Interaction-free information transmission by means of entangled photons
Interaction-free information transmission by means of entangled photons
By using a simple experiment, it is shown that the non-locality of quantum physics
permits an in principle instantaneous transmission of information without any
interaction between transmitter and receiver being necessary for this purpose.
Since no exchange of energy is required for the information transfer, the proposed
experiment is compatible with the special theory of relativity.
I.) Introduction
Within the context of classical physics, transmission of information is possible only
when energy is transmitted from the transmitter to the receiver, irrespective of how
the information is transmitted. Whether the information is written on a sheet of paper,
the latter is wound around a stone and the stone is thrown to the receiver, or light is
used for the information transmission, it is always necessary for energy to be
transmitted from the transmitter of the information to the receiver of the information.
Since, according to the special theory of relativity [1], the speed of light c has the
same value in all inertial systems and it is possible to accelerate material only to a
speed v, where v < c, within the context of classical physics information can thus be
transmitted only at a maximum of the speed of light.
However, it is not possible to conclude from this that information transmission is in
principle possible only at a maximum of the speed of light [2]. This can be looked at
simply. The special theory of relativity is based on two postulates: I.) All natural laws
are covariant with respect to the transformations between inertial systems. II.) The
speed of light c has the same value in all inertial systems. From the relativity
principle (Postulate I), it can merely be concluded that there must be a speed that is
invariant with respect to the permissible transformations between the inertial
systems. That this invariant speed, in which sense also always represents a basic
upper limit for an information transfer, cannot be derived from the relativity principle.
In addition, it is not possible to derive the practical value for this invariant speed from
the relativity principle. Only via Postulate II is this invariant speed assigned a
practical value – the speed of light c [3].
From the special theory of relativity, it is thus merely possible to derive the fact that
information which is transmitted by means of an energy transfer can be transferred at
a maximum of the speed of light. However, the fact that an information transfer is in
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Interaction-free information transmission by means of entangled photons
principle possible only at a maximum of the speed of light cannot be derived from the
special theory of relativity.
Quantum physics differs entirely in principle from classical physics. Within the
context of classical physics, a cause can be specified for each action. Within the
context of quantum physics, however, this is no longer the case. As early as 1935,
Einstein, Podolsky and Rosen (EPR) pointed out for the first time that quantum
systems can act on one another instantaneously over arbitrary distances and, within
the context of quantum physics, it is possible to specify neither a cause nor a
plausible explanation for these actions [4]. This non-locality, occurring within the
context of quantum physics, contradicts the locality concept of classical physics
(Einstein locality) [5]. Quantum physics thus cannot be traced back to a local theory
in the sense of the Einstein locality.
Since (EPR), the unspoken question is whether an instantaneous information transfer
is possible within the context of quantum physics. Since this question could not
previously be answered on the basis of general considerations, an experiment is
proposed below which, if successfully implemented, shows that this is possible.
The basis of the experiment described below is that, given sufficiently complex,
entangled quantum systems, there are a number of possibilities for a measurement
on a subsystem which change the basically accessible knowledge about the
entangled quantum system in a clearly distinguishable manner. This fact opens up
the possibility of a freely selectable, interaction-free state preparation of the quantum
system. A state analysis by means of beam splitters then permits interaction-free
information transmission.
II.) Interaction-free state preparation through measurement
An interaction-free state preparation through a measurement permits the
arrangement illustrated schematically in Fig. 1. The source (Q) is intended to
generate two linearly polarised photons that are identical in the energetic degree of
freedom simultaneously at freely selectable times. The two photons are intended to
be polarised at right angles to each other. The two possible polarisation directions
are designated by "H" for horizontal polarisation and by "V" for vertical polarisation.
The photon (P1) is to leave the source in the impulse mode "A", and the photon (P2)
is to leave the source in the impulse mode "B". The two photons are intended to be
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Interaction-free information transmission by means of entangled photons
entangled in the polarisation degree of freedom. The quantum system comprising the
two photons can then be assigned the pure state [6]
ΨQ = 1/21/2 |B>2 |A>1 ( |H>2 |V>1 + |V>2 |H>1)
(1)
Details relating to the implementation of an appropriate source (Q) are found in [6].
For the interaction-free state preparation, "Alice" has a polarising beam splitter (PTA)
and the detectors (DA1V) and (DA2H). The polarising beam splitter (PTA) is
arranged such that a photon with horizontal polarisation incident in the impulse mode
A is transmitted thereby and is detected by the detector (DA2H), and a photon with
vertical polarisation incident in the impulse mode A is reflected thereby and detected
by the detector (DA1V). For the state analysis "Bob" has a polarising beam splitter
(PTB) and the detectors (DB1V) and (DB2H). The polarising beam splitter (PTB) is
arranged such that a photon with horizontal polarisation incident in the impulse mode
B is transmitted thereby and is detected by the detector (DB2H), and a photon with
vertical polarisation incident in the impulse mode B is reflected thereby and is
detected by the detector (DB1V). The distance of the source (Q) from the two
detectors (DA1V) and (DA2H) should be the same but smaller than the distance of
the source (Q) from the beam splitter (PTB).
Alice then has two possible ways of carrying out a measurement. 1.) The polarising
beam splitter (PTA) is in the beam path, as described: Measurement "M1". 2.) Alice
removes (PTA) from the beam path: Measurement "M0".
If Alice carries out the measurement M1 on the photon (P1), then she certainly
knows which polarisation the photon (P2) has. If she obtains "V" as a result (detector
DA1V responds), then the photon (P2) is polarised horizontally. If she obtains "H" as
a result (detector DA2H responds), then the photon (P2) is polarised vertically. In
principle, it is impossible for Alice to predict individual measured results. As long as
Alice does not disclose the results, the photon incoming to Bob must be described by
means of the mixed state
|M1><M1| = 1/2 ( |B2,H2>< B2,H2| + | B2,V2>< B2,V2| )
(2)
where
|B2,H2>< B2,H2| and |B2,V2>< B2,V2|
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Interaction-free information transmission by means of entangled photons
designate the density operators assigned to the pure states
|B2,H2> = |B>2|H>2 and |B2,V2> = |B>2|V>2
(4)
The photons analysed by Bob are thus polarised horizontally in 50% of the cases
and vertically in 50% of the cases. Of course, it is also not possible for Bob to predict
when which result will occur.
If Alice carries out the measurement M0 on the photon (P1) (removes (PTA) from the
beam path), then all the photons will be detected by the detector (DA2H). However,
since no information about the polarisation of the photon (P1) is accessible during this
measurement, it is in principle impossible to predict which polarisation the photon
(P2) has. The pure state
Ψ M0 = 1/21/2|B>2 ( |H>2 + |V>2)
(5)
must therefore be assigned to the photon (P2). The photons analysed by Bob in this
case are thus polarised horizontally in 50% of the cases and vertically in 50% of the
cases. Of course, it is also not possible for Bob to predict which result will occur in
this case.
The fact that the statistical distribution of the results from Bob is independent of the
measurements from Alice, no matter how they are carried out, is not surprising since
Alice cannot force individual measured results. She can only take notice thereof.
However, it is astonishing that, by using this simple arrangement, both a pure state
and a mixed state can be generated without interaction (without any exchange of
energy). This opens up the possibility of interaction-free information transmission by
means of a symmetrical beam splitter if two sources are used. This will be shown
below.
III.) State analysis by means of beam splitters
A state analysis by means of a symmetrical beam splitter permits the arrangement
illustrated schematically in Fig. 2. The identically constructed sources (Q1) and (Q2)
are intended each to emit a pair of photons entangled in terms of polarisation degree
of freedom simultaneously at freely selectable times. The source (Q1) emits the
photons (P1) and (P3), and the source (Q2) emits the photons (P2) and (P4). All
four photons are intended to be identical in the energetic degree of freedom. The
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Interaction-free information transmission by means of entangled photons
photon (P1) is intended to reach the point (PA) on the detector (DA5) in the impulse
mode "A1" via the mirror (SP1), and the photon (P2) is intended to reach the point
(PA) on the detector (DA5) in the impulse mode "A0" via the mirror (SP2). The
photon (P3) is intended to reach the point (PB) on the symmetrical, ideally loss-free,
beam splitter (STB) in the impulse mode "B1" via the mirror (SP3), and the photon
(P4) is intended to reach the point (PB) on the symmetrical beam splitter (STB) in the
impulse mode "B0" via the mirror (SP4). The pairs of photons emitted by the sources,
after the four photons have passed the mirrors (SP1) to (SP4), can then be
described by the states
ΨQ1 = 1/21/2 |B1>3 |A1>1 ( |H>3 |V>1 + |V>3 |H>1)
(6)
ΨQ2 = 1/21/2 |B0>4 |A0>2 ( |H>4 |V>2 + |V>4 |H>2)
(7)
in a way analogous to (Eq. 1). For the overall system consisting of the four photons,
the result is thus the state
ΨQ1/2 =1/41/2|B0>4|B1>3|A0>2|A1>1 ( |H>3|V>1+|V>3|H>1) ( |H>4 |V>2+|V>4 |H>2)
(8)
since the two pairs of photons have been generated independently of one another.
If the detector (DA5) is removed from the beam path, then a photon in the impulse
mode (A1) can reach the polarising beam splitter (PTA2). The polarising beam
splitter (PTA2) is arranged such that a photon with horizontal polarisation incident in
the impulse mode (A1) is transmitted thereby and is detected by the detector
(DA2H), and a photon with vertical polarisation incident in the impulse mode (A1) is
reflected thereby and is detected by the detector (DA4V). When the detector (DA5) is
removed, a photon in the pulse mode (A0) can reach the polarising beam splitter
(PTA1). The polarising beam splitter (PTA1) is arranged such that a photon with
horizontal polarisation incident in the impulse mode (A0) is transmitted thereby and is
detected by the detector (DA1H), and a photon with vertical polarisation incident in
the impulse mode (A0) is reflected thereby and is detected by the detector (DA3V).
The distance of the source (Q1) from the point (PA) and the distance of the source
(Q2) from the point (PA) are intended to be equal, so that the photons (P1) and (P2)
are incident at the point (PA) at the same time. Furthermore, the distance of the
source (Q1) from the point (PB) and the distance of the source (Q2) from the point
(PB) are intended to be equal, so that the photons (P3) and (P4) are incident at the
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Interaction-free information transmission by means of entangled photons
point (PB) at the same time. The distance of the sources (Q1) and (Q2) from the
point (PB) is intended to be greater than the distance of the sources (Q1) and Q2)
from the detectors (DA1H), (DA2H), (DA3V) und (DA4V). Photons which are in the
impulse mode (B1) after the symmetrical beam splitter (STB) can be detected by
means of the detector (DB2). Photons which are in the impulse mode (B0) after the
symmetrical beam splitter can be detected by means of the detector (DB1).
By means of her arrangement, Alice now has two possible ways of carrying out a
measurement. 1.) The detector (DA5) is in the beam path, as described:
Measurement "MT0". 2.) Alice removes (DA5) from the beam path: Measurement
"MT1".
If Alice carries out the measurement MT0 on the photons (P1) and (P2) (detector
(DA5) is in the beam path, as described), then all the photons will be detected by the
detector (DA5). However, since during this measurement no information about the
polarisation of the photons (P1) and (P2) is accessible, it is in principle impossible to
predict which polarisation the photons (P3) and (P4) have. The photons (P3) and
(P4) must therefore be assigned the pure state
Ψ MT0 =1/41/2|B0>4|B1>3 ( |H>4 + |V>4) ( |H>3 + |V>3)
(9)
After both photons have passed the beam splitter, on account of the classical
properties of the symmetrical beam splitter, the expression
ΨST = 1/161/2( i|B0>4|B0>3 + i|B1>4|B1>3 + |B0>4|B1>3 - |B1>4|B0>3)
( |H>4 + |V>4) ( |H>3 + |V>3)
(10)
is obtained, since the classical beam splitter property acts only on the sub-space
generated by the state vectors |B0>4|B0>3, |B1>4|B1>3, |B0>4|B1>3 and |B1>4|B0>3 [6].
Since the photons (P3) and (P4) reach the beam splitter (STB) at the same time and,
in this case, it is in principle impossible to decide whether the photons (P3) and (P4)
were reflected or transmitted at the beam splitter (STB), the outgoing state at the
beam splitter must be symmetrical. Since the polarisation degree of freedom is
already symmetrical, it is only necessary for the impulse degree of freedom to be
made symmetrical [6]. The outgoing state at the beam splitter then results as
ΨOUT/MT0 = i/81/2( |B0>4|B0>3 + |B1>4|B1>3) ( |H>4 + |V>4) ( |H>3 + |V>3).
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Interaction-free information transmission by means of entangled photons
As is shown in [7], it is possible not only for the classical beam splitter property to be
described by means of a unitary operator but also the symmetrization in the impulse
degree of freedom. For this purpose, in [7] the unitary operator UNLB is introduced.
The photons detected by Bob by means of the detectors (DB1) and (DB2) are then
strongly correlated with regard to the impulse modes (both photons are either in the
impulse mode (B0) and detected by the detector (DB1) or are in the impulse mode
(B1) and detected by the detector (DB2)). Coincidence events (the detectors (DB1)
and (DB2) each detect a photon at the same time) do not occur.
If Alice carries out the measurement MT1 on the photons (P1) and (P2) (detector
(DA5) is removed from the beam path), then the photons (P1) and (P2) are analysed
by means of the detectors (DA1H), (DA2H), (DA3V) and (DA4V). Four results are
then possible:
A): The detectors (DA3V) and (DA4V) each detect a photon. In this case, Alice
knows that the photon (P3) and the photon (P4) are polarised horizontally.
B): The detectors (DA1H) and (DA2H) each detect a photon. In this case, Alice
knows that the photon (P3) and the photon (P4) are polarised vertically.
C): The detectors (DA1H) and (DA4V) each detect a photon. In this case, Alice
knows that the photon (P4) is polarised vertically and the photon (P3) is polarised
horizontally.
D): The detectors (DA3V) and (DA2H) each detect a photon. In this case, Alice
knows that the photon (P4) is polarised horizontally and the photon (P3) is polarised
vertically.
As long as Alice does not disclose her results, the photons (P3) and (P4) arriving at
Bob must be described by means of the mixed state
|MT1><MT1| = 1/4 ( |A >< A| + |B >< B| + |C >< C| + |D >< D| )
(12)
where
|A >< A|, |B >< B|, |C >< C| and |D >< D|
(13)
designate the density operators assigned to the pure states
|A> = |B0>4|B1>3|H>4|H>3
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Interaction-free information transmission by means of entangled photons
|B> = |B0>4|B1>3|V>4|V>3
|C> = |B0>4|B1>3|V>4|H>3
|D> = |B0>4|B1>3|H>4|V>3
(14)
Since, in case (A) and (B) the photons (P3) and (P4) reach the beam splitter (STB) at
the same time and, in these cases, it is in principle impossible to decide whether the
photons (P3) and (P4) were reflected or transmitted at the beam splitter (STB), and,
in case (C) and (D), because of the information available about the polarisation
degree of freedom, it is in principle possible to decide which photon was reflected at
the beam splitter (STB) and which was transmitted, following a short calculation, the
state
|MT1/OUT><MT1/OUT| = 1/4 ( |A/OUT><A/OUT| + |B/OUT><B/OUT|
+ |C/OUT><C/OUT| + |D/OUT>< D/OUT| )
(15)
is obtained for the outgoing photons (P3) and (P4) at the beam splitter (STB),
where
|A/OUT>< A/OUT|, |B/OUT>< B/OUT|, |C/OUT>< C/OUT|
and |D/OUT>< D/OUT|
(16)
designate the density operators assigned to the pure states
|A/OUT> = i/21/2( |B0>4|B0>3 + |B1>4|B1>3) |H>4|H>3
(17)
|B/OUT> = i/21/2( |B0>4|B0>3 + |B1>4|B1>3) |V>4|V>3
|C/OUT> = 1/41/2( i|B0>4|B0>3 + i|B1>4|B1>3 + |B0>4|B1>3 - |B1>4|B0>3) |V>4|H>3
|D/OUT> = 1/41/2( i|B0>4|B0>3 + i|B1>4|B1>3 + |B0>4|B1>3 - |B1>4|B0>3) |H>4|V>3
The photons detected by Bob by means of the detectors (DB1) and (DB2) are
then strongly correlated with regard to the impulse modes in only 75% of the
cases (both photons are either in the impulse mode (B0) and are detected by the
detector (DB1) or are in the impulse mode (B1) and are detected by the detector
(DB2)). In 25% of the cases, coincidence events (the detectors (DB1) and (DB2)
each at the same time detect a photon) then occur.
Thus, by using the frequency of coincidence events, Bob can detect whether Alice is
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Interaction-free information transmission by means of entangled photons
carrying out the measurement MT0 or MT1, although at no time is there any
exchange of energy (any interaction) between Alice and Bob.
For an interaction-free information transfer, it is thus only necessary for a protocol to
be agreed. For the purpose of transmission, for example, it can be agreed that the
measurement MT0 corresponds to a logic "0" and the measurement MT1
corresponds to a logic "1". It is further agreed that for each bit, exactly 1 µs
transmission time is used. The times at which Alice would like to transmit information
are precisely defined in advance. If the sources (Q1) and (Q2) each emit 1000 pairs
of photons per µs, for example, Bob can reliably decide within 1 µs whether Alice
wishes to transmit a logic "0" or a "1". Alice could then transmit data without
interaction at a data rate of up to 1 Mb/s.
IV.) Conclusions
For the success of the experiment proposed here for the interaction-free information
transmission, it is of critical importance for the formulation "knowledge which is in
principle present about the observed quantum system" to be interpreted correctly.
The generally acceptable idea here is that it does not matter at what time the "in
principle accessible knowledge about a quantum system" is present. Only what does
this mean in practical terms? The experiment observed here throws up the question
as to the location (area in space) at which the "in principle accessible knowledge
about a quantum system" must be present (in the sense of "in principle classically
retrievable"), and from which time? If knowledge is a classical property, knowledge
must always be represented by means of energy. If Alice knows, on the basis of her
measurement MT1, what polarisation the photons (P3) and (P4) have, and if the
distance between Alice and Bob is sufficiently great, this knowledge of Alice (if it is a
classical property) at the time at which the photons (P3) and (P4) reach the point PB
on the beam splitter (STB), is in principle never even present. If the experiment can
nevertheless
be
implemented
successfully,
this
shows
that
interaction-free
information transmission can take place instantaneously and knowledge is not a
classical property. The proposed experiment for interaction-free information
transmission can thus contribute to the clarification of entirely fundamental questions.
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PTB DB2H
DA2H PTA
DA1V
"Alice"
P1
"A"
Q
"B"
P2
DB1V
"Bob"
Interaction-free information transmission by means of entangled photons
Fig.1
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"B0"
SP4
Q2
P2
SP2
DA4V
PTA2
DA2H
DA1H
PTA1
DA5
DA3V
"Alice"
PA
SP1
"A0"
"A1"
P1
Q1
P3
P4
"B1"
SP3
PB
"Bob"
STB
DB1
DB2
Interaction-free information transmission by means of entangled photons
Fig.2
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Interaction-free information transmission by means of entangled photons
References:
[1]:
A. Einstein; Zur Elektrodynamik bewegter Körper [On the electrodynamics of
moving bodies]; Ann. D. Physik. 17, 891-921(1905).
[2]:
Maudlin, T.; Quantum non-locality and relativity: Metaphysical intimations of
modern physics; 3nd ed. Wily, Chichester, 2011.
[3]:
Roman U. Sexl and Helmuth K. Urbantke; Relativity, Groups, Particles:
Special Relativity and Relativistic Symmetry in Field and Particle Physics.
Springer, Vienna, New York, 2001.
[4]:
A. Einstein, B. Podolsky, N. Rosen; Phys. Rev. 47, 777 (1935).
[5]:
J. S. Bell. Speakable and unspeakable in quantum mechanics; Cambridge
University Press, Cambridge 1987.
[6]:
Markus Oberparleiter; Bosonische und Fermionische Zweiphotonenstatistik
am Strahlteiler [Boson and Fermi two-photon statistics at the beam
splitter]; Diplomarbeit, Universität Innsbruck, 1997.
[7]:
US Patent US 9,443,200 B2, Gerhart Schroff; 13. September 2016.
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