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I show any theory assuming state of an object, or even object’s existence, will be at odds with empirical evidence. I discuss QM relation with special relativity (SR). I argue all paradoxes are artifacts of factitious assumptions
Where unfathomable begins
Sergei Viznyuk
I show any theory assuming state of an object, or even object’s existence,
will be at odds with empirical evidence. I discuss QM relation with special
relativity (SR). I argue all paradoxes are artifacts of factitious assumptions
A dogmatic realist [1] would believe, at least on subconscious level, that information extracted
by the measurement pertains to an entity extraneous to that information, i.e., to some measured
object, which exists “out there”, beyond the tip of our noses, whether we measure it or not. Such
line of thought can only be a belief, since any attempt to prove the existence of extraneous entity
would have to attribute obtained information to that extraneous entity, i.e., the proof would involve
circular reasoning. Here I show such beliefs would also contradict some experiments
. Generally,
for any belief or theory, not equivalent to already existing objective facts, one can obtain empirical
evidence contrary to that belief. This principle is demonstrated in a double-slit experiment.
The measured by device expectation value of observable is given by Born rule:
 
, where density matrix represents information about state; is the operator matrix of observable
the device measures. The objectivity [2], signified by independence of objective facts on observer
[basis], dictates the classical information, such as event probabilities, is conserved upon observer
basis transformation. The unitarity, imposed [2] by objectivity, leads to conservation of quantum
, as manifested by no-hiding theorem [3]. Thus, without new measurement, quantum
and classical information are separately conserved. With measurement, the conserved property can
only be the sum of quantum and classical information, as measurement transforms quantum
information into classical
Consider a case when is the only device which performs measurement. If we theorize about
state of the object, the outcome of the theory may be an additional information, beyond what is
produced by device . This additional information is accounted for by density matrix . It
would lead to a different expectation value , i.e., the theory would generally contradict
experiment where device is the only source of information.
Consider a measurement in cardinality basis, i.e., a measurement of a qubit. The
measured by device expectation value is:
   
It was shown, e.g., that the assumption about radiation existing “out there”, in the open space, does not allow self -
consistent derivation of Planck’s radiation formula [31]. It also leads to zero-point energy paradox [33]: the gravity
from all zero-point energy modes would exceed the observed gravity by at least 58 orders of magnitude [32]
The term “quantum information” is widely used [35, 15, 3], but with no clear definition in sight. It appears a common
practice to write papers mentioning the term dozens of times, and not to bother defining it. I define quantum
information as the potential information, which can be converted into real, i.e., classical information, by the
measurement; the measurement being defined [5] as extraction of classical information
The distinction between quantum and classical information is the base of Bohr’s complementarity principle [30].
The wave-like behavior, associated with unitary transformation of density matrix, is said to complement the particle-
like outcomes of measurement events, delineating the boundary between quantum and classical physics [18, 2]
In order to have a room for conjecture, we deliberately choose device’ basis so it does not resolve
input states , i.e., . If we theorize that input states , correlate respectively with
states , of the object
, the predicted expectation value [4, 5] is different from :
   
The difference between and is especially pronounced if conjectured object states are
orthogonal: . This is the case when, e.g., input basis states  are chosen to correlate
respectively with the state of object’s existence , and non-existence .
The expectation value would be correct if and were not just conjectured object states,
but outcomes of an actual measurement
, performed in addition to the measurement by device .
The double-slit experiment is the canonical setup to confirm the above conclusion. Double-slit
generates a spatial qubit [6]. The device is the screen behind the slits. The input basis state is
that of a particle passing through left slit, and basis state is that of a particle passing through
right slit. In the absence of additional measurement, device measures expectation value . It
exhibits characteristic interference pattern due to   term. The expectation
would contradict any theory ascertaining the slit particle passed through, and, generally, any
theory whose predictions extend beyond information obtained
by device .
If, however, an additional measurement is performed, whose output correlates with state ,
and output correlates with state , the measured expectation value at the screen is given by .
The interference pattern is affected by term . A textbook example of a double-slit experiment
would include an additional measurement at the slits, to determine which slit particle passed
through. If measurement at the slits is accurate, the output states are orthogonal: , and
no interference pattern at the screen is observed [7, 8].
The expression goes beyond the case of interference decay. It may also involve a shift in
interference pattern, given , by ; combined with decay, if .
If  , no information is extracted. In this case just undergoes unitary transformation
The amount of quantum information, contained in state , which can be extracted per single
measurement event, in a limit of infinite size event sample, is evaluated using Von Neumann
entropy [9] as [10, 2]:
  
, where is the cardinality of measurement basis. The entropy of is, therefore, the amount of
already extracted information, per event. A known density matrix means the measurement has
been performed, either by preparation or by measuring device. For the finite size event sample, the
amount of extracted information is Boltzmann’s entropy , where is the
statistical weight of the sample [11].
Such correlation has earned a popular, albeit not informative name: entanglement
This expounds the falsehood of so-called Wigner’s friend paradox [28]. According to , the measurement
performed by Wigner’s friend, i.e., the measurement of , affects Wigner’s measurement of . The information
extracted by Wigner’s friend reduces the amount of information available for extraction by Wigner
The proof is rather trivial, as prediction of the theory can be mapped into , outcomes, leading to expectation
The shift in interference pattern here is a type of Aharonov-Bohm effect [29, 27]
For cardinality basis, is expressed in terms of the length of Bloch vector as [12]:
 
, where  
From ,, it follows, , or  are single parameters defining the amount of quantum
information in qubit. With measurement having output states ,  changes as
  
The parameter  equates to a coefficient of determination in statistics, as
a measure of how much the measurement of predetermines the measurement of .
The dependence of on product of entangled ancilla states creates an impression the
remote ancilla instantaneously affects the results of local measurement by device . There is
nothing in expression that prohibits instantaneous effect of the measurement of on the
measurement of . In varying forms, the expression is the root of continuing claims of QM
non-locality [13], and part of the problem reconciling QM with special relativity
. I reproduce the
issue in a thought experiment below.
Consider an experiment in which pairs  of left (L), and right (R) circular polarization-
entangled photons [14] are generated, by passing UV laser beam through SPDC crystal (Figure 1).
The polarizing beam splitters (PBS) separate vertical (V) and horizontal (H) polarizations down
two paths: for photon , and for photon . The paths are registered by
separate detectors, which I summarily reference as device . Path passes through  
which makes polarization of same as . The paths converge on screen to form
interference pattern.
Another part is the infinitely sharp boundary between the region of simultaneousness, in which no action could be
transmitted, and other regions, in which direct action from event to event could take place. Since an infinitely sharp
boundary means an infinite accuracy with respect to position in space and time, the momenta or energies must be
completely undetermined, or in fact arbitrarily high momenta and energies must occur with overwhelming probability.
Therefore, any theory which tries to fulfill the requirements of both special relativity and quantum theory will lead to
mathematical inconsistencies, to divergencies in the region of very high energies and momenta [1]
Figure 1
 
By virtue of entanglement, a photon registered at or points to the photon being in a
reciprocal or path, akin to detecting the slit particle passed through in a double-slit
experiment. The accurate detection of entangled photon at would lead to disappearance of
interference pattern at the screen . By modulating PBS in path , the experimenter can
communicate with observer , seemingly in violation of no-signaling theorem [15, 16], and in
violation of special relativity, since the distance between and device can be arbitrary large.
Even more paradoxical is the situation when length is longer than . In this case, the photon
is detected after photon has been registered by . Yet, the detection of photon at would
affect the interference pattern at , i.e., the measurement in the future would affect the
measurement in the past. Thus, in one thought experiment we find at least three paradoxes:
1. violation of special relativity
2. violation of causality [17]
3. violation of no-signaling theorem [15]
And yet, as I show below, none of these paradoxes is real. As all paradoxes, they are artifacts of
factitious assumptions. Let’s disassemble them.
One obvious assumption we made was that there are photons traveling
from SPDC down two
paths . Anyone who understands the base QM principles would know the photon comes into
being only as a measurement event [18, 19]. Until measurement there is no photon. A way to
describe the situation before measurement, i.e., to describe the measurement setup, is by
representing it as superposition of correlated radiation modes; with mode defined as possibility of
certain measurement outcome. Such superposition is what is referred to as quantum state. The
setup on Figure 1 is described, in two measurement eigenbases, as
 
, where are circular left and right polarization modes; 
. The expectation at the screen is:
 
, where  characterizes PBS efficiency. Eq. is same as ,
where      
, and .
While QM formalism above accurately predicts expectation value, it is seemingly at odds with
special relativity (SR), with an illusion of instantaneous effect of the measurement by detectors
on . Since SR is entirely in classical domain, for resolution, we should look at classical
information produced by the measurement.
A single measurement event is one of eigenstates of measuring device. The associated
eigenvalue is the device reading. In a limit of infinite number of measurement events, the event
sample is described by projection of quantum state on eigenspace of measuring device:
, where subscript indicates, the measurement basis is that of device .
The very word traveling implies intermediate measurement events, i.e., a trajectory
Since device readings are real-valued classical parameters, the eigenvalues of device operator
are real, i.e., operator is Hermitian. Below I show, the hermiticity condition is sufficient for
the measurements to comply with SR. In order to ascertain this in experiment on Figure 1, we
transform measurement basis from that of device to that of device . I designate this
transformation as : ; . The existence of such transformation indicates,
devices , belong to the same measurement context. In new basis,  becomes: .
Therefore, device operator intertwines with operator as
 
The irreducible representation of Hermitian operators , is [20]:
 
 
, where ;  are Pauli matrices. Representation , indicates the
measurement has an associated spacetime 4-vector
. The non-trivial intertwist between devices
, is possible only if matrix determinant of  is zero. With ,, this condition is
The above equates to: 
Eq.  splits into four relations between 4-vector components , signifying
different causal possibilities, as illustrated on Figure 2, where arrow base is the potential cause,
and arrow head is the possible effect; ; . The causal possibilities, as seen by
observer , are indicated by positive values of parameters ,  under  . Since 4-vectors
, are relative to observer, the observer is a key element of causal relationships.
With observer at device , . In this case  becomes
. Hence, relative
to device , the measurement by device is separated by time interval equal to distance
between and ; with associated speed limit . Not only QM is not at odds with SR, in
fact, SR is imposed by classicality of measurement results, which is one of QM base concepts.
If  in , we have additional conditions:
Spacetime as an entity emerges as encoding structure for classical information extracted by the measurement [2]
Figure 2
, and 
, from where it follows,
. Therefore, if , transformation has to be unitary,
i.e., a transformation of observer basis. Unitary transformations conserve information and
trivially comply with SR by . Consequently, Schrödinger equation, being an expression of
parameter-driven unitary transformation [2], also complies with SR, namely with unitary subgroup
of Lorentz transformations. Generally, however, Lorentz transformations, specifically boosts, do
not conserve quantum information. In one form or another, they imply measurement, i.e.,
extraction of information. The boost involves acceleration from to . It is done through the
action of a classical force, always accompanied by decoherence [21]. An assumption that one can
reconcile extraction of information, implied by the boost transformation, with unitarity, leads to a
collection of paradoxes, see, e.g., Lecture IV in [22].
As for the causality violation paradox, note, that we can remove device from measurement
setup on Figure 1 by increasing length to infinity. In this case, the interference pattern at the
screen would re-appear. The interference pattern re-appears when difference in lengths and
increases above coherence length, corresponding to [de]coherence time [23], i.e., the time it takes
to perform measurement
. If devices and are within coherence region, one cannot predict
which detector clicks first: or one of . There is no causal order in coherence region,
reflected by the fact that , are interchangeable in . A causal order would mandate
additional information beyond what is implanted in quantum state . The amount of information
extracted by single measurement event, equals Boltzmann’s entropy of event sample
 
  . It means, a single event in either device would not change the amount
of information available for extraction by another device. Thus, there is no causality violation as
there is no causal order in measurement events to begin with. It can be proven experimentally by
placing separate detectors in paths . These detectors would register random events with
 
probability each. There is no way to tell from these events if there is any measurement done
by detectors
. This is the essence of no-signaling theorem [15].
The causality arises when amount of classical information extracted by one device reduces the
amount of information available for extraction by another device. From above paragraph, it
follows, the causal relationship can only be between event samples, not between individual events.
The change in amount of information available for extraction by device due to the
measurement by device can be used for communication between experimenter controlling
device , and observer of device . The experimenter at device can modulate the amount of
Taking  and , from  we obtain: 
To confirm or deny the causal order, the experimenter can register the click time of each detector and mark points
on the screen , where photon has hit, with time of the corresponding click of detectors , . These marked
points on screen can be separated into two groups: one group of points for which detector clicked first, and second
group of points for which one of detectors clicked first. There is a causal order if group of points for which
detector clicked first exhibits interference pattern, while second group of points shows no interference pattern
extracted information
and thus affect the interference pattern at the screen . The measurement
by two devices is subject to constraint , i.e., there is no superluminal causal relationship.
Exactly this type of communication is used in common radio transmission, which also utilizes
shared entangled state. Instead of different polarization modes as in , radio transmission is
based on entanglement between modes of different frequency, with device the transmitter, and
device the receiver. The interference pattern at device is by time, instead of spatial coordinate.
The measurement transforms all or part of quantum information, implanted in quantum state,
into classical information, thus reducing the quantum state. It is the classical information which is
real, not the quantum state, which we devised only as a way to describe the measurement setup
[5]. In fact, we cannot even describe the measurement setup without measurement, since any such
description would require classical information, such as density matrix elements, which can only
be obtained through measurement, the preparation of quantum state being a form of measurement.
The conclusions which follow from above discussion:
1. One cannot derive new information from already existing information. New information
(knowledge) can only arise from new experience
2. There could be no theory which explains all the existing facts. Having such a theory means
being able to obtain new information (explanation), from existing information (facts). A
theory can only explain subset of existing facts, by correlating portions of existing
information. The correlation logic, i.e., the theory itself, is part of existing information. If
theory explains facts, congruent to information domain (questions), the output is
information domain (answers), while theory itself is congruent to information domain
(transformation logic). These domains are parts of existing information .
This principle is realized in del’s Incompleteness Theorem [24], and in Turing’s Halting
Problem [25]
3. One cannot ascribe any level of reality to an object, even its existence, outside of
measurement. Such attribution would mean creating information without measurement, out
of nothing. The only thing real is the information extracted by the measurement
4. The information is physical. The extracted information, in amount of ,
is persisted in some encoded form, i.e., it is physicalized in an encoding structure, such as
spacetime [2]. Each qubit of information is specified by a real-valued 4-vector, such as 4-
vector of spacetime, or energy-momentum 4-vector, or other equivalent representation.
What observer sees and feels are bits of encoded information
5. The information, being synonymous to objective facts, is absolute. In Wigner experiment,
the information extracted by Wigner’s friend affects the measurement performed by
Wigner, even though Wigner and his friend are spacetime-separated. Wigner experiment
is a form of double-slit experiment, where measurement at the slits is by friend, and
measurement at the screen is by Wigner. It demonstrates the absoluteness of classical
information, as information extracted by friend reduces information available for extraction
The experimenter can, e.g., modulate PBS efficiency in path , or modulate length in and out of coherence region
by Wigner. The information extracted by Wigner and his friend is part of the same
spacetime structure
6. All paradoxes are artifacts of false assumptions. I expounded the falsehood of Wigner’s
friend paradox. Other paradoxes can also be easily disassembled. Perhaps one of the most
chewed on paradoxes is the so-called black hole information paradox [26]. The paradoxical
here is the apparent loss of information about object falling into black hole, assuming
unitary dynamics of the whole system. This “paradox” is the perfect example of a falsehood
built into very statement of the problem. The phrase “falling into black hole” implies
knowledge of object’s coordinates, which, in its turn, implies measurements extracting this
information. The very fact of a measurement contradicts the assumption of unitarity. As
object falls into black hole, all information about object gets extracted
by the time object
reaches event horizon. The observer will not see anything actually ending up in black hole
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