Quantum Teleportation

QUANTUM TELEPORTATION

A SEMINAR REPORT

Submitted by

ANAND SHEKHAR

in partial fulfillment for award of the degree

of

BACHELOR OF TECHNOLOGY

in

COMPUTER SCIENCE & ENGINEERING

SCHOOL OF ENGINEERING

COCHIN UNIVERSITYUNIVERSITY OF SCIENCE &

TECHNOLOGY,KOCHI-682022

AUGUST 2008

DIVISION OF COMPUTER ENGINEERING

SCHOOL OF ENGINEERING

COCHIN UNIVERSITY OF SCIENCE AND TECHNOLOGY

KOCHI-682022

Certificate

Certified that this is a bonafide record of the seminar entitled

“QUANTUM TELEPORTATION”

done by the following student

ANAND SHEKHAR

of the VIIth semester, Computer Science and Engineering in the year 2008 in

partial fulfillment of the requirements to the award of Degree of Bachelor of

Technology in Computer Science and Engineering of Cochin University of

Science and Technology.

Mrs Sheikha Chenthara Dr. David Peter SSeminar Guide Head of Department

ACKNOWLEDGEMENT

I thank my seminar guide Mrs. Sheikha Chenthara, Lecturer, CUSAT, for her

proper guidance and valuable suggestions. I am greatly thankful to Mr. David Peter, the

HOD, Division of Computer Engineering & other faculty members for giving me an

opportunity to learn and do this seminar. If not for the above mentioned people, my

seminar would never have been completed successfully. I once again extend my sincere

thanks to all of them.

Anand Shekhar

i

ABSTRACTTeleportation - the transmission and reconstruction of objects over

arbitrary distances - is a spectacular process, which actually has been

invented by science fiction authors some decades ago. Unbelievable as it

seems in 1993 a theoretical scheme has been found by Charles Bennett that

predicts the existence of teleportation in reality - at least for quantum

systems. This scheme exploits some of the most essential and most

fascinating features of quantum theory, such as the existence of entangled

quantum states. Only four years after its prediction, for the first time

quantum teleportation has been experimentally realized by Anton Zeilinger ,

who succeeded in teleporting the polarization state of photons. Apart from

the fascination that arises from the possibility of teleporting particles,

quantum teleportation is expected to play a crucial role in the construction of

quantum computers in future.

Teleportation promises to be quite useful as an information

processing primitive, facilitating long range quantum communication and

making it much easier to build a working quantum computer.

ii

Table of contents

Chapter No.

Title Page No.

Abstract iiList of figures iv

1 Introduction 12 History 63 How quantum teleportation works

3.1 Bell-state measurements3.2 The teleporter3.3 Working3.4 Teleportation with squeezed light3.5 Fidelity(quantum vs classic)

88

11121415

4 Concept4.1 Description4.2 Entanglement swapping4.3 N-state particles4.4 Result4.5 Remarks

161617181922

5 General teleportation scheme5.1 General description5.2 Further details

232324

6 Applications6.1 Quantum information6.2 Quantum cryptography

262627

7 References 30

iii

List of figures

Sl. No.

Images Page No.

1.1 Researchers 2

1.2 Quantum Teleportation 3

1.3 Conventional method of transmission 5

3.1.1 Photons just before colliding 9

3.1.2 Photons reflected and transmitted 9

3.1.3 Photons are either transmitted or reflected 9

3.2.1 Photon being Teleported 11

3.3.1 Flowchart showing Teleportation 12

3.3.2 River Danube Experiment 12

3.4.1 Teleportation Apparatus 14

iv

Quantum Teleportation

1. INTRODUCTION

Teleportation - the transmission and reconstruction of objects over arbitrary

distances - is a spectacular process, which actually has been invented by science

fiction authors some decades ago. Unbelievable as it seems in 1993 a theoretical

scheme has been found by Charles Bennett that predicts the existence of teleportation

in reality - at least for quantum systems. This scheme exploits some of the most

essential and most fascinating features of quantum theory, such as the existence of

entangled quantum states. Only four years after its prediction, for the first time

quantum teleportation has been experimentally realized by Anton Zeilinger, who

succeeded in teleporting the polarization state of photons. Apart from the fascination

that arises from the possibility of teleporting particles, quantum teleportation is

expected to play a crucial role in the construction of quantum computers in future.

Quantum teleportation, or entanglement-assisted teleportation, is a

technique used to transfer information on a quantum level, usually from one particle

(or series of particles) to another particle (or series of particles) in another location via

quantum entanglement. It does not transport energy or matter, nor does it allow

communication of information at superluminal (faster than light) speed, but is useful

for quantum communication and computation.

More precisely, quantum teleportation is a quantum protocol by which a qubit

a (the basic unit of quantum information) can be transmitted exactly (in principle)

from one location to another. The prerequisites are a conventional communication

channel capable of transmitting two classical bits (i.e. one of four states), and an

entangled pair (b,c) of qubits, with b at the origin and c at the destination. (So whereas

b and c are intimately related, a is entirely independent of them other than being

initially colocated with b.) The protocol has three steps: measure a and b jointly to

yield two classical bits; transmit the two bits to the other end of the channel (the only

potentially time-consuming step, due to speed-of-light considerations); and use the

two bits to select one of four ways of recovering c. The upshot of this protocol is to

permute the original arrangement ((a,b),c) to ((b′,c′),a), that is, a moves to where c

was and the previously separated qubits of the Bell pair turn into a new Bell pair (b′,c

′) at the origin.

1Division of Computer Engineering

http://en.wikipedia.org/wiki/Qubit


Teleportation is the name given by science fiction writers to the feat of making

an object or person disintegrate in one place while a perfect replica appears

somewhere else. How this is accomplished is usually not explained in detail, but the

general idea seems to be that the original object is scanned in such a way as to extract

all the information from it, then this information is transmitted to the receiving

location and used to construct the replica, not necessarily from the actual material of

the original, but perhaps from atoms of the same kinds, arranged in exactly the same

pattern as the original. A teleportation machine would be like a fax machine, except

that it would work on 3-dimensional objects as well as documents, it would produce

an exact copy rather than an approximate facsimile, and it would destroy the original

in the process of scanning it. A few science fiction writers consider teleporters that

preserve the original, and the plot gets complicated when the original and teleported

versions of the same person meet; but the more common kind of teleporter destroys

the original, functioning as a super transportation device, not as a perfect replicator of

souls and bodies.

In 1993 an international group of six

scientists, including IBM Fellow

Charles H. Bennett, confirmed the

intuitions of the majority of science

fiction writers by showing that perfect

teleportation is indeed possible in

principle, but only if the original is

destroyed. In subsequent years, other

scientists have demonstrated

teleportation experimentally in a variety Fig 1.1 Researchers

of systems, including single photons, coherent light fields, nuclear spins, and trapped

ions. Teleportation promises to be quite useful as an information processing

primitive, facilitating long range quantum communication (perhaps unltimately

leading to a "quantum internet"), and making it much easier to build a working

quantum computer. But science fiction fans will be disappointed to learn that no one

expects to be able to teleport people or other macroscopic objects in the foreseeable

future, for a variety of engineering reasons, even though it would not violate any

fundamental law to do so.



In the past, the idea of teleportation was not taken very seriously by scientists,

because it was thought to violate the uncertainty principle of quantum mechanics,

which forbids any measuring or scanning process from extracting all the information

in an atom or other object. According to the uncertainty principle, the more accurately

an object is scanned, the more it is disturbed by the scanning process, until one

reaches a point where the object's original state has been completely disrupted, still

without having extracted enough information to make a perfect replica. This sounds

like a solid argument against teleportation: if one cannot extract enough information

from an object to make a perfect copy, it would seem that a perfect copy cannot be

made. But the six scientists found a way to make an end run around this logic, using a

celebrated and paradoxical feature of quantum mechanics known as the Einstein-

Podolsky-Rosen effect. In brief, they found a way to scan out part of the information

from an object A, which one wishes to teleport, while causing the remaining,

unscanned, part of the information to pass, via the Einstein- Podolsky-Rosen effect. In

brief, they found a way to scan out part of the information from an object A, which

one wishes to teleport, while causing the remaining, unscanned, part of the

information to pass, via the Einstein-Podolsky-Rosen effect, into another object C

which has never been in Contact with A.

Fig 1.2 Quantum Teleportation



Later, by applying to C a treatment depending on the scanned-out information,

it is possible to maneuver C into exactly the same state as A was in before it was

scanned. A itself is no longer in that state, having been thoroughly disrupted by the

scanning, so what has been achieved is teleportation, not replication.

As the figure above suggests, the unscanned part of the information is

conveyed from A to C by an intermediary object B, which interacts first with C and

then with A. What? Can it really be correct to say "first with C and then with A"?

Surely, in order to convey something from A to C, the delivery vehicle must visit A

before C, not the other way around. But there is a subtle, unscannable kind of

information that, unlike any material cargo, and even unlike ordinary information, can

indeed be delivered in such a backward fashion. This subtle kind of information, also

called "Einstein-Podolsky-Rosen (EPR) correlation" or "entanglement", has been at

least partly understood since the 1930s when it was discussed in a famous paper by

Albert Einstein, Boris Podolsky, and Nathan Rosen. In the 1960s John Bell showed

that a pair of entangled particles, which were once in contact but later move too far

apart to interact directly, can exhibit individually random behavior that is too strongly

correlated to be explained by classical statistics. Experiments on photons and other

particles have repeatedly confirmed these correlations, thereby providing strong

evidence for the validity of quantum mechanics, which neatly explains them. Another

well-known fact about EPR correlations is that they cannot by themselves deliver a

meaningful and controllable message. It was thought that their only usefulness was in

proving the validity of quantum mechanics. But now it is known that, through the

phenomenon of quantum teleportation, they can deliver exactly that part of the

information in an object which is too delicate to be scanned out and delivered by

conventional methods.



Fig 1.3 Conventional Method of Transmission

This figure compares conventional facsimile transmission with quantum

teleportation (see above). In conventional facsimile transmission the original is

scanned, extracting partial information about it, but remains more or less intact after

the scanning process. The scanned information is sent to the receiving station, where

it is imprinted on some raw material (eg paper) to produce an approximate copy of the

original. By contrast, in quantum teleportation, two objects B and C are first brought

into contact and then separated. Object B is taken to the sending station, while object

C is taken to the receiving station. At the sending station object B is scanned together

with the original object A which one wishes to teleport, yielding some information

and totally disrupting the state of A and B. The scanned information is sent to the

receiving station, where it is used to select one of several treatments to be applied to

object C, thereby putting C into an exact replica of the former state of A.



2. HISTORY

Teleportation is a term created by science fiction authors describing a process,

which lets a person or object disappear while an exact replica appears in the best

case immediately at some distant location. The first idea how the dream of

teleportation could be realized in practice might be the following: From a classical

point of view the object to be teleported can fully be characterized by its properties,

which can be determined by measurement. To create a copy of the object one does not

need the original parts and pieces, but all that is needed is to send the scanned

information to the place of destination, where the object can be reconstructed. Having

a closer look at that scheme, we realize that the weak point is the measuring process.

If we want to get a perfect replica of the object, it would be inevitable to determine

the states of molecules, atoms and electrons - in a word: we would have to measure

quantum properties. But according to Heisenberg’s uncertainty principle, these cannot

be determined with arbitrary precision not even in principle. We see that teleportation

is not practicable in this way. And even more: it seems as if the laws of quantum

mechanics prohibit any teleportation scheme in general.

It is the more surprising that in 1993 CharlesH. Bennett et al. have suggested

that it is possible to transfer the quantum state of a particle onto another provided one

does not get any information about the state in the course of this transformation. The

central point of Bennett’s idea is the use of an essential feature of quantum

mechanics: entanglement . Entanglement describes correlations between quantum

systems much stronger than any classical correlation could be. With the help of a so-

called pair of entangled particles it is possible to circumvent the limitations caused by

Heisen-berg’s uncertainty principle.

Quite soon after its theoretical prediction in 1997 Anton Zeilinger et al.

succeeded in the first experimental verification of quantum teleportation. By

producing pairs of entangled photons with the process of parametric down-conversion

and using two-photon interferometry for analyzing entanglement, they were able to

transfer a quantum property (the polarization state) from one photon to another.

Though the prediction and experimental realization of quantum teleportation

are surely a great success of modern physics, we should be aware of the differences

between the physical quantum teleportation and its science fiction counterpart. We



will see that quantum teleportation transfers the quantum state from one particle to

another, but doesn’t transfer mass. Furthermore the original state is destroyed in the

course of teleportation, which means that no copy of the original state is produced.

This is due to the no-cloning theorem, which says that it is impossible within quantum

theory to produce a clone of a given quantum system . Finally we will learn that

teleporting a quantum state has a natural speed limit. In the best case it is possible to

teleport at the speed of light - in accordance with Einstein’s theory of relativity.

The two parties are Alice (A) and Bob (B), and a qubit is, in general, a

superposition of quantum state labeled and . Equivalently, a qubit is a unit

vector in two-dimensional Hilbert space.

Suppose Alice has a qubit in some arbitrary quantum state . Assume that this

quantum state is not known to Alice and she would like to send this state to Bob.

Ostensibly, Alice has the following options:

1. She can attempt to physically transport the qubit to Bob.

2. She can broadcast this (quantum) information, and Bob can obtain the information

via some suitable receiver.

3. She can perhaps measure the unknown qubit in her possession. The results of this

measurement would be communicated to Bob, who then prepares a qubit in his

possession accordingly, to obtain the desired state. (This hypothetical process is called

classical teleportation.)

Option 1 is highly undesirable because quantum states are fragile and any

perturbation en route would corrupt the state.

The unavailability of option 2 is the statement of the no-broadcast theorem.

Similarly, it has also been shown formally that classical teleportation, aka.

option 3, is impossible; this is called the no teleportation theorem. This is another way

to say that quantum information cannot be measured reliably.

Thus, Alice seems to face an impossible problem. A solution was discovered

by Bennet et al. The parts of a maximally entangled two-qubit state are distributed to

Alice and Bob. The protocol then involves Alice and Bob interacting locally with the

qubit(s) in their possession and Alice sending two classical bits to Bob. In the end, the

qubit in Bob's possession will be in the desired state.




3. HOW QUANTUM TELEPORTATION WORKS

3.1 BELL-STATE MEASUREMENTS

In previous discussions we almost always talked about the spin state of

electrons, although we regularly pointed out that the same situations exist for the

polarization of light, albeit with a difference of a factor of 2 in the angles being used.

Here we will reverse the situation, and mostly talk about polarization states for

photons, although the arguments also apply to spin states of electrons.

The fact that we may talk about light polarization in almost the same way that we

discuss electron spin is not a coincidence. It turns out that photons have spins which

can exist in only two different states. And those different spins states are related to the

polarization of the light when we think of it as a wave.

Here we shall prepare pairs of entangled photons with opposite polarizations;

we shall call them E1 and E2. The entanglement means that if we measure a beam of,

say, E1 photons with a polarizer, one-half of the incident photons will pass the filter,

regardless of the orientation of the polarizer. Whether a particular photon will pass the

filter is random. However, if we measure its companion E2 photon with a polarizer

oriented at 90 degrees relative to the first, then if E1 passes its filter E2 will also pass

its filter. Similarly if E1 does not pass its filter its companion E2 will not.

Earlier we discussed the Michelson-Morley experiment, and later the Mach-Zehnder

interferometer. You will recall that for both of these we had half-silvered mirrors,

which reflect one-half of the light incident on them and transmit the other half without

reflection. These mirrors are sometimes called beam splitters because they split a light

beam into two equal parts.

We shall use a half-silvered mirror to perform Bell State Measurements. The

name is after the originator of Bell's Theorem.

We direct one of the entangled photons, say E1, to the

beam splitter.

Meanwhile, we prepare another photon with a

polarization of 450, and direct it to the same beam

splitter from the other side, as shown. This is the



photon whose properties will be transported; we label it K (for Kirk). We time it so

that both E1 and K reach the beam splitter at the same time.

Fig 3.1.1 Photons just before

colliding

The E1 photon incident from above will be

reflected by the beam splitter some of the time

and will be transmitted some of the time.

Similarly for the K photon that is incident from

below. So sometimes both photons will end up

going up and to the right as shown above.

Fig 3.1.2 Photons reflected and

transmitted

Similarly, sometimes both photons will end up going down and to the right.

But sometimes one photon will end up going

upwards and the other will be going downwards, as

shown. This will occur when either both photons

have been reflected or both photons have been

transmitted.

Thus there are three possible arrangements for the

photons from the beam splitter: both upwards, both

downwards, or one upwards and one downwards.

Fig 3.1.3 Photons are either

transmitted or reflected

Which of these three possibilities has occurred can be determined if we put

detectors in the paths of the photons after they have left the beam splitter.

However, in the case of one photon going upwards and the other going downwards,

we can not tell which is which. Perhaps both photons were reflected by the beam

splitter, but perhaps both were transmitted.

This means that the two photons have become entangled.



If we have a large beam of identically prepared photon pairs incident on the

beam splitter, the case of one photon ending up going upwards and the other

downwards occurs, perhaps surprisingly, 25% of the time.

Also somewhat surprisingly, for a single pair of photons incident on the beam

splitter, the photon E1 has now collapsed into a state where its polarization is -450, the

opposite polarization of the prepared 450 one. This is because the photons have

become entangled. So although we don't know which photon is which, we know the

polarizations of both of them.

The explanation of these two somewhat surprising results is beyond the level

of this discussion, but can be explained by the phase shifts the light experiences when

reflected, the mixture of polarization states of E1, and the consequent interference

between the two photons.

3.2 THE TELEPORTERNow we shall think about the E2 companion to E1.

25 percent of the time, the Bell-state measurement

resulted in the circumstance shown, and in these

cases we have collapsed the state of the E1 photon

into a state where its polarization is -450.



But since the two photon system E1 and E2 was

prepared with opposite polarizations, this means

that the companion to E1, E2, now has a

polarization of +450. Thus the state of the K photon

has now been transferred to the E2 photon. We

have teleported the information about the K photon

to E2.

Although this collapse of E2 into a 450 polarization state occurs instantaneously, we haven't achieved Fig 3.2.1 Photon being Teleported

teleportation until we communicate that the Bell-state measurement has yielded the

result shown. Thus the teleportation does not occur instantaneously.

Note that the teleportation has destroyed the state of the original K photon.

Quantum entanglements such as exist between E1 and E2 in principle are

independent of how far apart the two photons become. This has been experimentally

verified for distances as large as 10km. Thus, the Quantum Teleportation is similarly

independent of the distance.

The Original State of the Teleported Photon Must Be Destroyed

Above we saw that the K photon's state was destroyed when the E2 photon

acquired it. Consider for a moment that this was not the case, so we end up with two

photons with identical polarization states. Then we could measure the polarization of

one of the photons at, say, 450 and the other photon at 22.50. Then we would know the

polarization state of both photons for both of those angles.

As we saw in our discussion of Bell's Theorem, the Heisenberg Uncertainty

Principle says that this is impossible: we can never know the polarization of a photon

for these two angles. Thus any teleporter must destroy the state of the object being

teleported.

3.3 WORKINGBefore going further, here is how quantum teleportation works.



First, an entangled

state of ions A and B

is generated, then

the state to be

teleported -- a

coherent

superposition of

internal states -- is

created in a third

ion, P. Fig 3.3.1 Flowchart showing Teleportation

The third step is a joint measurement of P and A, with the result sent to the location of

ion B, where it is used to transform the state .

Now, let's look at the BBC News article.

Long distance teleportation is crucial if dreams of superfast quantum

computing are to be realised. When physicists say "teleportation", they are describing

the transfer of key properties from one particle to another without a physical link.

Researchers from the University of Vienna and the Austrian Academy of

Science used an 800m-long optical fibre fed through a public sewer system tunnel to

connect labs on opposite sides of the River Danube.

The link establishes a channel between the labs, dubbed Alice and Bob. This

enables the properties, or "quantum states", of light particles to be transferred between

the sender (Alice) and the receiver (Bob).

Fig 3.3.2 River Danube Experiment

This illustration

shows how the

experiment was

conducted.

In "Teleportation Takes Quantum Leap," National Geographic explains

why this experiment is a world's premiere.



"We were able to perform a quantum teleportation experiment for the first

time ever outside a university laboratory," said Rupert Ursin, a researcher at the

Institute for Experimental Physics at the University of Vienna in Austria.

The science is not new, said Mark Kuzyk, a physics professor at Washington

State University in Pullman. But this is the first time "researchers have demonstrated

that teleportation works in the kinds of real-life conditions that are found in telecom

applications."

Efficient long-distance quantum teleportation is crucial for quantum

communication and quantum networking schemes. Here we describe the high-fidelity

teleportation of photons over a distance of 600 metres across the River Danube in

Vienna, with the optimal efficiency that can be achieved using linear optics. Our

result is a step towards the implementation of a quantum repeater, which will enable

pure entanglement to be shared between distant parties in a public environment and

eventually on a worldwide scale.

3.4 TELEPORTATION WITH SQUEEZED LIGHT



We have implemented quantum teleportation with light beams serving as both

the entangled pair and the input (and output) state. Squeezed light is used to generate

the entangled (EPR) beams which are sent to Alice and Bob. A third beam, the input,

is a coherent state of unknown complex amplitude. This state is teleported to Bob

with a high fidelity only achievable via the use of quantum entanglement.

Teleportation

Apparatus

Entangled EPR beams

are generated by

combining two beams

of squeezed light at a

50/50 beamsplitter.

EPR beam 1 propagates

to Alice's sending

station, Fig

3.4.1 Teleportation

Apparatus

where it is combined at a 50/50 beamsplitter with the unknown input state, in this

case a coherent state of unknown complex amplitude. Alice uses two sets of balanced

homodyne detectors to make a Bell-state measurement on the amplitudes of the

combined state. Because of the entanglement between the EPR beams, Alice's

detection collapses Bob's field (EPR beam 2) into a state conditioned on Alice's

measurement outcome. After receiving the classical result from Alice, Bob is able to

construct the teleported state via a simple phase-space displacement of the EPR field

2.

3.5 FIDELITY(QUANTUM VS CLASSIC)



Quantum teleportation is theoretically perfect, yielding an output state which

equals the input with a fidelity F=1. In practice, fidelities less than one are realized

due to imperfections in the EPR pair, Alice's Bell measurement, and Bob's unitary

transformation. By contrast, a sender and receiver who share only a classical

communication channel cannot hope to transfer an arbitrary quantum state with a

fidelity of one. For coherent states, the classical teleportation limit is F=0.5, while for

light polarization states it is F=0.67. The quantum nature of the teleportation achieved

in this case is demonstrated by the experimentally determined fidelity of F=0.58,

greater than the classical limit of 0.5 for coherent states. Note that the fidelity is an

average over all input states and so measures the ability to transfer an arbitrary,

unknown superposition from Alice to Bob.

4. CONCEPT



Assume that Alice and Bob share an entangled qubit AB. That is, Alice has

one half, A, and Bob has the other half, B. Let C denote the qubit Alice wishes to

transmit to Bob.

Alice applies a unitary operation on the qubits AC and measures the result to

obtain two classical bits. In this process, the two qubits are destroyed. Bob's qubit, B,

now contains information about C; however, the information is somewhat

randomized. More specifically, Bob's qubit B is in one of four states uniformly chosen

at random and Bob cannot obtain any information about C from his qubit.

Alice provides her two measured qubits, which indicate which of the four states Bob

possesses. Bob applies a unitary transformation which depends on the qubits he

obtains from Alice, transforming his qubit into an identical copy of the qubit C.

4.1 DESCRIPTION

In the literature, one might find alternative, but completely equivalent,

descriptions of the teleportation protocol given above. Namely, the unitary

transformation that is the change of basis (from the standard product basis into the

Bell basis) can also be implemented by quantum gates. Direct calculation shows that

this gate is given by

where H is the one qubit Walsh-Hadamard gate and CN is the Controlled NOT gate.

4.2 ENTANGLEMENT SWAPPING



Entanglement can be applied not just to pure states, but also mixed states, or

even the undefined state of an entangled particle. The so-called entanglement

swapping is a simple and illustrative example.

If Alice has a particle which is entangled with a particle owned by Bob, and

Bob teleports it to Carol, then afterwards, Alice's particle is entangled with Carol's.

A more symmetric way to describe the situation is the following: Alice has one

particle, Bob two, and Carol one. Alice's particle and Bob's first particle are

entangled, and so are Bob's second and Carol's particle:

/ \ Alice-:-:-:-:-:-Bob1 -:- Bob2-:-:-:-:-:-Carol \___/

Now, if Bob performs a projective measurement on his two particles in the

Bell state basis and communicates the results to Carol, as per the teleportation scheme

described above, the state of Bob's first particle can be teleported to Carol's. Although

Alice and Carol never interacted with each other, their particles are now entangled.

4.3 N-STATE PARTICLES



One can imagine how the teleportation scheme given above might be extended

to N-state particles, i.e. particles whose states lie in the N dimensional Hilbert space.

The combined system of the three particles now has a N3 dimensional state space. To

teleport, Alice makes a partial measurement on the two particles in her possession in

some entangled basis on the N2 dimensional subsystem. This measurement has N2

equally probable outcomes, which are then communicated to Bob classically. Bob

recovers the desired state by sending his particle through an appropriate unitary gate.

4.4 RESULT



Suppose Alice has a qubit that she wants to teleport to Bob. This qubit can be

written generally as:

Our quantum teleportation scheme requires Alice and Bob to share a maximally

entangled state beforehand, for instance one of the four Bell states

,

,

,

.

Alice takes one of the particles in the pair, and Bob keeps the other one. The

subscripts A and B in the entangled state refer to Alice's or Bob's particle. We will

assume that Alice and Bob share the entangled state .

So, Alice has two particles (C, the one she wants to teleport, and A, one of the

entangled pair), and Bob has one particle, B. In the total system, the state of these

three particles is given by

Alice will then make a partial measurement in the Bell basis on the two qubits

in her possession. To make the result of her measurement clear, we will rewrite the

two qubits of Alice in the Bell basis via the following general identities (these can be

easily verified):



The three particle state shown above thus becomes the following four-term

superposition:

Notice all we have done so far is a change of basis on Alice's part of the

system. No operation has been performed and the three particles are still in the same

state. The actual teleportation starts when Alice measures her two qubits in the Bell

basis. Given the above expression, evidently the results of her (local) measurement is

that the three-particle state would collapse to one of the following four states (with

equal probability of obtaining each):

•

•

•

•

Alice's two particles are now entangled to each other, in one of the four Bell

states. The entanglement originally shared between Alice's and Bob's is now broken.

Bob's particle takes on one of the four superposition states shown above. Note how

Bob's qubit is now in a state that resembles the state to be teleported. The four

possible states for Bob's qubit are unitary images of the state to be teleported.

The crucial step, the local measurement done by Alice on the Bell basis, is done. It is

clear how to proceed further. Alice now has complete knowledge of the state of the

three particles; the result of her Bell measurement tells her which of the four states the

system is in. She simply has to send her results to Bob through a classical channel.

Two classical bits can communicate which of the four results she obtained.

After Bob receives the message from Alice, he will know which of the four

states his particle is in. Using this information, he performs a unitary operation on his

particle to transform it to the desired state :

• If Alice indicates her result is , Bob knows his qubit is already in the desired

state and does nothing. This amounts to the trivial unitary operation, the identity

operator.




• If the message indicates , Bob would send his qubit through the unitary gate

given by the Pauli matrix

to recover the state.

• If Alice's message corresponds to , Bob applies the gate

to his qubit.

• Finally, for the remaining case, the appropriate gate is given by

Teleportation is therefore achieved.

Experimentally, the projective measurement done by Alice may be achieved

via a series of laser pulses directed at the two particles.



4.5 REMARKS

After this operation, Bob's qubit will take on the state, and Alice's qubit

becomes (undefined) part of an entangled state. Teleportation does not result in the

copying of qubits, and hence is consistent with the no cloning theorem.

There is no transfer of matter or energy involved. Alice's particle has not been

physically moved to Bob; only its state has been transferred. The term "teleportation",

coined by Bennett, Brassard, Crépeau, Jozsa, Peres and Wootters., reflects the

indistinguishability of quantum mechanical particles.

The teleportation scheme combines the resources of two separately impossible

procedures. If we remove the shared entangled state from Alice and Bob, the scheme

becomes classical teleportation, which is impossible as mentioned before. On the

other hand, if the classical channel is removed, then it becomes an attempt to achieve

superluminal communication, again impossible.

For every qubit teleported, Alice needs to send Bob two classical bits of

information. These two classical bits do not carry complete information about the

qubit being teleported. If an eavesdropper intercepts the two bits, she may know

exactly what Bob needs to do in order to recover the desired state. However, this

information is useless if she cannot interact with the entangled particle in Bob's

possession.



5. GENERAL TELEPORTATION SCHEME

5.1 GENERAL DESCRIPTION

A general teleportation scheme can be described as follows. Three quantum

systems are involved. System 1 is the (unknown) state ρ to be teleported by Alice.

Systems 2 and 3 are in a maximally entangled state ω that are distributed to Alice and

Bob, respectively. The total system is then in the state

A successful teleportation process is a LOCC quantum channel Φ that satisfies

where Tr12 is the partial trace operation with respect systems 1 and 2, and denotes the

composition of maps. This describes the channel in the Schrodinger picture.

Taking adjoint maps in the Heisenberg picture, the success condition becomes

for all observable O on Bob's system. The tensor factor in is while

that of is .



5.2 FURTHER DETAILS

The proposed channel Φ can be described more explicitly. To begin

teleportation, Alice performs a local measurement on the two subsystems (1 and 2) in

her possession. Assume the local measurement have effects

If the measurement registers the i-th outcome, the overall state collapses to

The tensor factor in is while that of is .

Bob then applies a corresponding local operation Ψi on system 3. On the combined

system, this is described by

where Id is the identity map on the composite system .

Therefore the channel Φ is defined by

Notice Φ satisfies the definition of LOCC. As stated above, the teleportation is said to

be successful if, for all observable O on Bob's system, the equality

holds. The left hand side of the equation is:



where Ψi* is the adjoint of Ψi in the Heisenberg picture. Assuming all objects are

finite dimensional, this becomes

The success criterion for teleportation has the expression



6. APPLICATIONS

Teleporting the polarization state of a single photon a quarter of the time is a

long long way from reliably teleporting Captain Kirk. However, there are other

applications of the above sort of apparatus that may be closer to being useful.

6.1 QUANTUM INFORMATION

As you probably know, computers store information as sequences of 0's and

1's. For example, in the ASCII encoding the letter A is represented by the number 65.

As a binary number this is:

1,000,001

Inside the computer, there are transistors that are either on or off, and we

assign the on-state be 1 and the off state 0. However, the same information can be

stored in exactly the same way in any system that has two mutually exclusive binary

states.

For example, if we have a collection photons we could represent the 1's as

photons whose polarization is +450 and the 0's as polarizations of -450. We could

similarly use electrons with spin-up and spin-down states to encode the information.

These quantum bits of information are called qubits.

Above we were thinking about an apparatus to do Quantum Teleportation.

Now we see that we can think of the same apparatus as transferring Quantum

Information. Note that, as opposed to, say, a fax, when transferring Quantum

Information the original, the polarization of the K photon, is destroyed.



6.2 Quantum Cryptography

Cryptography depends on both the sender and receiver of the encrypted

information both knowing a key. The sender uses the key to encrypt the information

and the receiver uses the same key to decrypt it.

The key can be something very simple, such as both parties knowing that each

letter has been shifted up by 13 places, with letters above the thirteenth in the alphabet

rotated to the beginning. Or they can be very complex, such as a very very long string

of binary digits.

Here is an example of using binary numbers to encrypt and decrypt a message,

in this case the letter A, which we have seen is 1,000,001 in a binary ASCII encoding.

We shall use as the key the number 23, which in binary is 0,010,111. We will use the

key to encode the letter using a rule that if the corresponding bits of the letter and key

are the same, the result is a 1, and otherwise a 0.

A 1 0 0 0 0 0 1Key 0 0 1 0 1 1 1Encrypted 0 1 0 1 0 0 1

The encrypted value is 41, which in ASCII is the right parenthesis: )

To decrypt the message we use the key and the same procedure:

Encrypted 0 1 0 1 0 0 1Key 0 0 1 0 1 1 1A 1 0 0 0 0 0 1

Any classical encryption scheme is vulnerable on two counts:

• If the "bad guys" get hold of the key they too can decrypt the message. So-

called public key encryptation schemes reveals on an open channel a long

string of binary digits which must be converted to the key by means of a secret

procedure; here security is based on the computational complexity of

"cracking" the secret procedure.

• Because there are patterns in all messages, such as the fact that the letter e

predominates, then if multiple messages are intercepted using the same key the

bad guys can begin to decipher them.

To be really secure, then, there must be a unique secret key for each message. So

the question becomes how can we generate a unique key and be sure that the bad guys

don't know what it is.



To send a key in Quantum Cryptography, simply send photons in one of four

polarizations: -45, 0, 45, or 90 degrees. As you know, the receiver can measure, say,

whether or not a photon is polarized at 90 degrees and if it is not then be sure than it

was polarized at 0 degrees. Similarly the receiver can measure whether a photon was

polarized at 45 degrees, and if it is not then it is surely polarized at -45 degrees.

However the receiver can not measure both the 0 degree state and 45 degree state,

since the first measurement destroys the information of the second one, regardless of

which one is performed first.

The receiver measures the incoming photons, randomly choosing whether to

measure at 90 degrees or 45 degrees, and records the results but keeps them secret.

The receiver contacts the sender and tells her on an open channel which type of

measurement was done for each, without revealing the result. The sender tells the

receiver which of the measurements were of the correct type. Both the sender and

receiver keep only the qubits that were measured correctly, and they have now formed

the key.

If the bad guys intercept the transmission of photons, measure their polarizations,

and then send them on to the receiver, they will inevitably introduce errors because

they don't know which polarization measurement to perform. The two legitimate users

of the quantum channel test for eavesdropping by revealing a random subset of the

key bits and checking the error rate on an open channel. Although they cannot prevent

eavesdropping, they will never be fooled by an eavesdropper because any, however

subtle and sophisticated, effort to tap the channel will be detected. Whenever they are

not happy with the security of the channel they can try to set up the key distribution

again.

By February 2000 a working Quantum Cryptography system using the above

scheme achieved the admittedly modest rates of 10 bits per second over a 30 cm

length.

There is another method of Quantum Cryptography which uses entangled photons.

A sequence of correlated particle pairs is generated, with one member of each pair

being detected by each party (for example, a pair of photons whose polarisations are

measured by the parties). An eavesdropper on this communication would have to

detect a particle to read the signal, and retransmit it in order for his presence to remain

unknown. However, the act of detection of one particle of a pair destroys its quantum

correlation with the other, and the two parties can easily verify whether this has been



done, without revealing the results of their own measurements, by communication

over an open channel

7. REFERENCES



• http://www.primidi.com/2004/08/24.html

• http://www.sciam.com/article.cfm?id=why-teleporting-is-nothing-like-

star-trek

• http://www.upscale.utoronto.ca/GeneralInterest/Harrison/QuantTeleport/

QuantTeleport.html

• http://www.its.caltech.edu/~qoptics/teleport.html

• http://www.research.ibm.com/quantuminfo/teleportation/

• http://www.iop.org/EJ/article/1367630/9/7/211/njp7_7_211.html#nj248372

s4

• http://heart-c704.uibk.ac.at/publications/papers/nature04_riebe.pdf

• http://quantum.at/research/photonentangle/teleport/index.html

• http://www.quantum.physik.uni-

mainz.de/lectures/2004/ss04_quantenoptikseminar/quantumteleportation.pdf

• http://en.wikipedia.org/wiki/Quantum_teleportation


http://heart-c704.uibk.ac.at/publications/papers/nature04_riebe.pdf

http://www.iop.org/EJ/article/1367630/9/7/211/njp7_7_211.html#nj248372s4

http://www.iop.org/EJ/article/1367630/9/7/211/njp7_7_211.html#nj248372s4

http://www.research.ibm.com/quantuminfo/teleportation/

http://www.its.caltech.edu/~qoptics/teleport.html

http://www.upscale.utoronto.ca/GeneralInterest/Harrison/QuantTeleport/QuantTeleport.html

http://www.upscale.utoronto.ca/GeneralInterest/Harrison/QuantTeleport/QuantTeleport.html

http://www.sciam.com/article.cfm?id=why-teleporting-is-nothing-like-star-trek

http://www.sciam.com/article.cfm?id=why-teleporting-is-nothing-like-star-trek

http://www.primidi.com/2004/08/24.html

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