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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

1

. 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

information

2

, 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

3

.

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:

1

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]

2

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

3

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

4

, 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

5

, 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

6

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

7

.

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].

4

Such correlation has earned a popular, albeit not informative name: entanglement

5

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

6

The proof is rather trivial, as prediction of the theory can be mapped into , outcomes, leading to expectation

7

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

8

. 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

plate

which makes polarization of same as . The paths converge on screen to form

interference pattern.

8

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]

SPDC

Figure 1

PBS

PBS

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

9

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 .

9

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

10

. 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:

10

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

11

. 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

12

. 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

11

Taking and , from we obtain:

12

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

13

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 Gödel’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

13

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

14

by the time object

reaches event horizon. The observer will not see anything actually ending up in black hole

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14

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