Download THE ALMOST IMPOSSIBLE WORLDS IN QUANTUM INFORMATION

THE ALMOST IMPOSSIBLE WORLDS
IN QUANTUM INFORMATION
And their Influence on Reality
Vasil Penchev
[email protected], [email protected]
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:
𝟏 𝒒𝒖𝒃𝒊𝒕 = 𝜶|𝟎 + 𝜷|𝟏 , where 𝜶, 𝜷 are complex
numbers such that: 𝜶 𝟐 + 𝒃 𝟐 = 𝟏, and |𝟎 , |𝟏 are
two orthogonal subspaces of Hilbert space normable
to two orthonormal bases, e.g. two successive “axes”
such as: 𝒆𝒊𝒏𝝎 , 𝒆𝒊(𝒏+𝟏)𝝎
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 |𝟎 , |𝟏
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:
𝒃
𝟎
𝒂
∞
𝟎
𝒂
𝒙 𝒅𝒙 = 𝟎 or
𝒙 𝒅𝒙 > 𝟎, where 𝟎 𝒙 is a
probability density distribution such that:
∀𝒙: 𝒍𝒊𝒎 𝟎 𝒙 = 𝟎, 𝒂, 𝒃 is finite, and [𝒂, ∞) is
infinite: 𝒙 symbolizes an almost impossible world from
a compact set of ones, and 𝟎 𝒙 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:
𝒏𝒐𝒓𝒎𝒂𝒍𝒊𝒛𝒂𝒕𝒊𝒐𝒏
Spacetime
𝒅𝒆𝒐𝒓𝒅𝒆𝒓𝒊𝒏𝒈
𝒘𝒆𝒍𝒍−𝒐𝒓𝒅𝒆𝒓𝒊𝒏𝒈
probability
density
distribution
𝒅𝒆𝒏𝒐𝒓𝒎𝒂𝒍𝒊𝒛𝒂𝒕𝒊𝒐𝒏
Quantum information
[qubits]
Probability
density
distribution
"∞. 𝟎" = "𝟎. ∞"
Well-ordering
+
“Cure”:
Denormalization
Expanding space-time ("𝟎. ∞")
All quite impossible worlds ("∞. 𝟎")
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 
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 Bell’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
• 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 ℍ𝟏 , and the states of the other share
another one ℍ𝟐 :
• However that of the system of both ℍ cannot
be factorized to those of ℍ𝟏 and ℍ𝟐 . That is:
ℍ ≠ ℍ𝟏 ⨂ ℍ 𝟐
All almost
impossible
worlds
World 2
Some
other
quantum
system
State 2
ℍ
ℍ𝟐
ℍ𝟏
ℍ ≠ ℍ𝟏 ⨂ ℍ𝟐
The
common
quantum
system
of all
All almost
impossible states
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
Probability density distributions
Some
other
quantum
system
ℍ𝟐
A2(x2):
Some quantity (A2)
of the system
in some point (x2)
ℍ𝟏
The
common
quantum
system
of all
The conjugate (A1)
in some point (x1)
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
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 “socialism” 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
• 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
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
• The being of such a world should be
denominated as collective or holistic rather
than individual or separate
• Any almost impossible world considered
separately is quite impossible and thus it has
not any individual being
• 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.