a review on quantum teleportation based on: teleporting an ...kubarych/journal_pdfs/j... · quantum...

JOURNAL OF CHEMISTRY 571 VOLUME 1, NUMBER 1 DECEMBER 28, 2005

29

A review on quantum teleportation based on: Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels

Miri Shlomi Department of Chemistry, University of Michigan, 930 N. University Ave., Ann Arbor, MI 48109

(Received November 30, 2005; accepted December 14, 2005)

In 1993 Bennett et al discovered one of the

most astonishing features of Quantum Mechanics:

Quantum Teleportation. Teleportation, as science

fiction defined it was “scanning” an object

exactly, and then transferring it simultaneously to

another location. Most of the time, the initial

object being scanned was destroyed while the

exact replica appeared somewhere else. In the

past, however, this notion of teleportation was not

taken seriously by scientists as it was thought that

in order to make an exact copy of an object, the

Heisenberg principle would have to be violated.

The Heisenberg principle states that the more

accurately an object is scanned, the more it is

disturbed by the scanning process, until the point

where the object's original state has been

completely disrupted. Even at this point, not

enough information has been extracted in order to

make a perfect replica. This indeed sounds like a

solid argument against teleportation: if one cannot

extract enough information from an object to

make a perfect copy of it, then how could a

perfect copy be made? Nevertheless, the above

six scientists found a way around this logic, using

the astounding feature of quantum mechanics

known as the Einstein-Podolsky-Rosen effect or

quantum entanglement.

Entanglement is the process first described in

1935 by Einstein, Podolsky and Rosen. Using a

thought experiment, they found that two

entangled particles could interact with each other,

even though they were spatially separated; thus

making quantum mechanics a non-local theory.

This property actually made Einstein and others

dislike quantum mechanics. Einstein himself

referred to it as “spooky action at a distance”.

Since then quantum entanglement has been

experimentally observed many times. Therefore

Bennett et al ,using what they called an Einstein-

Podolsky-Rosen Channel showed that it is

possible to transmit a one qubit state from one

location (Alice) to another (Bob) sending 2 bits of

classical information. This is done without ever

finding out what really is. Suppose that Alice

has some particle in a certain quantum state and

she wants Bob, at a distant location, to have a

particle in that state. She could certainly send Bob

the particle directly. But suppose that the

communication channel between Alice and Bob is

not good enough to preserve the necessary

quantum coherence. Then Alice would have to

teleport her state. As mentioned above, there is no

measurement that Alice can do that will give Bob

sufficient information to reconstruct the state.

Quantum systems can be in a superposition of

several states, and a measurement on the quantum

system will force it into only one eigenstate of the

operator being measured—this is also known as

the projection postulate. Alice, on the other hand

can make her particle interact unitarily with

another system, or "ancilla", initially in a known

state0

a , in such a way that after the interaction

the original particle is left in a standard state0

,

while the ancilla is now in an unknown state a

containing complete information about .

A schematic figure of this procedure can be

seen in Fig. 1. To teleport object A, some

information from object A is scanned, while

causing the remaining unscanned part to be

transferred to C (through the EPR pair). This is

done without C ever being in direct contact with

A. Finally, depending on the scaned output, it is

possible to maneuver C into exactly the same

state A was in before it was scanned. A itself is no

longer in that state, hence teleportation is

achieved not replication! Indeed four years after


30

their paper was published, Bennett et al's protocol

was experimentally demonstrated1.

FIG. 1.

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

I. Quantum Teleportation

Now we turn to the mathematical derivation of

quantum teleportation. First it will be shown how

to teleport the quantum state of a spin-1

2

particle. Later, teleportation of more complicated

systems will be discussed.

The non-classical part of the teleportation

process will be transmitted first. To do so, two

spin- particles in an EPR singlet state are

prepared as follows:

( ) ( )23 2 3 2 3

1.

2= (1)

The subscript 2 and 3 refer to the particles in the

EPR pair(B and C in figure 1). Alice’s unknown

particle, whose state she wants to transfer,

will be labeled 1. Any of these 3 particles may be

1 D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl,

H. We-infurter, and A. Zeilinger, Nature 390, 575

(1997).

of different kinds as long as they are spin-

particles, for example electrons or protons.

Next one of the EPR pair is given to Alice,

while the other particle is given to Bob. At this

stage the entire system comprising Alice’s

particle and the EPR pair is in a pure product

state: 1

( )23

, that is at this stage Alice’s

particle (A in figure 1) and the EPR pair aren’t

entangled, nor do they have any classical

correlation between them.

The next step will be to entangle those three

particles. This will be done through a Von

Neumann measurement on the joint system

consisting of Alice’s particle and the EPR pair.

Von Neumann postulated that the state vector

evolves deterministically in a manner consistent

with Schrödinger's equation, until there is a

measurement, in which case there is a "collapse,"

which indeterministically alters the physical state

of the system. This is von Neumann's famous

"Projection Postulate."

Thus, von Neumann postulated that there were

two kinds of change that could occur in a state of

a physical system, one deterministic (Schrödinger

evolution), which occurs when the system is not

being measured, and one indeterministic

(projection or collapse), which occurs as a result

of measuring the system.

This measurement is performed in the Bell

operator basis consisting of the following four

states, which form a complete orthonormal basis

set:

( ) ( )12 1 2 1 2

1.

2

±= ±

( ) ( )12 1 2 1 2

1.

2

±= ±

The unknown state that is to be teleported can be

written as:

1 1 1,a b= +

(3)


31

With 2 2

1.a b+ =

The complete state of the three particles

before the measurement is, as mentioned earlier,

1

( )23

, or 123

.

( ) ( )123 1 2 3 1 2 3 1 2 3 1 2 3.

2 2

a b= +

(4)

If this equation is expressed in terms of the Bell

operator basis vectors ( )12

± and

( )12

±, then the following equation is obtained:

( ) ( ) ( ) ( )123 12 3 3 12 3 3 12 3 3 12 3 3

1.

2a b a b a b a b

+ += + + + + +

(5)

The four possible outcomes are equally likely to

be measured regardless of the state of particle 1.

The probability of obtaining any of those possible

outcomes is 1

4. Therefore after the Alice’s

measurement, Bob’s particle 3 will have been

projected into one of the four pure states of Eq.

(5). These are respectively:

3,

a

b

3

1 0,

0 1

3

0 1

1 0,

3

0 1

1 0 (6)

Now Alice can transmit to Bob the outcome of

her measurement. If she measured the first singlet

outcome, then Bob’s particle 3 is in the same state

as hers except for an irrelevant phase factor.

Hence, Bob needs not do anything to his particle

in order to create the desired replica. In any of the

other 3 cases Bob will need to apply any of the

unitary operators from equation (6).

Therefore an accurate teleportation can be

performed only when Alice communicates her

measurement outcome to Bob classically. The

classical message plays an important role in

teleportation. On may consider what will happen

if Bob becomes impatient and tries to guess

Alice’s result. Then Alice’s state 1

would be

reconstructed using (in the spin- state) a random

mixture of the four states of equation (6). This is a

maximally mixed state, and gives no information

about the initial state1

. Of course this could not

be no other way, because any correlation between

the input and guessed state will result in a

superluminal signal—an impossible event.

Alice, on the other hand, is left with particles

1 and 2 in either one of these states 12

± or

12

±, with no trace of her original

1 state.

This is a manifestation of the no-cloning theorem.

The term "cloning" in the quantum context, was

coined in a short paper by Wooters and Zurek.

The no cloning theorem describes the inability to

make separately measurable states. Or in other

words that measuring a state will destroy it. This

is another discrepancy from the classical world, as

can be seen in the following Fig. 2.

FIG. 2.

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

In Fig. 2, the difference between classical and

quantum “copies” is determined. In the classical

world, any number of scans can be made without

damaging the original. However, the copies are

never exact replicas of the original. In the

quantum world, copies cannot be made, as

mentioned previously, but the teleported object is

an exact replica of the original one.

Now what if Alice’s state was not a pure state,

but part of an entangled pair itself? Since

teleportation is a linear operation applied to the


32

quantum state1

, it will work on entangled

states as well as pure ones. For example if alice’s

1 was entangled with another particle 0 whose

state is0

. Then after teleportation particles 0

and 3 will be left in a singlet state, even though

they had originally belonged to uncorrelated

entangled states.

This logic can be extrapolated to systems

having N > 2 orthogonal states. Instead of an EPR

spin pair in a singlet state, Alice can use a pair of

N particles in a completely entangled state.

/

j

j j N

where j=0,1,…,N-1 labels the N elements of an

orthonormal basis for each of the N-state systems.

As before Alice performs a joint measurement on

particles 1 and 2. One of the measurements that

has the desired effect will be that whose

eigenvectors are defined by:

( )2 /mod /

ijn N

nm

j

e j j m N N= +

When Bob learns that Alice has received the

desired measurement, he can perform the

following unitary operation, thus giving his

particle Alice’s original state.

( )2 /mod /

ikn N

nm

k

U e k k m N N= +

As mentioned earlier quantum teleportation has

been realized experimentally. The next section

will discuss shortly the setup for such an

experiment.

FIG. 3.

(http://www.cco.caltech.edu/~qoptics/teleport.html)

II. A quantum teleportation experiment setting

In this experiment, entangled EPR beams are

generated by combining two beams of squeezed

light at a 50/50 beamsplitter. One of the beams is

propagated to Alice’s sending station, where it is

combined at a 50/50 beam splitter with the

unknown input state. Alice uses two sets of

balanced homodyne detectors, in order to make

the Bell-state measurement on the amplitude of

the combined state. Depending on which detector

is hit by the beam, Alice will tell Bob which

unitary transformation he has to apply to his

particle in order create a replica of Alice’s.

What do entangled photons look like?

"Photons emerging from type II down-conversion.

Photograph taken perpendicular to the

propagation direction. Photons are produced in

pairs.

A photon on the top circle is horizontally

polarized while it's exactly opposite partner in the

bottom circle is vertically polarized. At the

intersection points their polarizations are

undefined; all that is known is that they have to be

different, which results in entanglement."


33

FIG. 3. from Dik Bouwmeester, Jian-Wei Pan, Klaus

Mattle, Manfred Eibl, Harald Weinfurter & Anton Zeilinger

Nature, 390, 11 DECEMBER 1997.

III. Problems and potential for quantum teleportation

The most realistic application of quantum

teleportation outside the field of theoretical

physics is quantum computing. The main problem

with quantum teleportation is that almost any

interaction a quantum system has with its

environment constitutes a measurement. This

phenomenon, is called decoherence, and makes

further quantum calculation impossible. Thus,

when quantum teleportation is to be used, for

example, in quantum computing, the inner

workings of such a computer must somehow be

separated from its surroundings to maintain

coherence. But they must also be accessible so

that calculations can be loaded, executed and read

out.

http://english.pravda.ru/science/19/94/379/14789_

teleportation.html

Not there yet!

Scientific American, Anton Zeilinger, April 2000.

a review on quantum teleportation based on: teleporting an ...kubarych/journal_pdfs/j... · quantum...

Documents