THE ALMOST IMPOSSIBLE WORLDS IN QUANTUM INFORMATION

1. THE ALMOST IMPOSSIBLE WORLDS IN QUANTUM INFORMATION And their Influence on Reality

2. VasilPenchev vasildinev@gmail.com, vaspench@abv.bg 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

3. Worlds and the states of a quantum system • The many-worlds interpretation (Hugh Everett III, John Wheeler, Bryce DeWitt) of quantum mechanics can identify a possible state of a quantum system with a possible world one-to-one. Both can be considered as possible states of the universe • The world or the universe can be considered both as a coherent superposition of possible states and as a statistical ensemble of possible worlds, one of which the measurement choices randomly as a principle

4. The exact thesis of the many-worlds interpretation: Hilbert space implies: state = world “Hidden variables?” ??? No! The coherent superposition of unorderable states The well-ordered statistical ensemble of “worlds” Measurement Choice Hilbert space

5. The possibility and probability of a state or world • The quantity of the possibility of a state or world is the probability determined by the corresponding wave function associated with the state or with the world according to the many-worlds interpretation • Thus possibility is thought in the way of probability, i.e. both as a holistic whole of possibilities (a coherent state) and as a choice among an ensemble of possible worlds

6. The almost impossible worlds (states) • A state or world with converging to zero, but nonzero probability, can be designated as an almost impossible one unlike those with exactly zero probability, which are quite impossible • Thus a coherent superposition of those almost impossible states or an ensemble of those almost impossible worlds can have some nonzero probability

7. World and quantum state • The identification of ‘world’ and ‘quantum state’ can be generalized from the many-worlds interpretation to quantum mechanics at all thus: • Quantum mechanics does not differ any coherent superposition from any corresponding ensemble (set) both being described by one and the same mathematical formalism, that of Hilbert space • That is: It identifies any “much” with a one-to-one defined “many”. Those are the “much” of coherent states and the “many” of possible worlds in the case in question

8. The exact thesis of the “much – many” interpretation Hilbert space implies a common measure of “much” and “many” and thus “state = World” The unit of that common measure is a qubit: That is: , where are complex numbers such that: , and are two orthogonal subspaces of Hilbert space normable to two orthonormal bases, e.g. two successive “axes” such as:

9. The idea of quantum invariance • That identification of “much” and “many” originating from the common mathematical formalism of Hilbert space can be generalized as a special principle, that of quantum invariance • It involves the axiom of choice to guarantee a well-ordering of any quantum “much” leading to the corresponding “many” • Nevertheless the initial “much” excludes any well-ordering and thus the axiom of choice because of the absence of “hidden variables” (Neumann 1932; Kochen, Specker 1968)

10. A qubit (quantum bit) is the unit of the generalized quantum information: A qubit is isomorphic to a unit 3D ball, where are represented as any two orthogonal great circles The unit of information is a choice Classical information 1 bit = 1 binary (or any finite) choice Quantum information 1 qubit = 1 infinite choice 0 needs needn’t 1 The axiom of choice

11. The measurement between a world and a quantum state • The experimentally verifiable part of quantum mechanics cannot differ ‘world’ with probability one after measuring from ‘state’ with probability less than one before measuring • The world being randomly chosen among many ones absorbs all probability and thus possibility of them to acquire reality • The measurement is both the boundary and mediator between “virtuality” (as a whole of possibilities) and the reality of one among all

12. Measurement is a mapping of the measured into a well-ordering of results Choice The axiom of choice Measurement Well-ordering Information =Measurement A unit of information = A unit of measurement = = 1qubit 1 quantum measurement

13. Coherent superposition vs. statistical ensemble • Consequently quantum mechanics underlies and requires the identification of the coherent superposition of ‘states’ before measuring with the statistical ensemble of ‘worlds’ after measuring • The so-called many-worlds interpretation is not more than an example of a much more general principle contradicting the prejudices of “common sense” • That general principle called quantum invariance delivers the identity of coherent superposition and statistical ensemble in particular

14. Quantum information implies: “No hidden variables in quantum mechanics!” “Hidden variables” means: “Hidden variables” order in secret the coherent super- position of states before measurement The well-ordered statistical ensemble after measurement implies And thus: not true The mapping of “much” into “many” is an identity and that of a coherent state into a statistical ensemble as well However quantum information or measurement is just that mapping turning out to be trivial, empty, or “transparent”

15. Two kinds of impossible worlds • One can define ‘impossible world’ in quantum mechanics as a state of a quantum system having zero or converging to zero probability • Consequently one can distinguish two kinds of impossible worlds in quantum mechanics: quite impossible or almost impossible ones accordingly • For the absorption of possibilities (probabilities) they differ from each other in their influence on reality or in their capability to turn out transformed into reality: The latters unlike the formers have potency to became reality

16. Content logical contradictions Do not content logical contradictions The probability of a single quite impossible world to occur is exactly zero The probability of a single almost impossible world to occur converges to zero The probability of any collections of those is exactly zero, too The probability of big enough collections of those can be nonzero Their impossibility is logically necessary The impossibility of an almost impossible world is empirical: It occurs with zero probability: However this is a fact but not a logical corollary Their impossibility is “a priori” Their impossibility separately is “a posteriori” Quite impossible worlds Almost impossible worlds

17. The infinite set of impossible worlds • If the latter is the case, a consisting of those states set of infinite measure can have a finite nonzero probability There are two cases according to the corresponding integration ranges for compact sets. For example: or , where is a probability density distribution such that: is finite, and is infinite: symbolizes an almost impossible world from a compact set of ones, and is its probability density ‘almost zero’ for any world of those is ‘almost impossible’

18. Can reality consist of all quite impossible worlds? What happens with the quite impossible worlds then? They constitute ... reality caged in space-time: Indeed the well-ordering of time (or space-time) disciplines the quite impossible worlds curing them of contradictions so: The theses and antitheses of the contradictions are divided in different periods of time and that world is cured of the perfect impossibility: Thus it has become a “righteous citizen” of the universe, an almost impossible one

19. Relativity of the worlds One can easily obtain reality caged in space-time and all almost impossible worlds free of space-time thus: probability density distribution Space- time Quantum information [qubits]

20. Caging in space-time generates the non-zero probability density of reality Probability density distribution Quantum invariance Minkowski space Hilbert space + Well-ordering “Cure”: Denormalization All almost impossible worlds () Expanding space-time () All quite impossible worlds ()

21. The occurrence of the impossible • One of those “almost impossible” states will happen by the probability of the whole set after measuring the quantum system • One can utilize the metaphor of “jackpot” for measurement: All possible worlds being even almost impossible participate in the lottery “raffling reality”. Any possible world can win and turn into the real one according to the number of its “tickets” (its value of probability density), but even those “without money for tickets” wishing to participate and keeping the laws (i.e. the almost impossible worlds) have some chance to win

22. The winner in the quantum lottery of reality Needs The axiom of choice The possible worlds “with tickets”: Their chance separately to win is zero but proportional to the ratio of the numbers of “tickets” of each other The almost impossible worlds “without tickets”: Their chance collectively to win can be nonzero, and proportional to the number of unsold tickets The probability density distribution Converging to zero probability density Nonzero probability density Almost impossible worlds (a continuum) Possible worlds (a continuum)

23. The metamorphosis of an almost impossible world into a real one • Consequently that measurement can turn an almost impossible world into a real one • One can continue the metaphor: If any possible world is an individual participant, any almost impossible world is a collective sharer among an infinite number of ones. Its collective partakes as an individual participator. If that collective participator wins, a “secondary lottery” within the “primary one” will determine a single almost impossible world as the winner of the jackpot of reality

24. The democratic constitution of the universe: The universe is dominated by poor by honest (keeping the laws) “citizens” (the almost impossible worlds). The breaches the laws (the quite impossible worlds) are being removed automatically, deprived of any chance or ... constitute all possibleworlds

25. An example: tunnel junction • Tunnel junction is a phenomenon, which can illustrate this: • According to classical mechanics no particle can jump out of a potential well. Tunnel junction in quantum mechanics means that a quantum “particle” can do it with nonzero probability • Any state of that “particle”, which is out of the well, is almost impossible but such an almost impossible world can win reality after the measurement for the collective sited out of the well possesses a definite nonzero probability according the laws of quantum mechanics

26. The integral probability for the thing to be within the well < 1, e.g. =0,7 Energy Probability density The integral probability for the thing to be out of the well >0, e.g. =0,3 Almost impossible worlds A quantum anything, e.g. an electron A potential well Tunnel junction Position

27. The explanation of tunnel junction • The prerequisite for it to happen is the measured state to belong to a set of infinite measure: • It is implemented for any state out of the potential well according to quantum mechanics: • All probability within the well is rather less than one as to quantum particles commeasurable with the Planck constant, and a big enough ensemble of measurements can verify that “positions” out of the well occur

28. The integral probability for the thing to be within the well < 1, e.g. =0,7 Energy =1 Probability density The integral probability for the thing to be out of the well >0, e.g. =0,3 =0 (No) almost impossible worlds A classical body All classical is actual and ... grey  Position

29. (No) tunnel junction in the macroscopic world • Nothing like this can observed in the macroscopic world: no “unsold tickets” of the lottery of reality: • The probability of a macroscopic body to leap over the potential barrier is too small to be recorded experimentally • However special phenomena such as those in tunnel diodes or superconductors are based on the tunnel junction of the physical quantities in macroscopic bodies

30. Entanglement and quantum information • Quantum information is that part of quantum mechanics which studies the phenomena of entanglement • Entanglement is defined strictly as the relation of any Hilbert space to those Hilbert spaces, of which it cannot be a tensor product: • Since any quantum system and its subsystems correspond to some Hilbert spaces, entanglement can be interpreted physically in terms of those and it has originated historically from quantum mechanics

31. Classical correlations The violation of Bell’s inequalities World 1 World 2 A sufficient condition for: Quantum correlations Entanglement State 2 State 1 Quantum system 1 Quantum system 2

32. Entanglement World 1 World 2 A double interaction: both of worlds and states Quantum system 2 Quantum system 1 State 2 State 1 Each qubit is for two identical qubits: the one for the world, the other for the state Simultaneous correlation ofa pair of conjugates

33. The entanglement of a set of almost impossible states • A set of almost impossible states having nonzero common probability is entangled with another of any probability distribution: • Those almost impossible states share a Hilbert space , and the states of the other share another one : • However that of the system of both cannot be factorized to those of and . That is:

34. All almost impossible worlds World 2 The common quantum system of all Some other quantum system State 2 All almost impossible states

35. The deformation of probability distribution for entanglement • Entanglement will cause some restricting deformation of the probability density distribution • This is equivalent to the action of some physical force or to the interaction with some physical body or radiation correspondingly with some nonzero mass at rest or energy • The above case of entangled almost impossible states (worlds) can cause the same effect on the other entangled system as the action of some unknown force or interaction

36. Probability density distributions Some other quantum system The common quantum system of all The entangled quantity A12(x2) behaves as in some potential field A2(x2): Some quantity (A2) of the system in some point (x2) The conjugate (A1) in some point (x1)

37. The entangled almost impossible worlds as a physical force • Consequently the entangled almost impossible worlds can act as a physical force or interaction determining reality: • Any of those worlds is impossible separately: Nevertheless a nonzero-measure ensemble of them acts on reality and can be even the most significant force or interaction: • One can say figuratively that the impossible can be what determines the actual more than the actual itself or the possible

38. Five scenarios of how one can distribute the entire probability of reality Probability density distribution Scenarios Reality Irreality 0% 100% “Tyranny” (in classical physics) “Oligarchy” 100% 0% 100% 0% “Democracy”(in quantum mechanics) 100% “Socialism” 0% 0% 100% “Communism”

39. Entanglement and Lorentz invariance • Entanglement is not Lorentz invariant: • This allows the quantum correlation of any region within the light cone with any other sited outside of it (defined according to the special relativity) • Thus it can act at a distance, i.e. at any even at infinite distance. • (By the way Einstein rejecting it called it “spooky” being able to be due only to “God’s gambling in dice”. Nevertheless the experiments prove that is the case)

40. Space-time Entanglement depends only on the nonorthogonality of any Hilbert spaces associated with any quantum things anywhere Minkowskispace One might say that entanglement is implemented by the mediation of the universe as a whole Entanglement does not depend on the distance Entanglement is action at any distance (i.e. that action at a distance in classical physics)

41. The “dark” action of entanglement • It can act beyond the visible universe remains “dark” as “dark matter” or “dark energy” • The visible universe is the region of space-time and a segment of the light cone. It can be thought as a potential well having an infinite barrier for anything with nonzero energy within the light cone • Nevertheless the laws of quantum mechanics allow the tunnel junction across the light barrier and thus all region outside of it (i.e. outside of space-time) is a collection of almost (but not quite) impossible worlds

42. Space-time = an infinite potential well Entanglement ⇒Tunnel junction outside of it ⇒ Minkowski space Probability density distribution The universe as a heart of entanglement All possible worlds All almost impossible worlds

43. Space-time = The visible universe=???=Reality? Out of the space-time = The dark universe Minkowskispace Probability density distribution If “Yes!”: The “socialism” scenario is true 96% of the universe are dark! 4% are visible!

44. Entanglement as an explanation of “dark” phenomena It is a possible explanation for these mysterious phenomena discovered recentlyin physics and designated correspondingly as: • Dark matter: the matter of our galaxy, the Milky way, should be rather more than visible one not to break into parts due to the centrifugal forces according to its rotation speed • Dark energy: the energy necessary to maintain the observed acceleration of the universe expansion Both can be due to the entanglement with the region outside of the light cone

45. Probability density distribution Stage 4: Being The visible universe The dark universe “Space-time for the breaches!” Entanglement: God’s Love Stage 3: Jail for the breaches “Let there be the light cone!” 4% of breaches 96%OF RIGHTEOUS WORLDS Stage 2: The laws “Mathematics!” Stage 1: Nothing The Genesis of Being

46. Who acts: the whole or its element? • Furthermore, the action of almost impossible worlds cannot be ascribed to any separate entities but only to a whole of such ones • The action of any single impossible world is practically zero for its probability converges to zero • Nevertheless the integral action of all impossible worlds, to which that belongs, is finite • Consequently only the whole influences rather than any single element of it

47. The set is exactly equal to the sum of its elements A sum The interactions between the elements are exactly zero The set is more than the sum of its elements A system The interactions between the elements are more than zero A wholeness The sum of the elements is exactly zero and any element as well The set is exactly equal to the interactions between the elements. Mathematically it is an set of empty sets. Physically it is an entangled vacuum. Logically it is a set of almost impossible worlds It can be also interpreted as an empty set with some nonzero possibility to be chosen: Indeed, can an empty set be chosen due to the axiom of choice? Obviously, it should be able to

48. About the sense for an empty setto be chosen • The interpretation of the axiom of choice thus that an empty set can be chosen is logically consistent • The sense: You go shopping and buy that, that, that. All of them are chosen by you You go shopping and buy nothing. You have chosen an empty set A probability of that to happen can be assigned in both cases

49. The idea of holistic action • The almost impossible worlds can act only holistically and be discovered only by means of their effects indirectly remaining “dark” or “hidden” in principle • The almost impossible worlds can be thought as some holistic being, which is only as a whole, but not as a collection or as a set of any nonzero elements: • In that sense it can be thought as a set without elements and represented somehow in terms of set theory as an empty set, to which is assigned any finite probability of that to be chosen • On the base of that probability, its analog in quantum mechanics can act physically

50. h - the Planck constant t – time, ν- frequency • Their wholeness: • Probability = P≠0 • Energy = ??? Any element of entangled nothing: Probability → 0 Energy = 0 However the interpretation of is quite “dark” isQ, quantum information, i.e. the quantity of “much” (for ‘probability’) per a unit of “many” (for ‘time’) = qubits