hA , hA + |v A , v A ’ + (A ” B) ,

¦+ = e cos δ 2i sin δ hA , v A

2

(20.57)

√

i 2iφ0

¦’ hA , hA + |v A , v A ’ + (A ” B) .

= e i sin δ 2 cos δ hA , v A

2

(20.58)

Coincidence counts between the detectors at the output ports of the PBS will arise

from hA , v A , but not from hA , hA and |v A , v A . Since the coe¬cients depend on

the phase di¬erence δ, the two outcomes”coincidence counts or counts in one detector

only”can be separated by choosing δ to achieve destructive interference for one of the

terms. For example, adjusting L so that

(nh ’ nv ) ω

δ= L = nπ (20.59)

c

¾

Entanglement as a quantum resource

leads to the greatly simpli¬ed states

i 2iφ0 n

hA , hA + |v A , v A + (A ” B)

¦+ = e (’) (20.60)

2

and

i

¦’ = √ e2iφ0 (’)n+1 hA , v A + (A ” B) . (20.61)

2

In this case |¦’ produces coincidence counts between the h- and v-counters, while

|¦+ leads to two-photon counts in one or the other of the detectors.

The procedure outlined above constitutes a complete Bell measurement, but the

two photons must be hyperentangled. This Bell state analysis also makes substantial

demands on the photon counters. A demonstration experiment based on this scheme

has recently been carried out (Schuck et al., 2006). The result was that the four Bell

states could be identi¬ed with a probability in the range of 81%“89%. This is already

substantially greater than the 50% bound imposed by the no-go theorem for linear

optics, and further improvements of the experimental technique are to be expected.

20.4.2 Quantum teleportation

In quantum dense coding, the apparently arcane and counterintuitive property of

entanglement is precisely what allows Bob to transmit two classical bits of information

by means of local operations carried out on a single qubit. We next consider an even

more remarkable demonstration of the power of entanglement. In this scenario, Alice

has received a qubit in an unknown state |γ T ∈ HT ”where HT is the internal state

space of the qubit”and she wants to transmit this quantum information to Bob by

sending him two classical bits. This is the inverse of the quantum dense coding problem,

and the method used to accomplish this magic feat is called quantum teleportation

(Bennett et al., 1993).

If Alice were sent an unknown classical signal, she could simply make a copy and

send it to Bob, but the no-cloning theorem prohibits this action for an unknown

quantum signal. What, then, is Alice to do in the quantum case? Let us ¬rst consider

what can be done without the aid of any ancilla. In this situation, the only available

option is to measure the value of some observable OT = n · σT , where n is a unit

vector. Alice can measure OT and then tell Bob the components of n and the result,

(= ±1), of the measurement.

Bob™s task is to generate an approximation to the unknown state by using this

information. The only thing Bob knows is that the state |γ T has a nonvanishing

projection on the eigenstate | T of OT , so the best he can do is to prepare a qubit in

the mixed state

ρ = (1 + n · σB ) /2 ; (20.62)

see Ralph (2006) and Exercise 20.12. Under these circumstances, the average ¬delity

is 2/3. Since the attempt to send classical instructions for replicating |γ T does not

seem to be very promising, we next turn to the situation shown in Fig. 20.6, in which

Alice and Bob are supplied with an ancilla.

¾ Quantum information

1 photon

in same

unknown

state

OUT

Classical

channel

ALICE BOB

(Bell (local

state unitary

measurer) operations)

1 photon

in unknown

state

IN

Fig. 20.6 Schematic for quantum teleporta- 1 photon k) 1 photon k*

tion, in which an unknown polarization state

Source of

polarization-

of a photon entering Alice™s IN port is tele-

entangled

ported to become the same unknown polariza-

photons

tion state for the photon leaving Bob™s OUT

port.

A A generic teleportation model

In order to emphasize that the remarkable results of the following discussion apply to

all quantum systems, not just to photons, we will use the generic computational basis

de¬ned in eqn (20.13). In this notation, one example of an ancilla is provided by the

Bell state |Ψ’ AB de¬ned in eqn (20.15).

The complete three-particle system is described by the state

= Ψ’ AB |γ T

|˜ ABT

1

= √ [|0 A |1 B |γ ’ |1 |0 |γ T]. (20.63)

T A B

2

The order of the Hilbert-space vectors in the tensor product has no physical signi¬-

cance, so the three-particle state is equally well represented by

1 1

= √ |0 ’ √ |1

|˜ |γ |1 |γ |0 ∈ H A — HT — HB . (20.64)

AT B A T B A T B

2 2

The tensor products |u |γ (u = 0, 1) are given by

A T

|u |γ = γ0 |u, 0 + γ1 |u, 1 , (20.65)

A T AT AT

and the vectors |u, v AT are linear combinations of the Bell states spanning HA — HT ;

consequently”as one can show in Exercise 20.13”eqn (20.64) can be rewritten as

¾

Entanglement as a quantum resource

1’

|˜ Ψ AT {γ0 |0 B + γ1 |1 B }

=

AT B

2

1

+ Ψ+ AT {’γ0 |0 B + γ1 |1 B }

2

1

¦’ AT {γ1 |0 B + γ0 |1 B }

+

2

1

¦+ AT {’γ1 |0 B + γ0 |1 B } .

+ (20.66)

2

Having mastered this theory, Alice now performs a Bell measurement on her two

qubits. According to von Neumann, the result will be to project |˜ AT B onto one

of the four Bell states of HA — HT . Alice then sends Bob a message”of length two

bits”informing him which of the four possible outcomes actually occurred.

Bob, who has also learnt the theory, then knows that his qubit is in one of the states

shown in the four lines of eqn (20.66). For example, if Alice found |Ψ’ AT , then Bob

knows that his qubit is guaranteed to be in the original unknown state |γ T . The other

three states are related to the original state in one of three ways: (1) a phase-¬‚ip

(changing the relative phase of |0 B and |1 B by 180—¦ ); (2) a bit-¬‚ip (interchanging

|0 B and |1 B ); and (3) a combined phase- and bit-¬‚ip. In each of these cases, there

is a unitary operator”Upf for the phase-¬‚ip, Ubf for the bit-¬‚ip, and Upf Ubf for the

combination”that transforms the corresponding state into the state |γ B .

In an optical experiment, the unitary operators are realized by appropriate combi-

nations of beam splitters and phase shifters (Reck et al., 1994). By sending the photon

in his apparatus through the optical elements corresponding to the appropriate uni-

tary transformation, Bob can be sure that the qubit emitted from his OUT port is an

exact replica of the qubit given to Alice.

In this process, the only physical objects transferred from Alice to Bob are the

carriers of the two bits delivered through the classical channel. Consequently, the

teleportation process is limited by the speed of light, and it does not violate any

conservation laws.

This result raises several puzzles. The ¬rst is: What happened to the no-cloning

theorem? After all, we have just claimed that the procedure ends with Bob in posses-

sion of a perfect copy of the qubit sent to Alice. The answer is that the original qubit

no longer exists, so that the no-cloning theorem is not violated.

For any outcome of Alice™s Bell state measurement, the T -qubit is described by

the corresponding Bell state of HA — HT ; no information about the original state

|γ T is left in the A“T subsystem. In fact, any attempt on Alice™s part to ¬nd out

something about |γ T , before performing the Bell state measurement, would frustrate

the teleportation process. This is analogous to the destruction of the interference

pattern by any attempt to determine which pinhole a photon passes through in a

Young™s-type experiment. This leads to the very strange conclusion that neither Alice

nor Bob has any information about the mystery qubit |γ T , despite the fact that Bob

can be certain that he has a perfect copy.

An equally puzzling issue is the apparent discrepancy between the amount of in-

formation that is needed to specify |γ T and the two bits actually sent by Alice. To

see this explicitly, let us write

¾ Quantum information

θT θT

|γ |0 |1

+ eiφT sin

= cos , (20.67)

T T T

2 2

so that the state is represented by the point (θT , φT ) on the Poincar´ sphere. Precisely

e

specifying this point would require an in¬nite number of bits, and even a crude ap-

proximation would require many more than two bits. Thus it would seem that Alice

is getting an in¬nite return on her two bit investment.

The key to understanding this situation is that quantum results require careful

interpretation. In the present instance, the apparently in¬nite information carried by

|γ T is only potentially available. Measuring an observable OB = n · σB will provide

exactly one bit of information: the binary choice between the eigenvalues +1 and ’1.

This is, nevertheless, an amazing result. A potentially in¬nite number of bits have

been delivered by combining the entanglement resource with just two classical bits of

information.

Finally, there is a conceptual issue arising from the use of the word ˜teleportation™.

The question is: What has actually been transported? For this discussion, it is better

to replace the abstract formulation used above by a concrete example. Suppose that

the mystery qubit |γ T is a superposition of the states of a two-level atom, and that

the ancilla is an entangled state of a photon (sent to Alice) and an electron (sent to

Bob).

At the end of the process, Bob™s particle is described by the same superposition

as the one supplied to Alice, but the physical substrate is the two spin states of the

electron, not another two-level atom. For this example, one could argue that the term

quantum faxing might be more appropriate. It is true that quantum faxing”unlike

classical faxing”requires the destruction of the original information, but that is simply

the price that must be paid for working in the quantum domain.

A sceptically inclined onlooker might conclude that ˜teleportation™ is simply an-

other example of the irrationally exuberant terminology sometimes found in the ¬eld

of quantum information, but this would not be quite fair. Let us now consider a dif-

ferent example in which all three particles are photons. In this case, the photon in

Bob™s possession at the end is physically indistinguishable”at the most fundamental

level”from the original photon supplied to Alice; consequently, using the evocative

term ˜teleportation™ seems entirely reasonable.

B Teleportation of photons

Since this is a book on quantum optics, we will now concentrate on the three-photon

case. The only formal change in the theory is that the tensor products of states used

above are replaced by products of creation operators acting on the vacuum. Thus the

initial three-photon state is

= a† [γ] Ψ’

|˜ , (20.68)

T

ABT AB

where a† [γ] = γh a† h + γv a† v creates the unknown photon state in the T -channel,

T T

T

and the ancilla shared by Alice and Bob is given by the Bell state

1

Ψ’ = √ {|hA , vB ’ |vA , hB }

AB

2

¾

Entanglement as a quantum resource

1

= √ a† A h a† B v ’ a† A v a† B h |0 . (20.69)

2k k k k

The tensor product algebra used in the generic discussion is exactly mirrored by alge-

braic manipulations of the products of creation operators, so the theoretical argument,

as seen in Exercise 20.14, goes through as before.

The ¬rst laboratory demonstration of quantum teleportation for photons was car-

ried out by Bouwmeester et al. (1997). In this experiment a pulse of UV light produces

the ancillary photons in the A- and B-channels by down-conversion. The pulse is then

retrore¬‚ected to pass through the nonlinear crystal again, and thus produce another

pair of photons in the T - and T -channels. The T -channel photon is prepared in the

polarization state γ = (γh , γv ), and detection of the T photon signals that the mystery

photon is on the way.

In this proof-of-principle experiment the full Bell state analysis was replaced by

a simpler procedure in which the A“T pair is allowed to fall on the two input ports

of a beam splitter. The experimental arrangement can be extracted from Fig. 20.5

by changing B to T and omitting the birefringent elements and the polarizing beam

splitters.

The necessary two-photon interference e¬ects at the beam splitter will only occur

if the two wave packets overlap. In other words, it must not be possible to distinguish

the A- and T -wave packets by their arrival times. For this purpose, both photons

were sent through frequency ¬lters that narrowed their frequency spread and therefore

broadened their temporal spread. Of course, the ¬lters also cut down substantially on

the count rate, but this sort of trade-o¬ is a common feature of optical experiments.

As we have already seen, coincidence counts in the detectors in the A and B arms

of the apparatus signal that the Bell state |Ψ’ AT has been detected. Alice relays this

information to Bob, who then knows that the photon in the B-channel is in the same

polarization state as the photon that was sent to Alice. This will happen only one time

out of four, so the success rate for teleportation is less than 25%. In a later version of

this experiment (Pan et al., 2003) ¬delity in the successful cases exceeded 80%.

It should now be clear that Alice™s Bell state measurement poses substantial ex-

perimental di¬culties. In Section 20.4.1-B we presented a complete Bell state analysis

due to Kwiat and Weinfurter (1998), but their method avoids the no-go theorem by

relying on the hyperentanglement of down-converted photon pairs.

In a teleportation experiment, the photon state to be teleported and the two ancilla

photons are generated by independent sources; consequently, the photon in the T -

channel is only entangled with the ancilla photons in the A- and B-channels to the

minimal extent required by Bose statistics. Thus the no-go theorem limits any linear

optical scheme for discriminating between the photonic Bell states {|Ψ± AT , |¦± AT }

in a teleportation experiment to a 50% success rate.

This limitation on the success rate does not, however, mean that only one Bell state

can be detected. A three-Bell-state analyzer (van Houwelingen et al., 2006)”employing

only linear optics and no additional ancillary photons”and a four-Bell-state analyzer

(Walther and Zeilinger, 2005)”depending on additional ancillary photons”have both

been experimentally demonstrated.

The obstacles presented by the no-go theorem for linear optics suggest exploiting

¿¼ Quantum information

nonlinear optical e¬ects. An experiment of this kind has been performed (Kim et al.,

2001) by using sum-frequency generation (SFG)”the inverse of down-conversion”in

type-I and type-II crystals. This technique permits a full Bell state analysis, but the

e¬ciency is strongly limited by the weakness of the SFG e¬ect and the necessity of

ensuring a good overlap between the spatial modes. The observed ¬delity of F = 0.83

is a convincing demonstration of quantum teleportation, but the low count rate means

that this method is not yet useful for quantum communication protocols.

20.5 Quantum computing

The ¬rst proposals for quantum computing were independently made in 1982 by Be-

nio¬ (1982) and Feynman (1982). Benio¬ presented a quantum version of a Turing

machine that would operate without dissipation of energy, while Feynman was inter-

ested in the possible use of a quantum computer to simulate the behavior of other

quantum systems.

These papers excited a substantial amount of interest at the time, but the rapid

growth in this ¬eld was ¬rst stimulated by the work of Deutsch and Jozsa (1992),

Grover (1997), and Shor (1997).

Deutsch and Jozsa demonstrated a quantum algorithm for a certain decision prob-

lem that is guaranteed to be exponentially faster than any classical algorithm.

Grover showed that a quantum computer could search a database of length N

√

in a time”i.e. a number of steps”proportional to N . The optimum time for a

classical search strategy is proportional to N , so Grover™s work constitutes a rigorous

demonstration of a problem of practical interest for which a quantum computer is

superior to any classical computer.

Shor™s work concerned the problem of ¬nding the prime factors of an integer N .

The most e¬cient known classical algorithm, the number ¬eld sieve, requires a time

t ∼ exp 2 (ln N )1/3 (ln ln N )2/3 to ¬nd the factors. This time grows faster than any

power of ln N , and it is ¬rmly believed”but not proven”that all classical factoriza-

tion algorithms share this property. Shor demonstrated a quantum algorithm with a

factorization time t ∼ (ln N )3 , i.e. it is only polynomial in ln N . The appearance of a

quantum computer would therefore be very bad news for those using trapdoor codes

that depend on the di¬culty of factoring large integers.

The Grover and Shor algorithms are quite complicated, and in any case are be-

yond the purview of this book. For the general topic of quantum computing, we will

restrict ourselves to a very brief discussion of the prevailing generic model. More de-

tailed descriptions can be found in several texts, e.g. Nielsen and Chuang (2000). This

introduction will be followed by a brief discussion of a proposed all-optical scheme.

For topics like this that are the subject of current investigations the best strategy is

to consult recent review articles, e.g. Ralph (2006).

20.5.1 A generic model for quantum computers

Feynman™s original proposal was motivated by the extreme computational demands

of quantum theory. Consider, for example, a very simple classical system composed of

N bits. In this case there are 2N possible states, each labeled by an N -digit binary

number.

¿½

Quantum computing

By contrast, the states of a quantum system consisting of N qubits occupy a

Hilbert space of dimension 2N . The number of basis vectors is the same as the number

of classical states, but the superposition principle requires the inclusion of all possible

linear combinations of the basis vectors.

As we have seen in Section 18.7.2, the density matrix for this system has O 22N

elements. For a system of modest size, e.g. N = 100, the dimension of the quantum

state space is O 1030 . Simulating this system on a classical computer is possible in

principle, but the memory and running time needed make it impossible in practice.

This prompted Feynman to consider replacing the classical computer by a quantum

computer.

Generally speaking, a quantum computer is any device that employs speci¬cally

quantum e¬ects, such as entanglement, to accomplish a computational task. The stan-

dard conceptual model currently in use includes a collection of N qubits called a quan-

tum register, which is initially in some state |Λin , and a unitary transformation Ualg

that implements the algorithm.

Since unitary transformations are invertible, this scheme represents a reversible

quantum computer. The unitary transformation is expressed as the product of a

set of standard transformations, called quantum gates, that operate on a few qubits

at a time. The result of the computation is read out by performing measurements

on some or all of the qubits. The corresponding theoretical operation is the projec-

tion of the output state Ualg |Λin onto the basis vector describing the measurement

outcome.

A Quantum parallelism

The procedure outlined above has two crucial features related to the unitary trans-

formation and the measurement step respectively. The unitary transformation is in-

vertible, so it preserves the enormous amount of information in the state vector. This

property, which is called quantum parallelism, o¬ers the possibility of converting

the high dimension of the Hilbert space from a di¬culty into an advantage.

The measurement step renders the outcome probabilistic; there is no way of pre-

dicting which of the possible measurement outcomes will occur. Running the algorithm

twice will in general produce di¬erent results. Furthermore, the reduction of the state

vector accompanying the measurement destroys all the information associated with

the measurement outcomes that did not occur.

Successful quantum algorithms”such as those of Grover and Shor”are cleverly

contrived to achieve good results in spite of the evident tension between the unitary

algorithm and the reductive measurement. For example, Shor™s algorithm does not

always result in factorization, but it does succeed with high probability.

A simple example illustrating quantum parallelism is provided by the following toy

problem which employs a variant of the Deutsch“Jozsa algorithm. Consider a function,

f (x), where x ranges over {0, 1} and f (x) can only have the values 0 or 1. There are

exactly four such functions, so a classical algorithm for f (x) must be provided with

two bits of data to specify which function is to be evaluated.

The computer and the algorithm are shrouded in secrecy inside a black box, but

we are allowed to submit values of x in order to get f (x). If we want to know both

¿¾ Quantum information

f (0) and f (1), then we must either run the algorithm twice”once for each input”or

else run two identically programmed computers in parallel.

As an alternative, suppose there is a hidden quantum computer with a two-qubit

register. In this situation, programming the computer to yield a given set of values

f = (f (0) , f (1)) is the same as the quantum dense coding problem. In Section 20.4.1

we saw that it is always possible to devise a set of unitary operations that convert a

known initial state into one of the Bell states. We may as well simplify this part of

the problem by assuming that the initial state of the quantum register is itself a Bell

state, e.g. the initial state |˜ of the dense coding discussion is replaced by |¦+ .

In accord with the usual conventions in the ¬eld of quantum information processing,

we will also assume that the unitary operators act on the ¬rst, rather than the second,

qubit. If we associate the possible functions f with operators U f according to the

encoding scheme