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 S
Seminar 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

ABSTRACT
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.

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 Title Page
No. No.
Abstract ii
List of figures iv
1 Introduction 1
2 History 6
3 How quantum teleportation works 8
3.1 Bell-state measurements 8
3.2 The teleporter 11
3.3 Working 12
3.4 Teleportation with squeezed light 14
3.5 Fidelity(quantum vs classic) 15

4 Concept 16
4.1 Description 16
4.2 Entanglement swapping 17
4.3 N-state particles 18
4.4 Result 19
4.5 Remarks 22

5 General teleportation scheme 23
5.1 General description 23
5.2 Further details 24

6 Applications 26
6.1 Quantum information 26
6.2 Quantum cryptography 27

7 References 30

iii

List of figures
Sl. Images Page
No. 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.

1
Division of Computer Engineering


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.

2


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

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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.

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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.

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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

6


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.

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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

8


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.

9


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 TELEPORTER
Now 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.

10


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 WORKING
Before going further, here is how quantum teleportation works.

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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).
This illustration
shows how the
experiment was
conducted.

Fig 3.3.2 River Danube Experiment
In "Teleportation Takes Quantum Leap," National Geographic explains
why this experiment is a world's premiere.

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"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

13


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)

14


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

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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

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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

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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

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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):

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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.

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• 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.

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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.

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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 .

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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:

24


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

25


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.

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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 1000001
Key 0010111
Encrypted 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 1
Key 0010111
A 1000001
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.

27


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

28


done, without revealing the results of their own measurements, by communication
over an open channel

7. REFERENCES

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• 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

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