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For non-orthogonal qubits, |γ and |ζ , equating the two results leads to
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† .
Ҡ = (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.

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