Beyond and across space: entanglement

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All those studies in quantum mechanics and the theory of quantum information reflect on the philosophy of space and its cognition Space is the space of realizing choice Space unlike Hilbert space is not able to represent the states before and after choice or their unification in information However space unlike Hilbert space is: The space of all our experience, and thus The space of any possible empirical knowledge

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Beyond and across space: From space to entanglement and back

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Vasil Penchev Bulgarian Academy of Science: Institute for the Study of Societies and Knowledge: Dept. of Logical Systems and Models vasildinevgmail.com July 7 2015 13:00 Lecture hall: 1/2 floor “Velika dvorana” Split Teslina 12 The Fourth Physics Philosophy Conference: "Time Space and Space-Time" University of Split Croatia July 6-7 2015

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I History I History

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Einstein Podolsky and Rosen 1935 hey suggested a thought experiment in order to demonstrate that quantum mechanics was ostensibly incomplete ne can try to complement and elucidate its sense newly by Einstein’s criticism 1909 1910 commented by Haubold Mathai and Saxena 2004 about the quantity of thermodynamic probability W and “Boltzmann’s principle” i.e. the proportionality of entropy S and log W: . k – the Boltzmann constant

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Furthermore they showed: f the mathematical formalism of quantum mechanics had been granted as complete it would imply instein called it “spooky” robability “W” implies some uncertainty lack of knowledge about the macrostate the one “ocular” in terms of the microstates the other “ocular” hus it reproduces “binocularly” the cognitive space of possible solutions after which that space can be merely observed and the “events” in it described in “Gedankenexperiments”.

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The “spooky” acTion: ince that kind of action contradicted the principle of physics ostensibly quantum mechanics should be incomplete in their opinion ne can say that quantum mechanics turns out to be a thermodynamic theory seen “binocularly” in that space his originates from its fundamental principles formulated yet by Bohr: nlike classical mechanics it is a “binocular” or “dualistic” theory about both quantum entities and “apparatus” and thus about both microstate and macrostate implying a fundamental counterpart of W

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Edwin Schrödinger 1935 e also pointed out that quantum mechanics implies some special kind of interactions between quantum systems « » called by him ollowing Einstein’s tradition of “Gedankenexperiments” let us begin shrink the “apparatus” more and more he shrink of the apparatus causes some diminution of all microstates and the microstates remain constant his results into increasing W thermodynamic probability and decreasing S entropy

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John Bell 1964 e suggested a real experiment t was apt to distinguish quantitatively and observably between: he classical case without that “spooky action at a distance” and he quantum one involving a special kind of correlation between physical systems hen the size of the macrostates becomes commeasurable with that of the microstates W begins to converge to 1 and S to 0 his happens when the size of the apparatus has become commeasurable with that of the measured quantum entities

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Bell’s inequalities 1964 his kind of correlation quantum correlations can exceed the maximally possible limit of correlation in classical physics hat exceeding the so-called violation of Bell’s inequalities can be measured experimentally hus one can test the incompleteness of quantum mechanics according to the literal EPR argument hen their sizes become equal to each other W is just 1 and S is 0 “ ne microstate one macrostate one theory probability one but zero freedom entropy” rather “totalitarian”: Classical mechanics is deterministic

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Aspect Grangier and Roger 1981 1982 heir experiments as well as all later ones showed unambiguously that the forecast quantum correlations are observable phenomena ome would stop the “thought experiment” here. Not we he apparatus continues to shrink and its size is already less than that of the measured entities he microstate is correspondingly bigger than that of the macrostate and W 1: an extraordinary kind of probability and S changes sign from plus to minus transforming itself into negative

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II Concepts II Concepts

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«Spooky» quantum mechanics hus that “spooky action at a distance” exists and thus quantum mechanics should be complete he case of probability bigger than 1 can be equivalently represented as that of negative probability if one considers the system of two independent events the probability of the one of which is negative Penchev 2012 he negative probability implies the complex values of entropy: he room of the macrostate is already so tiny that a part of the microstate is already forced to go out in the space of the microstate

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Entanglement he new phenomenon was called “entanglement” and a separate branch of quantum mechanics he theory of quantum information studying that kind of phenomena has appeared and blossomed out since the 90th of the past century ts probability is negative and its entropy is complex adding some purely imaginary entropy for the parts of the microstate remained outside of the macrostate his is the world of quantum information and entanglement

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Entanglement and space he concept of entanglement restricts that of space hat restriction refers to the coherent states in quantum mechanics et us exchange the inscriptions “MACROSTATE” and “MICROSTATE” to each other: uddenly we turn out to be in the starting point of the “Gedankenexpereiment” i.e. in our world his is the quantum world if one exchanges the inscriptions “MACROSTATE” and “MICROSTATE” owever one cannot even exchange them but may look to the sky at night and to see the “microstates” as big as stars and nebulas …

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Space versus coherent state pace is a well-ordered set of points in relation to any observer or reference frame in it oherent state in quantum mechanics is the whole of those points: t is inseparable and thus unorderable in principle oth concepts of space and coherent state are initial elements of cognition mutually restricting their applicability n the ground of that “Gedankenexperiment” one can reflect Einstein’s criticism to both “Boltzmann’s principle” and quantum mechanics newly

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Experience and science « pace» refers to our everyday experience and he concepts of coherent state and entanglement to scientific cognition in an area inaccessible to our senses he quantity of our “ignorance” W 1 − about any physical quantity of any microstate makes physical sense in quantum mechanics as the thermodynamic probability W of the conjugate of the physical quantity at issue he necessary condition is: 1 − ≅ 1 − − which is true only if ≅ 0 i.e. the “size” of the microstate is much much less than that of the microstate: ight the case in quantum mechanics

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The limits of «space» he concept of space should be limited to the relations between physical bodies of commeasurable mass owever the above thought experiment demonstrates that quantum mechanics should be approximately valid if Boltzmann principle holds and the Boltzmann – Gibbs – Shannon definition of entropy is relevant n fact the theorem about the absence of hidden variables Neumann 1932 Cochen and Specker 1968 demonstrate that quantum mechanics is complete: hus Boltzmann’s principle and entropy should be only approximately valid right just to that limit of macrostates much much bigger than microstates

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De Broglie wave 1925 he concept of space is being diluted gradually to and the beyond the limits de Broglie wave can be attached to any physical entity according to quantum mechanics he theorems about the absence of hidden variables in quantum mechanics Neumann 1932 Kochen and Specker 1968 can be interpreted as both: bsolute exact coincidence of model and reality and nversing the relation between the model and reality in comparison to classical physics ere is how:

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The period of de Broglie wave ts period is reciprocal to its mass or energy ne can interpret this period as the length of the present moment specific to the corresponding physical entity of this mass energy he model in quantum mechanics equates the degree of our ignorance about any physical quantity i.e. the mismatch of the model to reality to its conjugate owever the conjugate is merely another physical quantity and therefore EPR’s “element of reality” uantum mechanics transforms our ignorance in a exactly measurable quantity though in another experiment

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III Interpretations III Interprettaions

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For the mass of an observer uman beings are granted as observers in space he range of masses comparable with their mass or energy determines fussily and roughly a domain ithin its scale the concept of space is just applicable fter the difference between the model and reality is included in both model and reality this implies formally their necessary coincidence his corresponds rather directly to the axiom of choice in mathematics

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The measure of an oBserver’s mass f the masses energies of the interacting physical entities are commeasurable they can share approximately a common enough present hen one can postulate that “ridiculous principle”: here is a special theory right quantum mechanics which is always and forever true i.e. in any reality instein’s general relativity seems to be an apparent exclusion of the “ridiculous principle” though

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The masses of the apparatus and quantum entity f their masses energies are incommensurable the lengths of their present moments the corresponding periods of de Broglie waves are also incommensurable ust that is the case in quantum mechanics ndeed it studies the system of a macroscopic device which measures one or more microscopic quantum systems nd vice versa: if the “ridiculous principle” holds even to it entanglement and gravitation should be linked to each other

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A point on a segment he present of the entity of much bigger mass energy can be idealized as a point t should be somewhere on the segment representing the length of the present of the entity with much less mass energy he relation or even ratio of the macro- and microstate as is variable in our “Gedanken- experiment” hen energy conservation should be generalized to action conservation for the essentially different “lengths of now”

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Future and the past within the present he present of the measured quantum systems is an approximately common segment t will include also as the past as future rather than only the present of the device uantum mechanics is forced to invent the relevant way to describe both quantitatively and uniformly future and the past along with the present lassical mechanics is restricted only to the present

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The past and future of the device he past of the device is all points of the segment which are before the point of the present ts future will be those after this point owever the way of being for both sub-segments above is radically different even opposed to each other he points of the past are always a well-ordered series n the contrary the points of future constitute an inseparable coherent whole

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Time space and coherent state he concept of coherent state in quantum mechanics refers to both future and past as well as to the present of the investigated system though owever the interpretation of «coherent state» is absolutely different: It is: nseparable in future well-ordered series in the past statistical ensemble of states and the choice of a trajectory in the present « pace» will refer only to its present shared by both apparatus and quantum entity

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Unforecastable future ndeed the future of any entity is unorderable in principle ust this property is rigorously and thus quantitatively represented by the concept of coherent state ny wave function is some state of some quantum system t means the so-called superposition of all possible states of the system at issue as to future

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the always well-ordered past owever the past of any entity is always the well-ordered series of all past moments in time he concept of wave function needs a not less relevant interpretation as to the past hen it is equivalent to a transfinite series of bits i.e. to an “infinitely long” tape of a Turing machine he concept of Hilbert space is just that relevant mathematical structure which is able to describe uniformly both unorderable future and well- ordered the past

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The two elements: future and the past herefor the description in quantum mechanics has to provide the invariance to both unorderable future and well-ordered past ne can thought of them as two opposite media being reconciled by the “phase transition” of the present ilbert space is: hat manages to provide the relevant tool for a general theory of phase transition nother viewpoint to phase space

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mathematics enters ... econciling both “elements” means: he so-called well-ordering theorem equivalent to the axiom of choice is necessarily involved owever we have already demonstrated: ilbert space is able to represent both “elements” and thus even the phase transition of the present between them uniformly ilbert space is invariant to the axiom of choice in the sense above

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The present between the inconsistent two elements he present always is situated and intermediates between the past and future hoice in the present is just what transforms future into the past hen: ilbert space consists of choices in final analysis nd the phases of choice are: uture before choice he past after choice he present the choice properly

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Hilbert space as the space of information o Hilbert space consists of choices in final analysis nformation is the quantity of choices hat implies is: ilbert space is the space of information he units of information are: it : the choice between two equally probable alternatives classical information about finite entities ubit : the choice between infinite alternatives quantum information about infinite entities

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The service of space pace in turn is what makes possible choice and thus the transformation of future into the past pace unlike Hilbert space refers only to the present pace unlike Hilbert space represents any motion only continuously evertheless space and Hilbert space are topologically equivalent by virtue of the Poincaré conjecture proved by Grigori Perelman 20022003 n fact this is implied by that Hilbert space is the mathematical structure unifying the description of both continuous even smooth and discrete motion

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The unity of time and what entanglement serves ntanglement transcending space should be defined as temporal interaction t involves future and the past of the macroscopic device hus it demonstrates quantum correlation lassical correlation is only within space and thus the present uantum correlation is in Hilbert space adding correlation due to future he past being already well-ordered seems not to allow of any correlation in principle

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The temporal secret of entanglement ny classical correlation should refer only to the present of the correlating entities hus refers only to the space in which they are and which they share ny quantum correlation transcends the present and space It involves future and the past nly so it can exceed the maximal possible bound of any classical correlations

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Quantum information ntanglement involves the concept of quantum information uantum information as well as its unit of qubit is shared by both single Hilbert space and two or more entangled Hilbert spaces he former is the case where the system is considered as a whole he latter is the case where the system is considered as composed by subsystems he two cases are equivalent to each other

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Quantum information as a generalization t is a generalization of the classical concept of information n it the units of elementary finite choice are merely substituted by ones among an infinite set of alternatives n fact the fundamental equation of quantum mechanics the Schrödinger equation means: nergy of quantum information is conserved in the course of time: from future via the present to the past

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Conclusions: Conclusions:

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The reflection on philosophy of space and time ll those studies in quantum mechanics and the theory of quantum information reflect on the philosophy of space and its cognition pace is the space of realizing choice pace unlike Hilbert space is not able to represent the states before and after choice or their unification in information owever space unlike Hilbert space is: he space of all our experience and thus he space of any possible empirical knowledge

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What should space be philosophically pace should be discussed as: “transcendental screen” necessary condition of visualization or objectification n it all phenomena are represented by masses comparable with those of observers granted as human beings ilbert space relevant to physical reality anyway can be exhaustedly projected on the screen of space as well-ordered series of “frames” Bergson 1908 ntanglement seems to be gravity after that projection

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The limits of our sensual experience ur sensual experience as well as classical physics observes and studies only phenomena within the framework of space herefore it cannot transcend its limits owever our knowledge is able to transcend them by means of: oing consistent any series of “frames” in space dding those elements hidden for sensual experience but necessary for the series of frames to “make sense” to be consistent

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The breakthrough of quantum theory uantum theories can also transcend those limits he general quantum principle of knowledge is: ilbert space to be restored by any empirical or experimental series of frames in space hus quantum mechanics allows of interpreting space newly: t is the domain of interaction of bodies of both commeasurable mass and energy hus it is the area of choice transforming future into the past

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References quantum mechanics: spect A Grangier P. Roger G. 1981 Experimental tests of realistic local theories via Bell’s theorem. Physical Review Letters 477 460-463. spect A Grangier P. Roger G. 1982 Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedanken Experiment: A New Violation of Bell’s Inequalities. Physical Review Letters 492 91-94. ell J. 1964 On the Einstein ‒ Podolsky ‒ Rosen paradox. Physics New York 1 3 195-200. roglie L. de 1925 Recherches sur la th éorie des quanta Researches on the quantum theory Thesis Paris 1924. Annales de Physique Paris 10- ème s érie 3 22-128.

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References quantum mechanics: instein A. Podolsky B. Rosen N. 1935 Can Quantum-Mechanical Description of Physical Reality Be Considered Complete Physical Review 47 10 777-780. ochen S. and E. Specker 1968 The problem of hidden variables in quantum mechanics. Journal of Mathematics and Mechanics 17 1: 59-87. eumann J. von 1932 Mathematische Grundlagen der Quantenmechanik Berlin: Verlag von Julius Springer. chr ödinger E. 1935 Die gegenw ärtige situation in der Quantenmechanik. Die Naturwissenschaften 2348 807-812 2349 823-828 2350 844-849.

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instein A. Theorie der Opaleszenz von homogenen Fl üssigkeiten und Fl üssigkeitsgemischen in der N ähe des kritischen Zustandes. Annalen der Physik Leipzig 33: 1275–1298 1910. instein A. Zum gegenw ärtigen Stand des Strahlungsproblems. Physikalische Zeitschrift 10: 185–193 1909. aubold H. J. A. M. Mathai R. K. Saxena. Boltzmann-Gibbs Entropy Versus Tsallis Entropy: Recent Contributions to Resolving the Argument of Einstein Concerning “Neither Herr Boltzmann nor Herr Planck has Given a Definition of W” Astrophysics and Space Science 2903-4: 241-245 2004. enchev V . A Philosophical View on the Introduction of Negative and Complex Probability in Quantum Information. Philosophical Alternatives 20121: 63-78. References einsTein’s Thermodynamics:

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Other References: ergson H. 1908 L évolution cr éatrice. Paris: Félix Alcan erelman G. 2002 The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159 . erelman G. 2003 Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109 . erelman G. 2003 Finite extinction time for the solutions to the Ricci flow on certain three- manifolds. arXiv:math.DG/0307245 .

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