ńņš. 24 |

(= Ā±1), of the measurement.

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

information. The only thing Bob knows is that the state |Ī³ 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

Ļ = (1 + n Ā· ĻB ) /2 ; (20.62)

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

is 2/3. Since the attempt to send classical instructions for replicating |Ī³ 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.

Ā¾ 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 Aliceā™s IN port is tele-

entangled

ported to become the same unknown polariza-

photons

tion state for the photon leaving Bobā™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

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

Bell state |ĪØā’ AB deļ¬ned in eqn (20.15).

The complete three-particle system is described by the state

= ĪØā’ AB |Ī³ T

|Ī˜ ABT

1

= ā [|0 A |1 B |Ī³ ā’ |1 |0 |Ī³ T]. (20.63)

T A B

2

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

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

1 1

= ā |0 ā’ ā |1

|Ī˜ |Ī³ |1 |Ī³ |0 ā H A ā— HT ā— HB . (20.64)

AT B A T B A T B

2 2

The tensor products |u |Ī³ (u = 0, 1) are given by

A T

|u |Ī³ = Ī³0 |u, 0 + Ī³1 |u, 1 , (20.65)

A T AT AT

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

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

Ā¾

Entanglement as a quantum resource

1ā’

|Ī˜ ĪØ AT {Ī³0 |0 B + Ī³1 |1 B }

=

AT B

2

1

+ ĪØ+ AT {ā’Ī³0 |0 B + Ī³1 |1 B }

2

1

Ī¦ā’ AT {Ī³1 |0 B + Ī³0 |1 B }

+

2

1

Ī¦+ AT {ā’Ī³1 |0 B + Ī³0 |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 |Ī˜ AT B onto one

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

bitsā”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 |ĪØā’ AT , then Bob

knows that his qubit is guaranteed to be in the original unknown state |Ī³ T . The other

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

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

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

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

combinationā”that transforms the corresponding state into the state |Ī³ 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 ļ¬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 Aliceā™s Bell state measurement, the T -qubit is described by

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

|Ī³ T is left in the Aā“T subsystem. In fact, any attempt on Aliceā™s part to ļ¬nd out

something about |Ī³ 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

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

nor Bob has any information about the mystery qubit |Ī³ 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 |Ī³ T and the two bits actually sent by Alice. To

see this explicitly, let us write

Ā¾ Quantum information

ĪøT ĪøT

|Ī³ |0 |1

+ eiĻT sin

= cos , (20.67)

T T T

2 2

so that the state is represented by the point (ĪøT , ĻT ) on the PoincarĀ“ sphere. Precisely

e

specifying this point would require an inļ¬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 inļ¬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 inļ¬nite information carried by

|Ī³ T is only potentially available. Measuring an observable OB = n Ā· ĻB will provide

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

This is, nevertheless, an amazing result. A potentially inļ¬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 ā˜teleportationā™.

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 |Ī³ 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, Bobā™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 faxingā”unlike

classical faxingā”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 ā˜teleportationā™ is simply an-

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

Bobā™s possession at the end is physically indistinguishableā”at the most fundamental

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

term ā˜teleportationā™ 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

= aā [Ī³] ĪØā’

|Ī˜ , (20.68)

T

ABT AB

where aā [Ī³] = Ī³h aā h + Ī³v aā 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

ĪØā’ = ā {|hA , vB ā’ |vA , hB }

AB

2

Ā¾

Entanglement as a quantum resource

1

= ā aā A h aā B v ā’ aā A v aā 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 ļ¬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

retroreļ¬‚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 Ī³ = (Ī³h , Ī³v ), 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 Aā“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 eļ¬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 ļ¬lters that narrowed their frequency spread and therefore

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

the count rate, but this sort of trade-oļ¬ 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 |ĪØā’ 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) ļ¬delity in the successful cases exceeded 80%.

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

perimental diļ¬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 {|ĪØĀ± AT , |Ī¦Ā± 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)ā”employing

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

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

been experimentally demonstrated.

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

ĀæĀ¼ Quantum information

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

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

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

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

ensuring a good overlap between the spatial modes. The observed ļ¬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 ļ¬rst proposals for quantum computing were independently made in 1982 by Be-

nioļ¬ (1982) and Feynman (1982). Benioļ¬ 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 ļ¬eld was ļ¬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

ā

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

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

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

superior to any classical computer.

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

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

t ā¼ exp 2 (ln N )1/3 (ln ln N )2/3 to ļ¬nd the factors. This time grows faster than any

power of ln N , and it is ļ¬rmly believedā”but not provenā”that all classical factoriza-

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

factorization time t ā¼ (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 diļ¬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

Feynmanā™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.

ĀæĀ½

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 speciļ¬cally

quantum eļ¬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 |Īin , 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 |Īin 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, oļ¬ers the possibility of converting

the high dimension of the Hilbert space from a diļ¬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 diļ¬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 algorithmsā”such as those of Grover and Shorā”are cleverly

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

algorithm and the reductive measurement. For example, Shorā™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 Deutschā“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

ĀæĀ¾ Quantum information

f (0) and f (1), then we must either run the algorithm twiceā”once for each inputā”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 |Ī˜ of the dense coding discussion is replaced by |Ī¦+ .

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

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

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

encoding scheme

10 10

U (0,0) = , U (1,1) = ,

0 ā’1

01

(20.70)

01 01

(0,1) (1,0)

U ,U ,

= =

ā’1 0

10

then it is easy to verify that

U (1,1) Ī¦+ = Ī¦ā’ , U (0,1) Ī¦+ = ĪØ+ , U (1,0) Ī¦+ = ĪØā’ . (20.71)

After the programmer supplies the two bits needed to choose the operator U f ā”

i.e. the one that gives the same output as the classical computerā”the output of the

computation is obtained by performing a Bell state measurement. If the result is |ĪØ+ ,

then f = (0, 1), etc. The important point is that it is only necessary to run the quantum

algorithm once to get both values f (0) and f (1). Thus quantum parallelism gives the

same result as classical parallelism, but the work of the two classical computers is done

by one quantum computer.

Quantum logic gatesā—

B

The description of the simple quantum computer given in the last section ļ¬ts con-

veniently with the discussion of quantum dense coding in Section 20.4.1, but it does

not have the form commonly used in the quantum computing literature. The usual

procedure is to express the operator Ualg as the product of a standard set of unitary

operators, called quantum logic gates, that typically act on one or two qubits out

of the N qubits in the register. Since the output of each gate serves as input to the

next, the collection of gates can be visualized as a quantum circuit.

Classical computers employ operations on single bits and pairs of bits, and it has

been shown that the most general computation can be performed by means of a single

kind of two-bit gate combined with a collection of single-bit gates. An analogous result

holds for quantum computers, so we only need to consider a single kind of two-qubit

gate.

A one-qubit logic gate is completely speciļ¬ed by its action on the basis vectors |0

and |1 ; for example, the X gate is deļ¬ned by X |0 = |1 , and X |1 = |0 . This is

analogous to the classical NOT gate that interchanges 0 (false) and 1 (true). There

ĀæĀæ

Quantum computing

are also useful one-qubit gates that do not have classical analogues, such as the Z

gate: Z |0 = |0 , Z |1 = ā’ |1 , and the Hadamard gate:

1 1

H |0 = ā {|0 + |1 } , H |1 = ā {|0 ā’ |1 } .

2 2

These gates can all be expressed as 2 Ć— 2 matrices, andā”as seen in Exercises 20.15

and 20.16ā”they are also related to rotations on the PoincarĀ“ sphere.

e

An important two-qubit gate is the controlled-NOT (C-NOT) gate, deļ¬ned by

CNOT |a, b = |a, b ā• a (a, b = 0, 1) , (20.72)

where ā• represents addition modulo 2. The ļ¬rst and second qubits in the two-qubit

state |a, b = |a |b are conventionally called the control qubit and the target qubit

respectively.

Thus the C-NOT gate has the following eļ¬ects. (1) The control qubit is left un-

changed. (2) The target qubit is ļ¬‚ipped if the control qubit is 1, and left alone if the

control qubit is 0. A convenient graphical notation for these standard gates is shown

in Fig 20.7.

Another useful two-qubit gate is the controlled-sign or controlled-phase gate

deļ¬ned by

ab

CS |a, b = (ā’1) |a, b (a, b = 0, 1) . (20.73)

This operation does nothing unless both the control and target qubits are |1 , in which

case it multiplies the two-qubit state by ā’1.

C Quantum circuitsā—

In Section 20.5.1-A we ļ¬‚outed the convention that the register always begins in a

standard state, e.g. |Īin = |0, 0 . It is easy to verify that |Ī¦+ = CNOT H |0, 0 , i.e.

the initial state used in the previous discussion is built up from the standard state by

applying a Hadamard gate followed by a controlled-NOT gate.

Inspection of eqn (20.70) shows that the operator U (0,1) leading to the outcome

|ĪØ+ is an X gate, so the result f = (0, 1) is achieved by the unitary transformation

|ĪØ+ = U (0,1) |Ī¦+ = XCNOT H |0, 0 . The corresponding quantum circuit diagram,

shown in Fig. 20.8, is to be read from left to right. Other examples are considered in

Exercise 20.17.

Fig. 20.7 Graphical representations of quantum logic gates: (a) a generic one-qubit gate,

and (b) a controlled-NOT gate, with control qubit |a and target qubit |b .

Āæ Quantum information

Fig. 20.8 Quantum circuit diagram for

the program implemented by the sequence:

Hadamard gate, controlled-NOT gate, and X

gate.

Quantum computing experimentsā—

20.5.2

Experimental realizations of the idealized devices discussed above must overcome a

number of very serious diļ¬culties. To begin with, the qubits must be controllable

to one part in 104 by means of analog pulses (Berggren, 2004). This is an especially

acute problem if the qubits are carried by photons. The dissipative interaction of the

qubits with the environment poses a still more daunting obstacle, since the resulting

decoherence will destroy the entangled state.

Decoherence can be reduced by clever design, but it is impossible to eliminate it

altogether. This fact has necessitated the introduction of error-correction protocols,

ļ¬rst by Shor (1995), and later by Bennett et al. (1996) and Knill and Laļ¬‚amme (1997).

A common feature of these schemes is the use of a large number of ancillary qubits to

guarantee the accuracy of the computation.

The necessity of error correction is a strong contributor to estimates that something

like 106 qubits would be needed for a computation of practical interest (Berggren,

2004). Experiments performed to date only involve a few qubits, but scalability, i.e.

the potential for extending a scheme to a very large number of qubits, is a primary

concern.

The ļ¬rst experimental demonstrations of quantum computing (Chuang et al., 1998;

Vandersypen et al., 2001) used the method of bulk quantum computation (Knill et al.,

1998), in which a large number of qubitsā”provided by spin-1/2 nuclei in moleculesā”

are manipulated in parallel by nuclear magnetic resonance (NMR) techniques. This

approach is adequate for proof-of-principle demonstrations but cannot be used for

register sizes much greater than ten.

In order to achieve scalability, subsequent proposals have concentrated on vari-

ous solid-state systems, e.g. nuclear spins of donor atoms in Si (Kane, 1998), elec-

tron spins in quantum dots (Loss and DiVincenzo, 1998; Petta et al., 2005), qubits

formed by counter-circulating persistent currents in Josephson junction circuits (Mooij

et al., 1999), electron-spin-resonance transistors (Vrijen et al., 2000), and electron spins

bound to deep donor states in Si (Stoneham et al., 2003).

The physical system of greatest interest for usā”the photonā”is conspicuously ab-

sent from this list of candidates for quantum computers. The reason is that a two-qubit

logic gate, such as the C-NOT gate discussed in Section 20.5.1-B, can only produce an

entangled stateā”in our terminology a dynamically entangled stateā”of two photons

by means of photonā“photon coupling, i.e. an optical nonlinearity.

As suggested by Milburn (1989), one way to do this would be to induce a cross-

Kerr couplingā”see Section 13.4.3ā”between two optical modes. Unfortunately, the

materials provided by nature have Ļ(3) s that are orders-of-magnitude too small to

accomplish the desired eļ¬ects. Increasing the length of the nonlinear region does not

Āæ

Quantum computing

help, because the accompanying linear absorption will defeat the purpose of the device.

Another possibility is to trap an atom in a very small, high ļ¬nesse cavity, but this

approach has not yet been successful. This situation led to the general feeling that

large-scale quantum computing by optical means is not a practical possibility.

Two-photon logic gates with linear opticsā—

20.5.3

The consensus view that optical methods are not suitable for quantum computing

was challenged by the work of Knill, Laļ¬‚amme, and Milburn (KLM) (Knill et al.,

2001), who showed that quantum algorithms could be implemented by combining

single-photon sources, photon detectors, and passive linear optical elements.

Their scheme eliminated the need for strong optical nonlinearities in the manipula-

tion of photonic qubits. This is a complex and rapidly evolving subject, so we will only

sketch the ļ¬rst step in its development. More details can be found in recent review

articles, e.g. Ralph (2006).

One possible designā”adapted from the work of Hofmann and Takeuchi (2002)ā”for

a two-photon logic gate utilizing only linear optics and photon detection is shown in

Fig. 20.9.

This is a four channel/eight port device; the four input ports are the Control-

in port, the Target-in port, and the unused ports of beam splitters 1 and 3, that

Control-in

Vac-1

Loss-1 D

L

Target-in

D L

L D

Control-out

L

Vac-2

D

Loss-2

Target-out

Fig. 20.9 Schematic for a nondeterministic control-NOT gate. The polarizing beam splitters

transmit v-polarized light and reļ¬‚ect h-polarized light at 90ā—¦ . The half-wave plate (hwp) at

the control input is aligned at Ļ‘ = 0, while the hwps at the target input and output ports

are aligned at Ļ‘ = ā’22.5ā—¦ , where Ļ‘ is the angle between the h-polarization and the fast axis;

see Exercise 20.8. The beam splitters are asymmetric.

Āæ Quantum information

communicate with the vacuum channels Vac-1 and Vac-2. The beam splitters are

asymmetric, with scattering laws of the general form

ā

ā

a1 = 1 ā’ Ra1 ā’ Ra2 ,

ā ā (20.74)

Ra1 + 1 ā’ Ra2 ,

a2 =

worked out in Exercise 8.1. All three beam splitters are assumed to have the same

reļ¬‚ectivity R, and the sign factors = Ā±1 are chosen to accomplish the design objec-

tives. The half-wave plate at the control input is a Z gate, and the half-wave plates at

the target ports are Hadamard gates.

The use of passive optical elements ensures that photon number is conserved, so for

two incident photonsā”one in the control channel and one in the target channelā”we

can be sure that exactly two photons will be emitted. However, the mixing occurring at

the beam splitters implies that the output state will be a superposition of all possible

two-photon states in the output channels: Control-out, Target-out, Loss-1, and Loss-2.

The central beam splitter is particularly important in this regard, since photons

are incident from both sides. As we have seen in Section 10.2.1, this is precisely the

situation required for the strictly quantum interference eļ¬ects associated with diļ¬erent

Feynman paths having the same end point.

The key to the operation of this gate is postselection, i.e. discarding all outcomes

that do not satisfy a chosen criterion. In the present case, the ļ¬rst part of the criterion

is that detectors in the Control-out and Target-out channels should eventually register

a coincidence count. The states that can contribute to such a coincidence event are su-

perpositions of the coincidence basis states {|hC , hT , |hC , vT , |vC , hT , |vC , vT }.

Satisfying the condition (20.72) for a control-NOT gate further requires that ex-

actly one member of the coincidence basis occurs in the output state for each of the four

possible input states. A rather lengthy calculation, outlined in Exercise 20.18, shows

that this goal can only be reached for the value R = 1/3 and asymmetry parameters

satisfying 2 = ā’ 3 = 1 .

With these values, the operation of the gate is given by

1 1

CNOT |hC , hT = |hC , hT + Ā· Ā· Ā· , CNOT |hC , vT = |hC , vT + Ā· Ā· Ā· ,

3 3 (20.75)

1 1

CNOT |vC , hT = |vC , vT + Ā· Ā· Ā· , CNOT |vC , vT = |vC , hT + Ā· Ā· Ā· ,

3 3

where ā˜Ā· Ā· Ā· ā™ contains the terms that are not in the subspace spanned by the coincidence

basis. The target photon polarization is unchanged if the control photon is h-polarized

but ļ¬‚ipped (h ā” v), when the control photon is v-polarized. With the identiļ¬cation

h ā” 0 and v ā” 1, this is the photonic version of eqn (20.72).

A simple modiļ¬cation of the design in Fig. 20.9 yields the gate action

1 1

CS |hC , hT = |hC , hT + Ā· Ā· Ā· , CS |hC , vT = |hC , vT + Ā· Ā· Ā· ,

3 3 (20.76)

1 1

CS |vC , hT = |vC , hT + Ā· Ā· Ā· , CS |vC , vT = ā’ |vC , vT + Ā· Ā· Ā· ,

3 3

which satisļ¬es the deļ¬nition (20.73) of a controlled-sign gate.

Āæ

Quantum computing

The postselection criterion picks out the appropriate outcomes, but the probability

2

for successful operation is (1/3) = 1/9. Eight times out of nine, the two photons are

emitted into the wrong channels, e.g. one photon into Loss-2 and one into Control-1,

or two photons into Control-out, etc. The success or failure of the gate is heraldedā”

i.e. the outcome is knownā”by the presence or absence of a coincidence between the

Control-out and Target-out channels. Additional checks could be made by detecting

photons emitted into the loss channels, or by discriminating between one- or two-

photon events in the control and target channels.

This gate has been experimentally realized (Oā™Brien et al., 2003) by using down-

conversion to produce the input photons and quantum state tomography to verify

that the output states agreed with the theoretical model. In this approach, the neces-

sity for dynamically coupling the two photons has been avoided by incorporating the

coincidence measurement as part of the action of the device.

Linear optical quantum computingā—

20.5.4

The quantum logic gate discussed above does provide a nontrivial two-qubit operation,

but it fails the scalability test. A device containing many such gates, each with a success

probability of 1/9, would almost never work. In their general scheme, KLM answered

this objection by making use of the ideas involved in quantum teleportation.

The use of quantum teleportation to carry out general quantum computations was

ļ¬rst suggested by Gottesman and Chuang (1999), and KLM showed that a so-called

teleportation gate could be realized with a high probability of success, given a

suļ¬ciently complex entangled state.

The KLM approach avoids the failure mode associated with a vanishingly small

success probability, but the resources required are too large for practical scalability.

For example, the number of Bell pairsā”i.e. pairs of photons described by a Bell

stateā”needed to implement a single controlled-sign gate with success probability of

95% is of the order 10 000 (Ralph, 2006).

This resource cost can be greatly reduced by using parity-state encoding (Hayes

et al., 2004). Single-qubit parity states are the alternative basis states,

ā

|Ā± = (|0 Ā± |1 ) / 2 , (20.77)

that satisfy X |Ā± = Ā± |Ā± . In parity-state encoding, the logical 0 and 1 are represented

by n-qubit states:

1

|0 (n) = ā ā—n |+ j + ā—n |ā’ j ,

j=1 j=1

2

(20.78)

1

(n)

= ā ā—j=1 |+ j ā’ ā—j=1 |ā’ j .

|1 n n

2

A clever application of this encoding scheme reduces the overhead cost to the order of

100 Bell pairs per gate.

An alternative schemeā”which actually amounts to a fundamentally diļ¬erent model

for quantum computingā”grew out of a theoretical proposal by Raussendorf and Briegel

(2001). In the standard model sketched in Section 20.5.1, an algorithm is represented

as a sequence of unitary operators that are physically realized by quantum logic gates.

Āæ Quantum information

The logic gates produce a sequence of entangled states that ends in the desired ļ¬nal

state, which is measured to produce the computational result.

In the new model, the entanglement resource is prepared beforehand, in the form

of a highly entangled, multi-qubit, initial state. The nature of this state is most easily

understood by visualizing the qubits as spin-1/2 particles attached to the sites of a

lattice and interacting through nearest-neighbor coupling. A cluster is a collection

of occupied lattice points such that each pair of sites is connected by jumps across

nearest-neighbor links. Each qubit is initially prepared in the parity state |+ and a

cluster state is generated by pair-wise entanglement of the initial qubits.

As an example, consider a one-dimensional lattice with three occupied sites, 1, 2,

3, so that the initial state is |+ 1 |+ 2 |+ 3 . The corresponding cluster state can be

generated by successive application of controlled-sign gates as follows:

(1,2) (2,3)

|Ī¦lin3 = CS |+ CS |+ |+ , (20.79)

1 2 3

(i,j)

where CS acts on two-qubit states |a i |b j . Carrying out the gate operations, with

the aid of the deļ¬nitions (20.73) and (20.77), leads to the explicit expression

1

|Ī¦lin3 = ā [|+ |0 |+ + |ā’ |1 |ā’ 3 ] (20.80)

1 2 3 1 2

2

for the cluster state. The cluster states needed for nontrivial calculations generally

involve clusters on two-dimensional lattices and many more qubits.

The cluster state provides the essential substrate for the computation, but the al-

gorithm itself is deļ¬ned by combining two further elements: (1) a sequence of local

measurements (von Neumann measurements on individual qubits); and (2) classi-

cal feedforward. The latter term means that the result of one measurement in the

sequence can be used to determine the choice of the measurement basis used in a

subsequent measurement.

These two elements can replace any of the operations considered in Section 20.5.1.

For example, any unitary operation on a single qubit can be simulated by means of a

four-qubit cluster state and three measurements. In general, one-qubit measurements

are used to imprint the initial data onto the cluster state, and then process it to yield

the ļ¬nal result.

The use of irreversible measurements as an integral part of the algorithm, rather

than just the ļ¬nal readout step, has led to the name one-way quantum computing

for this approach. For a suļ¬ciently large cluster state, it has been shown that these

elements are suļ¬cient to implement a universal quantum computer. The diļ¬erent

structures of reversible and one-way computing make comparisons a bit diļ¬cult, but

the current estimate is that one-way computing requires roughly 60 Bell pairs per

two-qubit gate operation.

Highly entangled states of many atoms have been experimentally produced by

precise control of the interactions between neutral atoms bound by dipole forces to

the sites of an optical lattice (Mandel et al., 2003), but we are more interested in

optical realizations of cluster states. Walther et al. (2005) demonstrated a one-way

version of a simple example of Groverā™s search algorithm.

Āæ

Exercises

In their experiment, a four-photon cluster state was directly produced by down-

conversion techniques. Four-qubit cluster states have also been produced by entangling

EPR pairs with a controlled-sign gate (Kiesel et al., 2005), and by a technique called

type-I qubit fusion (Zhang et al., 2006), which combines Bell states by mixing at a

beam splitter and postselection. One-way quantum computing may, therefore, be a

promising application of quantum optics to quantum computing.

20.6 Exercises

20.1 Variable retarder plate

Design a variable retarder plate by joining two identical, thin, right-angle prisms along

their hypotenuses. Sketch the appropriate arrangement and carry out the following.

(1) Assuming that the light passes through the central part of the retarder, show how

the optical path length can be adjusted by sliding the prisms along their common

hypotenuse.

(2) Express the optical path length in terms of the index of refraction and the geo-

metrical parameters of the device. Assign numerical values for a practical design.

(3) Calculate the optical path lengths required to obtain the phase shifts Īø = ā’Ļ/2

and Īø = Ļ/2.

20.2 Modiļ¬ed beam splitter

Consider the modiļ¬ed beam splitter pictured in Fig. 20.1.

(1) Derive eqn (20.5) for the scattering matrix.

(2) For general values of Īø and Īø , use the scattering matrix to express the output

quadratures X1 , Y1 , X2 , Y2 in terms of the input quadratures X1 , Y1 , X2 , Y2 .

Calculate the variances of the output quadratures. Explain why the values Īø =

ā’Ļ/2 and Īø = Ļ/2 are particularly useful.

(3) If no variable retarder plates are available, i.e. Īø = Īø = 0, how can the operation

of the SQLG be changed to achieve the same noiseless splitting of the input signal

X1 .

20.3 Bell states

Consider the Bell states deļ¬ned in eqn (20.15).

(1) Show that the Bell states are mutually orthogonal and all normalized to unity.

(2) Explainā”ideally without any further algebraā”why the Bell states form a basis

for Ha ā— Hb .

20.4 No-cloning theorem for photons

Consider cloning the one-photon states |Ī³ = Ī“ā |0 and |Ī¶ = Z ā |0 , where

Ī¶ks aā .

Zā = ks

s

Ā¼ Quantum information

(1) Derive the commutator

Z, Ī“ā = ā—

Ī¶ks Ī³ks ā” (Ī¶, Ī³) .

ks

(2) Adapt the general proof for the no-cloning theorem to show that the cloning

assumption (20.25), and the corresponding assumption for |Ī¶ , cannot be satisļ¬ed

for all choices of the operators Ī“ā and Z ā .

20.5 Cloning a known state

For the device in Section 20.2.1 that clones a known state, assume the model interaction

Hint = gĻā’ aā + aā + HC, between the two-level atom and the ļ¬eld. For the initial

kh kv

state |1kh , use ļ¬rst-order, time-dependent perturbation theory to calculate the change

in the initial state vector and thus derive eqn (20.26).

BuĖekā“Hillery QCMā—

20.6 z

Use the explicit expressions (20.31) and (20.32) to evaluate the reduced qubit density

operators Ļab , Ļa , and Ļb . Use the results to calculate the ļ¬delities for the clones a and

b.

Photon cloning machineā—

20.7

Consider the photon cloning machine described in Section 20.2.2-B.

(1) Denote the polarization basis for the kn -mode (n = 1, 2) by {eh (kn ) , ev (kn )}.

For a rotation of each basis around kn by the angle Īø, i.e.

eh (kn ) = cos Īø eh (kn ) + sin Īø ev (kn ) ,

ev (kn ) = ā’ sin Īø eh (kn ) + cos Īø ev (kn ) ,

derive the corresponding transformation of the creation operators aā n h , aā n v and

k k

show that the Hamiltonian in eqn (20.33) has the same form in the new basis.

(2) The (2, 0)-events in which two photons are present in the k1 v-mode are counted

by letting the output fall on a beam splitter with detectors at each output port.

A coincidence count shows that two photons were present. For an ideal balanced

beam splitter and 100% detectors, show that the probability of a coincidence count

is 1/2. Use this to explain the discrepancy between eqn (20.42) and the baseline

data in Fig. 20.3.

20.8 Wave plates

A polarization-dependent retarder plate (wave plate) is made from an anisotropic

crystal, with indices of refraction nF and nS for light polarized along the fast axis eF

and the slow axis eS respectively (Saleh and Teich, 1991, Sec. 6.1-B).

Consider a classical ļ¬eld with amplitude E = Eh eh + Ev ev propagating in the

z-direction, that falls on a retarder plate of thickness āz lying in the (x, y)-plane.

Ā½

Exercises

(1) By discarding an overall phase factor show that the output ļ¬eld E = Eh eh + Ev ev

is related to the input ļ¬eld by col (Eh ,Ev ) = TĪ¾ (Ļ‘) col (Eh ,Ev ), where the Jones

matrix TĪ¾ (Ļ‘) is given by

ā’ sin Ļ‘ cos Ļ‘ 1 ā’ eiĪ¾

cos2 Ļ‘ + sin2 Ļ‘eiĪ¾

TĪ¾ (Ļ‘) = ,

ā’ sin Ļ‘ cos Ļ‘ 1 ā’ eiĪ¾ sin2 Ļ‘ + cos2 Ļ‘eiĪ¾

Ļ‘ is the angle between eh and eF , and Ī¾ = (nS ā’ nF ) Ļāz/c.

(2) Evaluate the Jones matrix for Ī¾ = Ļ/2 (the quarter-wave plate) and Ī¾ = Ļ (the

half-wave plate).

(3) For Ļ‘ = 0 and a 45ā—¦ -polarized input, i.e. Eh = Ev , what is the output polarization

state? Answer the same question if Ļ‘ = Ļ/4 and the input ļ¬eld is h-polarized.

20.9 Quantum dense coding

The unitary operators used by Bob for quantum dense coding are deļ¬ned by

U Ļ‘Ī»/4 , Ļ‘Ī»/2 = TĻ/2 Ļ‘Ī»/4 TĻ Ļ‘Ī»/2 , where TĪ¾ (Ļ‘) is given by the result of the previ-

ous exercise. As explained in the text, this operator only acts on the second argument

of |sA , sB .

(1) For the general state

|Ī˜ = chh |hA , hB + chv |hA , vB + cvh |vA , hB + cvv |vA , vB

determine the expansion coeļ¬cients for which U (0, 0) |Ī˜ = |ĪØ+ .

(2) Find three other sets of values Ļ‘Ī»/4 , Ļ‘Ī»/2 such that U Ļ‘Ī»/4 , Ļ‘Ī»/2 |Ī˜ is equal

(up to a phase factor) to the remaining Bell states.

20.10 Bell states incident on a balanced beam splitter

For the Bell states in eqns (20.52) and (20.53) use the method described in Section

8.4.1 to show that the scattered states produced by a balanced beam splitter are

ĪØā’ = ĪØā’ ,

1

= ā |hA , vA + (A ā” B) ,

ĪØ+

2

i

Ī¦Ā± = {|hA , hA Ā± |vA , vA } + (A ā” B) .

2

20.11 Rotated polarization basis

Consider the 45ā—¦ -rotated polarization basis deļ¬ned by eqn (20.48).

(1) Derive

ā ā

aā = a ā ā’ a ā / 2 , a ā = a ā + a ā / 2 ,

Ī³v

Ī³v Ī³v

Ī³h Ī³h Ī³h

where Ī³ ā {A, B}.

Ā¾ Quantum information

(2) Show that

1 1

ā’ ā hA , v A ,

|hA , hA = hA , hA + |v A , v A

2 2

1 1

+ ā hA , v A .

|vA , vA hA , hA + |v A , v A

=

2 2

(3) Starting with eqn (20.56), derive eqns (20.57) and (20.58).

20.12 Insuļ¬cient information

Consider Aliceā™s attempt to give Bob instructions for making an approximate copy of

her unknown qubit |Ī³ .

(1) Given the unit vector n and the eigenvalue of n Ā· Ļ, explain why Bobā™s best

estimate for the unknown state |Ī³ is given by eqn (20.62).

(2) Why cannot Alice get more information for Bob by making further measurements?

(3) Suppose that the sender of Aliceā™s qubit, who does know the state |Ī³ , is willing

to send her an endless stream of qubits, all prepared in the same state. Aliceā™s

research budget, however, limits her to a ļ¬nite number of measurements. Can

Alice supply Bob with enough information to permit an exact reproduction (up

to an overall phase) of |Ī³ ?

20.13 Teleportation of qubits

(1) Express the basis states |u, v AT (u, v = 0, 1) as linear combinations of the Bell

states, and then derive eqn (20.66).

(2) Show that the Pauli matrices are unitary as well as hermitian, and use this fact

to construct unitary operators for the phase-ļ¬‚ip and the bit-ļ¬‚ip.

(3) Suppose that Alice does her Bell state measurement, but that Eve intercepts the

message to Bob. Calculate the reduced density operator ĻB that Bob must use in

this circumstance, and comment on the result.

(4) Now suppose that Alice misunderstands the theory, and thinks that she should

make a measurement that projects onto the basis vectors |u, v AT . After Alice

tells Bob which of the four possibilities occurred, what information does Bob have

about his qubit?

20.14 Teleportation of photons

Consider the application of the teleportation protocol to photons.

(1) Write out the explicit expressions for the Bell states in the Aā“T subsystem.

(2) Derive the photonic version of eqn (20.66).

(3) Give explicit forms for the action of the unitary transformations Upf (phase-ļ¬‚ip)

and Ubf (bit-ļ¬‚ip) on the creation operators.

Āæ

Exercises

Quantum logic gatesā—

20.15

(1) Show that the X, Z, and Hadamard gates are unitary operators.

(2) Use the representation |Ī³ = Ī³0 |0 + Ī³1 |1 of a general qubit to express all three

gates as 2 Ć— 2 matrices. Explain the names for the X and Z gates by relating them

to Pauli matrices.

(3) For a spin-1/2 particle, the operator for a rotation through the angle Ī± around the

axis directed along the unit vector u is (Bransden and Joachain, 1989, Sec. 6.9)

Ī±

Ī±

ā’ i sin uĀ·Ļ.

Ru (Ī±) = cos

2 2

Combine this with the PoincarĀ“-sphere representation

e

Īø Īø

|Ī³ = cos |0 + eiĻ sin |1

2 2

for qubits to show that the X, Z, and Hadamard gates are respectively given by

iRux (Ļ), iRuz (Ļ), and iRh (Ļ), where ux , uy , uz are the coordinate unit vectors

ā ā

and h = ux / 2 + uz / 2.

(4) Show that the control-NOT operator CNOT , deļ¬ned by eqn (20.72), is unitary.

Use the basis {|0, 0 , |0, 1 , |1, 0 , |1, 1 } to express CNOT as a 4 Ć— 4 matrix.

Single-photon gatesā—

20.16

Identify the polarization states of a single photon with the logical states by |h ā” |0

and |v ā” |1 . Use the results of Exercise 20.8 to show that the Z and Hadamard gates

can be realized by means of half-wave plates.

Quantum circuitsā—

20.17

Work out the gates required for the outcomes |Ī¦ā’ and |ĪØā’ in the computation

discussed in Section 20.5.1-A and draw the corresponding quantum circuit diagrams.

Controlled-NOT gateā—

20.18

For the nondeterministic control-NOT gate sketched in Section 20.5.3, use the notation

aCh , aCv , aT h , aT v for the control and target modes and b1h , b2h for h-polarized

vacuum ļ¬‚uctuations in the Vac-1 and Vac-2 channels. Devise a suitable notation for

the operators associated with the internal lines in Fig. 20.9, and carry out the following

steps.

(1) Write out the scattering relations for each of the optical elements in the gate. For

this purpose it is useful to impose a consistent convention for assigning the Ā± s

ā

to the asymmetric beam splitters, e.g. assign ā’ R for reļ¬‚ection from the lower

surface of a beam splitter.

(2) Explain why the vacuum v-polarizations b1v , b2v can be omitted.

(3) Use the scattering relations to eliminate the internal variables and thus ļ¬nd the

overall scattering relations (aCh , aCv , . . .) ā’ (aCh , aCv , . . .) which deļ¬ne the ele-

ments of the scattering matrix for the gate.

Quantum information

(4) Employ the general result (8.40) to determine the action of the gate on each input

state in the coincidence basis, and thus show that

1 1

|hC , hT ā’ ā’ 3 ) R |hC , hT ā’ 3 ) R |hC , vT + Ā·Ā·Ā· ,

( ( +

1 2 1 2

2 2

1 1

|hC , vT ā’ ā’ 3 ) R |hC , hT

1 ( 2 ā’ 3 ) R |hC , vT + Ā· Ā· Ā· ,

1( 2 + +

2 2

1 1

|vC , hT ā’ [(2 ā’ 2 3 ) R ā’ 1] |vC , hT + [1 ā’ (2 + 2 3 ) R] |hC , vT + Ā· Ā· Ā· ,

2 2

1 1

|vC , vT ā’ [1 ā’ (2 + 2 3 ) R] |vC , hT + [(2 ā’ 2 3 ) R ā’ 1] |hC , vT + Ā· Ā· Ā· .

2 2

Determine the value of R and the assignment of the s needed to deļ¬ne a control-

NOT gate.

Appendix A

Mathematics

A.1 Vector analysis

Our conventions for elementary vector analysis are as follows. The unit vectors cor-

responding to the Cartesian coordinates x, y, z are ux , uy , uz . For a general vector

v, we denote the unit vector in the direction of v by v = v/ |v|.

The scalar product of two vectors is a Ā· b = ax bx + ay by + az bz , or

3

aĀ·b= ai b i , (A.1)

i=1

where (a1 , a2 , a3 ) = (ax , ay , az ), etc. Since expressions like this occur frequently, we will

use the Einstein summation convention: repeated vector indices are to be summed

over; that is, the expression ai bi is understood to imply the sum in eqn (A.1). The

summation convention will only be employed for three-dimensional vector indices. The

cross product is

(a Ć— b)i = ijk aj bk , (A.2)

where the alternating tensor is deļ¬ned by

ijk

ā§

āŖ1 ijk is an even permutation of 123 ,

āØ

= ā’1 ijk is an odd permutation of 123 , (A.3)

ijk

āŖ

ā©

0 otherwise .

A.2 General vector spaces

A complex vector space is a set H on which the following two operations are deļ¬ned.

(1) Multiplication by scalars. For every pair (Ī±, Ļ), where Ī± is a scalar, i.e. a complex

number, and Ļ ā H, there is a unique element of H that is denoted by Ī±Ļ.

(2) Vector addition. For every pair Ļ, Ļ of vectors in H there is a unique element of H

denoted by Ļ + Ļ.

The two operations satisfy (a) Ī±(Ī²Ļ) = (Ī±Ī²) Ļ, and (b) Ī± (Ļ + Ļ) = Ī±Ļ + Ī±Ļ. It

is assumed that there is a special null vector, usually denoted by 0, such that Ī±0 = 0

and Ļ + 0 = Ļ. If the scalars are restricted to real numbers these conditions deļ¬ne a

real vector space.

Mathematics

Ordinary displacement vectors, r, belong to a real vector space denoted by R3 . The

set Cn of n-tuplets Ļ = (Ļ1 , . . . , Ļn ), where each component Ļi is a complex number,

deļ¬nes a complex vector space with component-wise operations:

Ī±Ļ = (Ī±Ļ1 , . . . , Ī±Ļn ) ,

(A.4)

Ļ + Ļ = (Ļ1 + Ļ1 , . . . , Ļn + Ļn ) .

Each vector in R3 or Cn is speciļ¬ed by a ļ¬nite number of components, so these spaces

are said to be ļ¬nite dimensional.

The set of complex functions, C (R), of a single real variable deļ¬nes a vector space

with point-wise operations:

(Ī±Ļ) (x) = Ī±Ļ (x) , (A.5)

(Ļ + Ļ) (x) = Ļ (x) + Ļ (x) , (A.6)

where Ī± is a scalar, and Ļ (x) and Ļ (x) are members of C (R). This space is said to

be inļ¬nite dimensional, since a general function is not determined by any ļ¬nite set

of values.

For any subset U ā‚ H, the set of all linear combinations of vectors in U is called

the span of U, written as span (U). A family B ā‚ H is a basis for H if H = span (B),

i.e. every vector in H can be expressed as a linear combination of vectors in B. In this

situation H is said to be spanned by B.

A linear operator is a rule that assigns a new vector M Ļ to each vector Ļ ā H,

such that

M (Ī±Ļ + Ī²Ļ) = Ī±M Ļ + Ī²M Ļ (A.7)

for any pair of vectors Ļ and Ļ, and any scalars Ī± and Ī². The action of a linear operator

M on H is completely determined by its action on the vectors of a basis B.

A.3 Hilbert spaces

A.3.1 Deļ¬nition

An inner product on a vector space H is a rule that assigns a complex num-

ber, denoted by (Ļ, Ļ), to every pair of elements Ļ and Ļ ā H, with the following

properties:

(Ļ, Ī±Ļ + Ī²Ļ) = Ī± (Ļ, Ļ) + Ī² (Ļ, Ļ) , (A.8a)

ā—

(Ļ, Ļ) = (Ļ, Ļ) , (A.8b)

(Ļ, Ļ) < ā ,

0 (A.8c)

(Ļ, Ļ) = 0 if and only if Ļ = 0 . (A.8d)

An inner product space is a vector space equipped with an inner product. The

inner product satisļ¬es the Cauchyā“Schwarz inequality:

2

|(Ļ, Ļ)| (Ļ, Ļ) (Ļ, Ļ) . (A.9)

Two vectors are orthogonal if (Ļ, Ļ) = 0. If F is a subspace of H, then the orthogonal

complement of F is the subspace Fā„ of vectors orthogonal to every vector in F.

Hilbert spaces

The norm Ļ of Ļ is deļ¬ned as Ļ = (Ļ, Ļ), so that Ļ = 0 implies Ļ = 0.

Vectors with Ļ = 1 are said to be normalized. A set of vectors is complete if

the only vector orthogonal to every vector in the set is the null vector. Each complete

set contains a basis for the space. A vector space with a countable basis set, B =

Ļ(1) , Ļ(2) , . . . , is said to be separable. The vector spaces relevant to quantum theory

are all separable. A basis for which Ļ(n) , Ļ(m) = Ī“nm holds is called orthonormal.

Every vector in H can be uniquely expanded in an orthonormal basis, e.g.

ā

Ļn Ļ(n) ,

Ļ= (A.10)

n=1

where the expansion coeļ¬cients are Ļn = Ļ(n) , Ļ .

A sequence Ļ 1 , Ļ 2 , . . . , Ļ k , . . . of vectors in H is convergent if

Ļ k ā’ Ļ j ā’ 0 as k, j ā’ ā . (A.11)

A vector Ļ is a limit of the sequence if

Ļ k ā’ Ļ ā’ 0 as k ā’ ā . (A.12)

A Hilbert space is an inner product space that contains the limits of all convergent

sequences.

A.3.2 Examples

The ļ¬nite-dimensional spaces R3 and CN are both Hilbert spaces. The inner product

for R3 is the familiar dot product, and for CN it is

N

ā—

(Ļ, Ļ) = Ļn Ļn . (A.13)

n=1

If we constrain the complex functions Ļ (x) by the normalizability condition

ā

dx |Ļ (x)|2 < ā , (A.14)

ā’ā

then the Cauchyā“Schwarz inequality for integrals,

ā ā ā

2

ā— 2 2

dx |Ļ (x)| dx |Ļ (x)| ,

dxĻ (x) Ļ (x) (A.15)

ā’ā ā’ā ā’ā

is suļ¬cient to guarantee that the inner product deļ¬ned by

ā

dxĻ ā— (x) Ļ (x)

(Ļ, Ļ) = (A.16)

ā’ā

makes the vector space of complex functions into a Hilbert space, which is called

L2 (R).

Mathematics

A.3.3 Linear operators

Let A be a linear operator acting on H; then the domain of A, called D (A), is the

subspace of vectors Ļ ā H such that AĻ < ā. An operator A is positive deļ¬nite

0 for all Ļ ā D (A), and it is bounded if AĻ < b Ļ , where b is a

if (Ļ, AĻ)

constant independent of Ļ. The norm of an operator is deļ¬ned by

AĻ

A = max for Ļ = 0 , (A.17)

Ļ

so a bounded operator is one with ļ¬nite norm.

If AĻ = Ī»Ļ, where Ī» is a complex number and Ļ is a vector in the Hilbert space,

then Ī» is an eigenvalue and Ļ is an eigenvector of A. In this case Ī» is said to belong

to the point spectrum of A. The eigenvalue Ī» is nondegenerate if the eigenvector

Ļ is unique (up to a multiplicative factor). If Ļ is not unique, then Ī» is degenerate.

The linearly-independent solutions of AĻ = Ī»Ļ form a subspace called the eigenspace

for Ī», and the dimension of the eigenspace is the degree of degeneracy for Ī». The

continuous spectrum of A is the set of complex numbers Ī» such that: (1) Ī» is not

an eigenvalue, and (2) the operator Ī» ā’ A does not have an inverse.

The adjoint (hermitian conjugate) Aā of A is deļ¬ned by

ā—

Ļ, Aā Ļ = (Ļ, AĻ) , (A.18)

and A is self-adjoint (hermitian) if D Aā = D (A) and (Ļ, AĻ) = (AĻ, Ļ). Bounded

self-adjoint operators have real eigenvalues and a complete orthonormal set of eigen-

vectors. For unbounded self-adjoint operators, the point and continuous spectra are

subsets of the real numbers. Note that Ļ, Aā AĻ = (Ļ, Ļ), where Ļ = AĻ, so that

Ļ, Aā AĻ 0, (A.19)

i.e. Aā A is positive deļ¬nite.

A self-adjoint operator, P , satisfying

P2 = P (A.20)

is called a projection operator; it has only a point spectrum consisting of {0, 1}.

Consider the set of vectors P H, consisting of all vectors of the form P Ļ as Ļ ranges

over H. This is a subspace of H, since

Ī±P Ļ + Ī²P Ļ = P (Ī±Ļ + Ī²Ļ) (A.21)

shows that every linear combination of vectors in P H is also in P H. Conversely, let S

be a subspace of H and Ļ(n) an orthonormal basis for S. The operator P , deļ¬ned

by

Ļ(n) , Ļ Ļ(n) ,

PĻ = (A.22)

n

is a projection operator, since

P 2Ļ = Ļ(n) , Ļ P Ļ(n) = Ļ(n) , Ļ Ļ(n) = P Ļ . (A.23)

n n

Thus there is a one-to-one correspondence between projection operators and subspaces

of H. Let P and Q be projection operators and suppose that the vectors in P H are

Hilbert spaces

orthogonal to the vectors in QH; then P Q = QP = 0 and P and Q are said to be

orthogonal projections. In the extreme case that S = H, the expansion (A.10)

shows that P is the identity operator, P Ļ = Ļ.

A self-adjoint operator with pure point spectrum {Ī»1 , Ī»2 , . . .} has the spectral

resolution

A= Ī»n Pn , (A.24)

n

where Pn is the projection operator onto the subspace of eigenvectors with eigenvalue

Ī»n . The spectral resolution for a self-adjoint operator A with a continuous spectrum

is

A = Ī» dĀµ (Ī») , (A.25)

where dĀµ (Ī») is an operator-valued measure deļ¬ned by the following statement:

for each subset ā of the real line,

P (ā) = dĀµ (Ī») (A.26)

ā

ā’1

is the projection operator onto the subspace of vectors Ļ such that (Ī» ā’ A) Ļ <ā

for all Ī» ā ā (Riesz and Sz.-Nagy, 1955, Chap. VIII, Sec. 120).

/

A linear operator U is unitary if it preserves inner products, i.e.

(U Ļ, U Ļ) = (Ļ, Ļ) (A.27)

for any pair of vectors Ļ, Ļ in the Hilbert space. A necessary and suļ¬cient condition

for unitarity is that the operator is norm preserving, i.e.

(U Ļ, U Ļ) = (Ļ, Ļ) for all Ļ if and only if U is unitary . (A.28)

The spectral resolution for a unitary operator with a pure point spectrum is

eiĪøn Pn , Īøn real ,

U= (A.29)

n

and for a continuous spectrum

eiĪø dĀµ (Īø) , Īø real .

U= (A.30)

A linear operator N is said to be a normal operator if

N, N ā = 0 . (A.31)

The hermitian and unitary operators are both normal. The hermitian operators N1 =

N + N ā /2 and N2 = N ā’ N ā /2i satisfy N = N1 + iN2 and [N1 , N2 ] = 0. Normal

operators therefore have the spectral resolutions

N= (xn P1n + iyn P2n ) , [P1n , P2m ] = 0 (A.32)

n

Ā¼ Mathematics

for a point spectrum, and

N= x dĀµ1 (x) + i y dĀµ2 (y) , dĀµ1 (x) , dĀµ2 (y) = 0 (A.33)

ā1 ā1

for a continuous spectrum.

A.3.4 Matrices

A linear operator X acting on an N -dimensional Hilbert space, with basis f (1) , . . . ,

f (N ) , is represented by the N Ć— N matrix

Xmn = f (m) , Xf (n) . (A.34)

The operator and its matrix are both called X. The matrix for the product XY of

two operators is the matrix product

N

(XY )mn = Xmk Ykn . (A.35)

k=1

The determinant of X is deļ¬ned as

X1n1 Ā· Ā· Ā· XN nN ,

det (X) = (A.36)

n1 Ā·Ā·Ā·nN

n1 Ā·Ā·Ā·nN

where the generalized alternating tensor is

ā§

āŖ1 n1 Ā· Ā· Ā· nN is an even permutation of 12 Ā· Ā· Ā· N ,

āØ

ā’1 n1 Ā· Ā· Ā· nN is an odd permutation of 12 Ā· Ā· Ā· N , (A.37)

n1 Ā·Ā·Ā·nN =

āŖ

ā©

n1 Ā·Ā·Ā·nN 0 otherwise .

The trace of X is

N

Tr X = Xnn . (A.38)

n=1

The transpose matrix X T is deļ¬ned by Xnm = Xmn . The adjoint matrix X ā

T

ā ā—

is the complex conjugate of the transpose: Xnm = Xmn . A matrix X is symmetric if

X = X T , self-adjoint or hermitian if X ā = X, and unitary if X ā X = XX ā = I, where I

is the N Ć— N identity matrix. Unitary transformations preserve the inner product. The

hermitian and unitary matrices both belong to the larger class of normal matrices

deļ¬ned by X ā X = XX ā .

A matrix X is positive deļ¬nite if all of its eigenvalues are real and non-negative.

This immediately implies that the determinant and trace of the matrix are both non-

negative. An equivalent deļ¬nition is that X is positive deļ¬nite if

Ļā XĻ 0 (A.39)

for all vectors Ļ. For a positive-deļ¬nite matrix X, there is a matrix Y such that

X = Y Y ā .

The normal matrices have the following important properties (Mac Lane and Birk-

hoļ¬, 1967, Sec. XI-10).

Ā½

Fourier transforms

Theorem A.1 (i) If f is an eigenvector of the normal matrix Z with eigenvalue z,

then f is an eigenvector of Z ā with eigenvalue z ā— , i.e. Zf = zf ā’ Z ā f = z ā— f .

(ii) Every normal matrix has a complete, orthonormal set of eigenvectors.

Thus hermitian matrices have real eigenvalues and unitary matrices have eigenvalues

of modulus 1.

A.4 Fourier transforms

A.4.1 Continuous transforms

In the mathematical literature it is conventional to denote the Fourier (integral)

transform of a function f (x) of a single, real variable by

ā

dxf (x) eā’ikx ,

f (k) = (A.40)

ā’ā

so that the inverse Fourier transform is

ā

dk

f (k) eikx .

f (x) = (A.41)

2Ļ

ā’ā

The virtue of this notation is that it reminds us that the two functions are, generally,

drastically diļ¬erent, e.g. if f (x) = 1, then f (k) = 2ĻĪ“ (k) .

On the other hand, the is a typographical nuisance in any discussion involving

many uses of the Fourier transform. For this reason, we will sacriļ¬ce precision for

convenience. In our convention, the Fourier transform is indicated by the same letter,

and the distinction between the functions is maintained by paying attention to the

arguments.

The Fourier transform pair is accordingly written as

ā

dxf (x) eā’ikx ,

f (k) = (A.42)

ā’ā

ā

dk

f (k) eikx .

f (x) = (A.43)

2Ļ

ā’ā

This is analogous to the familiar idea that the meaning of a vector V is independent

of the coordinate system used, despite the fact that the components (Vx , Vy , Vz ) of

V are changed by transforming to a new coordinate system. From this point of view,

the functions f (x) and f (k) are simply diļ¬erent representations of the same physical

quantity. Confusion is readily avoided by paying attention to the physical signiļ¬cance

of the arguments, e.g. x denotes a point in position space, while k denotes a point

in the reciprocal space or k-space.

If the position-space function f (x) is real, then the Fourier transform satisļ¬es

ā—

f ā— (k) = [f (k)] = f (ā’k) . (A.44)

When the position variable x is replaced by the time t, it is customary in physics to

use the opposite sign convention:

Ā¾ Mathematics

ā

dxf (x) eiĻt ,

f (Ļ) = (A.45)

ā’ā

ā

dĻ

f (k) eā’iĻt .

f (t) = (A.46)

2Ļ

ā’ā

Fourier transforms of functions of several variables, typically f (r), are deļ¬ned

similarly:

d3 rf (r) eā’ikĀ·r ,

f (k) = (A.47)

d3 k

(k) eikĀ·r ,

f (r) = 3f (A.48)

(2Ļ)

where the integrals are over position space and reciprocal space (k-space) respectively.

If f (r) is real then

f ā— (k) = f (ā’k) . (A.49)

Combining these conventions for a spaceā“time function f (r, t) yields the transform

pair

dtf (r, t) eā’i(kĀ·rā’Ļt) ,

d3 r

f (k, Ļ) = (A.50)

d3 k dĻ

f (k, Ļ) ei(kĀ·rā’Ļt) .

f (r, t) = (A.51)

3 2Ļ

(2Ļ)

The last result is simply the plane-wave expansion of f (r, t). If f (r, t) is real, then

the Fourier transform satisļ¬es

f ā— (k, Ļ) = f (ā’k, ā’Ļ) . (A.52)

Two related and important results on Fourier transformsā”which we quote for the

one- and three-dimensional casesā”are Parsevalā™s theorem:

dĻ ā—

dtf ā— (t) g (t) = f (Ļ) g (Ļ) , (A.53)

2Ļ

d3 k ā—

ā—

3

d rf (r) g (r) = 3 f (k) g (k) , (A.54)

(2Ļ)

and the convolution theorem:

dt f (t ā’ t ) g (t )

h (t) = if and only if h (Ļ) = f (Ļ) g (Ļ) , (A.55)

dĻ

f (Ļ ā’ Ļ ) g (Ļ )

h (Ļ) = if and only if h (t) = f (t) g (t) , (A.56)

2Ļ

d3 r f (r ā’ r ) g (r )

h (r) = if and only if h (k) = f (k) g (k) , (A.57)

d3 k

(k ā’ k ) g (k ) if and only if h (r) = f (r) g (r) .

h (k) = 3f (A.58)

(2Ļ)

These results are readily derived by using the delta function identities (A.95) and

(A.96).

Āæ

Fourier transforms

A.4.2 Fourier series

It is often useful to simplify the mathematics of the one-dimensional continuous trans-

form by considering the functions to be deļ¬ned on a ļ¬nite interval (ā’L/2, L/2) and

imposing periodic boundary conditions. The basis vectors are still of the form

uk (x) = C exp (ikx), but the periodicity condition, uk (ā’L/2) = uk (L/2), restricts k

to the discrete values

2Ļn

(n = 0, Ā±1, Ā±2, . . .) .

k= (A.59)

L

ā

Normalization requires C = 1/ L, so the transform is

L/2

1

dxf (x) eā’ikx ,

fk = ā (A.60)

L ā’L/2

and the inverse transform f (x) is

1

f (x) = ā fk eikx . (A.61)

L k

The continuous transform is recovered in the limit L ā’ ā by ļ¬rst using eqn (A.60)

to conclude that ā

Lfk ā’ f (k) as L ā’ ā , (A.62)

and writing the inverse transform as

ā

1

Lfk eikx .

f (x) = (A.63)

L

k

The diļ¬erence between neighboring k-values is āk = 2Ļ/L, so this equation can be

recast as

āk ā dk

Lfk eikx ā’ f (k) eikx .

f (x) = (A.64)

2Ļ 2Ļ

k

In Cartesian coordinates the three-dimensional discrete transform is deļ¬ned on a

rectangular parallelepiped with dimensions Lx , Ly , Lz . The one-dimensional results

then imply

1

d3 rf (r) eā’ikĀ·r ,

fk = ā (A.65)

VV

where the k-vector is restricted to

2Ļny 2Ļnz

2Ļnx

ux + uy + uz , (A.66)

k=

Lx Ly Lz

and V = Lx Ly Lz . The inverse transform is

1

f (r) = ā fk eikĀ·r , (A.67)

V k

and the integral transform is recovered by

Mathematics

ā

V fk ā’ f (k) as V ā’ ā . (A.68)

The sum and integral over k are related by

d3 k

1

ā’ , (A.69)

(2Ļ)3

V

k

which in turn implies

3

V Ī“k,k ā’ (2Ļ) Ī“ (k ā’ k ) . (A.70)

A.5 Laplace transforms

Another useful ideaā”which is closely related to the one-dimensional Fourier trans-

formā”is the Laplace transform deļ¬ned by

ā

dt eā’Ī¶t f (t) .

f (Ī¶) = (A.71)

0

In this case, we will use the standard mathematical notation f (Ī¶), since we do not use

Laplace transforms as frequently as Fourier transforms. The inverse transform is

Ī¶0 +iā

dĪ¶ Ī¶t

f (t) = e f (Ī¶) . (A.72)

2Ļi

Ī¶0 ā’iā

The line (Ī¶0 ā’ iā, Ī¶0 + iā) in the complex Ī¶-plane must lie to the right of any poles

in the transform function f (Ī¶).

The identity

df

(Ī¶) = Ī¶ f (Ī¶) ā’ f (0) (A.73)

dt

is useful in treating initial value problems for sets of linear, diļ¬erential equations. Thus

to solve the equations

dfn

ńņš. 24 |