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 Does information exist physically? Now we have to make clear the terms of that question. To begin with "exist". It is usually to say: There is a tree. There are honesty. Hence, was is there exists. If anything is nowhere, it does not exist. Mathematical term about the place where something is there, i.e. there is something, or something exists, is "set". Accordingly, if anything is over there, in that set, it exists. Then does a set exist? Yes, if it is a part from another set. Does every set exist? No, we cannot support that every set exists, because we can know no way the set of all the sets exist. Was we do not know, we are able to postulate. But if this is the case, the set of all the sets must comprise itself, it has to belong to itself. Consequently, a set that does not belong to itself is necessary to belong to some set that belongs to itself. It means the set of all the sets that do not belong to themselves does not exist. And it is the essence of ZermeloFraenkel's way out of Russell's paradox. Because of that, information could exist if it belongs to some set we are able to determine. On interesting in physically existing of information,

we have already suggested implicitly information belongs to the set of physical things, i.e. to the φυσις, but hot to the τεχνη, to the things create by themselves, but not to those are created by men.On the contrary, the notion of information arises in the domain of technics, by coding, transporting, and decoding massages. Hence, it is very important to make clear the term "information". There is a chain of definitions connecting it consecutively with the notions of negentropy, entropy, probability, measure, and set. On the other hand, the concept of entropy springs up in the classical thermodynamics and not till later arises the statistical thermodynamics that interprets it as distribution of energy in subsystems. Here, the chain of definitions associates the idea of entropy with those of temperature, volume, pressure, work, movement, statistical ensemble of particles. Moreover, here arises generalization of entropy as a measure of any distribution unevenness. Furthermore, the term of information is an usual word. It means something is known about another, there is knowledge about it. We associate information with facts, data,communication, cognition, truth. 

 For an analogy between distillation formation of pure entangled states and the Carnot cycle, see note 30.

None of the previously mentioned uses of "information" claims it to be an element of the universe such as fire, ground, water, air for the ancient Greeks, or energy and the four interactions for the contemporary physics. But it is a semi-philosophical and semi-scientific conception called synergetics that postulates existing of fundamental relation of energy and information. Nevertheless synergetics is not a science, or at least an universally recognized science. However all of those meanings of "information" are irrelevant, or partly relevant to the term used in our initial question: Does information exist physically? Rather they constitute background of a new understanding of information that is being formed by quantum mechanics in the last decade of XX century. It is connected with phenomena of nonlocality, entanglement, decoherence, einselection, quantum teleportation, quantum cryptography, quantum cloning, quantum tomography, quantum computing, Bell's inequalities, Einstein– Podolsky–Rosen's states, etc. That field of scientific researches is united by title "quantum information" and it is in full force progressing one. In that realm, the question of physically existing information means there are phenomena of interactions, but without any exchange of energy (consequently, of matter). Nevertheless we affirm those phenomena are physical on the grounds of

correlations between them (Bell's inequalities) of observables (resp. measurables). HISTORY OF QUANTUM INFORMATION The most publications on quantum information begin by almost ritual quoting of Einstein-PodolskyRosen's article Can quantum-mechanical description of physical reality be considered complete?.(Here is me after them.) The task of paper was to demonstrate incompleteness of quantum mechanics by Gedanken experiment. One possible contemporary commentary is: the three authors have discovered an absolutely new physical phenomenon as a consequence of quantum mechanics principles and have suggested a gedanken experiment... but not to verify their own theory, rather to refute the quantum mechanics completeness by reductio ad absurdum and that absurd is their own Gedanken experiment theory that however is true. Too much irony of Geschichte! The cause even such an innovator as Einstein to startle is the necessity to be accepted existing of physical interaction without any exchange of energy. It seems as though



 Phys. Rev., 1935, 47, 777-780.  On contemporary interpretation see: Yu Shi. Early Gedanken experiments of quantum mechanics Annalen der Physik, 9, revisited. 2000, 8, 637-648.   I.e. Heidegger's fate–history.

microobjects inform themselves how they ought to hold so as not to break Heisenberg's uncertainty. Schrödinger (1935) invented his dead-and–alive cat to illustrate the quantum mechanics principle of superposition and points to the peculiar features of what he called entangled states (verschränkte Zustande" in his original words). "Einstein, Podolsky and Rosen, and Schrödinger showed that when the superposition principle is applied to multiparticle systems, situations even more bizarre (i.e. counter to our classical intuition) than two-slit experiment occur. A first example is the infamous quantum measurement problem: the superposition principle predicts that if result 1 is possible with the system+measurement apparatus final state represented by Ψ1 and result 2 is possible with final state Ψ2, then, in principle, Ψ1+Ψ2 represents also a possible state, although 

 It is very interesting to discuss relation between quantum information and the fourth Heisenberg's uncertainty about time. See for example: L. de Broglie. Les incertudes d'Heinsenberg et l'interprétation probabiliste de la mécanique ondulatiore. Paris: Bordas, 1982, chapter XVI. The law of energy conservation is deduced from inertial systems invariancy concerning time. One hypothetical law of information conservation should deduce from time direction reversing invariancy.   In: Quantum Theory and Measurement, ed. J.A.Wheeler and W.H.Zurek, Princeton University Press, New York, (1983

this is never observed. This measurement problem suffers from a serious limitation: while quantum theory admits Ψ1+Ψ2 as a state, it predicts that the situation represented by Ψ1+Ψ2 is in practice unobservable. This is due to the unavoidable interaction with the environment, which hides the correlations predicted by Ψ1+Ψ2. This "hiding" is called decoherence." It is an usual contemporary representation of Schrödinger's idea. According to direction of this paper, we would suggest a set theory interpretation of the principle of superposition based on notion of quaset "for a collection of elements which may be indistinguishable from one another". So, the 

 N. Gisin, V. Scarany, W. Tittel, and H. Zbinden. Optical tests of quantum nonlocality: from EPR–Bell tests towards experiments with moving observers. Annalen der Physik. Vol. 9 (2000) No 11-12, p. 832.  M.L.Dalla Chiara and G.Toraldo di Francia. Individuals, kinds and names in physics. In: Bridging the gap, eds. G. Corsi, M.L.Dalla Chiara, and G.C. Ghirardi,Boston studies in the philosophy of science, Vol. 140, Dordrecht, 1991, p. 268. Also: "Yet the electrons of an atom, taken as a whole, possess some properties that are characteristic of a set. For instance, they have a cardinality, even if we cannot count (or well– order) them, hence cannot make an ordinal number to correspond to each electron. How can we establish the cardinality? We can follow two different ways: one is theoretical, the other one is experimental. Take the case of a helium atom. Theoretically, we can establish that the electrons are two, because the wave function depends on the six coordinates x1, y1, z1, x2, y2, z2; we can therefore say that

superposition of quantum states may be represented by as a superposition of all the well-orders of a set with a cardinal number. The inconspicuousness of indistinguishable elements is interpreted as equality of all the well–orders, i.e. none of them may be preferred. The einselection, resp. the decoherency have to be interpreted as an one alternative choice of well-order. For instance, the well-order I is (1,2) and well–order II is (2,1). The "einselection" I is realized by the choice of 1 as the well–order I first element and it determines "non–locally" 2 as its second element. Consequently, the unobservable coherent state (equivalent of a quaset) is the both wellorders I+II (Schrödinger's dead–and–alive cat) superposition and the environment orders that coherent state by means of decoherency. 

the wave function has the same degrees of freedom as a system of two classical particles. Experimentally, we can ionise the atom (by bombardment or other means) and extract two separate electrons..." (p.268). "We thus arrive at a situation, which is usually believed to be impossible in classical semantics: different extensions can correspond to one and the same intension. Of course, the reverse situation of one and the same extension corresponding to different intensions is trivially possible, as in classical semantics (for instance, instead of giving the mass of a particle, one could give its rest energy). (P. 270)  See: R. Omnès. The interpretation of Quantum mechanics (Princeton University Press, Princeton, 1994). "In this formalism, one describes a quantum

Bell's inequalities arise to convert EinsteinBohr's dispute concerning Einstein–Podolsky– Rosen's gedanken experiment from metaphysical into scientific one."Bell's inequalities are traditionally derived from an assumption about the existence of local hidden variables." "...Bell demonstrated that the attempt to complete the theory with so–called hidden variables and maintaining the locality condition leads to statistical predictions for measurements along nonorthogonal bases that differ from those given



system in terms of an exhaustive set of possible histories, which must satisfy a decoherence or consistency criterion. Histories which satisfy this criterion do not interfere with each other, and may be assigned probabilities that obey the usual classical probability sum rules. Both quantum trajectories and consistent histories describe a quantum system in terms of alternative possible evolutions; they thus bear a certain resemblance to each other. What is more, sets of histories corresponding to possible records of a "classical" measurement device will always decohere. Thus, there will be a set of consistent histories that correspond to the quantum trajectories of a continuously measured system."(Todd A. Brun. Continuous measurements, quantum trajectories, and decoherent histories. Philosophical Review A, 61, 4 (2000), 042107.)  J. S. Bell. On the Einstein–Podolsky-Rosen paradox, Physics, vol. 1 (1964) 195–200.  R. F. Werner and M.M.Wolf. Bell's inequalities for states with positive partial transpose. Physical Review A. Vol. 61 (2000), N 6, 062102

by standard quantum theory." "Let a and a' denote possible measurements on A with results α and α', respectively, and similarly b and b'. Then (α,β) and (α,β') denote possible joint results when a is measured simultaneously with b and b', respectively. Now it seems almost obvious that if the objects A and B are spatially separated, then results on A do not depend on what is measured on B. This locality assumption implies that if result α is registered, this does not depend on which measurement b or b' was performed on B, and vice– versa. Hence, ascribing to the results the values +1,–1, they satisfy the following inequality: α.β + α.β' + α'.β – α'.β' = α(β+β')+α'(β–β')≤ 2 Consequently, the locality assumption implies that the expectation values E(a,b)= Mean{α.β} satisfy the CHSH-Bell inequality: E(a,b)+ E(a,b')+E(a',b)–E(a',b')≤ 2 But, according to quantum mechanics, this is not so! All entangled pure states violate the



 W. Tittel, J. Brendel, N. Gisin, and H. Zbinden. Long–distance Bell–type tests using energy–time entangled photons. Physical Review A. Vol. 59 (1999), N 6, 4150. 

inequality for some properly chosen measurements." The essence of Bell's inequalities is related to Einstein notion of "realism"; that an object has "objective properties" whether they are measured or not. Bell's inequalities reflect constrains on the statistics of any three local properties of a collection of objects. These constrains must be obeyed if the three properties can be independently known for each object. Consider a set of objects, each characterized by three two– valued (or dichotomic) properties a, b, and c. Then, grouping the objects as a function of two (out of the three) properties (for instance grouping together objects having property a but not b), it is easy to build up a simple inequality relating the number of objects in various groups defined by different pairs of properties. For n(a, not b)≤ example, ≤ n(a,not c)+n(notb,c). While such an inequality only refers to the simultaneous specification of any pair of properties, its satisfaction depends on the existence of a probability distribution for all three. Thus, even when the three properties cannot be accessed at the same time (for whatever reason) the equation above still holds provided that there exists such 

 N.Gisin, V.Scarani, W.Tittel, and H.Zbinden. Optical tests of quantum nonlocality: from EPR-Bell tests towards experiments with moving observers. Annalen der Physik. Vol. 9 (2000), N 11–12, p. 833.

an objective description of each object using three parameters a, b, and c; therefore the equation above provides a straightforward test of "local realism" (i.e. the combination of objectivity and locality). As confirmed experimentally, such equalities can be violated in quantum mechanics. It is the uncertainty principle (implying that the simultaneous perfect knowledge of two conjugate observables is impossible) which is at the root of such a violation. If we return to conception of quaset, then the choice of any well–order is nonlocal one. So is possible to formulate the assumption that spatial separability (and may be the mathematical notion of separability) means absence of well–order in a sense. Bell's inequalities are probably only a corollary, that may be deduced from the conjecture of spatial inseparability. Kochen-Specker theorem generalizes Bell's theorem. Bell's theorem excludes local hidden variables. 

 N.J.Cerf and C. Adami. Entropic Bell inequalities. Physical Review A. Vol. 55, 5 (1997), 3371.  See: N.J.Cerf and C. Adami. Quantum extension of conditional probability. Physical Review A. 60 (1999) 2, 893–898; N.J.Cerf, C.Adami, and R.M.Gingrich. Reduction criterion for separability. Physical Review A. 60 (2000) 2, 898–910. Ping-Xing Chen and Lin–Mei Liang, Cheng–Zu Li, Ming–Qiu Huang. Necessary and sufficient condition of separability of any system. Physical Review A. Vol. 63 (2001), 5, 052306(5). 

The Kochen-Specker theorem excludes noncontextual hidden variables. In local hidden variables theories the predetermined results for a given measurement are independent of which measurement are performed at spacelike separation. In non– contextual hidden–variable theories the predetermined results are independent of any measurement that are performed jointly. However A.Kent is of the opinion that noncontextual hidden variable theories cannot be excluded by the theoretical arguments of the Kochen–Specker theorem. In the early 80's Aspect and co-workers performed experiments confirming the predictions of quantum mechanics by using the entanglement of the polarization of two photons. A few years ago, W. 

 C. Simon, C. Brukner, and A. Zeilinger. Hidden– Variable Theorems for Real Experiments. Physical Review Letters. Vol. 86, N 20, (2001), 4427.   A. Kent. Noncontextual Hidden Variables and Physical Measurements. Physical Review Letters. Vol. 83, N 19, 1999, 3755(3).  A.Aspect, P.Grangier, and G. Roger. Experimental tests of realistic local theories via Bell's theorem,Phys.Rev.Lett. 47 (1981) 460-463; A. Aspect, P.Grangier and G. Roger. Experimental Realization of Einstein-Pododlsky-Rosen-Bohm Gedanken experiment: A New Violation of Bell's Inequalities, Phys. Rev. Lett. 49 (1982) 91.  "Bell's inequality has already been tested experimentally with photons, i.e. massless particles. To date, however, no experiments have be reported for massive particles such as electrons."

Tittel et al. demonstrated that the experiment can also be performed by photons that propagated 8 and 9 km and it showed the robustness of entanglement and its potential for applications in quantum communication. Furthermore, Greenberger, Horne and Zeilinger (GHZ) have shown that such correlations, in the case of three or more entangled particles, lead to a strikingly more direct refutation of the argument of Einstein,Pododlsky, and Rosen on the possibility of introducing elements of reality to complete quantum mechanics than considerations involving only pairs of particles."That is, the increasing number of particles, in this case, does not bring us closer to the classical realm, as it often supposed, but rather, makes the discrepancies between the quantum and the



(Shiro Kawabata. Test of Bell's Inequality using the Spin Filter Effect in Ferromagnetic Semiconductor Microstructures. Journal of the Physical Society of Japan. Vol. 70, No 5, May, 2001, p. 1210.  W. Tittel, J. Brendel, N. Gisin, and N.Zbinden, Violation of Bell inequalities by photons more than 10 km apart, Phys. Rev. Lett. 81 (1998) 3563–3566; W.Tittel, J.Brendel, N.Gisin, and H.Zbinden. Long– distance Bell–type tests using energy–time entangled photons, Phys. Rev. A 59 (1999) 41504163.  D.M.Greenberger, M.A.Horne, and A. Zeilinger, in Bell's Theorem and the conception of the Universe, edited by M. Kafatos (Dordrecht, 1989).

classical more profound." The GHZ proof of Bell's theorem provided an "all versus nothing" refutation of EPR elements of reality but required three or more spacelike separated observers. Such a greater contradiction between EPR local elements of reality and quantum mechanics than that predicted by Bell's theorem is possible to be proved in the case of two observers. Theory of measurements has a special status in quantum information (as well as in quantum mechanics). Unlike classical mechanics, in quantum mechanics it cannot be assumed that the effect of the measurement on the system can be made arbitrarily small. It is necessary to supplement quantum theory with additional postulates, describing the measurement. One such additional postulate is von Neumann's state reduction (or projection) postulate. The important consequence of von Neumann's projection postulate is the quantum Zeno effect. The effect is so called in allusion to the paradox stated by the Greek philosopher Zeno (or Zenon) of Elea. The 

 M. Zukowski and D.Kaszlikowski. Greenberger– Horne–Zeilinger paradoxes with symmetric multiport beam splitters. Physical Review A. 59 (1999), 5, 3200(4).  A. Cabello. "All versus Nothing" Inseparability for Two Observers. Physical Review Letters. Vol. 87, N 1 (2001), 010403(4).  J. von Neumann. Mathematisch Grundlagen der Quantenmechanik (Springer, Berlin, 1932).

observable Zeno effect would be: "frequent observations slowed down the decay. An unstable particle would never decay when continuously observed." The Zeno effect has been experimentally proved. However it can demonstrate that "under suitable conditions, the decay of some quantum systems from an initial state might be accelerated by frequent interrogations when the system remain in that state. We call this phenomenon a quantum anti–Zeno effect." Fisher, Gutiérrez–Medina, and Raizen report the first observation of both the Zeno and anti–Zeno effects by repeated measurements during the nonexponential period of an unstable quantum system. "Recently, Bennett, Brassard, Crepeau, Jozsa, Peres, and Wootters (BBCJPW) proposed a gedanken experiment which they called "quantum teleportation". Many proposals were suggested and 

 J. Ruseckas and B. Kaulakys. Real measurements and the quantum Zeno effect. Physical Review A, Vol. 63 (2001), 6, 062103  W. M. Itano, D.Heinzen, J.J. Bollinger, and D.J.Wineland, Phys. Rev. A 41, 2295 (1990)  M.Lewenstein and K. Rzazewski. Quantum anti–Zeno effect. Physical Review A, Vol. 61 (2000), 2, 022105.  M.C. Fischer, B Gutiérrez–Medina, and M.G.Raizen. Observation of the Quantum Zeno and Anti–Zeno Effects in an Unstable System. – Physical Review Letters.Vol. 87, N 4, 040402(4).  Phys. Rev. Lett. 70, 1895 (1993)

a few real experiments were performed inspired by Quantum teleportation the BBCJPW work". transports the quantum state of a system and/or its correlations to another system. The state is disintegrated in one place and a perfect replica appears at a distant site. The state or its complete description is never located between the two sites during the transportation. The teleportation procedure, apart from quantum channels (prepared in advance), requires telegraphing a surprisingly small amount of information between the two sites. Telegraphing classical information cannot be instantaneous. Therefore teleportation cannot be instantaneous either. This, however, is not surprising, since quantum states can carry information and special relativity does not allow instantaneous transmission of signals.



 B. Bouwmeester, J.W.Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, Nature (London), 390, 575 (1997); J.W.Pan, D. Bouwmeester, H. Weinfurter, and A. Zeilinger, Phys. Rev. Lett. 80, 3891 (1998); D. Boschi, S. Branca, F. De Martini, L. Hardi, and S. Popesku, Phys. Rev. Lett. 80, 1121 (1997). For the general principle of quantum teleportation, see: Y.H.Shi. Quantum entanglement and quantum teleportation. Annalen der Physik, 10, (2001), 1–2, 28–34.   L. Vaidman and N. Yoran. Methods for reliable teleportation. Physical Review A. 59 (1999), 1, 116.

All the nonlocal operations, e.g. purification, storage, compression, tomography, and probabalistic implementation rely on the same phenomenon of entanglement. Though nothing was made mention of quantum cloning,quantum computing, quantum 

 W. Dür and J.I.Cirac. Nonlocal operations: Purification, storage, compression, tomography, and probabalistic implementation. Physical Review A. 64 (2001), 1, 012317(14).  "The No–Cloning theorem states that laws of quantum mechanics forbid one to design an apparatus which is always successful in making an exact copy of an unknown quantum state. The fact that it is possible to make either imperfect copies with probability one or perfect copies with probability less than one is by now well established, and upper bounds for these scenarios have been derived." (D. Brub, J. Calsamiglia and N. Lütkenhaus. Quantum cloning and distributed measurements. Physical Review A, 63, 4, (2001), 042308.   The quantum computer is distinguished from the classical computer by the capabilities of operating quantum mechanically on superposition of quantum states and of exploiting resulting interference effects. Owing to these capabilities, quantum computers can outperform classical computers in solving classically intractable problems or finding tractable solutions more rapidly. The implementation of quantum computing is based on two fundamental operations. One is the single–qubit rotation, and the other is two–qubit operation.(Feng Mang, Zhu Xi–Wen, Gao Ke–Lin, Shi Lei. Quantum Computation by Pairing Trapped Ultracold Ions. Chinese Physical Letters. Vol. 18, N 6 (2001),718–720. And: C. Zalka. Grover's quantum searching algorithm is optimal. Physical Review A. Vol. 60, N 4 (1999), 2746(6).) 

cryptography and others, it is clear that quantum information is the frontier of quantum mechanics and physics. PHILOSOPHY AND QUANTUM INFORMATION Philosophy strives for discovering of the world elements. But this long cherished aspiration have been fulfilled by physics of today. Sensuous intuition has not gained the world essence and its place is taken by mathematical one. Until recently, physics has taken for granted that energy is the very initial element. Matter has been necessary energy, but not vice versa. Everything that exists physically is energy. Energy is the predicate of a φυσις set. Since the last decade of XX century, information call in question the position of energy. As 

 "Do quantum information phenomena provide objective evidence for the existence of "natural" ontology inherent in quantum formalism? It is interesting that out of the recently discovered quantum effects, only quantum cryptography provides the answer "yes"."(R.Horodecki, M. Horodecki, and P. Horodecki. Balance of information in bipartite quantum–communication systems: Entanglement-energy analogy. Physical Review A, 63, 2, (2001) 022310– 6.)  For "interaction-free measurement" see: P. Horodecki. "Interaction–free" interaction: Entangling evolution coming from the possibility of detection. Physical Review A, 63, 2, (2001), 022108(5).  "In spite of many wonderful experimental and theoretical results on entanglement, there are still difficulties in understanding its many faces.

though we have to replace "energy" by "information" everywhere in the last paragraph: energy is necessary information, but not vice versa. But the energy conservation law is the very core of the contemporary physics paradigm. Consequently, the first question is: Does an 

This seems to be a reflection of the basic difficulties inherent im the interpretation of the quantum formalism as well as quantum–classical hybridism in our perception of Nature. To overcome the latter, it has been postulated that the existence of a unitary information field is a necessary condition of any communication (or correlation). In addition, the information interpretation of the quantum wave function has also been considered. It rests on the generic information paradigm, according to which the notion of information represents a basic category, and it can be defined independently of probability itself. It implies that Nature is an unbroken entity. However, according to the double, hylemorphic nature of the unitary information field, there are two mutually coupled levels of physical reality in logical Nature: (informational), due to the potential field of alternatives, and energetic, due the field of activities (events). From the point of view of the generic information paradigm, the quantum formalism is merely a set of extremely useful informational algorithms describing the above complementary aspects of the same, truly existing, unitary information field. It leads in a natural way to an analogy between information (entanglement) and energy being nothing but a reflection of unity of Nature." (R.Horodecki, M. Horodecki, and P. Horodecki. Balance of information in bipartite quantum–communication systems: Entanglement-energy analogy. Physical Review A, 63, 2, (2001) 022310–1.)

information conservation law exist? To be answered, it is necessary an information physical quantity definition, resp. to define an information operator in quantum mechanics. "... as there are experimental implementations of quantum information protocol, it follows that quantum information is objective and can provide a natural ontological basis for interpretation of quantum mechanics. Quantum states carry two complementary kinds of information, the "classical" information, involving quantum measurements, and "quantum information, which cannot be cloned. Note that this is consistent with an information interpretation proposed earlier of the wave function in terms of objective 

 "To determine the balance of information in the closed system U, we adopt two basic postulates: (i) entanglement is a form of quantum information corresponding to internal energy; (ii) sending qubits corresponds to work. In accordance with the postulate (i), the information is a physical quantity that, in particular, should be conserved in closed quantum systems, similar to energy. The second postulate allows us to deal with communication processes (in thermodynamic, work is functional of processes). (R.Horodecki, M. Horodecki, and P. Horodecki. Balance of information in bipartite quantum–communication systems: Entanglement-energy analogy. Physical Review A, 63, 2, (2001) 022310–2.)  M. Hall. Universal geometric approach to uncertainty, entropy, and information. Physical Review A, 59, 4 (1999), 2602(14). 

information content. On the other hand, it contradicts the Copenhagen interpretation, according to which the wave functions have no objective meaning and only reality is the result of a measurement. It is remarkable that the above information interpretation of quantum states is compatible with the above-mentioned unitary information field concept, which rests on the assumption that information is physical and can be defined independently of probability itself" Cerf and Adami showed that the violation of Bell's inequalities in quantum mechanics is directly connected to the existence of negative quantum entropies, a feature which is classically forbidden. Thus, Bell's inequality themselves may be considered as a testifying to the physically existing information (if the last is negative quantum entropy). The phenomena of nonlocality and entanglement are by now well established. However it seems that only a shadow of information may be seen in (i.e. 

 Proceedings of the R. Horodecki, in International Conference on Problems in Quantum Physics II: Gdansk'87 (World Scientific, Singapore, 1990)  R.Horodecki, M. Horodecki, and P. Horodecki. Balance of information in bipartite quantum– communication systems: Entanglement-energy analogy. Physical Review A, 63, 2, (2001) 022310–6.  N.J.Cerf and C.Adami. Entropic Bell inequalities. Physical Review A. Vol. 55, N 5 (1997), 3371(4).

by means of) Hilbert's spaces. The moment is ripe for revolutionary volte-faces. The theory of quantum mechanics is so less relevant to that of quantum information as the classical physics to the first. The mathematical formalism of quantum mechanics is a small part of the contemporary mathematics. The last is based on the set theory and the notion of function realized as an one-one, or many–one, but not many–many correspondence. It is almost surely that Zermelo's choice axiom have to be significantly reduced or even abandoned. A well–order set may be interpreted as a collection of reference frames, but it is doubtful whether it might connect any reference frame with a microobject. For Galileo's or Einstein's conception of reference frame is essentially macroscopic one. The possibility to separate such a system is equivalent of that to be set elements distinguishable, or of separability. However a collection of microobjects is not set theory one, because it does not consist of indistinguishable elements. It seems that quantum information will give rise to one "quantum" set theory as quantum mechanics has provoked appearance of quantum logic. The very notion of physically existing information as well as the rest quantum information concepts (entanglement, decoherence, etc.) may be defined in terms of transitions from an indistinguishable (and unobservable) state to distinguishable one.

Probably, the physical quantity of information have to be the frequency or qubits. The exposed above point of view is compatible with the consistent histories program, as developed by Gell–Mann and Hartle following pioneering work by Griffiths and Omnès. The central idea of the consistency scheme is that under certain conditions it is possible to assign probabilities to generalized histories of a system. In normal quantum theory such histories are represented by time–ordered strings of propositions; however the scheme allows for much more general histories in which there is no a priori notion of time ordering. These generalized histories are expected to play a key role in application of the formalism 

 "The central position of entanglement in quantum information theory, and its usefulness in applications, has led to considerable efforts being devoted to finding a suitable measure of how much entanglement a quantum systems contains. This problem has been solved completely for bipartite pure state, and the accepted measure is the subsystem von Neumann entropy, conventionally taken to the base 2, so that a maximally entangled state of a pair of two–level quantum systems, or qubits, possesses one unit of entanglement. This fundamental unit is known as an ebit. The production of entanglement requires the transmission of quantum information between systems. Conversely, the transmission of quantum information between systems can be used to establish entanglement between systems." (A.Chefles, C.Gilson, S. Barnett. Entanglement, information, and multiparticle quantum operations. Physical Review A, 63, 3,(2001) 032314.)

to quantum gravity. "In the generalized version of the history scheme ... the central mathematical ingredients are a set of histories 83 (or, more propositions accurately, the set of about histories) and an associated set of decoherence functions ', with the pair (83, ') being regarded as the analog in the history theory of the pair (/, 6) in standard quantum theory, where / is the lattice of propositions and 6 is the spaces of states on/." Which is the information operator in "classical" quantum mechanics? There exists many reasons it to be identified as a time operator, however: "As Pauli showed,one cannot have a time operator if the hamiltonian of the system is bounded from above or below." It follows from existing a time operator that the energy conservation law is violated."But is this really new? Already in the old EPR paradox, the momentum correlation corresponds to kinetic–energy correlation: the 

 C. Isham,N.Linden. Information entropy and the space of decoherence functions in generalized quantum theory. Physical Review A. Vol. 55, N 6 (1997), 4030. W. Pauli, in Handbook of Physics, edited by H. Geiger and K. Schell (Springer–Verlag, Berlin, 1933), Vol. 24, Pt. 1; another edition, edited by S. Flügge (Springer Verlag, Berlin, 1958), Vol. V/1, p. 60. J.Oppenheim, B. Reznik, and W.G.Unruh. Time–of– arrival states. Physical review A, 59, 3 (1999), 1804.

kinetic energy measured on one particle determines the energy of the other particle. Hence, the distant particle (i.e. not in direct contact with the measurement apparatus) can end in states with different energies. Where does this energy come from or go to?" But such a violation is possible perhaps in the framework of a more general law of information conservation. It is known that Bohr and co–workers admit a violation of the energy conservation law. Pauli refutes such a possibility and predicts neutrino existing. However some bizarre its properties may be explained by neutrino existing as pure information (resp. violating the energy conservation law). Moreover, the questionable information operator have to be interpreted as the origin of time, but as ceaseless repeated acts. Thus, the time itself will be a result of an extraction of a well–order (i.e. history extraction) from the quantum indistinguishability. If we have moved to the domain of philosophical speculation, then we may say that substance of information is related to the time being understood philosophically. If everything is information, then everything is time in the final analysis. The things themselves consist of time 

 N. Gisin, V. Scarany, W. Tittel, and H. Zbinden. Optical tests of quantum nonlocality: from EPR–Bell tests towards experiments with moving observers. Annalen der Physik. Vol. 9 (2000) No 11-12, p. 839.

and if information substance.

is

substance,

so

time

is