Rγ |Rζ γ |ζ = 1 . (20.23)

The inner product Rγ |Rζ automatically satis¬es | Rγ |Rζ | 1, and we can

always choose |γ and |ζ so that | γ |ζ | < 1; therefore, there are states |γ and |ζ

for which eqn (20.23) cannot be satis¬ed. This contradiction proves the theorem.

This elegant proof shows that the impossibility of perfect cloning of unknown, and

hence arbitrary, states is a fundamental feature of quantum theory; indeed, the only

requirement is that quantum operations are represented by unitary transformations.

In this respect it is similar to the Heisenberg uncertainty principle, for which the sole

requirement is the canonical commutation relation [q, p] = i .

We should emphasize, however, that this argument only excludes universal clon-

ing machines, i.e. those that can clone any given state. This leaves open the possibility

that speci¬c states could be cloned. In fact the argument does not prohibit the cloning

of each member of a known set of mutually orthogonal states.

The application of this theorem in the context of quantum optics raises some

problems. The proof rests on the assumption that the qubits are distinguishable and

localizable, but photons are indistinguishable, massless bosons that cannot be precisely

localized and are easily created and destroyed. Thus it is not immediately obvious that

the proof of the no-cloning theorem given above applies to photons.

A second problem arises from the observation that stimulated emission”which

produces new photons with the same wavenumber and polarization as the incident

photons”would seem to provide a ready-made copying mechanism. Why is it that

stimulated emission is not a counterexample to the no-cloning theorem? In the follow-

ing paragraphs we will address these questions in turn.

Since photons are indistinguishable bosons, we cannot add any identifying subscript

to a photonic qubit |γ , and the two-qubit space is the two-photon Fock space, H(2) .

The simplest way to de¬ne a photonic qubit is to choose a speci¬c wavevector k, and

set |γ = “† |0 , where

γks a† .

“† = (20.24)

ks

s

Since s takes on two values, the state |γ quali¬es as a qubit.

½¼ Quantum information

Cloning this qubit can only mean that a second photon is added in the same mode;

therefore, the cloning transformation (20.19) for this case would be

1

U “† |0 |R = √ “†2 |0 |Rγ . (20.25)

anc anc

2

By contrast to the distinguishable qubit model, the polarization state is not imposed

on an existing photon in a blank state; instead, a new photon is created with the same

polarization as the original. Despite this signi¬cant physical di¬erence, a similar proof

of the no-cloning theorem can be constructed by following the hints in Exercise 20.4.

The proof of the no-cloning theorem”either the standard version starting with

eqn (20.19) or the photonic version treated in Exercise 20.4”does not suggest any

speci¬c mechanism that prevents cloning. Finding a mechanism of this sort for photons

turns out to be related to the second problem noted above. Could stimulated emission

provide a cloning method?

The discussion of stimulated emission starts with a photon incident on an atom in

an excited state. In this case, the nonzero ratio A/B = k 3 /π 2 = 0 of the Einstein

A and B coe¬cients provides the essential clue: stimulated emission is unavoidably

accompanied by spontaneous emission. Since the spontaneously emitted photons have

random directions and polarizations, they will violate the cloning assumptions (20.25).

This argument eliminates cloning machines based on excited atoms, but what about

parametric ampli¬ers, such as the traveling-wave OPA in Section 16.3.2, in which

there are no population inversions and, consequently, no excited atoms? This possible

loophole was closed by the work of Milonni and Hardies (1982), in which it is shown

that stimulated emission is necessarily accompanied by spontaneous emission, even in

the absence of inverted atoms.

In the context of quantum optics, the impossibility of perfect, universal cloning

can therefore be understood as a consequence of the unavoidable pairing of stimulated

and spontaneous emission.

The no-cloning theorem does not exclude devices that can clone each member of a

known set of orthogonal states. For example, two orthogonal polarization states can be

cloned by exploiting stimulated emission. For this purpose, suppose that the sum over

polarizations in eqn (20.24) refers to the linear polarization vectors eh (horizontal)

and ev (vertical ).

The cloning device consists of a trap containing a single excited atom, followed by

a polarizing beam splitter. The PBS is oriented so that h- and v-polarized photons

are sent through ports 1 and 2 respectively. For an initial state |1kh , the ¬rst-order

perturbation calculation suggested in Exercise 20.5 shows that the combination of

stimulated and spontaneous emission produces an output state proportional to

√

2 |2kh + |1kh , 1kv . (20.26)

Since the PBS sends the unwanted v-polarized photon through port 2, the only two-

photon state emitted through port 1 is the desired cloned state |2kh . The argument

is symmetrical under the simultaneous exchange of h with v and port 1 with port 2;

therefore, the device is equally good at cloning v-polarized photons.

½½

Quantum cloning

This design produces perfect clones of each state in the basis, but only if the

basis is known in advance, so that the PBS can be properly oriented. As usual, the

experimental realization is a di¬erent matter. This idea depends on having detectors

that can reliably distinguish between one and two photons in a given mode, but such

detectors are”to say the least”very hard to ¬nd.

Since classical theory is an approximation to quantum theory, we are left with a

¬nal puzzle: How is it that the no-cloning theorem does not prohibit the everyday

practice of amplifying and copying classical signals? To understand this, we observe

that for an incident state with ni photons, the total emission probability for the

ampli¬er is proportional to ni +1, where ni and 1 respectively correspond to stimulated

and spontaneous emission.

If ni = 1, the two processes are equally probable, but if ni 1, then stimulated

emission dominates the output signal. Thus the classical copying process can achieve

its aim, despite the fact that it cannot create a perfect clone of the input.

Quantum cloning machines—

20.2.2

The ideal cloning operation in eqn (20.19) would”if only it were possible”produce

an exact copy of a qubit without damaging the original. In their seminal paper on

imperfect cloning, Hillery and Buˇek posed two questions: (1) How close can one come

z

to perfect cloning? (2) What happens to the original qubit in the process?

Attempts to answer these questions have generated a large and rapidly developing

¬eld of research. In the remainder of this section, we will give a very brief outline of the

basic notions, and discuss one optical implementation. For those interested in a more

detailed account, the best strategy is to consult a recent review article, e.g. Scarani

et al. (2005) or Fan (2006).

A Cloning distinguishable qubits—

The unattainable ideal of perfect cloning is replaced by the idea of a quantum cloning

machine (QCM), which consists of a chosen ancillary state |R anc in Hanc and a

unitary transformation U acting on H = HQ — Hanc . We will only discuss the simplest

case of 1 ’ 2 cloning, for which the action of U on the initial state |γ, B, R de¬nes

the cloned state

|γ; γ ≡ U |γ, B, R . (20.27)

In general, the vector |γ; γ represents an entangled state of the ancilla and the two

qubits, so the state of the qubits alone is described by the reduced density operator

ρab = Tranc |γ; γ γ; γ| , (20.28)

where the trace is de¬ned by summing over a basis for the ancillary space Hanc . The

states of the individual qubits are in turn represented by the reduced density operators

ρa = Trb ρab and ρb = Tra ρab . (20.29)

The task is to choose |R anc and U to achieve the best possible result, as opposed

to imposing the form of |γ; γ a priori. This e¬ort clearly depends on de¬ning what is

meant by ˜best possible™.

½¾ Quantum information

Of the many available measures of success, the most commonly used is the ¬delity:

Fa (γ) = a γ |ρa | γ , Fb (γ) = b γ |ρb | γ , (20.30)

a b

which measures the overlap between the mixed state produced by the cloning operation

and the original pure state. A QCM is said to be a universal QCM if the ¬delities

are independent of |γ , i.e. the machine does equally well at cloning every state.

A nonuniversal QCM is called a state-dependent QCM. The QCM is a sym-

metric QCM if the ¬delities of the output states are equal, i.e. Fa (γ) = Fb (γ), and

it is an optimal QCM if the ¬delities are as large as quantum theory allows.

The unitary operator U for a QCM is linear, so its action on the general input

state |γ, B, R is completely determined by its action on the special states |0, B, R

and |1, B, R , where 0 and 1 label the computational basis vectors de¬ned by eqn

(20.13). For the Buˇek“Hillery QCM, the ancilla consists of a single qubit, |R anc =

z

R0 |0 anc + R1 |1 anc , and the transformation U is de¬ned by

2 1+

|0 a |0 b |1 anc ’

U |0, B, R = Ψ ab |0 anc , (20.31)

3 3

2 1+

|1 a |1 b |1 anc +

U |1, B, R = ’ Ψ ab |1 anc . (20.32)

3 6

The Bell state |Ψ+ ab is de¬ned in eqn (20.15). In Exercise 20.6, these explicit

expressions are used to evaluate the reduced density operators ρa and ρb which yield

the ¬delities Fa (γ) = Fb (γ) = 5/6. Thus the Buˇek“Hillery QCM is universal and

z

symmetric. It has also been shown”see the references given in Scarani et al. (2005)”

that it is optimal.

Cloning photons—

B

In order to carry out an actual experiment, the abstractions of the preceding discus-

sion must be replaced by real hardware. Furthermore, the application of these ideas

in quantum optics also requires a more careful use of the theory. Both of these con-

siderations are illustrated by an experimental demonstration of a cloning machine for

photons (Lamas-Linares et al., 2002).

The basic idea, as shown in Fig. 20.2, is to use stimulated emission in a type

II down-conversion crystal, which is adjusted so that the down-converted photons

propagating along certain directions are entangled in polarization (Kwiat et al., 1995b).

1

Pump

Fig. 20.2 Schematic for a photon cloning ma-

2

chine. The type II down-converter produces

1

nondegenerate signal and idler modes with

Down-

wavevectors k1 (mode 1) and k2 (mode 2). The

converter

photons are entangled in polarization.

½¿

Quantum cloning

The pump beam and the single photon to be injected into the crystal are both derived

from a Ti“sapphire laser producing 120 fs pulses. The pump is created by frequency-

doubling the laser beam, and the single-photon state is generated by splitting o¬ a

small part of the beam, which is then attenuated below the single-photon level.

With this method, there is still a small probability that two photons could be

injected. If no down-conversion occurs, the transmitted two-photon state will appear as

a false count for cloning. These false counts can be avoided by triggering the detectors

for the k1 -photons with the detection of the conjugate k2 -photon, which is a signature

of down-conversion.

To model this situation, we ¬rst pick a pair of orthogonal linear polarizations, eh

and ev , for each of the wavevectors. The production of polarization-entangled signal

and idler modes is then described by the interaction Hamiltonian

HSS = „¦P a† 1 v a† 2 h ’ a† 1 h a† 2 v + HC . (20.33)

k k k k

By following the hints in Exercise 20.7, one can show that this Hamiltonian is invariant

under joint and identical rotations of the two polarization bases around their respective

wavevectors.

The cloning e¬ect is consequently independent of the polarization of the input

photon; that is, this should be a universal QCM. It is therefore su¬cient to con-

sider a particular input state, say |1k1 v = a† 1 v |0 , which evolves into |• (t) =

k

exp (’iHSS t/ ) |1k1 v . The relevant time t is limited by the pulse duration of the

pump, which satis¬es „¦P tP 1; therefore, the action of the evolution operator can

be approximated by a Taylor series expansion of the exponential in powers of „¦P t:

|• (t) ≈ {1 ’ iHSS t/ + · · · } |1k1 v

√

= |1k1 v ’ i„¦P t 2 |2k1 v , 1k2 h ’ |1k1 v , 1k1 h , 1k2 v + · · · . (20.34)

This result for |• (t) displays the probabilistic character of this QCM; the most

likely outcome is that the injected photon passes through the crystal without producing

a clone. The cloning e¬ect occurs with a probability determined by the ¬rst-order

√

term in the expansion. The factor 2 in the ¬rst part of this expression represents the

enhancement due to stimulated emission.

According to von Neumann™s projection rule, the detection of a trigger photon with

wavevector k2 and either polarization leaves the system in the state

P2 |• (t)

|• (t) = , (20.35)

red

• (t) |P2 | • (t)

where

P2 = |1k2 v 1k2 v | + |1k2 h 1k2 h | (20.36)

is the projection operator describing the reduction of the state associated with this

measurement. Combining eqns (20.34) and (20.36) yields

2 1

|• (t) = ’i |2k1 v , 1k2 h ’ |1k1 v , 1k1 h , 1k2 v . (20.37)

red

3 3

½ Quantum information

The probability of detecting two photons in the mode k1 v is 2/3 and the probability

of detecting one photon in each of the modes k1 v and k1 h is 1/3. The factor of two

between the probabilities is also a consequence of stimulated emission.

The indistinguishability of the photons guarantees that the QCM is symmetric, but

it also prevents the de¬nition of reduced density operators like those in eqn (20.29).

In this situation, the cloning ¬delity can be de¬ned as the probability that an output

photon with wavevector k1 has the same polarization as the input photon. This hap-

pens with unit probability for the ¬rst term in |• (t) red and with probability 1/2 in

the second term; therefore, the ¬delity is

2 1 1 5

— (1) + —

F= = . (20.38)

3 3 2 6

The theoretical model for this QCM therefore predicts that it is universal, symmetric,

and optimal.

In the experiment, the incident photon ¬rst passes through an adjustable optical

delay line, which is used to control the time lapse ∆T between its arrival and that of

the laser pulse that generates the down-converted photons. Stimulated emission should

only occur when the photon wave packet and the pump pulse overlap. The results of

the experiment, which are shown in Fig. 20.3, support this prediction.

The number of counts, N (2, 0), with two photons in the mode k1 v and no photon in

the mode k1 h is shown in Fig. 20.3 as a function of the distance c∆T , for three di¬erent

polarization states”curves (a)“(c)”of the injected photon. As expected, there is a

pronounced peak at zero distance. The corresponding plots (d)“(f) of N (1, 1)”the

number of counts with one photon in each polarization mode”show no such e¬ect.

The experimental ¬delity can be derived from the ratio

Npeak (2, 0)

R= (20.39)

Nbase (2, 0)

between the peak value and the base value of the N (2, 0) curve. At maximum overlap

between the incident single-photon wave packet and the pump pulse (c∆T = 0), the

probability of the (2, 0)-con¬guration is

Npeak (2, 0)

P (2, 0) = , (20.40)

Npeak (2, 0) + Npeak (1, 1)

which becomes

R

P (2, 0) = (20.41)

R + Npeak (1, 1) /Nbase (2, 0)

when expressed in terms of R.

The base values Nbase (1, 1) and Nbase (2, 0) represent the situation in which there

is no overlap between the single-photon wave packet and the pump pulse. In this case,

the detection of the original photon and a down-converted photon in the spatial mode

k1 are independent events. Down-conversion produces k1 v and k1 h photons with equal

½

Quantum cloning

Linear 0o

(a) (d)

1200

500

1000

Counts (100s)

400

800

300

600

200

400

N(2,0) N(1,1)

100 200

0 0

’100 ’50 ’100 ’50

0 50 100 0 50 100

Linear 45o

(b) (e)

1000

500

Counts (100s)

800

400

600

300

400

200

N(2,0) N(1,1)

200

100

0 0

’100 ’50 0 50 100 ’100 ’50 0 50 100

Left circular

(c) (f)

800

400

Counts (100s)

600

300

400

200

N(2,0) N(1,1)

200

100

0

0

’100 ’50 ’100 ’50

0 50 100 0 50 100

Position (µm) Position (µm)

Fig. 20.3 Plots (a)“(c) show N (2, 0) as a function of c∆T for linear at 0—¦ (vertical), linear

at 45—¦ , and left circular polarizations respectively. Plots (d)“(f) show N (1, 1) for the same

polarizations. (Reproduced from Lamas-Linares et al. (2002).)

probability; therefore, the probability that the polarizations of the two k1 -photons are

the same is 1/2. This implies that

Nbase (1, 1) = Nbase (2, 0) . (20.42)

The apparent disagreement between eqn (20.42) and the data in the plot pairs (a) and

(d), (b) and (e), and (c) and (f) is an artefact of the detection method used to count

the (2, 0)-con¬gurations; see Exercise 20.7.

The data show that Npeak (1, 1) = Nbase (1, 1); therefore

R

P (2, 0) = , (20.43)

R+1

½ Quantum information

and

1

P (1, 1) = . (20.44)

R+1

Applying the argument used to derive eqn (20.38) leads to

1 1 R + 1/2

R

—1+ — = . (20.45)

F=

R+1 R+1 2 R+1

The data yield essentially the same ¬delity, F = 0.81 ± 0.01, for all polarizations. This

is close to the optimal value F = 5/6 0.833; consequently, this QCM is very nearly

universal and optimal.

20.3 Quantum cryptography

The history of cryptography”the art of secure communication through the use of

secret writing or codes”can be traced back at least two thousand years (Singh, 1999),

and the importance of this subject continues to increase. In current practice, the

message is expressed as a string of binary digits M , and then combined with a second

string, known as the key, by an algorithm or cipher. The critical issue is the possibility

that the encrypted message could be read by an unauthorized person.

For most applications, it is su¬cient to make this task so di¬cult that the message

remains con¬dential for as long as the information has value. The commonly employed

method of public key cryptography enforces this condition by requiring the solution

of a computationally di¬cult problem, e.g. factoring a very large integer. This kind of

encryption is not provably secure, since it is subject to attack by cryptanalysis, e.g.