Published on July 20, 2014
Quantum Information: Quantum Information A s the Substance of the World Vasil Penchev: Vasil Penchev [email protected] , [email protected] http:// www.scribd.com/vasil7penchev http://www.wprdpress.com/vasil7penchev CV: http:// old-philosophy.issk-bas.org/CV/cv-pdf/V.Penchev-CV-eng.pdf ___________ 5 Sep 14 17:00 Calouste Gulbenkian Foundation and School of Human and Social Sciences of Nova University in Lisbon, on 5 and 6 September 2014 The thesis: The thesis The concept of matter in physics can be considered as a generalized form of information, that of quantum information involved by quantum mechanics Even more, quantum information is a generalization of classical information: So, information either classical or quantum is the universal fundament of all in the world In particular, the ideal or abstract objects also share information (the classical one) in their common base PowerPoint Presentation: Quantum information Material world Ideal world The substance of all M atter Energy Mass Abstractions Abstract objects Information Information, mass, and energy: Information, mass, and energy Thus information can be seen as the universal substance of the world and therefore, as the relevant generalization of the notions of mass and energy in physics particularly That common base of the world originates from time, which is common for consciousness and the world and thus for the abstract and concrete objects accordingly Consequently the quantity of information is a number assigned to anything to describe the course of time from the unorderable future to the well-ordered past by the choice of the present Measurement and interpretation: Measurement and interpretation Language as ontology Measurement Interpretation Understanding Reality TIME Future present past The unity of mass and energy: The unity of mass and energy Contemporary physics introduces the notion of matter and quantity of mass as a form of energy according to Einstein’s famous equation ( ) Another possible step would be the equivalent of that quantity of mass-energy to information, and more especially to quantum information The equivalent mass-energy of classical information would be exactly zero PowerPoint Presentation: E=mc 2 Q=f(E) “ Q ” means quantum information “ f ” means some unknown function ? ? The substance of the physical world: The substance of the physical world So the nonzero quantity of mass-energy is shared by the physical world and thus by any item within it The physical world and all entities within it (the concrete objects) share that quantity of matter The ideal world of abstract objects also share that quantity, however, it is necessarily zero for each item within it: The abstract objects cannot be distinguished from each other in their mass-energy as it is always zero PowerPoint Presentation: Quantum information Material world Ideal world The substance of all M atter Energy Mass Abstractions Abstract objects Information ≠ 0 ≠ 0 Mass = 0 Energy = 0 Information and thermodynamic entropy: Information and thermodynamic entropy The dimensionless physical quantity of thermodynamic entropy shares the same or similar mathematical formula as information, and more especially that of classical information Indeed any statistical ensemble consists of many physical items, which are always a finite number though perhaps very large: Classical information (or the physical quantity of entropy) reflects order (respectively disorder) of that finite set of items whatever be: Its quantity is a number unambiguously assigned to that set PowerPoint Presentation: . . . . . . . . . . . . . . A statistical ensemble: A huge but finite number of material items Quantum information corresponds to the substance of any entity Information (entropy) corresponds to the order (disorder) of all entities A n ordinal number Finite Infinite Entropy vs. mass and energy: Entropy vs. mass and energy However it always refers to some statistical ensembles of material (energetic) entities and thus the demarcation between mass (energy) and information is conserved Thus entropy is a quantity concerning a set which is always finite One can suggest that the mass-energy of any element of it represents the information of some relevant infinite set Even more, the quantum information involved by quantum mechanics can be supposedly interpreted as that information of infinite sets PowerPoint Presentation: The “river” between infinity and finiteness Here is the state of finiteness And here is the state of infinity Inhabitants: Quantum information , material items having nonzero mass-energy Inhabitants: Classical information , ideal or abstract items having zero mass-energy Information is the only bridge The quantity of information: The quantity of information Information can be considered as a quantity describing the degree of ordering (or disordering, or complexity) of any collection If the collection is a finite set, the corresponding information should be classical Accordingly, if the collection is infinite, the corresponding information should be quantum Even more, if the collection is infinite, it can be as a set as anything else: Nevertheless its information can be defined as quantum in both cases PowerPoint Presentation: The “river” between infinity and finiteness Here is the state of finiteness And here is the state of infinity Quantum information: Well-ordering of infinite sets: i.e. transfinite ordinals Classical information: Well-ordering of finite sets: i.e. finite ordinals Thus information is well-ordering in both cases, on both sides of the boundary Thus information is well-ordering in both cases, on both sides of the boundary Physical being as quantum information: Physical being as quantum information The concept of quantum information allows for any physical entity to be interpreted as some nonzero quantity of quantum information That quantity of quantum information in turn can be interpreted as an infinite number, i.e. as a transfinite ordinal number One can construct a one-to-one mapping between the set of values of quantum information and the set of “small” transfinite ordinal numbers (either in Neumann, or in Cantor-Russell) PowerPoint Presentation: Any physical item Some corresponding nonzero quantity of quantum information A corresponding transfinite ordinal number Some corresponding mathematical item The “river” between physics and mathematics 1:1 1:1 Quantum information as generalized information: Quantum information as generalized information Quantum information can be seen as that generalization of information, which is relevant to infinite collections The classically defined information can refer only to finite ones Thus quantum information corresponds to that generalization of information, which passes from finite to infinite sets Indeed the information of any infinite set can be represented as the ordinal number of this set PowerPoint Presentation: 1 n ω ω +1 ω +n ω ω “Little” transfinite ordinals All is quantum information Classical information “Bits” “ Qubits ” 1 n The most entangled qubit Ω ... ... ... ... ... ... Mass and energy as quantum information: Mass and energy as quantum information On the other hand, the quantities of mass and energy are interpretable as some nonzero amount of quantum information If this is indeed so, just one transfinite ordinal number will correspond to any value of mass-energy That is just the case for the mass-energy corresponds to the operator transforming the zero wave function into the given one: The latter indicates exactly one value of quantum information and thus exactly one transfinite ordinal number PowerPoint Presentation: ω ω +1 A rather conventional boundary: Between finiteness and infinity 1 That qubit is absolutely extraordinary and even unique This qubit is absolutely extraordinary and even unique because it determines the “Standard model” of the universe It defines a privileged reference frame: that of the “Big Bang” Matter as the information of infinite collections: Matter as the information of infinite collect ions Consequently, the physical concept of mass, energy and matter can be interpreted as the notion of information in the case of quantum information, i.e. of an infinite collection Indeed: A transfinite ordinal number corresponds to the infinite set in question. Further, a value of quantum information corresponds to that ordinal number; a wave function, to that value of quantum information; a value of mass-energy, to that wave function That is: A transfinite ordinal number corresponds to any given value of mass-energy in final analysis PowerPoint Presentation: The information of an infinite set A transfinite ordinal A value of quantum information A wave function A value of mass-energy Infinity and matter: Infinity and matter Furthermore, the mathematical analysis of the relation between infinity and finiteness can be transferred to elucidate the essence of matter even in an ontological sense Indeed if matter refers to mass-energy, and a transfinite ordinal number equivalent to some value of quantum information refers to that value of mass-energy, then matter should refer immediately to infinity Even more, any quantity of matter (such as matter- energy) should distinguish different “infinities” (infinite sets) from each other PowerPoint Presentation: Infinity as a whole (“actual infinity”) Infinity as a process (“potential infinity”) A variety of different infinities A well-ordering of different transfinite ordinals: ω+1; ω+2; ... Different cardinal numbers Different cardinal numbers A well-ordering of different infinite cardinals: A definite and FINITE value of mass-energy ? ℵ₀; ℵ 1 ; ℵ 2 ... Choice and information: Choice and information Further , the notion of choice grounds that of information Information can be thought as a few equated relations: Between two orders in the present, or as the steps (choices) to the latter order starting from the former one Between two well-orderings in the past, or as the corresponding steps (choices) as above Between two possible states in the future, or corresponding steps (choices) PowerPoint Presentation: Future Past Choice Information The leap of the present An order Another order A well-ordering Another well-ordering A possibility Another possibility Time and information: Time and information Even more, information can express equivalently any heterogeneous temporal relation This means: One member of the relation belongs to a given temporal modus (past, present, future), but the other one to some different temporal modus: For example: The relation between a well-ordering in the past and a possible state in the future PowerPoint Presentation: Future Past Choice Information The leap of the present An order Another order A well-ordering Another well-ordering A possibility Another possibility An order A possibility A well-ordering The unit of information as an elementary choice: The unit of information as an elementary choice The quantity of information can be seen as the quantity of elementary choices or that of units of choice, which are also units of information Indeed the elementary choices can be equated to the number of minimal discrete steps necessary to achieve a given order, well-ordering, or future state from another For example, a bit (i.e. a Binary digIT ) of information is the minimal discrete step between the two binary digits, such as “0” and “1”, or the choice between them PowerPoint Presentation: 0 1 One bit One elementary choice One elementary step Ordering Ordering Quantum information and the axiom of choice: Quantum information and the axiom of choice The generalization of information through the boundary of infinity requires the axiom of choice in order to be legitimated the notion of choice as to infinity Any choice of an element of any finite set is guaranteed by the implicit definition of an element of a finite set in set theory In a sense, the axiom of choice confirms that any element of a infinite set should be able to be chosen just so as the element of any finite set can be chosen in our experience or experiments and thus separated from the rest ones of the set PowerPoint Presentation: 1 n ω ω +1 ω +n ω ω Classical information “Bits” “ Qubits ” 1 n Ω ... ... ... ... ... ... No need of the axiom of choice One needs the axiom of choice ... ... The sense of the axiom of choice: The choice is uniform from the finiteness to the infinity The notion of infinite choice: The notion of infinite choice One can demonstrate that quantum mechanics involves and even develops implicitly the concept of choice as to infinity The same makes also set theory (the so-called paradox of Skolem based on the axiom of choice ) One can define the infinite generalization of a bit (i.e. an elementary choice between two equiprobable alternatives) as the elementary choice between infinitely many alternatives Further, one can demonstrate that this is equivalent to a quantum bit ( qubit ) involved by quantum mechanics and information PowerPoint Presentation: Quantum mechanics Physics Mathematics The foundation of the physical world The foundation of mathematics S et theory The bridge between the two domains The axiom of choice The choice in an infinite set ! Choice in ontology: Choice in ontology Thus the understanding of matter as information elucidates how choice underlies matter and even ontology at all Indeed the choice is naturally thought as “subjective”: This means that the choice is proper (i.e. inherent) as to ‘subject ’ However now, quantum m echanics and the theory of quantum information are able to demonstrate that the choice is not less objective, too: Indeed it underlies quantum information and by thus, all ‘objects’ and the physical worl d PowerPoint Presentation: Subject Object Ontology The dualism of classical philosophy Two fundamental approaches in philosophy: ! Information Choice The concept of quantum information: The concept of quantum information The concept of quantum information can be introduced in different ways. Some of them (and the most important ones in the present context) are: By Hilbert space By wave functions By operators in Hilbert space By a quantum Turing machine (tape) By quantum computations PowerPoint Presentation: Quantum information in terms of Hilbert space Wave functions Turing machines Quantum computations One and the same ! Quantum information by Hilbert space: Quantum information by Hilbert space One of them defines it by means of Hilbert space: Thus any point in it can be considered as a certain value of the quantity of quantum information However , any point in Hilbert space is equivalent to a wave function, i.e. to a state of some quantum system Consequently, the quantity of quantum information is able to feature unambiguously any quantum system, i.e. any physical system PowerPoint Presentation: Hilbert space as the free variable of quantum information Any point in it as the value of that free variable A wave function A state of a quantum system Mathematics Physics = = = = ! A quantum bit (qubit): A quantum bit ( qubit ) The notion of quantum bit (or ‘ qubit ’) underlies quantum information A quantum bit is usually defined as the normed superposition of any two subspaces of Hilbert space orthogonal to each other: That is: where are two complex numbers such that + , and are any two orthonormal vectors (e.g. the orthonormal bases of any two subspaces) in any vector space (e.g. Hilbert space, Euclidean space, etc.) Quantum “Turing tape”: Quantum “Turing tape” Furthermore, Hilbert space can be represented as a “tape” of qubits : Given any point in (complex) Hilbert space as a vector one can replace any successive couple of its components such as ( , with a single corresponding qubit such that: ; if are not both 0 . However if both are 0 one needs to add conventionally the center ( to conserve the mapping of Hilbert space and an infinite qubit tape to be one-to-one PowerPoint Presentation: Components “Axes” Hilbert space Quantum Turing tape 1 ... n n+1 ... The/No last cell ... ... ........ Bit and qubit: Bit and qubit Then if any bit is an elementary binary choice between two disjunctive options usually designated by “0” and “1”, any qubit is a choice between a continuum of disjunctive options as many (or “much”) as the points of the surface of the unit ball: Indeed a qubit is equivalently representable as a unit ball in Euclidean space and two points, the one chosen within the ball, and the other on its surface, i.e. as a mapping of a unit ball onto its surface (or any other unit sphere) PowerPoint Presentation: β 1 α 1 defines a point of the unit ball and define a point of the unit sphere + are two complex numbers: , are two orthonormal vectors or a basis such as two orthogonal great circles of the unit bal l The qubit as a unit ball Choice, information, and computation : Choice, information, and computation Thus the concept of choice is the core of computation and information The choice unifies them for: Information and computation mean one and the same from two different viewpoints: Information is a relation between an order, well-ordering, or possible state and another Computation is the same relation represented as a process, i.e. step by step That is: The computation can be considered as the constructive analog of the information PowerPoint Presentation: Infinity as a whole “actual infinity” Infinity as a process “potential infinity” A relation between two infinities (as possible states as orderings) Information A process from one to another infinity (only as well-orderings) Quantum computation = = The boundary between classical and quantum information: The boundary between classical and quantum information Both classical and quantum information possess a corresponding constructive “twin” (computation) What demarcates the classical and quantum case is the bound between a finite vs. infinite number of the alternatives of the corresponding choice Consequently, the limit between finiteness and infinity is therefore the boundary between the classical and quantum information PowerPoint Presentation: Finiteness Infinity Mathematics Informatics Quantum information Reality Physics Classical i nformation Ideality Information as the number of primary choices: Information as the number of primary choices The concept of information can be interpreted as the quantity of the number of primary choices It can be visualized as the minimal length of the Turing tape, either finite or transfinite (quantum), necessary for this information to be written For example, the quantity of classical information is equivalent to that number written in binary notation on the Turing tape (e.g. as the ultimate result of processing by a Turing machine) PowerPoint Presentation: 1 n ω ω +1 ω +n ω ω Classical information “Bits” “ Qubits ” 1 n Ω ... ... ... ... ... ... ... ... Information as the number correspondingly Of finite choices (such as bits) → Of infinite or transfinite choices (such as qubits ) → The concept of Turing machine as a bridge between computation and information: The concept of Turing machine as a bridge between computation and information Furthermore, the Turing machine either classical or quantum as a model links computation to information directly: Indeed the processing of that information written on a Turing tape, either finite or transfinite, is the computation of a Turing machine, correspondingly either classical or quantum In other words, computation is always the constructive analog of information not less that information is the “actual” or “non-constructive” analog of computation PowerPoint Presentation: A “classical” Turing machine A quantum Turing machine 1 ... n n+1 ... The last cell A classical Turing tape of bits: A quantum Turing tape of qubits : 1 ... n n+1 ... The/No last cell The list of all operations on a cell: 1. Write! 2. Read! 3. Next! 4. Stop! Information “bit by bit”: Information “bit by bit” This equivalence of information and computation allows of discussing the information “step by step” or “one by one”: The quantity of information can be thought as the sum of the change bit by bit or qubit by qubit , i.e. as the change of number written by two or infinitely many digits The only difference between the two cases is the number of different digits: two (or equivalently any finite number) versus infinite ones Information is classical in the former case and quantum in the latter PowerPoint Presentation: 1 n ω +1 ω +n Classical information “Bits” “ Qubits ” 1 n ... ... ... ... ... ... A classical Turing machine → A quantum Turing machine → A choice among infinite alternatives: A choice among infinite alternatives The generalization from information to quantum information can be interpreted as the corresponding generalization of ‘choice ’: From the choice between two (or any finite number of) disjunctive alternatives to infinitely many alternatives Even more, after choice and thus the axiom of choice has been involved, the distinction between many alternatives and a continuum of alternatives (much alternatives) becomes problematic PowerPoint Presentation: 0 1 0 1 One bit ( a finite choice ) One qubit ( an infinite choice ) Choice Well-ordering Information within a sell : Information within a sell Thus, the distinction between the classical and quantum case can be limited within any cell of an algorithm or ( qu )bit of information: Any bit or qubit is a digit of the written information: If the system of notation contents of a finite number of “digits” (or the number of alternatives of an elementary choice), the corresponding written information is classical If not (i.e. the “digits” are infinitely many or much), the information is quantum as in the physical world PowerPoint Presentation: ... ... ........ An example of a binary number (i.e. a well-ordered series of bits and binary values): 1001001011111000001010 An example of a decimal number (i.e. a well-ordered series of digits and decimal values): 7370450982347936121450 An example of a number in a counting system of “infinite base” (i.e. a well-ordered series of qubits and infinite values): PowerPoint Presentation: Components “Axes” Hilbert space Quantum Turing tape 1 ... n n+1 ... The/No last cell ... ... ........ Physical processes as quantum computations: Physical processes as quantum computations Therefore any physical process can be interpreted as a quantum computation : Indeed: Any physical process is a change of a relevant wave function: The change of a wave function is a change of the relevant value of quantum information The change of quantum information is a change of the relevant transfinite ordinal number The change of an ordinal number is a computation PowerPoint Presentation: Different ways to represent a physical process as different, but equivalent changes Of wave functions: Of values of quantum information: Of transfinite ordinal numbers: Thus as a quantum computation → → → → Physical states and quantities in terms of Hilbert space: Physical states and quantities in terms of Hilbert space Quantum mechanics states that any physical state or its change is a self- adjoint operator in Hilbert space as any physical system can be considered as a quantum one Consequently the states can be interpreted as the quantities to the state of absolute rest Then the boundary between ’quantum states’ and ‘quantum quantities’ is relative: Indeed any point in Hilbert space can be interpreted as the relevant operator transforming the zero point into the point in question PowerPoint Presentation: The temporal interpretation of the definition of physical quantity (A) in quantum mechanics Future past present Ψ (x) Ψ *( x) ∫ Ψ (x) d x x ! The universe as a quantum computer: The universe as a quantum computer Consequently all physical process can be interpreted as the calculation of a single computer and thus the universe as it The link between the physical world and the computations underlain it is quantum information In other words, the quantum information can be interpreted as a “two faced J anus”, the one face of whom is the physical world in space-time, and the other one is the quantum computer of the universe Its “display” is the present PowerPoint Presentation: The physical processes in space-time Computations of a quantum computer What the universe is ! Physical processes as changes of information: Physical processes as changes of information Consequently all physical processes are informational in fact : The boundary between the physical and the informational can be represented temporally: The physical is all till now: This is the current computational result as well all previous results constituting a well-ordering The informational includes the physical: However it includes the future as a single coherent state as well as its gradual transformation in the present and past of the physical: Time and entanglement: Time and entanglement TIME Future present past A coherent state A few entangled states A few well-ordered series in time Quantum mechanics as a universal doctrine : Quantum mechanics as a universal doctrine Indeed quantum mechanics is the universal doctrine about the physical world and any physical process can be interpreted as a quantum one If any physical process is quantum in the above sense and thus involves quantum information and its changes, quantum mechanics implies that the physical world at all, the universe as a whole should be understood as an immense computer, namely a quantum one PowerPoint Presentation: All is quantum information because: Quantum mechanics offers an universal scientific theory: Any physical item and process are quantum in final analysis Thus all should be quantum information and/ or its processing Quantum processes in terms of quantum information: Quantum processes in terms of quantum information Any quantum process is informational in terms of a generalized kind of information: quantum information The course of time implies this: The absolutely coherent future containing all possibilities of development is being transformed into the well-ordered series of events in the past by mediation of the present, within which the choice necessary for that transformation is being accomplished PowerPoint Presentation: Future past present ! The principles of least action and most probability: 1 st stage: 2 nd stage: Physical motion: A leap from past into future A choice of a smooth trajectory from future to past, “from the end to the beginning” Physical processes are informational: Physical processes are informational Consequently the quantity of quantum information as a measure of choices is implied by the course of time Thus all physical processes are informational for any physical process is necessarily in the course of time Even more, one may say that the physical being without any exceptions is created by time: “Time and Being” PowerPoint Presentation: Sein und Zeit Zeit Sein choice and information Quantum information Quantum information as the base: Quantum information as the base Quantum information is the real fundament of the world If the time is what creates the world, the quantum information is what builds the world Indeed the world in all variety needs freedom and choice to be able to be created: It is impossible both in the absolutely rigorously ordered and thus lifeless past and in the quite chaotic and elemental future: The world is possible only in the thin strip, the precinct or the phase transition between them: the present PowerPoint Presentation: Future past present The world Time creates the world Quantum information builds the world The world All wave functions as values of quantum information: All wave functions as values of quantum information Indeed all physical states in the world are wave functions and thus they are different values of quantum information They are also points in Hilbert space and not less different transfinite ordinals If the Hilbert space is a “twofaced Janus” with finite arithmetic as the one, and geometry as the other, then the transfinite arithmetic should correspond to geometry Hilbert space allows of visualizing the transfinite ordinal numbers as geometrical points PowerPoint Presentation: Hilbert space Geometry Arithmetic The quantum quantities as the changes of quantum information : The quantum quantities as the changes of quantum information All physical quantities in the world are a certain kind of changes of wave functions and thus of quantum information Thus they can be as interpreted as a “distraction” of transfinite ordinal numbers If that difference is a finite, the information is classical If that difference is transfinite, the information is quantum: There is some physical change PowerPoint Presentation: 1 ω +n ω Classical information “Bits” “ Qubits ” n 0 _ = _ = = _ = = n Quantum information = a physical change Classical information = an ideal change Quantum information as the matter of the physical world: Quantum information as the matter of the physical world Consequently, one can certainly state that the physical world consists only of quantum information This means that matter is interpreted as the stuff, by which any standalone item in the world is build, or as the energy describing the motions, interactions and changes of those items Furthermore, the concept of information is more extensive that than of the matter, which is only “stuff & energy” PowerPoint Presentation: Staff Energy Matter Information Ideality Reality Time Being A local viewpoint A global viewpoint The philosophical map of concepts Quantum information and matter: Quantum information and matter Quantum information is the substance of the physical world, its “matter ” However quantum information being a generalization of classical information and therefore including it as the particular case as to finite sets underlies also the abstract or ideal world Thus quantum information underlies both the material and ideal world, i.e. anything existing Indeed anything, which is being now, requires or make some choices, the corresponding quantity of which is that of information PowerPoint Presentation: Staff Energy Matter Quantum information Ideality Reality The choices of time Another map of concepts Classical information Information and the unity of physics and mathematics: Information and the unity of physics and mathematics The conception of quantum information unifies physics and mathematics and thus the material and the ideal world as well as the concrete and abstract objects : Indeed mathematics studies an ideal world and abstract objects in it while physics the material world and concrete objects in it: After information underlies both worlds and their objects, the study of information can unify the corresponding areas of cognition: mathematics and physics PowerPoint Presentation: Staff Energy Matter Quantum information Mathematics Physics The choices of time The union of physics and mathematics Classical information Quantum information and the unity of the material and ideal world: Quantum information and the unity of the material and ideal world As information is a dimensionless quantity equally well referring both to a physical entity or to a mathematical class, it can serve as a “bridge ”: Between physics and mathematics And thus a bridge between the material and ideal world A bridge between the concrete and abstract objects This bridge connects the areas of infinity and finiteness over the gap of ... complementarity PowerPoint Presentation: The bridge over the gap Information and quantum information Mathematics Physics The material world The ideal world Abstract objects Concrete objects ... of complementarity Finiteness Infinity Quantum information and unity of the concrete and abstract objects: Quantum information and unity of the concrete and abstract objects In fact, quantum information being a generalized kind of information is just what allows of the physical and mathematical, the concrete and abstract to be considered as two interpretations of the underlying quantum information: If the information refers to the order of finite sets, it is classical If the sets in question are infinite, it is quantum However the underlying mathematical structure in both cases can be or is one and the same: Hilbert space PowerPoint Presentation: Finite sets Infinite sets Classical information Quantum information The fundament Conclusions:: Conclusions: What is is being now Thus all which is is in time The fundamental structure of all is defined between the relation between the future and the past in the present by means of choices Choice is the only way for the unorderable future to be transformed into the well-ordered past Choice can happen only at the boundary of the future and the past, i.e. the present More conclusions:: More conclusions: Information corresponds to the amount of choices necessary for an order to be transformed into another Thus the quantity of information is the quantity of elementary choices Classical information refers to finite sets Quantum information can be interpreted as the corresponding generalization referring to infinite sets Further conclusions:: Further conclusions: The unit of classical information, i.e. a bit, means the elementary choice between two equiprobable alternatives Analogically, the unit of quantum information, i.e a quantum bit (a qubit ) means the elementary choice between an infinite set of alternatives Quantum information is equivalent to Hilbert space. Any point in it, i.e. a wave function, is a value of quantum information Ultimate conclusions:: Ultimate conclusions: Quantum information unifies: The material and ideal world The concrete and abstract objects Physics and mathematics Thus quantum information is the fundamental substance of anything, which is being So, quantum information is correlative to time in a philosophical sense: It allows of understanding how time underlies being References I:: References I: Aspect, Alain & Grangier , Philippe & Roger , Gérard ( 1981) “Experimental Tests of Realistic Local Theories via Bell’s Theorem,” Physical Review Letters 47(7): 460-463. Aspect, Alain & Grangier , Philippe & Roger, Gérard (1982 ) “Experimental Realization of Einstein- Podolsky -Rosen- Bohm Gedanken Experiment: A New Violation of Bell’s Inequalities,” Physical Review Letters 49(2): 91-94 . Banach, Stefan and Alfred Tarski (1924) “Sur la decomposition des ensembles de points en parties respectivement congruentes,” Fundamenta Mathematicae 6, (1): 244-277. Bell, John (1964) “On the Einstein ‒ Podolsky ‒ Rosen paradox,” Physics (New York ) 1(3): 195-200 . Boolos , George (1987) “The Consistency of Frege's Foundations of Arithmetic,” in : Thomson, J. (ed .) On Beings and Sayings: Essays in Honor of Richard Cartwright . Cambridge, MA: MIT Press, pp. 3-20. Broglie, Louis de (1925) “ Recherches sur la théorie des quanta,” Annales de Physique (Paris, 10-ème série ) 3: 22-128 . Cantor, Georg (1897) “ Beitrage zur Begrundung der transfiniten Mengenlehre ( Zweiter Artikel ),” Mathematische Annalen 49(2): 207-246 . Deutsch, David (1985) “Quantum theory, the Church-Turing principle and the universal quantum computer,” Proceedings of the Royal Society of London A 400: 97-117. Deutsch, David (1989) “Quantum computational networks,” Proceedings of the Royal Society of London , Volume A 425: 73-90. Einstein, Albert & Podolsky , Boris & Rosen, Nathan (1935) “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” Physical Review , 47(10): 777-780 . Gödel , Kurt (1931) “Über formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I,“ Monatshefte der Mathematik und Physik 38(1 ): 173-198 . Heidegger, Martin (1927) Sein und Zeit . Halle: Max Niemeyer. Kochen , Simon and Specker , Ernst (1968) “The problem of hidden variables in quantum mechanics,” Journal of Mathematics and Mechanics 17(1): 59-87. References II:: References II: Linnebo , Øystein (2010) “Pluralities and Sets,” Journal of Philosophy 107(3): 144-164. Neumann , Johan von (1923) “ Zur Einführung der trasfiniten Zahlen ,” Acta litterarum ac scientiarum Ragiae Universitatis Hungaricae Francisco- Josephinae , Sectio scientiarum mathematicarum 1(4): 199–208. Neumann, Johan. von (1932 ) Mathematische Grundlagen der Quantenmechanik . Berlin: Springer, pp. 167-173 (Chapter IV.2). Popper, Karl. (1935) Logik der forschung : zur erkenntnistheorie der modernen naturwissenschaft . Wien: Springer . Schrödinger , E (1935) “Die gegenwärtige situation in der Quantenmechanik ”, Die Naturwissenschaften 23(48), 807-812; 23(49), 823-828, 23(50), 844-849. Skolem , Thoralf (1922) “ Einige Bemerkungen zur axiomatischen Begründung der Mengenlehre . ‒ In: T. Skolem ,” in Selected works in logic (ed. E. Fenstad ), Oslo: Univforlaget (1970). Turing, Allen (1937) “On computable numbers, with an application to the Entscheidungsproblem ,” Proceedings of London Mathematical Society , series 2 42(1 ): 230-265 . Whitehead, Alfred North and Russell, Bertrand. (any edition) Principia Mathematica , Vol. 2(*153), Vol. 3(*251 ). Yao, Andrew (1993). “Quantum circuit complexity,” in Proceedings of the 34th Annual Symposium on Foundations of Computer Science , pp. 352–361 Zermelo , Ernst (1904) “ Beweis , dass jede Menge wohlgeordnet werden kann ,” Mathematische Annalen 59(4 ): 514–16. Zermelo , Ernst (1908) “ Untersuchungen über die Grundlagen der Mengenlehre I,” Mathematische Annalen 65(2): 261-281. PowerPoint Presentation: Muito obrigado pela sua atenção ! PowerPoint Presentation: Congratulamo-nos com suas perguntas e comentários !