Teleportation. 2 bits Teleportation BELL MEASUREMENT.

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

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

1| | |

2a b

1| | |

2a b

1| | |

2a b

1| | |

2a b

Aharonov-Bohm Effect:

SOLENOID

Aharonov-Bohm Effect

1| | |

2a b

1| | |

2a b

1| | |

2a b

1| | |

2a b


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.


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


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

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


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


LOCAL SPIN MEASUREMENTS RELATIVE PHASE


Ax Bx

A B

x

t

0

1 0h E E h B INSTEAD OF

REALISTIC EXPERIMENT: TWO-LEVEL ATOM | |

| |

z

z

e

g


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


†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


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


(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.

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)

FxV



FxV

c

Vd

d EdF

d

V

V



c

Vd

d EdF

d

V

V



Fx

c

Vd

d EdF

d

V

V



Teleportation. 2 bits Teleportation BELL MEASUREMENT.

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