teleportation. 2 bits teleportation bell measurement
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Teleportation
2 bits
Teleportation
BELL MEASUREMENT
Bell States
1 2 1 2
1 2 1 2
1 2 1 2
1 2 1 2
1
21
21
21
2
1 2EPR
EPR
EPR z
EPR
EPR
x
y
spin rotation
one spin rotation
2 bits
BELL MEASUREMENT
1 1 2 3 2 3
1
2
1 2 1 2 3 3
1 2 1 2 3 3
1 2 1 2 3 3
1 2 1 2 3 3
The EPR-Bohm State
1 2 1 2 1 2 1 2
10, 0
2x x z zEPR
1 2EPR
David Bohm
iBELL
EPR
x
x
3i Teleportation
3i
The EPR State
1 2 1 20, 0EPR q q p p
1q 2q
1q a 2q aEPR
The EPR State
1 2 1 20, 0EPR q q p p
1q 2q
1q EPR 1
* q
The EPR State
1q 2q
EPR
1 2 1 20, 0EPR q q p p
,a bBELL
EPR
,a b
Continuous Variables Teleportation
unknown *
ab
ab
ab
shift , kick a b
( )ibqe q a
L. Vaidman,PRA 49, 1473 (1994)
1 2 1 2, ,a b q q a pBELL p b
1 2EPR
spin measurement
The EPR State = teleportation machine of a known spin up to a flip
1 2EPR
spin measurement
The EPR State = teleportation machine of a known spin up to a flip
1 2EPR
spin measurement
The EPR State = teleportation machine of a known spin up to a flip
Many-Worlds Interpretation
1 2EPR
spin measurement
mixture of and
In the Universe is not moved from Alice to Bob
But in Teleportation it is moved!
Teleportation
In all worlds!
mixture of and and and But after rotation we get
The information sent is only about in which world we are
i
Local Bell measurements split the nonlocal world and thebranching is the carier of the huge amount of information.
We cannot measure (scan) Ψ Too much information to send We cannot clone Ψ
We do not scan Ψ
Why teleportation is possible?
We do not clone Ψ
We cannot measure (scan) Ψ Too much information to send We cannot clone Ψ
We do not scan Ψ
Why teleportation is possible?
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| | |
2a b
1| | |
2a b
1| | |
2a b
1| | |
2a b
Aharonov-Bohm Effect:
SOLENOID
Aharonov-Bohm Effect
Aharonov-Bohm Effect
1| | |
2a b
1| | |
2a b
1| | |
2a b
1| | |
2a b
Aharonov-Bohm Effect
1| | |
2a b
1| | |
2a b
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.
Aharonov-Bohm Effect
21| | |
2
ia e b
2ie
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.
21| | |
2
ia e b
2ie
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| | |
2a b
1| | |
2a b
Aharonov-Bohm Effect
A
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| | |
2a b
1| | |
2a b
Aharonov-Bohm Effect
A
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.
Paradox I
1| | |
2a b
1| | |
2ia e b
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.
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.
Paradox II
1| | |
2a b
1| | |
2ia e b
1| | |
2ia e b
ie
1| | |
2ia e b
The relative phase is observable locally
1 1| | | |1 | 0 | 0 |1
2 2i i
A B A Ba e b e
| |1 Aa | |1 Bb
Ax Bx
A B
x
1| | |
2ia e b
1| | | |
2i
A B A Be
The relative phase is observable locally
EPR correlations are observable locally
Ax Bx
A B
x
t
0
1| | | | |
2i
EPR A B A Be
A B
RESULTS OF LOCAL MEASUREMENTS RELATIVE PHASE
(| ,| ) 0,z A z Bprob 2|1 |
(| ,| ) 0,...8
i
x A x B
eprob
1 1| | | |1 | 0 | 0 |1
2 2i i
A B A Ba e b e
| |1 Aa | |1 Bb
Ax Bx
A B
x
t
0
PHOTON QUANTUM WAVE EPR
1| | |
2ia e b
Ax Bx
A B
x
t
0
1 1| | | | | |
2 2i i
A B A Ba e b e
PHOTON QUANTUM WAVE EPR
Ax Bx
A B
x
t
0
1 1| | | | | |
2 2i i
A B A Ba e b e
h B
B
B
PHOTON QUANTUM WAVE EPR
Ax Bx
A B
x
t
0
1 1| | | | | |
2 2i i
A B A Ba e b e
h B †ˆ ˆ| | | |A BH H a a
B
B
1 1|1 | 0 | 0 |1 | | | | | |
2 2i i
A B A B A B A B A Be e
PHOTON QUANTUM WAVE EPR
Ax Bx
A B
x
t
0
1 1| | | | | |
2 2i i
A B A Ba e b e
B
B
h B †ˆ ˆ| | | |A BH H a a
1 1|1 | 0 | 0 |1 | | | | | |
2 2i i
A B A B A B A B A Be e
LOCAL SPIN MEASUREMENTS RELATIVE PHASE
PHOTON QUANTUM WAVE EPR
Ax Bx
A B
x
t
0
1 0h E E h B INSTEAD OF
REALISTIC EXPERIMENT: TWO-LEVEL ATOM | |
| |
z
z
e
g
PHOTON QUANTUM WAVE EPR
INSTEAD OF A SPIN IN THE MAGNETIC FIELD
Ax Bx
A B
x
t
0
REALISTIC EXPERIMENT: TWO-LEVEL ATOM INSTEAD OF A SPIN IN THE MAGNETIC FIELD
| |
| |
z
z
e
g
PHOTON QUANTUM WAVE EPR
†ˆ ˆ| | | |A BH H a g e a e g
1 1|1 | 0 | 0 |1 | | | | | |
2 2i i
A B A B A B A B A Be g g e g e g e
†ˆ ˆ| | | |A BH H a a
1 1|1 | 0 | 0 |1 | | | | | |
2 2i i
A B A B A B A B A Be e
1 0h E E h B INSTEAD OF
Ax Bx
A B
x
t
0
REALISTIC EXPERIMENT: TWO-LEVEL ATOM INSTEAD OF A SPIN IN THE MAGNETIC FIELD
| |
| |
z
z
e
g
PHOTON QUANTUM WAVE EPR
†ˆ ˆ| | | |A BH H a g e a e g
1 1|1 | 0 | 0 |1 | | | | | |
2 2i i
A B A B A B A B A Be g g e g e g e
†ˆ ˆ| | | |A BH H a a
1 1|1 | 0 | 0 |1 | | | | | |
2 2i i
A B A B A B A B A Be e
1 0h E E h B INSTEAD OF
†ˆ ˆ| | | |H a g e a e g
HOW TO MAKE THE ANALOG OF THE SPIN MEASUREMENTS ON THE ATOM?
1 1| (| | ), | (| | )
2 2x xe g g e ARE NOT MEASURABLE DIRECTLY
ROTATION IN | |e g SPACE
COUPLING H TO A COHERENT STATE | , | | 1 | |2| |
!
n
e nn
†ˆ | |a ROTATION:
| |
| |
cos(| | ) sin(| | )
sin(| | ) cos(| | )i
i
t t
t t
REALISTIC EXPERIMENT: TWO-LEVEL ATOM INSTEAD OF A SPIN IN THE MAGNETIC FIELD
| |
| |
z
z
e
g
PHOTON QUANTUM WAVE EPR
(RABI OSCILLATIONS):
1| | |
2ia e b
| a |b
The relative phase of a photon is observable locally
L. Hardy, Phys. Rev. Lett. (1994)
1| | |
2ia e b
2| |
4 2( )
| '!
nn
e an
| a |b
2| |
4 2( )
| '!
nn
e bn
The relative phase of a photon is observable locally
1| | |
2ia e b
2| |
2 | '!
nn
e an
| a |b
2| |2 | '
!
nn
e bn
The relative phase of a charged pion is observable locallyY. Aharonov, and L. Susskind, Phys. Rev. 155, 1428 (1967)
2| |2| |
!
n
e q nen
This is a gedanken experiment because such a coherent state is unstable
| a |b
The relative phase of an electron is not observable locally
But it is observable, if we have a positron in a superposition with a known phase.
Y. Aharonov, and L. Vaidman, PRA 61, 2108 (2000)
1| | |
2ia e b
1| | ' | '
2a b
| 'b | 'a
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.
Paradox II
1| | |
2a b
1| | |
2ia e b
1| | |
2ia e b
ie
1| | |
2a b
1| | |
2ia e b
1| | |
2ia e b
ie
The key to the resolution of the paradox is that the measuring device measuring relative phase “feels” the Aharonov-Bohm effect too.
ie
1| | |
2ia e b
ie
2| |
2 | '!
nn
e an
2| |
2 | '!
nnie e b
n
The key to the resolution of the paradox is that a measuring device measuring relative the phase “feels” the Aharonov-Bohm effect too.
2 2| | | | 2
2 2 | | ( 2 ) 1| ' | ' | ' | '
! ! ! 2
ie
ni nn nn n i
ee a e b e a e b
n n n
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.
Paradox II - resolution
1| | |
2a b
1| | |
2ia e b
1| | |
2ia e b
ie
1| | |
2a b
1| | |
2a b
A
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.
1| | |
2a b
1| | |
2a b
A
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.
Paradox I
1| | |
2a b
1| | |
2ia e b
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.
LINE OF CHARGE
NEUTRON
Aharonov-Bohm Effect Aharonov-Casher Effect
SOLENOID
ELECTRON
The Aharonov-Casher Effect is dual to the Aharonov-Bohm Effect
due to symmetry in electron neutron interaction
The motion of the electron should be identical to the motion of the neutron, but the neutron feels force!?
ELECTRON
The motion of the electron is identical to the motion of the neutron
Paradox III The motion of the electron inside the interferometer is the same with or without the solenoid
ELECTRON
NEUTRON
LINE OF CHARGE
NEUTRON
0F
0F
AC dual to AB
Neutron slows down
Neutron accelerates
Fx Fx
Fx Fx
LINE OF CHARGE
NEUTRON
0F
It is a current loopI
The force exerted on the neutron
N
S
A neutron is not two magnetic monopoles
The model of the magnetic moment of a neutron
T.H. Boyer, Am .J. Phys. 56, 688 (1988)
c
Vd
A moving current loop has an electric dipole moment
The inhomogeneous electric field exerts force on the dipole
EdF
E
d
Fx
d
V
The force exerted on the neutron
V
T.H. Boyer, Am .J. Phys. 56, 688 (1988)
The forces exerted on the neutron can give energy for nothing!
Fx Fx
Fx Fx
Paradox IV (Aharonov)
Paradox IV (Aharonov)
The forces exerted on the neutron can give energy for nothing!
FxV
Paradox IV (Aharonov)
The forces exerted on the neutron can give energy for nothing!
FxV
c
Vd
d EdF
d
V
V
The forces exerted on the neutron can give energy for nothing!
Paradox IV (Aharonov)
c
Vd
d EdF
d
V
V
The forces exerted on the neutron can give energy for nothing!
Paradox IV (Aharonov)
Fx
c
Vd
d EdF
d
V
V
The forces exerted on the neutron can give energy for nothing!
Paradox IV (Aharonov)
Fx
c
Vd
d EdF
d
V
V
The forces exerted on the neutron can give energy for nothing!
Paradox IV (Aharonov)
Fx
c
Vd
d EdF
d
V
V
The forces exerted on the neutron can give energy for nothing!
Paradox IV (Aharonov)
Fx
c
Vd
d EdF
d
V
V
The forces exerted on the neutron can give energy for nothing!
Paradox IV (Aharonov)