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>

However, this review paper is about the relationship of special relativity and quantum information theory, and it is not entirely clear whether Hewitt really has the same notions in mind as Peres and Terno.⏎

==What is Information?==

===Information about What?===

Chris Fuchs [2004] says that quantum information is “the potential consequences of our experimental interventions into nature.”

===Interventions and observations===
[[Werner Heisenberg]] in a quote that he attributed to [[Albert Einstein]] many years after the fact stated [Heisenberg 1971]:

:It is quite wrong to try founding a theory on observable magnitudes alone. In reality the very opposite happens. It is the theory which decides what we can observe. You must appreciate that observation is a very complicated process. The phenomenon under observation produces certain events in our measuring apparatus. As a result, further processes take place in the apparatus, which eventually and by complicated paths produce sense impressions and help us to fix the effects in our consciousness.  Along this whole path—from the phenomenon to its fixation in our consciousness—we must be able to tell how nature functions, must know the natural laws at least in practical terms, before we can claim to have observed anything at all. Only theory, that is, knowledge of natural laws, enables us to deduce the underlying phenomena from our sense impressions. When we claim that we can observe something new, we ought really to be saying that, although we are about to formulate new natural laws that do not agree with the old ones, we nevertheless assume that the existing laws—covering the whole path from the phenomenon to our consciousness—function in such a way that we can rely upon them and hence speak of “observation.”

===Incompleteness===
Chris Fuchs [2004] states:

:He [Einstein] was the first person to say in absolutely unambiguous terms why the quantum state should be viewed as information (or, to say the same thing, as a representation of one’s beliefs and gambling commitments, credible or otherwise).  His argument was simply that a quantum-state assignment for a system can be forced to go one way or the other by interacting with a part of the world that should have no causal connection with the system of interest. The paradigm here is of course the one well known through the [[EPR paradox|Einstein, Podolsky, Rosen [1935] paper]], but simpler versions of the train of thought had a long pre-history with Einstein himself (see [Fine 1986] and [Jammer 1985]).

:The best was in essence this. Take two spatially separated systems ''A'' and ''B'' prepared in some entangled quantum state |&psi;<sup>''AB''</sup>>. By performing the measurement of one or another of two observables on system ''A'' alone, one can immediately write down a new state for system ''B''. Either the state will be drawn from one set of states {|&phi;<sub>i</sub><sup>''B''</sup>>} or another {|&eta;<sub>i</sub><sup>''B''</sup>>}, depending upon which observable is measured. The key point is that it does not matter how distant the two systems are from each other, what sort of medium they might be immersed in, or any of the other fine details of the world. Einstein concluded that whatever these things called quantum states be, they cannot be “real states of affairs” for system ''B'' alone. For, whatever the real, objective state of affairs at ''B'' is, it should not depend upon the measurements one can make on a causally unconnected system ''A''.

:Thus one must take it seriously that the new state (either a |&phi;<sub>i</sub><sup>''B''</sup>> or |&eta;<sub>i</sub><sup>''B''</sup>>) represents information about system ''B''. In making a measurement on ''A'', one learns something about B, but that is where the story ends. The state change cannot be construed to be something more physical than that.  More particularly, the final state itself for ''B'' cannot be viewed as more than a reflection of some tricky combination of one’s initial information and the knowledge gained through the measurement.  Expressed in the language of Einstein, the quantum state cannot be a “complete” description of the quantum system.
:...
:The last seventeen years have given confirmation after confirmation that the [[Bell's theorem|Bell inequality]] (and several variations of it) are indeed violated by the physical world. The [[No-communication theorem|Kochen-Specker no-go theorems]] have been meticulously clarified to the point where simple textbook pictures can be drawn of them [ [[Asher Peres]] 1993]. Incompleteness, it seems, is here to stay: The theory prescribes that no matter how much we know about a quantum system—even when we have maximal information about it—there will always be a statistical residue. There will always be questions that we can ask of a system for which we cannot predict the outcomes. In quantum theory, maximal information is simply not complete information [Caves and Fuchs 1996]. But neither can it be completed. As [[Wolfgang Pauli]] once wrote to Markus Fierz in [1954], “The well-known ‘incompleteness’ of quantum mechanics (Einstein) is certainly an existent fact somehow-somewhere, but certainly cannot be removed by reverting to classical field physics.” Nor, I would add, will the mystery of that “existent fact” be removed by attempting to give the quantum state anything resembling an ontological status.

:The complete disconnectedness of the quantum-state change rule from anything to do with [[spacetime]] considerations is telling us something deep: The quantum state is information. Subjective, incomplete information. Put in the right mindset, this is not so intolerable. It is a statement about our world. There is something about the world that keeps us from ever getting more information than can be captured through the formal structure of quantum mechanics.

==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.
==Reference==
*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.

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