Teleportation
Teleportation
2 bits
BELL MEASUREMENT
Bell States
 
 
 
 
1
2
1
2
1
2
1
2




1
1
1
1
  
2
  
2
  
2
  
2
one spin rotation
1
1
1
1




2
2
2
2




 EPR
z


 EPR
 EPR
x
 EPR
y

EPR
1
2
spin rotation

 
 
 

1
1
1
1




2
2
2
2
 
 
 
 
1
1
1
1




2
2
2
2








2 bits
BELL MEASUREMENT
 

2
1
1
 
1


2
  
3
2

3





3
3
3
3
 
 
 
 
3
3
3
3




The EPR-Bohm State
David Bohm
EPR   1x   2 x  0,  1z   2 z  0 

1

2
1
 
2
EPR
1
2
1

2

Teleportation
i3

x
i3
BELLi

x
EPR
The EPR State
EPR  q1  q2  0, p1  p2  0
q1  a
q1
EPR
q2  a
q2
The EPR State
EPR  q1  q2  0, p1  p2  0
  q1 
q1
  q1 
*
EPR
q2
The EPR State
EPR  q1  q2  0, p1  p2  0
EPR
q1
q2
Continuous Variables
Teleportation
L. Vaidman,PRA 49, 1473 (1994)

a, b
b
BELLa,b
unknown

*
b
a

EPR

b
a
BELLa,b  q1  q2  a,
shift
a , kick b
p1  p2  b
  e (q  a)
ibq
a
The EPR State = teleportation machine
of a known spin up to a flip
EPR
q, f
1
spin measurement
2
The EPR State = teleportation machine
of a known spin up to a flip
q, f
EPR
q, f
1
spin measurement
2
The EPR State = teleportation machine
of a known spin up to a flip
q, f
EPR
q, f
1
spin measurement
2
Many-Worlds Interpretation
In the Universe q, f is not moved from Alice to Bob
But in Teleportation it is moved!
q, f
EPR
q, f
2
1
spin measurement
mixture of
and

Teleportation
i
q, f
In all worlds!
q, f
mixture of
and
and
and

But after rotation we get
q, f
The information sent is only about in which world we are
Local Bell measurements split the nonlocal world and the
branching is the carier of the huge amount of information.
Why teleportation is possible?
We cannot measure (scan) Ψ
Too much information to send
We cannot clone Ψ
We do not scan Ψ
We do not clone Ψ
Why teleportation is possible?
We cannot measure (scan) Ψ
Too much information to send
We cannot clone Ψ
We do not scan Ψ
We do not clone Ψ
Most of information is in branching of the world
Paradoxes
in the context of the Aharonov-Bohm
and the Aharonov-Casher effects
Mach Zehnder Interferometer
Mach Zehnder Interferometer
Mach Zehnder Interferometer
Mach Zehnder Interferometer

|  
|  
1
| a | b 
2

|  
1
| a | b 
2


1
| a | b 
2
|  
1
| a | b 
2
Aharonov-Bohm Effect:
Aharonov-Bohm Effect
SOLENOID
Aharonov-Bohm Effect
Aharonov-Bohm Effect

|  
1
| a | b 
2

1
|  
| a | b 
2

|  
1
| a | b 
2

|  
1
| a | b 
2
Aharonov-Bohm Effect
The solenoid causes a relative phase, but the time when
the phase is gained depends on the choice of gauge, and
therefore, it is unobservable.
|  


1
| a | b 
2
|  
1
| a | b 
2
Aharonov-Bohm Effect
The solenoid causes a relative phase, but the time when
the phase is gained depends on the choice of gauge, and
therefore, it is unobservable.
e
i
q
2
q
i

1 
2
|  
|
a


e
|
b



2

q
Aharonov-Bohm Effect
The solenoid causes a relative phase, but the time when
the phase is gained depends on the choice of gauge, and
therefore, it is unobservable.
q
e
i
q
2
q
i

1 
2
|  
 | a   e | b 
2

Aharonov-Bohm Effect
The solenoid causes a relative phase, but the time when
the phase is gained depends on the choice of gauge, and
therefore, it is unobservable.
A

|  
1
| a | b 
2



|  
1
| a | b 
2
Aharonov-Bohm Effect
The solenoid causes a relative phase, but the time when
the phase is gained depends on the choice of gauge, and
therefore, it is unobservable.
A

|  
1
| a | b 
2



|  
1
| a | b 
2
Paradox I
At every place on the paths of the wave packets of the
electron there is no observable action, but nevertheless,
the relative phase is obtained.
|  
1
| a | b 
2
|  
1
| a   ei | b 

2
Paradox II
The relative phase is observable locally, therefore the
time of change of the relative phase can be observed, in
contradiction with the fact that it is a gauge dependent
property.
ei
|  
1
| a | b 
2
|  
1
| a   ei | b 

2
|  
1
| a   ei | b 

2
The relative phase is observable locally
|  
1
| a  ei | b 

2
The relative phase is observable locally
|  
1
| a  ei | b 

2
A
| a |1 A
B
| b |1 B
xA
xB
1
1
i
|  
|
a


e
|
b


|1 A | 0 B ei | 0 A |1 B 



2
2
1
i
|

|


e
| A | B 

A
B
2
EPR correlations are observable locally
x
|  EPR 
t
1
| A | B ei | A | B 

2
A
B
0
A
B
xA
RESULTS OF LOCAL MEASUREMENTS
xB

prob(|z  A ,|z  B )  0,
|1  ei |2
prob(| x  A ,| x  B ) 
 0,...
8
RELATIVE PHASE 
x
PHOTON QUANTUM WAVE  EPR
|  
t
A
1
| a  ei | b 

2
B
0
| a |1 A
| b |1 B
xA
xB
1
1
i
|  
|
a


e
|
b


|1 A | 0 B ei | 0 A |1 B 



2
2
x
PHOTON QUANTUM WAVE  EPR
t
1
1
i
|
a


e
|
b


| A | B ei | A | B 



2
2
A
B
0
xA
xB
x
PHOTON QUANTUM WAVE  EPR
1
1
i
|
a


e
|
b


| A | B ei | A | B 



2
2
t
A
B
0
B
B
xA
h   B
xB
x
PHOTON QUANTUM WAVE  EPR
1
1
i
|
a


e
|
b


| A | B ei | A | B 



2
2
t
A
B
0
B
B
xA
h   B
xB
x
H A  H B  aˆ † || aˆ ||
1
1
i
i
|1

|
0


e
|
0

|1

|

|


|

|


e
| A | B 



A
B
A
B
A
B
A
B
2
2
PHOTON QUANTUM WAVE  EPR
1
1
i
|
a


e
|
b


| A | B ei | A | B 



2
2
t
A
B
0
B
B
xA
h   B
xB
x
H A  H B  aˆ † || aˆ ||
1
1
i
i
|1

|
0


e
|
0

|1

|

|


|

|


e
| A | B 



A
B
A
B
A
B
A
B
2
2
LOCAL SPIN MEASUREMENTS

RELATIVE PHASE 
PHOTON QUANTUM WAVE  EPR
t
REALISTIC EXPERIMENT: TWO-LEVEL ATOM
INSTEAD OF A SPIN IN THE MAGNETIC FIELD
| z  | e
| z  | g 
A
B
0
xA
h  E1  E0
INSTEAD OF
h   B
xB
x
PHOTON QUANTUM WAVE  EPR
t
REALISTIC EXPERIMENT: TWO-LEVEL ATOM
INSTEAD OF A SPIN IN THE MAGNETIC FIELD
| z  | e
| z  | g 
A
B
0
xA
h  E1  E0
H A  H B  aˆ † | g e | aˆ | e g |
INSTEAD OF
h   B
xB
x
H A  H B  aˆ † || aˆ ||
1
1
i
i
|1

|
0


e
|
0

|1

|
g

|
g


|
e

|
g


e
| g  A | e B 



A
B
A
B
A
B
A
B
2
2
1
1
i
i
|1

|
0


e
|
0

|1

|

|


|

|


e
| A | B 


A
B
A
B
A
B
A
B
2
2
PHOTON QUANTUM WAVE  EPR
t
REALISTIC EXPERIMENT: TWO-LEVEL ATOM
INSTEAD OF A SPIN IN THE MAGNETIC FIELD
| z  | e
| z  | g 
A
B
0
xA
h  E1  E0
H A  H B  aˆ † | g e | aˆ | e g |
INSTEAD OF
h   B
xB
x
H A  H B  aˆ † || aˆ ||
1
1
i
i
|1

|
0


e
|
0

|1

|
g

|
g


|
e

|
g


e
| g  A | e B 



A
B
A
B
A
B
A
B
2
2
1
1
i
i
|1

|
0


e
|
0

|1

|

|


|

|


e
| A | B 


A
B
A
B
A
B
A
B
2
2
PHOTON QUANTUM WAVE  EPR
REALISTIC EXPERIMENT: TWO-LEVEL ATOM
INSTEAD OF A SPIN IN THE MAGNETIC FIELD
| z  | e
| z  | g 
HOW TO MAKE THE ANALOG OF THE SPIN MEASUREMENTS ON THE ATOM?
1
| x  
(| e | g  ),
2
ROTATION IN
1
| x  
(| g  | e )
2
| e  | g 
SPACE
(RABI OSCILLATIONS):
COUPLING H TO A COHERENT STATE
| ,
n
|  e 
| n
n!
H  aˆ | ge | aˆ | e g |

| | 1
 |2|
†
aˆ † |      |  
ARE NOT MEASURABLE DIRECTLY
ROTATION:
 cos(|  | t )
 i 
 | | sin(|  | t )
sin(|  | t ) 

cos(|  | t ) 

i| |
The relative phase of a photon is observable locally
|  
| a
1
| a  ei | b 

2
| b
The relative phase of a photon is observable locally
L. Hardy, Phys. Rev. Lett. (1994)
|  
| a
e
| |2
 4

( 2 )n
n!
1
| a  ei | b 

2
| a ' 
n
e
| |2
 4

( 2 )n
n!
| b ' 
n
| b
The relative phase of a charged pion is observable locally
Y. Aharonov, and L. Susskind, Phys. Rev. 155, 1428 (1967)
|  
| a
e
| |2
 2


n
|     e 
| q  ne
n!
2
 |2|
n
n!
1
| a  ei | b 

2
| a ' 
n
e
| |2
 2

n
n!
| b ' 
n
| b
This is a gedanken experiment because
such a coherent state is unstable
The relative phase of an electron is not observable locally
Y. Aharonov, and L. Vaidman, PRA 61, 2108 (2000)
| a
| a '
1
|  
| a  ei | b 

2
|  
1
| a ' | b ' 
2
| b
| b '
But it is observable, if we have a positron in a superposition with
a known phase.
Paradox II
The relative phase is observable locally, therefore the
time of change of the relative phase can be observed in
contradiction with the fact that it is gauge dependent
property.
ei
|  
1
| a | b 
2
|  
1
| a   ei | b 

2
|  
1
| a   ei | b 

2
The key to the resolution of the paradox is that the measuring
device measuring relative phase “feels” the Aharonov-Bohm
effect too.
ei e  i
|  
1
| a | b 
2
|  
1
| a   ei | b 

2
|  
1
| a   ei | b 

2
The key to the resolution of the paradox is that a measuring device
measuring relative the phase “feels” the Aharonov-Bohm effect too.
e

| |2
2

n
n!
| a ' 
n
e
i
|  
e

e

| |2
 2

| |2
2

n
e

n!
i
| b ' 
1
| a   ei | b 

2
n
n
i
n


n
i

2
|e  |
e

2



(
2

)
1
n
n


 2
| |
i


|
a
'

e
|
b
'


e
|
a
'


e
| b ' 





 n!





n!
n!  2

 

n
Paradox II - resolution
The relative phase of the measuring device which measures the
relative phase of the particle also depends on the chosen gauge. In
fact, local outcomes are not influenced by the solenoid, only their
interpretation is. Even the interpretation is gauge dependent.
ei
|  
1
| a | b 
2
|  
1
| a   ei | b 

2
|  
1
| a   ei | b 

2
Paradox II - resolution
The relative phase of the measuring device which measures the
relative phase of the particle also depends on the chosen gauge. In
fact, local outcomes are not influenced by the solenoid, only their
interpretation is. Even the interpretation is gauge dependent.
A

|  
1
| a | b 
2



|  
1
| a | b 
2
Paradox II - resolution
The relative phase of the measuring device which measures the
relative phase of the particle also depends on the chosen gauge. In
fact, local outcomes are not influenced by the solenoid, only their
interpretation is. Even the interpretation is gauge dependent.
A

|  
1
| a | b 
2



|  
1
| a | b 
2
Paradox I
At every place on the paths of the wave packets of the
electron there is no observable action, but nevertheless,
the relative phase is obtained.
|  
1
| a | b 
2
|  
1
| a   ei | b 

2
The Aharonov-Casher Effect is dual
to the Aharonov-Bohm Effect
due to symmetry in electron neutron interaction
Aharonov-Bohm Effect
ELECTRON
Aharonov-Casher Effect
NEUTRON
SOLENOID
LINE OF CHARGE
The motion of the electron should be identical to the
motion of the neutron, but the neutron feels force!?
Paradox III
The motion of the electron inside the interferometer is the same with or without
the solenoid

ELECTRON
ELECTRON
AC dual to AB
F 0

NEUTRON
F 0
NEUTRON
LINE OF CHARGE
The motion of the electron is identical
to the motion of the neutron
F 0
NEUTRON
LINE OF CHARGE
Neutron slows down
Fx
Fx
Fx
Neutron accelerates
Fx
The force exerted on the neutron
T.H. Boyer, Am .J. Phys. 56, 688 (1988)
The model of the magnetic moment  of a neutron

N

A neutron is not two magnetic monopoles
S
It is a current loop

I
The force exerted on the neutron
T.H. Boyer, Am .J. Phys. 56, 688 (1988)
A moving current loop has an
electric dipole moment

 
 V 
d
c
d
V
Fx
The inhomogeneous electric field
exerts force on the dipole


  

F  d  E
V
d
E
Paradox IV
(Aharonov)
The forces exerted on the neutron can give energy for nothing!
Fx
Fx
Fx
Fx
Paradox IV
(Aharonov)
The forces exerted on the neutron can give energy for nothing!
Paradox IV
(Aharonov)
The forces exerted on the neutron can give energy for nothing!
Fx
V
Paradox IV
(Aharonov)
The forces exerted on the neutron can give energy for nothing!
Fx
V


  

F  d  E
 
 V 
d
c
d
V
d
V
Paradox IV
(Aharonov)
The forces exerted on the neutron can give energy for nothing!
Fx


  

F  d  E
 
 V 
d
c
d
V
d
V
Paradox IV
(Aharonov)
The forces exerted on the neutron can give energy for nothing!
Fx


  

F  d  E
 
 V 
d
c
d
V
d
V
Paradox IV
(Aharonov)
The forces exerted on the neutron can give energy for nothing!
Fx


  

F  d  E
 
 V 
d
c
d
V
d
V
Paradox IV
(Aharonov)
The forces exerted on the neutron can give energy for nothing!
Fx


  

F  d  E
 
 V 
d
c
d
V
d
V
Paradox IV
(Aharonov)
The forces exerted on the neutron can give energy for nothing!


  

F  d  E
 
 V 
d
c
d
V
d
V