Beyond and across space: entanglement

Beyond and across space:
From space to entanglement
and back

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

III Interpretations
III Interprettaions

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

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

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

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

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

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

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

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

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

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

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

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

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

The service of space
pace in turn is what makes possible choice and thus
the transformation of future into the past
pace unlike Hilbert space refers only to the present
pace unlike Hilbert space represents any motion
only continuously
evertheless space and Hilbert space are
topologically equivalent by virtue of the Poincaré
conjecture proved by Grigori Perelman (2002;2003)
n fact, this is implied by that Hilbert space is the
mathematical structure unifying the description of
both continuous (even smooth) and discrete motion

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

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

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

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

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

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

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

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

References
(quantum mechanics):
spect, A, Grangier, P., Roger, G. (1981) Experimental
tests of realistic local theories via Bell’s theorem.
Physical Review Letters, 47(7), 460-463.
spect, A, Grangier, P., Roger, G. (1982) Experimental
Realization of Einstein-Podolsky-Rosen-Bohm
Gedanken Experiment: A New Violation of Bell’s
Inequalities. Physical Review Letters, 49(2), 91-94.
ell, J. (1964) On the Einstein ‒ Podolsky ‒ Rosen
paradox. Physics (New York), 1 (3), 195-200.
roglie, L. de (1925) Recherches sur la théorie des quanta
(Researches on the quantum theory), Thesis (Paris), 1924.
Annales de Physique (Paris, 10-ème série) 3, 22-128.

References
(quantum mechanics):
instein, A., Podolsky, B., Rosen, N. (1935) Can
Quantum-Mechanical Description of Physical Reality
Be Considered Complete? Physical Review, 47 (10),
777-780.
ochen, S. and E. Specker (1968) The problem of
hidden variables in quantum mechanics. Journal of
Mathematics and Mechanics, 17 (1): 59-87.
eumann, J. von (1932) Mathematische Grundlagen
der Quantenmechanik, Berlin: Verlag von Julius
Springer.
chrödinger, E. (1935) Die gegenwärtige situation in
der Quantenmechanik. Die Naturwissenschaften,
23(48), 807-812; 23(49), 823-828; 23(50), 844-849.

instein, A. Theorie der Opaleszenz von homogenen
Flüssigkeiten und Flüssigkeitsgemischen in der Nähe des
kritischen Zustandes. Annalen der Physik (Leipzig) 33: 1275–1298
(1910).
instein, A. Zum gegenwärtigen Stand des Strahlungsproblems.
Physikalische Zeitschrift 10: 185–193 (1909).
aubold, H. J. A. M. Mathai, R. K. Saxena. Boltzmann-Gibbs
Entropy Versus Tsallis Entropy: Recent Contributions to
Resolving the Argument of Einstein Concerning “Neither Herr
Boltzmann nor Herr Planck has Given a Definition of W”?
Astrophysics and Space Science, 290(3-4): 241-245 (2004).
enchev, V. A Philosophical View on the Introduction of
Negative and Complex Probability in Quantum Information.
Philosophical Alternatives, 2012(1): 63-78.
References
(einsTein’s Thermodynamics):

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

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Beyond and across space: entanglement

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