THE ALMOST IMPOSSIBLE WORLDS IN QUANTUM INFORMATION (And their Influence on Reality)

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

Vasil Penchev vasildinev@gmail.com, vaspench@abv.bg http://www.scribd.com/vasil7penchev http://www.wprdpress.com/vasil7penchev CV: http://old-philosophy.issk-bas.org/CV/cvpdf/V.Penchev-CV-eng.pdf

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-toone. 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

The exact thesis of the many-worlds interpretation:

The coherent The well-ordered superposition Measurement statistical ensemble Choice of unorderable of “worlds� states

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

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

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-toone defined “many”. Those are the “much” of coherent states and the “many” of possible worlds in the case in question

The exact thesis of the “much – many� interpretation

The unit of that common measure is a qubit: That is: đ?&#x;? đ?’’đ?’–đ?’ƒđ?’Šđ?’• = đ?œś|đ?&#x;Ž + đ?œˇ|đ?&#x;? , where đ?œś, đ?œˇ are complex numbers such that: đ?œś đ?&#x;? + đ?’ƒ đ?&#x;? = đ?&#x;?, and |đ?&#x;Ž , |đ?&#x;? are two orthogonal subspaces of Hilbert space normable to two orthonormal bases, e.g. two successive “axesâ€? such as: đ?’†đ?’Šđ?’?đ??Ž , đ?’†đ?’Š(đ?’?+đ?&#x;?)đ??Ž

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)

A qubit (quantum bit) is the unit of the generalized quantum information:

A qubit is isomorphic to a unit 3D ball, where |đ?&#x;Ž , |đ?&#x;? are represented as any two orthogonal great circles

Classical information 1 bit = 1 binary (or any finite) choice

0 1

Quantum information 1 qubit = 1 infinite choice

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

Measurement is a mapping of the measured into a well-ordering of results

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

Quantum information implies: “No hidden variables in quantum mechanics!”

“Hidden variables” means: The well-ordered statistical ensemble after measurement

“Hidden variables” order in secret the coherent superposition of states before measurement

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”

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

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

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: đ?’ƒ đ?&#x;Ž đ?’‚

∞ đ?&#x;Ž đ?’‚

đ?’™ đ?’…đ?’™ = đ?&#x;Ž or đ?’™ đ?’…đ?’™ > đ?&#x;Ž, where đ?&#x;Ž đ?’™ is a probability density distribution such that: ∀đ?’™: đ?’?đ?’Šđ?’Ž đ?&#x;Ž đ?’™ = đ?&#x;Ž, đ?’‚, đ?’ƒ is finite, and [đ?’‚, ∞) is infinite: đ?’™ symbolizes an almost impossible world from a compact set of ones, and đ?&#x;Ž đ?’™ is its probability density ‘almost zero’ for any world of those is ‘almost impossible’

Can reality consist of all quite impossible worlds?

What happens with the quite impossible worlds then? They constitute ... reality caged in spacetime: 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

Relativity of the worlds One can easily obtain reality caged in space-time and all almost impossible worlds free of space-time thus: đ?’?đ?’?đ?’“đ?’Žđ?’‚đ?’?đ?’Šđ?’›đ?’‚đ?’•đ?’Šđ?’?đ?’?

Spacetime

đ?’…đ?’†đ?’?đ?’“đ?’…đ?’†đ?’“đ?’Šđ?’?đ?’ˆ đ?’˜đ?’†đ?’?đ?’?−đ?’?đ?’“đ?’…đ?’†đ?’“đ?’Šđ?’?đ?’ˆ

probability density distribution

đ?’…đ?’†đ?’?đ?’?đ?’“đ?’Žđ?’‚đ?’?đ?’Šđ?’›đ?’‚đ?’•đ?’Šđ?’?đ?’?

Quantum information [qubits]

Probability density distribution

"∞. đ?&#x;Ž" = "đ?&#x;Ž. ∞" Well-ordering + “Cureâ€?: Denormalization Expanding space-time ("đ?&#x;Ž. ∞")

All quite impossible worlds ("∞. đ?&#x;Ž")

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

The winner in the quantum lottery of reality The possible worlds “with The almost impossible tickets”: Their chance worlds “without tickets”: separately to win is zero Their chance collectively to but proportional to the win can be nonzero, and ratio of the numbers of proportional to the number “tickets” of each other of unsold tickets The probability density distribution

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

The democratic constitution of the universe:

Reality Almost impossible worlds Slightly possible worlds Possible worlds Very possible worlds 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 possible worlds

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

The integral probability for the thing to be within the well < 1, e.g. = 0,7

Energy Probability density

Almost impossible worlds

The integral probability for the thing to be out of the well >0, e.g. = 0,3 A quantum anything, e.g. an electron

A potential well

Position

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

The integral probability for the thing to be within the well < 1, e.g. = 0,7

Energy Probability density

(No) almost impossible worlds

=1 The integral probability for the thing to be out of the well >0, e.g. = 0,3

=0

A classical body

All classical is actual and ... grey ď &#x160; Position

(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

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

Classical correlations The violation of Bellâ&#x20AC;&#x2122;s inequalities World 1 World 2 A sufficient condition for:

Quantum correlations

State 2

Quantum system 2

State 1

Quantum system 1

World 2

Quantum system 2

State 2

A double interaction: both of worlds and states

World 1

Quantum system 1

Each qubit is for two identical State 1 qubits: the one for the world, the other for the state Simultaneous correlation of a pair of conjugates

The entanglement of a set of almost impossible states â&#x20AC;˘ A set of almost impossible states having nonzero common probability is entangled with another of any probability distribution: â&#x20AC;˘ Those almost impossible states share a Hilbert space â&#x201E;?đ?&#x;? , and the states of the other share another one â&#x201E;?đ?&#x;? : â&#x20AC;˘ However that of the system of both â&#x201E;? cannot be factorized to those of â&#x201E;?đ?&#x;? and â&#x201E;?đ?&#x;? . That is: â&#x201E;? â&#x2030; â&#x201E;?đ?&#x;? â¨&#x201A; â&#x201E;? đ?&#x;?

All almost impossible worlds

World 2 Some other quantum system State 2

â&#x201E;?đ?&#x;?

â&#x201E;? â&#x201E;?đ?&#x;?

â&#x201E;? â&#x2030; â&#x201E;?đ?&#x;? â¨&#x201A; â&#x201E;?đ?&#x;?

The common quantum system of all

All almost impossible states

The deformation of probability distribution for entanglement â&#x20AC;˘ Entanglement will cause some restricting deformation of the probability density distribution â&#x20AC;˘ 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 â&#x20AC;˘ 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

Probability density distributions

Some other quantum system

â&#x201E;?đ?&#x;?

A2(x2): Some quantity (A2) of the system in some point (x2)

â&#x201E;?đ?&#x;?

The common quantum system of all

The conjugate (A1) in some point (x1)

The entangled almost impossible worlds as a physical force â&#x20AC;˘ Consequently the entangled almost impossible worlds can act as a physical force or interaction determining reality: â&#x20AC;˘ 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: â&#x20AC;˘ One can say figuratively that the impossible can be what determines the actual more than the actual itself or the possible

Reality 100%

“Tyranny” (in classical physics) “Oligarchy”

100% ≪100%

≫0%

“Democracy” (in quantum mechanics)

≪100%

“Socialism”

100%

“Communism”

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)

Space-time

Entanglement depends only on the nonorthogonality of any Hilbert spaces associated with any quantum things anywhere

Minkowski space 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)

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

Space-time = an infinite potential well

Minkowski space Probability density distribution

All almost impossible worlds

Space-time = The visible universe =???= Reality?

Minkowski space Probability density distribution

The â&#x20AC;&#x153;socialismâ&#x20AC;? scenario is true

Entanglement as an explanation of “dark” phenomena It is a possible explanation for these mysterious phenomena discovered recently in 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

Probability density distribution

The visible universe The dark universe

Stage 4: Being

Stage 3: Jail for the breaches Stage 2: The laws Stage 1: Nothing

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

The set is exactly equal to the sum of its elements The interactions between the elements are exactly zero The set is more than the sum of its elements The interactions between the elements are more than zero 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

About the sense for an empty set to be chosen â&#x20AC;˘ The interpretation of the axiom of choice thus that an empty set can be chosen is logically consistent â&#x20AC;˘ 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

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

h - the Planck constant t – time, ν- frequency Any element of entangled nothing:

However the interpretation of is quite dark

is , quantum information, i.e. the quantity of “much” (for ‘probability’) per a unit of “many” (for ‘time’) =

Thus: Its being is not actual since it is nothing However: It can act just as anything from reality Furthermore: It is physically the most power force forming reality

The modality of almost impossible worlds The being of the almost impossible worlds in quantum mechanics and information can be distinguished as a separate modality possessing unique features: • Taken separately, any almost impossible world neither exist nor can exist, but yet no prohibition to exist • Nevertheless it can turn out a real one after measurement thanks to the axiom of choice, and • It can impact reality in a sense

An almost impossible world can become a real one • Any almost impossible world can be transformed into a real one after measurement like a possible one though its probability is zero practically • However that kind of transformation requires it to be somehow chosen among a continuum of similar worlds and thus, the axiom of choice • Fortunately the axiom of choice and thus its choice is guaranteed by quantum invariance discussed above. Measurement is what implements the choice practically

An almost impossible world can act on reality • An almost impossible world can act on reality as an element of a set of infinite measure only holistically and thus remaining invisible, “dark” • One can distinguish an almost impossible world from a quite impossible world only logically after both cannot have any individual verification experimentally • Physically it means that an almost (unlike quite) impossible world is a bound nothing or nonbeing.

Its being consists in its infinite connectivity and can be physically interpreted as the entangled vacuum

‘Dark’ modality • Said metaphorically, that modality of an almost impossible world can be called ‘dark’ • It is an allusion to the concepts of “dark matter” or “dark energy” in physics recently: There it designates very powerful physical action originating from some missing or unknown source, as if from the physical nonbeing • Analogically the term of “dark modality” leads to that action, which comes from nonbeing or from the transcendent. Nevertheless it can be founded scientifically

Collective or holistic being â&#x20AC;˘ The being of such a world should be denominated as collective or holistic rather than individual or separate â&#x20AC;˘ Any almost impossible world considered separately is quite impossible and thus it has not any individual being â&#x20AC;˘ However it has that potency to unite with infinitely many other similar almost impossible worlds, which acquire a common collective being jointly

End

Thank you for your kind attention!

References: • DeWitt, Bryce and John Wheeler (eds.) 1968. The EverettWheeler Interpretation of Quantum Mechanics. Battelle Rencontres: 1967 Lectures in Mathematics and Physics.

New York: W.A.Benjamin.

• Everett III Hugh1957. „Relative state” Formulation of Quantum Mechanics,” Reviews of Modern Physics. Vol. 29, No 3 (July 1957), 454-462.

• Kochen, Simon and Ernst Specker 1968. “The problem of hidden variables in quantum mechanics,” Journal of Mathematics and Mechanics. 17 (1): 59-87.

• Neumann, Johan von 1932. Mathematische Grundlagen der Quantenmechanik, Berlin: Verlag von Julius Springer.