(1) Use eqn (19.50) to expand |hA , vB and |vA , hB in terms of |hA , βB and vA , β B .

(2) Evaluate the reduced states |Ψ’ yes and |Ψ’ no .

(3) Calculate the conditional probabilities p(Ayes |», ±, β, Byes ) and p(Ayes |», ±, β,

Bno ).

(4) Calculate the joint probability p (Ayes , Byes |Ψ’ , ±, β ).

(5) If |Ψ’ is replaced by |φ = |hA , vB , is outcome independence still violated?

19.4 Violation of Bell™s inequality

(1) Carry out the calculations needed to derive eqns (19.58) and (19.59).

(2) If |Ψ’ is replaced by |φ = |hA , vB , is the Bell inequality still violated?

20

Quantum information

Quantum optics began in the early years of the twentieth century, but its applications

to communications, cryptography, and computation are of much more recent vintage.

The progress of communications technology has made quantum e¬ects a matter of

practical interest, as evidenced in the discussion of noise control in optical transmission

lines in Section 20.1. The issue of inescapable quantum noise is also related to the

di¬culty”discussed in Section 20.2”of copying or cloning quantum states.

Other experimental and technological advances are opening up new directions for

development in which the quantum properties of light are a resource, rather than a

problem. Streams of single photons with randomly chosen polarizations have already

been demonstrated as a means for the secure transmission of cryptographic keys, as

discussed in Section 20.3. Multiphoton states o¬er additional options that depend on

quantum entanglement, as shown by the descriptions of quantum dense coding and

quantum teleportation in Section 20.4. This set of ideas plays a central role in the

closely related ¬eld of quantum computing, which is brie¬‚y reviewed in Section 20.5.

20.1 Telecommunications

Optical methods of communication”e.g. signal ¬res, heliographs, Aldis lamps, etc.”

have been in use for a very long time, but high-speed optical telecommunications

are a relatively recent development. The appearance of low-loss optical ¬bers and

semiconductor lasers in the 1960s and 1970s provided the technologies that made new

forms of optical communication a practical possibility.

The subsequent increases in bandwidth to 104 GHz and transmission rates to the

multiterabit range have led”under the lash of Moore™s law”to substantial decreases

in the energy per bit and the size of the physical components involved in switching

and ampli¬cation of signals. An inevitable consequence of this technologically driven

development is that phenomena at the quantum level are rapidly becoming important

for real-world applications.

Long-haul optical transmission lines require repeater stations that amplify the

signal in order to compensate for attenuation. This process typically adds noise to the

signal; for example, erbium doped ¬ber ampli¬ers (EDFA) degrade the signal-to-noise

ratio by about 4 dB. Only 1 dB arises from technical losses in the components; the

remaining 3 dB loss is due to intrinsic quantum noise.

Thus quantum noise is dominant, even for apparently classical signals containing

a very large number of photons. Similar e¬ects arise when the signal is divided by a

passive device such as an optical coupler. Future technological developments can be

¼¾ Quantum information

expected to increase the importance of quantum noise; therefore, we devote Sections

20.1.2 and 20.1.3 to the problem of quantum noise management.

Optical transmission lines—

20.1.1

Let us consider an optical transmission line in which the repeater stations employ

phase-insensitive ampli¬ers. For phase-insensitive input noise, the input and output

signal-to-noise ratios are de¬ned by

2

bγ (ω)

[SNR]γ = (γ = in, out) , (20.1)

Nγ (ω)

where Nin and Nout are the noise in the input and output respectively. The relation

between the input and output signal-to-noise ratios is obtained by combining eqns

(16.36) and (16.150) to get

2 2

bout (ω) bin (ω) [SNR]in

[SNR]out = = = . (20.2)

Nout (ω) Nin (ω) + A (ω) 1 + A (ω) /Nin (ω)

The most favorable situation occurs when the input noise strength has the standard

quantum limit value 1/2. In this case one ¬nds

[SNR]out 1 1 1

’.

= (20.3)

2 “ 1/G (ω)

[SNR]in 1 + 2A (ω) 2

The inequality follows from eqn (16.151) and the ¬nal result represents the high-gain

limit. The decibel di¬erence between the signal-to-noise ratios is therefore bounded by

[SNR]out

’10 log 2 ≈ ’3 .

d = 10 log (20.4)

[SNR]in

In other words, the quantum noise added by a high-gain, phase-insensitive ampli¬er

degrades the signal-to-noise ratio by at least three decibels. This result holds even for

strong input ¬elds containing many photons. For example, if the input is described by

the multi-mode coherent state de¬ned by eqns (16.98)“(16.100), then the input noise

strength is Nin (ω) = 1/2. In this case the inequality (20.4) is valid for any value of

2

the e¬ective classical intensity |βin (t)| , no matter how large.

This result demonstrates that high-gain, phase-insensitive ampli¬ers are intrinsi-

cally noisy. This noise is generated by fundamental quantum processes that are at

work even in the absence of the technical noise”e.g. insertion-loss noise and Johnson

noise in the associated electronic circuits”always encountered in real devices.

Reduction of ampli¬er noise—

20.1.2

In the discussion of squeezing in Section 15.1 we have seen that quantum noise can be

unequally shared between di¬erent ¬eld quadratures by using nonlinear optical e¬ects.

This approach”which can yield essentially noise-free ampli¬cation for one quadrature

by dumping the unwanted noise in the conjugate quadrature”is presented in the

present section.

¼¿

Telecommunications

There is an alternative scheme, based on the special features of cavity quantum

electrodynamics, in which the signal propagates through a photonic bandgap. This

is a three-dimensional structure in which periodic variations of the refractive index

produce a dispersion relation that does not allow propagating solutions in one or more

frequency bands”the bandgaps”so that vacuum ¬‚uctuations and the associated noise

are forbidden at those frequencies (Abram and Grangier, 2003).

In the discussion of linear optical ampli¬ers in Chapter 16, we derived the in-

equality (16.147) which shows that the ampli¬er noise for a phase-conjugating am-

pli¬er is always larger than the vacuum noise, i.e. Namp > 1/2. On the other hand,

the noise added by a phase-transmitting ampli¬er can be made as small as desired

by allowing G (ω) to approach unity. Thus noise reduction can be achieved with a

phase-transmitting ampli¬er, provided that we are willing to give up any signi¬cant

ampli¬cation.

Achieving noise reduction by giving up ampli¬cation scarcely recommends itself

as a useful strategy for long-haul communications, so we turn next to phase-sensitive

ampli¬ers. In this case, the lower bound (16.147) on ampli¬er noise is replaced by the

ampli¬er uncertainty principle (16.169). The resemblance between eqn (16.169) and

the standard uncertainty principle for canonically conjugate variables is promising,

since the latter is known to allow squeezing.

Furthermore, the ampli¬er uncertainty principle has the additional advantage that

the lower bound is itself adjustable; indeed, it can be set to zero. Even when this is

not possible, the noise in one quadrature can be reduced at the expense of increasing

the noise in the conjugate quadrature. We ¬rst demonstrate two examples in which

the ampli¬er noise actually vanishes, and then discuss what can be achieved in less

favorable situations.

The phase-sensitive, traveling-wave ampli¬er described in Section 16.3.2 is intrinsi-

cally noiseless, so the lower bound of the ampli¬er uncertainty principle automatically

vanishes. For applications requiring the generally larger gains possible for regenerative

ampli¬ers, the phase-sensitive OPA presented in Section 16.2.2 can be modi¬ed to

provide noise-free ampli¬cation.

For the phase-sensitive OPA, the ampli¬er noise comes from vacuum ¬‚uctuations

entering the cavity through the mirror M2, as shown in Fig. 16.2. Thus the ampli¬er

noise would be eliminated by preventing the vacuum ¬‚uctuations from entering the

cavity. In an ideal world, this can be accomplished by making M2 a perfect re¬‚ector,

i.e. setting κ2 = 0 in eqns (16.47)“(16.49). Under these circumstances, eqn (16.49)

reduces to · (ω) = 0, so that the ampli¬er is noiseless.

In these examples, the noise vanishes for both of the principal quadratures, i.e.

A1 (ω) = A2 (ω) = 0. According to eqn (16.167), this means that the signal-to-noise

ratio is preserved by ampli¬cation. This is only possible if the lower bound in eqn

(16.170) vanishes, and this in turn requires G1 (ω) G2 (ω) = 1. Consequently, the price

for noise-free ampli¬cation is that one quadrature is attenuated while the other is

ampli¬ed.

In the real world”where traveling-wave ampli¬ers may not provide su¬cient gain

and there are no perfect mirrors”other options must be considered. The general idea is

to achieve high gain and low noise for the same quadrature. For this purpose, the signal

¼ Quantum information

should be carried by modulation of either the amplitude or the phase of the chosen

c c

quadrature, e.g. Xin (ω); and the input noise should be small, i.e. ∆Xin (ω) 1/2.

In the high-gain limit, the lower bound in eqn (16.169) is proportional to

G1 (ω) G2 (ω); consequently, the ampli¬er noise in the conjugate quadrature is nec-

essarily large. This is not a problem as long as the noisy quadrature is strongly rejected

by the detectors in use. The degree to which these objectives can be attained depends

on the details of the overall design.

20.1.3 Reduction of branching noise

The information encoded in an optical signal is often intended for more than one

recipient, so that it is necessary to split the signal into two or more identical parts,

usually by means of a directional coupler. These junction points”which are often

called optical taps”may also be used to split o¬ a small part of the signal for

measurement purposes.

Whatever the motive for the tap, it is in e¬ect a measurement of the radiation

¬eld. A measurement of any quantum system perturbs it in an uncontrollable fashion;

consequently, the optical tap must add noise to the signal. A succession of taps will

therefore degrade the signal, even if there is no associated ampli¬er noise.

In fact, we have already met with this e¬ect, in the guise of the partition noise

at a beam splitter. The explanation that partition noise arises from vacuum ¬‚uctu-

ations entering through the unused port of the beam splitter suggests that injecting

a squeezed vacuum state into the unused port might help with the noise problem.

This idea was initially proposed in 1980 (Shapiro, 1980) and experimentally realized

in 1997 (Bruckmeier et al., 1997). We discuss below a simple model that illustrates

this approach.

The idea is to add two elements, shown in Fig. 20.1, to the simple beam splitter

described in Section 8.4: (1) a squeezed-light generator (SQLG); and (2) a pair of

variable retarder plates (see Exercise 20.1). The SQLG, which is the essential part

of the modi¬ed beam splitter, injects squeezed light into the previously unused input

port 2. The function of the variable retarder plates, which are placed at the input port

2 and the output port 2 , is to simplify the overall scattering matrix.

The phase transformations, a2 ’ eiθ a2 and a2 ’ eiθ a2 , imposed by the retarder

plates are more usefully described as rotations of the input and output quadratures

through the angles θ and θ . Combining the phase transformations with eqn (8.63)”as

outlined in Exercise 20.2”yields the scattering matrix

√ √

i Reiθ

T

S = √ iθ √ i(θ+θ ) . (20.5)

i Re Te

√

The phases of the beam splitter coe¬cients have been chosen so that t = T is real and

√

r = i R is pure imaginary, where T and R are respectively the intensity transmission

and re¬‚ection coe¬cients.

The SQLG is designed to emit a squeezed state for the quadrature

1 ’iβ

e a2 + eiβ a† ,

X2 = (20.6)

2

2

¼

Telecommunications

Fig. 20.1 Modi¬ed beam splitter for noiseless

branching. The OPA injects squeezed light into

port 2 and the phase plates are used to obtain

a convenient form for the scattering matrix.

so an application of eqn (15.39) produces the variances

e’2r e2r

V (X2 ) = , V (Y2 ) = , (20.7)

4 4

where Y2 is the conjugate quadrature and r is the magnitude of the squeezing para-

meter. For the special values θ = ’π/2 and θ = π/2 the input“output relations for

the amplitude quadratures are

√ √

X1 T √R X1

√

= , (20.8)

’R

X2 X2

T

where the quadratures for the input channel 1, and the output channels 1 and 2 are

de¬ned by the angle β used in eqn (20.6).

Let us now specialize to a balanced beam splitter, and assume that the signal is

carried by X1 . The squeezed state satis¬es X2 = 0; consequently, the two output

signals have the same average:

1

X1 = X2 = √ X1 . (20.9)

2

Since the input signal X1 and the SQLG output are uncorrelated, the variances of the

output signals X1 and X2 are also identical:

¼ Quantum information

e’r

1

V (X1 ) = V (X2 ) = V (X1 ) + . (20.10)

2 8

The 50% reduction of the output variances compared to the input variance does

not mean that the output signals are quieter; it merely re¬‚ects the reduction of the

√

amplitudes by the factor 1/ 2. This can be seen by de¬ning the signal-to-noise ratios,

2 2

| Xm | | Xm |

SNR (Xm ) = , SNR (Xm ) = (m = 1, 2) , (20.11)

V (Xm ) V (Xm )

and using the previous results to ¬nd

SNR (X1 )

SNR (X1 ) = SNR (X2 ) = . (20.12)

1 + e’r / [4V (X1 )]

In the limit of strong squeezing, this coupler almost exactly preserves the signal-

to-noise ratio of the input signal. Consequently, the output signals are faithful copies

of the input signal down to the level of the quantum ¬‚uctuations. The injection of the

squeezed light into port 2 has e¬ectively diverted almost all of the partition noise into

the unobserved output quadratures Y1 and Y2 .

This scheme succeeds in splitting the signal without adding any noise, but at the

cost of reducing the intensity of the output signals by 50%. This drawback can be

overcome by inserting a noiseless ampli¬er, e.g. the traveling-wave OPA described in

Section 16.3.2, prior to port 1 of the beam splitter. The gain of the ampli¬er can

be adjusted so that each of the split signals has the same strength and the same

signal-to-noise ratio as the original signal.

20.2 Quantum cloning

At ¬rst glance, it may seem that the noiseless beam splitter of Section 20.1.3 produces

a perfect copy or clone of the input signal. This impression is misleading, since only the

expectation values and variances of the particular input quadrature X1 are faithfully

copied; indeed, the variance of the output conjugate quadrature Y1 is much larger than

the variance of Y1 .

This observation suggests a general question: To what extent does quantum theory

allow cloning? In the following section, we will review the famous no-cloning theorem

(Dieks, 1982; Wootters and Zurek, 1982), which outlaws perfect cloning of an unknown

quantum state.

We should note that this work was not done to answer the question we have

just raised. It was a response to a proposal by Herbert (1982) for a superluminal

communications scheme employing EPR correlations. The connection between no-

cloning and no-superluminal-signaling is a recurring theme in later work (Ghirardi

and Weber, 1983; Bussey, 1987).

The no-cloning theorem quickly became an important physical principle which

was, for example, used to argue for the security of quantum cryptography (Bennett

and Brassard, 1984). The ¬nal step in the initial development was the extension of the

result from pure to mixed states (Barnum et al., 1996).

¼

Quantum cloning

This was immediately followed by the work of Buˇek and Hillery (1996) who began

z

the investigation of imperfect cloning. We will study the degree of cloning allowed by

quantum theory in Section 20.2.2.

In the study of quantum information, the systems of interest are usually described

by states in a ¬nite-dimensional Hilbert space Hsys . For the special case of two-state

systems”e.g. a two-level atom, a spin-1/2 particle, or the two polarizations of a

photon”Hsys is two-dimensional, and a vector |γ in Hsys is called a qubit. The

generic description of qubits employs the so-called computational basis {|0 , |1 }

de¬ned by

σz |0 = |0 , σz |1 = ’ |1 . (20.13)

In this notation a general qubit is represented by |γ = γ0 |0 + γ1 |1 .

In the more general case, dim (Hsys ) = d > 2, the state is called a qudit. We will

follow the usual convention by referring to the systems under study as qubits, but

it should be kept in mind that many of the results also hold in the general ¬nite-

dimensional case. In the interests of simplicity, we will only treat closed systems un-

dergoing unitary time evolution.

For the applications considered below, it is often necessary to consider one or more

ancillary (helper) systems in addition to the system of interest. The reservoirs used

in the treatment of dissipation in Chapter 14 are an example of ancillary systems or

ancillas. In that case the unitary evolution of the closed sample“reservoir system was

used to derive the dissipative equations by tracing over the ancilla degrees of freedom.

Another common theme in this ¬eld is the assumption that the total system consists

of a family of distinguishable qubits. The Hilbert space H for the system is H =

HQ — Hanc , where Hanc and HQ are respectively the state spaces for the ancillas and

the family of qubits. This abstract approach has the great advantage that the results

do not depend on the speci¬c details of particular physical realizations, but there are,

nevertheless, some implicit physical assumptions involved.

If the qubits are particles, then”as we learnt in Section 6.5.1”the Hilbert space

HQ for two qubits is

(Hsys — Hsys )sym for bosons ,

HQ = (20.14)

(Hsys — Hsys )asym for fermions .

For massive particles”e.g. atoms, molecules, quantum dots, etc.”a way around this

complication is to choose an experimental arrangement in which each particle™s center-

of-mass position can be treated classically. In these circumstances, as we saw in Section

6.5.2, the symmetrization or antisymmetrization normally required for identical par-

ticles can be ignored. In this model, a qubit located at ra is described by a copy of

Hsys , called Ha . The vectors |γ a in Ha represent the internal states of the qubit.

For a family of two qubits, located at ra and rb , the space HQ is the unsymmetrized

tensor product: HQ = Ha — Hb . The Bell states, ¬rst de¬ned in Section 13.3.5 for

photons, are represented by

¼ Quantum information

1

Ψ± = √ {|1, 0 ± |0, 1 ab } ,

ab

ab

2

(20.15)

1

±

= √ {|0, 0 ± |1, 1 ab } ,

¦ ab

ab

2

where

≡ |u |v b .

|u, v (20.16)

ab a

More features of the Bell states can be found in Exercise 20.3.

In the general case of qubits located at r1 , . . . , rN the qubit space is

N

HQ = Ha , (20.17)

a=1

and a generic state is denoted by |u1 , . . . , uN 12···N . When no confusion will result, the

notation is simpli¬ed by omitting the subscripts on the kets, e.g. |u, v ab ’ |u, v . The

application of these ideas to photons requires a bit more care, as we will see below.

20.2.1 The no-cloning theorem

For closed systems, we can assume that every physically permitted operation is de-

scribed by a unitary transformation U acting on the Hilbert space H describing the

qubits and the ancillas. To set the scene for the cloning discussion, we assume that

there is a set of qubits, |B b , all in the same (internal) blank state |B , and a cloning

device which is initially in the ready state |R anc ∈ Hanc .

If we only want to make one copy”this is called 1 ’ 2 cloning”the total initial

state is

|γ, B, R ≡ |γ a — |B b — |R anc = |γ a |B b |R anc . (20.18)

The cloning assumption is that there is a unitary operator U such that

U |γ, B, R = |γ, γ, Rγ = |γ |γ |Rγ , (20.19)

a b anc

where |Rγ anc is the state of the cloner after it has cloned the state |γ a . In this

approach, cloning is not the creation of a new particle, but instead the imposition of

a speci¬ed internal state on an existing particle.

After this preparation, the no-cloning theorem can be stated as follows (Scarani

et al., 2005).

Theorem 20.1 There is no quantum operation that can perfectly duplicate an un-

known quantum state.

We will use a proof given by Peres (1995, Sec. 9-4) that exhibits a contradiction

following from the assumption that a cloning operation does exist, i.e. that there is a

unitary operator satisfying eqn (20.19).

¼

Quantum cloning

Since the cloning device is supposed to work in the absence of any knowledge of

the initial state, it must be possible to use U to clone a di¬erent state |ζ , so that

U |ζ, B, R = |ζ, ζ, Rζ . (20.20)

A direct use of the unitarity of U yields

γ, γ, Rγ | ζ, ζ, Rζ = γ, B, R| ζ, B, R = γ |ζ , (20.21)

where we have imposed the convention that the initial states |R anc , |γ a , |B b , and

|ζ a are all normalized and that the inner product between internal states does not

depend on the location of the qubit.

Using the explicit tensor products in eqns (20.19) and (20.20) produces the alter-

native form

2

γ, γ, Rγ | ζ, ζ, Rζ = Rγ |Rζ γ |ζ . (20.22)