<<

. 24
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>>

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
10 10
U (0,0) = , U (1,1) = ,
0 ’1
01
(20.70)
01 01
(0,1) (1,0)
U ,U ,
= =
’1 0
10
then it is easy to verify that

U (1,1) ¦+ = ¦’ , U (0,1) ¦+ = Ψ+ , U (1,0) ¦+ = Ψ’ . (20.71)

After the programmer supplies the two bits needed to choose the operator U f ”
i.e. the one that gives the same output as the classical computer”the output of the
computation is obtained by performing a Bell state measurement. If the result is |Ψ+ ,
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 gates—
B
The description of the simple quantum computer given in the last section ¬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 speci¬ed by its action on the basis vectors |0
and |1 ; for example, the X gate is de¬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
¿¿
Quantum computing

are also useful one-qubit gates that do not have classical analogues, such as the Z
gate: Z |0 = |0 , Z |1 = ’ |1 , and the Hadamard gate:
1 1
H |0 = √ {|0 + |1 } , H |1 = √ {|0 ’ |1 } .
2 2
These gates can all be expressed as 2 — 2 matrices, and”as seen in Exercises 20.15
and 20.16”they are also related to rotations on the Poincar´ sphere.
e
An important two-qubit gate is the controlled-NOT (C-NOT) gate, de¬ned by

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

where • represents addition modulo 2. The ¬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 e¬ects. (1) The control qubit is left un-
changed. (2) The target qubit is ¬‚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
de¬ned by
ab
CS |a, b = (’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 ’1.

C Quantum circuits—
In Section 20.5.1-A we ¬‚outed the convention that the register always begins in a
standard state, e.g. |Λin = |0, 0 . It is easy to verify that |¦+ = 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
|Ψ+ is an X gate, so the result f = (0, 1) is achieved by the unitary transformation
|Ψ+ = U (0,1) |¦+ = 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 .
¿ 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 experiments—
20.5.2

Experimental realizations of the idealized devices discussed above must overcome a
number of very serious di¬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,
¬rst by Shor (1995), and later by Bennett et al. (1996) and Knill and La¬‚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 ¬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 qubits”provided by spin-1/2 nuclei in molecules”
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 us”the photon”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 state”in our terminology a dynamically entangled state”of two photons
by means of photon“photon coupling, i.e. an optical nonlinearity.
As suggested by Milburn (1989), one way to do this would be to induce a cross-
Kerr coupling”see Section 13.4.3”between two optical modes. Unfortunately, the
materials provided by nature have χ(3) s that are orders-of-magnitude too small to
accomplish the desired e¬ects. Increasing the length of the nonlinear region does not
¿
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 ¬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 optics—
20.5.3
The consensus view that optical methods are not suitable for quantum computing
was challenged by the work of Knill, La¬‚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 ¬rst step in its development. More details can be found in recent review
articles, e.g. Ralph (2006).
One possible design”adapted from the work of Hofmann and Takeuchi (2002)”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 re¬‚ect h-polarized light at 90—¦ . The half-wave plate (hwp) at
the control input is aligned at ‘ = 0, while the hwps at the target input and output ports
are aligned at ‘ = ’22.5—¦ , where ‘ is the angle between the h-polarization and the fast axis;
see Exercise 20.8. The beam splitters are asymmetric.
¿ Quantum information

communicate with the vacuum channels Vac-1 and Vac-2. The beam splitters are
asymmetric, with scattering laws of the general form


a1 = 1 ’ Ra1 ’ Ra2 ,
√ √ (20.74)
Ra1 + 1 ’ Ra2 ,
a2 =
worked out in Exercise 8.1. All three beam splitters are assumed to have the same
re¬‚ectivity R, and the sign factors = ±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 photons”one in the control channel and one in the target channel”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 e¬ects associated with di¬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 ¬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 = ’ 3 = 1 .
With these values, the operation of the gate is given by
1 1
CNOT |hC , hT = |hC , hT + · · · , CNOT |hC , vT = |hC , vT + · · · ,
3 3 (20.75)
1 1
CNOT |vC , hT = |vC , vT + · · · , CNOT |vC , vT = |vC , hT + · · · ,
3 3
where ˜· · · ™ 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 ¬‚ipped (h ” v), when the control photon is v-polarized. With the identi¬cation
h ” 0 and v ” 1, this is the photonic version of eqn (20.72).
A simple modi¬cation of the design in Fig. 20.9 yields the gate action
1 1
CS |hC , hT = |hC , hT + · · · , CS |hC , vT = |hC , vT + · · · ,
3 3 (20.76)
1 1
CS |vC , hT = |vC , hT + · · · , CS |vC , vT = ’ |vC , vT + · · · ,
3 3
which satis¬es the de¬nition (20.73) of a controlled-sign gate.
¿
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 heralded”
i.e. the outcome is known”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 (O™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 computing—
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
¬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
su¬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 pairs”i.e. pairs of photons described by a Bell
state”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,

|± = (|0 ± |1 ) / 2 , (20.77)

that satisfy X |± = ± |± . In parity-state encoding, the logical 0 and 1 are represented
by n-qubit states:
1
|0 (n) = √ —n |+ j + —n |’ j ,
j=1 j=1
2
(20.78)
1
(n)
= √ —j=1 |+ j ’ —j=1 |’ 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 scheme”which actually amounts to a fundamentally di¬erent model
for quantum computing”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.
¿ Quantum information

The logic gates produce a sequence of entangled states that ends in the desired ¬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)
|¦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 de¬nitions (20.73) and (20.77), leads to the explicit expression
1
|¦lin3 = √ [|+ |0 |+ + |’ |1 |’ 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 de¬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 ¬nal result.
The use of irreversible measurements as an integral part of the algorithm, rather
than just the ¬nal readout step, has led to the name one-way quantum computing
for this approach. For a su¬ciently large cluster state, it has been shown that these
elements are su¬cient to implement a universal quantum computer. The di¬erent
structures of reversible and one-way computing make comparisons a bit di¬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 Grover™s search algorithm.
¿
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 θ = ’π/2
and θ = π/2.

20.2 Modi¬ed beam splitter
Consider the modi¬ed beam splitter pictured in Fig. 20.1.
(1) Derive eqn (20.5) for the scattering matrix.
(2) For general values of θ and θ , 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 θ =
’π/2 and θ = π/2 are particularly useful.
(3) If no variable retarder plates are available, i.e. θ = θ = 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 de¬ned in eqn (20.15).
(1) Show that the Bell states are mutually orthogonal and all normalized to unity.
(2) Explain”ideally without any further algebra”why the Bell states form a basis
for Ha — Hb .

20.4 No-cloning theorem for photons
Consider cloning the one-photon states |γ = “† |0 and |ζ = Z † |0 , where

ζks a† .
Z† = ks
s
¼ Quantum information

(1) Derive the commutator

Z, “† = —
ζks γks ≡ (ζ, γ) .
ks

(2) Adapt the general proof for the no-cloning theorem to show that the cloning
assumption (20.25), and the corresponding assumption for |ζ , cannot be satis¬ed
for all choices of the operators “† and Z † .

20.5 Cloning a known state
For the device in Section 20.2.1 that clones a known state, assume the model interaction
Hint = gσ’ a† + a† + HC, between the two-level atom and the ¬eld. For the initial
kh kv
state |1kh , use ¬rst-order, time-dependent perturbation theory to calculate the change
in the initial state vector and thus derive eqn (20.26).

Buˇek“Hillery QCM—
20.6 z
Use the explicit expressions (20.31) and (20.32) to evaluate the reduced qubit density
operators ρab , ρa , and ρb . Use the results to calculate the ¬delities for the clones a and
b.

Photon cloning machine—
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 θ, i.e.

eh (kn ) = cos θ eh (kn ) + sin θ ev (kn ) ,
ev (kn ) = ’ sin θ eh (kn ) + cos θ ev (kn ) ,

derive the corresponding transformation of the creation operators a† n h , a† 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 ¬eld with amplitude E = Eh eh + Ev ev propagating in the
z-direction, that falls on a retarder plate of thickness ∆z lying in the (x, y)-plane.
½
Exercises

(1) By discarding an overall phase factor show that the output ¬eld E = Eh eh + Ev ev
is related to the input ¬eld by col (Eh ,Ev ) = Tξ (‘) col (Eh ,Ev ), where the Jones
matrix Tξ (‘) is given by

’ sin ‘ cos ‘ 1 ’ eiξ
cos2 ‘ + sin2 ‘eiξ
Tξ (‘) = ,
’ sin ‘ cos ‘ 1 ’ eiξ sin2 ‘ + cos2 ‘eiξ

‘ is the angle between eh and eF , and ξ = (nS ’ nF ) ω∆z/c.
(2) Evaluate the Jones matrix for ξ = π/2 (the quarter-wave plate) and ξ = π (the
half-wave plate).
(3) For ‘ = 0 and a 45—¦ -polarized input, i.e. Eh = Ev , what is the output polarization
state? Answer the same question if ‘ = π/4 and the input ¬eld is h-polarized.

20.9 Quantum dense coding
The unitary operators used by Bob for quantum dense coding are de¬ned by
U ‘»/4 , ‘»/2 = Tπ/2 ‘»/4 Tπ ‘»/2 , where Tξ (‘) 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

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

determine the expansion coe¬cients for which U (0, 0) |˜ = |Ψ+ .
(2) Find three other sets of values ‘»/4 , ‘»/2 such that U ‘»/4 , ‘»/2 |˜ 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

Ψ’ = Ψ’ ,
1
= √ |hA , vA + (A ” B) ,
Ψ+
2
i
¦± = {|hA , hA ± |vA , vA } + (A ” B) .
2
20.11 Rotated polarization basis
Consider the 45—¦ -rotated polarization basis de¬ned by eqn (20.48).
(1) Derive
√ √
a† = a † ’ a † / 2 , a † = a † + a † / 2 ,
γv
γv γv
γh γh γh

where γ ∈ {A, B}.
¾ Quantum information

(2) Show that

1 1
’ √ hA , v A ,
|hA , hA = hA , hA + |v A , v A
2 2
1 1
+ √ 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 Insu¬cient information
Consider Alice™s attempt to give Bob instructions for making an approximate copy of
her unknown qubit |γ .

(1) Given the unit vector n and the eigenvalue of n · σ, explain why Bob™s best
estimate for the unknown state |γ 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 Alice™s qubit, who does know the state |γ , is willing
to send her an endless stream of qubits, all prepared in the same state. Alice™s
research budget, however, limits her to a ¬nite number of measurements. Can
Alice supply Bob with enough information to permit an exact reproduction (up
to an overall phase) of |γ ?

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-¬‚ip and the bit-¬‚ip.
(3) Suppose that Alice does her Bell state measurement, but that Eve intercepts the
message to Bob. Calculate the reduced density operator ρB 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 A“T subsystem.
(2) Derive the photonic version of eqn (20.66).
(3) Give explicit forms for the action of the unitary transformations Upf (phase-¬‚ip)
and Ubf (bit-¬‚ip) on the creation operators.
¿
Exercises

Quantum logic gates—
20.15
(1) Show that the X, Z, and Hadamard gates are unitary operators.
(2) Use the representation |γ = γ0 |0 + γ1 |1 of a general qubit to express all three
gates as 2 — 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 ± around the
axis directed along the unit vector u is (Bransden and Joachain, 1989, Sec. 6.9)
±
±
’ i sin u·σ.
Ru (±) = cos
2 2
Combine this with the Poincar´-sphere representation
e
θ θ
|γ = cos |0 + eiφ sin |1
2 2
for qubits to show that the X, Z, and Hadamard gates are respectively given by
iRux (π), iRuz (π), and iRh (π), where ux , uy , uz are the coordinate unit vectors
√ √
and h = ux / 2 + uz / 2.
(4) Show that the control-NOT operator CNOT , de¬ned by eqn (20.72), is unitary.
Use the basis {|0, 0 , |0, 1 , |1, 0 , |1, 1 } to express CNOT as a 4 — 4 matrix.

Single-photon gates—
20.16
Identify the polarization states of a single photon with the logical states by |h ” |0
and |v ” |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 circuits—
20.17
Work out the gates required for the outcomes |¦’ and |Ψ’ in the computation
discussed in Section 20.5.1-A and draw the corresponding quantum circuit diagrams.

Controlled-NOT gate—
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 ¬‚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 ± s

to the asymmetric beam splitters, e.g. assign ’ R for re¬‚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 ¬nd the
overall scattering relations (aCh , aCv , . . .) ’ (aCh , aCv , . . .) which de¬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 ’ ’ 3 ) R |hC , hT ’ 3 ) R |hC , vT + ··· ,
( ( +
1 2 1 2
2 2
1 1
|hC , vT ’ ’ 3 ) R |hC , hT
1 ( 2 ’ 3 ) R |hC , vT + · · · ,
1( 2 + +
2 2
1 1
|vC , hT ’ [(2 ’ 2 3 ) R ’ 1] |vC , hT + [1 ’ (2 + 2 3 ) R] |hC , vT + · · · ,
2 2
1 1
|vC , vT ’ [1 ’ (2 + 2 3 ) R] |vC , hT + [(2 ’ 2 3 ) R ’ 1] |hC , vT + · · · .
2 2
Determine the value of R and the assignment of the s needed to de¬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 · b = ax bx + ay by + az bz , or

3
a·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 — b)i = ijk aj bk , (A.2)

where the alternating tensor is de¬ned by
ijk

§
⎪1 ijk is an even permutation of 123 ,

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

©
0 otherwise .

A.2 General vector spaces
A complex vector space is a set H on which the following two operations are de¬ned.

(1) Multiplication by scalars. For every pair (±, ψ), where ± is a scalar, i.e. a complex
number, and ψ ∈ H, there is a unique element of H that is denoted by ±ψ.
(2) Vector addition. For every pair ψ, φ of vectors in H there is a unique element of H
denoted by ψ + φ.

The two operations satisfy (a) ±(βψ) = (±β) ψ, and (b) ± (ψ + φ) = ±ψ + ±φ. It
is assumed that there is a special null vector, usually denoted by 0, such that ±0 = 0
and ψ + 0 = ψ. If the scalars are restricted to real numbers these conditions de¬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 ψ = (ψ1 , . . . , ψn ), where each component ψi is a complex number,
de¬nes a complex vector space with component-wise operations:
±ψ = (±ψ1 , . . . , ±ψn ) ,
(A.4)
ψ + φ = (ψ1 + φ1 , . . . , ψn + φn ) .

Each vector in R3 or Cn is speci¬ed by a ¬nite number of components, so these spaces
are said to be ¬nite dimensional.
The set of complex functions, C (R), of a single real variable de¬nes a vector space
with point-wise operations:
(±ψ) (x) = ±ψ (x) , (A.5)
(ψ + φ) (x) = ψ (x) + φ (x) , (A.6)
where ± is a scalar, and ψ (x) and φ (x) are members of C (R). This space is said to
be in¬nite dimensional, since a general function is not determined by any ¬nite set
of values.
For any subset U ‚ H, the set of all linear combinations of vectors in U is called
the span of U, written as span (U). A family B ‚ 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 ψ to each vector ψ ∈ H,
such that
M (±ψ + βφ) = ±M ψ + βM φ (A.7)
for any pair of vectors ψ and φ, and any scalars ± and β. 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 De¬nition
An inner product on a vector space H is a rule that assigns a complex num-
ber, denoted by (φ, ψ), to every pair of elements φ and ψ ∈ H, with the following
properties:
(φ, ±ψ + βχ) = ± (φ, ψ) + β (φ, χ) , (A.8a)

(φ, ψ) = (ψ, φ) , (A.8b)
(φ, φ) < ∞ ,
0 (A.8c)
(φ, φ) = 0 if and only if φ = 0 . (A.8d)
An inner product space is a vector space equipped with an inner product. The
inner product satis¬es the Cauchy“Schwarz inequality:
2
|(φ, ψ)| (φ, φ) (ψ, ψ) . (A.9)
Two vectors are orthogonal if (φ, ψ) = 0. If F is a subspace of H, then the orthogonal
complement of F is the subspace F⊥ of vectors orthogonal to every vector in F.
Hilbert spaces


The norm ψ of ψ is de¬ned as ψ = (ψ, ψ), so that ψ = 0 implies ψ = 0.
Vectors with ψ = 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 =
φ(1) , φ(2) , . . . , is said to be separable. The vector spaces relevant to quantum theory
are all separable. A basis for which φ(n) , φ(m) = δnm holds is called orthonormal.
Every vector in H can be uniquely expanded in an orthonormal basis, e.g.

ψn φ(n) ,
ψ= (A.10)
n=1

where the expansion coe¬cients are ψn = φ(n) , ψ .
A sequence ψ 1 , ψ 2 , . . . , ψ k , . . . of vectors in H is convergent if

ψ k ’ ψ j ’ 0 as k, j ’ ∞ . (A.11)

A vector ψ is a limit of the sequence if

ψ k ’ ψ ’ 0 as k ’ ∞ . (A.12)

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

A.3.2 Examples
The ¬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

(ψ, φ) = ψn φn . (A.13)
n=1

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

dx |ψ (x)|2 < ∞ , (A.14)
’∞

then the Cauchy“Schwarz inequality for integrals,
∞ ∞ ∞
2
— 2 2
dx |ψ (x)| dx |φ (x)| ,
dxψ (x) φ (x) (A.15)
’∞ ’∞ ’∞

is su¬cient to guarantee that the inner product de¬ned by

dxψ — (x) φ (x)
(ψ, φ) = (A.16)
’∞

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 ψ ∈ H such that Aψ < ∞. An operator A is positive de¬nite
0 for all ψ ∈ D (A), and it is bounded if Aψ < b ψ , where b is a
if (ψ, Aψ)
constant independent of ψ. The norm of an operator is de¬ned by

A = max for ψ = 0 , (A.17)
ψ
so a bounded operator is one with ¬nite norm.
If Aψ = »ψ, where » is a complex number and ψ is a vector in the Hilbert space,
then » is an eigenvalue and ψ is an eigenvector of A. In this case » is said to belong
to the point spectrum of A. The eigenvalue » is nondegenerate if the eigenvector
ψ is unique (up to a multiplicative factor). If ψ is not unique, then » is degenerate.
The linearly-independent solutions of Aψ = »ψ form a subspace called the eigenspace
for », and the dimension of the eigenspace is the degree of degeneracy for ». The
continuous spectrum of A is the set of complex numbers » such that: (1) » is not
an eigenvalue, and (2) the operator » ’ A does not have an inverse.
The adjoint (hermitian conjugate) A† of A is de¬ned by

ψ, A† φ = (φ, Aψ) , (A.18)
and A is self-adjoint (hermitian) if D A† = D (A) and (φ, Aψ) = (Aφ, ψ). 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 ψ, A† Aψ = (φ, φ), where φ = Aψ, so that
ψ, A† Aψ 0, (A.19)
i.e. A† A is positive de¬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 ψ as ψ ranges
over H. This is a subspace of H, since
±P φ + βP χ = P (±φ + βχ) (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 φ(n) an orthonormal basis for S. The operator P , de¬ned
by
φ(n) , ψ φ(n) ,
Pψ = (A.22)
n
is a projection operator, since
P 2ψ = φ(n) , ψ P φ(n) = φ(n) , ψ φ(n) = P ψ . (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 ψ = ψ.
A self-adjoint operator with pure point spectrum {»1 , »2 , . . .} has the spectral
resolution
A= »n Pn , (A.24)
n

where Pn is the projection operator onto the subspace of eigenvectors with eigenvalue
»n . The spectral resolution for a self-adjoint operator A with a continuous spectrum
is
A = » dµ (») , (A.25)

where dµ (») is an operator-valued measure de¬ned by the following statement:
for each subset ∆ of the real line,

P (∆) = dµ (») (A.26)


’1
is the projection operator onto the subspace of vectors ψ such that (» ’ A) ψ <∞
for all » ∈ ∆ (Riesz and Sz.-Nagy, 1955, Chap. VIII, Sec. 120).
/
A linear operator U is unitary if it preserves inner products, i.e.

(U ψ, U φ) = (ψ, φ) (A.27)

for any pair of vectors ψ, φ in the Hilbert space. A necessary and su¬cient condition
for unitarity is that the operator is norm preserving, i.e.

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

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

eiθn Pn , θn real ,
U= (A.29)
n

and for a continuous spectrum

eiθ dµ (θ) , θ real .
U= (A.30)

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

N, N † = 0 . (A.31)
The hermitian and unitary operators are both normal. The hermitian operators N1 =
N + N † /2 and N2 = N ’ N † /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
¼ Mathematics

for a point spectrum, and

N= x dµ1 (x) + i y dµ2 (y) , dµ1 (x) , dµ2 (y) = 0 (A.33)
∆1 ∆1

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 — 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 de¬ned as
X1n1 · · · XN nN ,
det (X) = (A.36)
n1 ···nN
n1 ···nN

where the generalized alternating tensor is
§
⎪1 n1 · · · nN is an even permutation of 12 · · · N ,

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

©
n1 ···nN 0 otherwise .
The trace of X is
N
Tr X = Xnn . (A.38)
n=1

The transpose matrix X T is de¬ned by Xnm = Xmn . The adjoint matrix X †
T
† —
is the complex conjugate of the transpose: Xnm = Xmn . A matrix X is symmetric if
X = X T , self-adjoint or hermitian if X † = X, and unitary if X † X = XX † = I, where I
is the N — N identity matrix. Unitary transformations preserve the inner product. The
hermitian and unitary matrices both belong to the larger class of normal matrices
de¬ned by X † X = XX † .
A matrix X is positive de¬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 de¬nition is that X is positive de¬nite if
φ† Xφ 0 (A.39)
for all vectors φ. For a positive-de¬nite matrix X, there is a matrix Y such that
X = Y Y †.
The normal matrices have the following important properties (Mac Lane and Birk-
ho¬, 1967, Sec. XI-10).
½
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 † with eigenvalue z — , i.e. Zf = zf ’ Z † f = z — 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

dxf (x) e’ikx ,
f (k) = (A.40)
’∞

so that the inverse Fourier transform is

dk
f (k) eikx .
f (x) = (A.41)

’∞

The virtue of this notation is that it reminds us that the two functions are, generally,
drastically di¬erent, e.g. if f (x) = 1, then f (k) = 2πδ (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 sacri¬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

dxf (x) e’ikx ,
f (k) = (A.42)
’∞

dk
f (k) eikx .
f (x) = (A.43)

’∞

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 di¬erent representations of the same physical
quantity. Confusion is readily avoided by paying attention to the physical signi¬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 satis¬es

f — (k) = [f (k)] = f (’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:
¾ Mathematics


dxf (x) eiωt ,
f (ω) = (A.45)
’∞


f (k) e’iωt .
f (t) = (A.46)

’∞

Fourier transforms of functions of several variables, typically f (r), are de¬ned
similarly:

d3 rf (r) e’ik·r ,
f (k) = (A.47)

d3 k
(k) eik·r ,
f (r) = 3f (A.48)
(2π)
where the integrals are over position space and reciprocal space (k-space) respectively.
If f (r) is real then
f — (k) = f (’k) . (A.49)
Combining these conventions for a space“time function f (r, t) yields the transform
pair

dtf (r, t) e’i(k·r’ωt) ,
d3 r
f (k, ω) = (A.50)

d3 k dω
f (k, ω) ei(k·r’ωt) .
f (r, t) = (A.51)
3 2π
(2π)
The last result is simply the plane-wave expansion of f (r, t). If f (r, t) is real, then
the Fourier transform satis¬es
f — (k, ω) = f (’k, ’ω) . (A.52)
Two related and important results on Fourier transforms”which we quote for the
one- and three-dimensional cases”are Parseval™s theorem:
dω —
dtf — (t) g (t) = f (ω) g (ω) , (A.53)

d3 k —

3
d rf (r) g (r) = 3 f (k) g (k) , (A.54)
(2π)
and the convolution theorem:
dt f (t ’ t ) g (t )
h (t) = if and only if h (ω) = f (ω) g (ω) , (A.55)


f (ω ’ ω ) g (ω )
h (ω) = if and only if h (t) = f (t) g (t) , (A.56)

d3 r f (r ’ r ) g (r )
h (r) = if and only if h (k) = f (k) g (k) , (A.57)

d3 k
(k ’ k ) g (k ) if and only if h (r) = f (r) g (r) .
h (k) = 3f (A.58)
(2π)
These results are readily derived by using the delta function identities (A.95) and
(A.96).
¿
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 de¬ned on a ¬nite interval (’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 (’L/2) = uk (L/2), restricts k
to the discrete values
2πn
(n = 0, ±1, ±2, . . .) .
k= (A.59)
L

Normalization requires C = 1/ L, so the transform is
L/2
1
dxf (x) e’ikx ,
fk = √ (A.60)
L ’L/2

and the inverse transform f (x) is
1
f (x) = √ fk eikx . (A.61)
L k

The continuous transform is recovered in the limit L ’ ∞ by ¬rst using eqn (A.60)
to conclude that √
Lfk ’ f (k) as L ’ ∞ , (A.62)
and writing the inverse transform as

1
Lfk eikx .
f (x) = (A.63)
L
k

The di¬erence between neighboring k-values is ∆k = 2π/L, so this equation can be
recast as
∆k √ dk
Lfk eikx ’ f (k) eikx .
f (x) = (A.64)
2π 2π
k

In Cartesian coordinates the three-dimensional discrete transform is de¬ned on a
rectangular parallelepiped with dimensions Lx , Ly , Lz . The one-dimensional results
then imply
1
d3 rf (r) e’ik·r ,
fk = √ (A.65)
VV
where the k-vector is restricted to
2πny 2πnz
2πnx
ux + uy + uz , (A.66)
k=
Lx Ly Lz
and V = Lx Ly Lz . The inverse transform is
1
f (r) = √ fk eik·r , (A.67)
V k

and the integral transform is recovered by
Mathematics


V fk ’ f (k) as V ’ ∞ . (A.68)

The sum and integral over k are related by

d3 k
1
’ , (A.69)
(2π)3
V
k

which in turn implies
3
V δk,k ’ (2π) δ (k ’ k ) . (A.70)

A.5 Laplace transforms
Another useful idea”which is closely related to the one-dimensional Fourier trans-
form”is the Laplace transform de¬ned by

dt e’ζt f (t) .
f (ζ) = (A.71)
0

In this case, we will use the standard mathematical notation f (ζ), since we do not use
Laplace transforms as frequently as Fourier transforms. The inverse transform is
ζ0 +i∞
dζ ζt
f (t) = e f (ζ) . (A.72)
2πi
ζ0 ’i∞

The line (ζ0 ’ i∞, ζ0 + i∞) in the complex ζ-plane must lie to the right of any poles
in the transform function f (ζ).
The identity
df
(ζ) = ζ f (ζ) ’ f (0) (A.73)
dt
is useful in treating initial value problems for sets of linear, di¬erential equations. Thus
to solve the equations
dfn

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