JOURNAL OF MODERN OPTICS,
2000,
VOL.
47,
NO.
2/3, 233 ± 246
Optim al qu an tu m c lon in g an d u n iv e rsal NOT w ith ou t
qu an tu m gate s
CHRIST OPH S IMON, GREGOR WEIHS and ANT ON
ZEILINGER
Institut fuÈ r Experimentalphysik, UniversitaÈt Wien, Boltzmanngasse 5,
A-1090 Wien, Austria
(Received 30 April 1999 )
Abstrac t.
We present ways of realizing quantum cloning via stimulated
emission. Universality of the cloning procedure is achieved by choosing
systems that have appropriate symmetries. We ® rst discuss a scheme based
on certain three-level systems, e.g. atoms in a cavity. Our numerical results
show that this scheme approaches optimal cloning for short interaction times.
T hen we demonstrate that optimal universal cloning can be realized using
parametric down-conversion. At the same time, our down-conversion scheme
also implements the optimal universal NOT operation. We conclude with some
remarks on cloning and superluminal signalling, using our cloner as an
illustrative example.
1.
In trod u c tion
An ideal quantum cloning machine would be a device that, given a quantum
system in an arbitrary state, produces an arbitrary number of perfect copies of that
system. Such a device would allow the exact determination of the quantum state of
a single system, which would even make superlumin al communication possible [1].
I n 1982, Wootters and Zurek [2] and Dieks [3] showed that the existence of such a
device is forbidden by the linearity of quantum mechanics ; perfect copying of
general quantum systems is impossible.
Non- perfect copying, or cloning, though, is possible. In their 1996 seminal
paper, BuzÏ ek and Hillery [4] constructed a `universal quantum copying machine’
that produces two clones of a single qubit. Since then, quantum cloning has been
extensively studied theoretically. T he BuzÏek± Hillery cloner was shown to be
optimal by Bruû et al. [5]. More general qubit cloners producing M clones starting
from N qubits were constructed by Gisin and Massar [6] and their optim ality
shown ® rst by Gisin and Massar, and then, more generally, by Bruû et al. [7], who
derived bounds for the ® delity of N to M cloners.
Gisin [8] showed that in the case of the production of two clones the bound for
the ® delity can be derived from the condition that no superluminal communication
is possible.
T he optimal cloning map for general d-dimensional systems was found by
Werner [9], and optimal cloning machines (i.e. unitary transformations realizing
this map) for d-dimensional systems were constructed by BuzÏek and Hillery [10].
Journal of Modern Optics ISSN 0950± 0340 print/ISS N 1362± 3044 online # 2000 T aylor & Francis Ltd
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234
C. S imon et al.
All devices proposed so far make use of quantum gates. T his means that their
practical realization is not to be expected in the immediate future. On the other
hand, cloning was originally discussed in the context of stimulated emission [1, 2].
It was realized that perfect copying is made impossible by the unavoidable
presence of spontaneous emission [11, 12]. In the light of the newer developments,
the question arises quite naturally, how high the ® delity of cloning via stimulated
emission can be. In this paper we show that optimal cloning can indeed be realized
in this way.
An important requirement an optim al quantum cloning machine has to satisfy,
is universality, i.e. it has to produce copies of equal ® delity for all possible input
states. T his can be achieved for stimulated emission, if the cloning system is
symmetric under general unitary transformations of the system that is to be
cloned. T o be more speci® c, imagine that one would like to clone a general
qubit represented by the polarization state of a photon. Consider a population
inverted medium whose initial state and interaction Hamiltonian with the
electromagnetic ® eld are invariant under general polarization transformations
so that it can emit photons of any polarization with the same probability. If
a photon enters such a medium, it stimulates the emission of photons of the
same polarization. T he photons in the ® nal state can be considered as clones of
the original incoming photon. Of course, whenever there is stimulated emission,
there is also spontaneous emission (of photons of the wrong polarization, in
our example), which limits the possible ® delity of the clones. T he interesting
question is, how small the deteriorating e€ ect of spontaneous emission can be,
i.e. whether optimal cloning in the above sense is still possible. We will see that
the answer is yes.
T he ® delity of an optimal universal symmetrical cloner [7] that produces M
identical clones starting from N qubits is
F opt …M ; N † ˆ
NM ‡ N ‡ M
:
M …N ‡ 2 †
…1 †
T his ® delity is de® ned as hÁj»out jÁi, where jÁi is the state of one of the original
qubits and »out is the reduced density matrix of one of the clones. In our case this
will be compared to the relative frequency of photons of the right polarization in
the ® nal state. T he concepts of relative frequency and of ® delity as de® ned above
are completely equivalent for symmetrical states (i.e. for states that are invariant
under permutation of the particles ). Note that in the limit of M, the number of
clones, tending towards in® nity, Fopt tends towards …N ‡ 1 †=…N ‡ 2 †, which is the
optimum ® delity for state estimation. In this limit, there is no better way of cloning
than trying to guess the state as well as possible (based on an optimal measurement
[13]) , and then producing an in® nity of systems in the guessed state. T his paper is
organized as follows. In section 2 we describe a scheme for universal cloning that is
based on stimulated emission in three- level systems. In section 3 we show how
optimal universal cloning can be realized with parametric down-conversion, and
we show that the optimal universal NOT operation is implemented at the same
time. We also make some remarks on non- universal cloning. In section 4 we
discuss the connections between cloning and superluminal signalling. In particular, we analyse in detail how a simple scheme for superlumin al signalling based
on our down-conversion cloner fails.
Optimal quantum cloning and universal NOT
Figure 1.
235
Level structure of systems used for cloning.
2.
Clon in g w ith an in v e rte d m e d iu m
T he ® rst possible practical realization that we want to discuss is close in spirit
to Herbert’ s original idealized laser gain tubes. T he inverted medium consists of
an ensemble of three- level systems. T hese systems have a ground level g and two
degenerate upper levels e1 and e2 . T hese levels are connected by two modes of the
electromagnetic ® eld, a 1 and a 2 (see ® gure 1 ). T hese two modes de® ne the Hilbert
space of our qubits, i.e. we want to clone general superposition states
…¬a 1y ‡ ­ a 2y†j0 i. Note that we are talking about photons and polarization in order
to be speci® c, but one is free to think of other systems (e.g. phonons ) and other
degrees of freedom, as long as they are described by the same formalism. Similarly,
we will refer to the three- level systems as atoms [12]. T he interaction Hamiltonian
has the following form:
N
H int ˆ
X
K ˆ1
® …¼K1 a 1 ‡ ¼K2 a 2 † ‡ h :c:;
‡
‡
…2 †
where ¼‡1 …2† ˆ je1 …2†ihgj, its complex conjugate is denoted by ¼¡1…2†, and the index
K refers to the K th atom.
T his Hamiltonian is invariant under simultaneous unitary transformations of
the vectors …a 1y; a 2y† and …je1 i; je2 i†. Furthermore, we require each atom to be
initially in a mixed state
» ˆ 12…je1 ihe1 j ‡ je2 ihe2 j†;
…3 †
which is invariant under the same unitary transf ormations. T he invariance of the
Hamiltonian and the initial state together ensure the universality of the cloning
procedure. Consider an incident photon in a general superposition state
…¬a 1y ‡ ­ a 2y†j0 i. T ogether with the orthogonal one- photon state this de® nes a
new basis in the 1± 2 space, which is connected to the original one by a unitary
transf ormation. If the atomic states are now rewritten in the basis that is connected
to the original one by the same unitary transformation, then the interaction
Hamiltonian and initial state of the atoms look exactly the same as in the original
basis. T heref ore it is su cient to analyse the perf ormance of the cloner for
one basis. In our calculations, we considered an incoming one- photon state
jÁi i ˆ a1yj0 i.
We have performed numerical computations for systems of a few (up to N ˆ 6)
atoms. From the form of the Hamiltonian (2 ) it is clear that an atom can only
be de-excited via emission of a photon in the corresponding mode (1 or 2 ) and
only be excited via absorption of a photon of the appropriate type, i.e. the
number of photons plus excited atoms is a conserved quantity for both modes.
L et us denote these quantities by N 1 and N 2 . Because of their conservation, the
whole Hilbert space is decomposable into invariant subspaces, ranging from
236
C. S imon et al.
…N1 ˆ N ‡ 1 ; N 2 ˆ 0 † to …N 1 ˆ 1 ; N 2 ˆ N †. N 1 cannot be zero, because there is
one photon of type 1 coming in. For the case N ˆ 6 the dimensions of these
subspaces are 643, 495, 256 (each occurring twice), and 64. T his gives the
dimensions of the sub-matrices of the block- diagonal Hamiltonian that have to
be exponentiated in order to calculate the time development operator
U ˆ exp …¡iHt†. T he time development of the density matrix of the atom± photon
system is obtained by successive application of U for a certain time step t, starting
from the initial density matrix
»i ˆ
1
«N
je1 e1 j je2 e2 j
2 N … ih ‡ ih †
«a1yj0ih0ja1 :
…4 †
T he probability of ® nding k `right’ and l `wrong’ photons, denoted by p …k ; l †,
was calculated for all possible values of k and l and for di€ erent values of ® t, and
from it the overall `® delity’
f clones ˆ
N ‡1 N
X
X
k ˆ0 l ˆ0
p …k ; l †
k
k ‡l
;
…5 †
which is just the relative frequency of right photons averaged over all possible ® nal
states. T his was compared to the mean ® delity of an ensemble of optimal cloners
producing the same distribution of numbers of clones,
f opt ˆ
where
N ‡1
X
X
n ˆ1
p …n † ˆ
p…n †Fopt …n ; 1 †;
k‡l ˆn
p …k ; l †
…6 †
…7 †
is the probability to ® nd exactly n photons in the output state of our cloner, and
Fopt …n ; 1 † ˆ 23 ‡
1
:
3n
…8 †
We also compare our cloner to a machine that does not a€ ect the one incoming
photon and just produces additional photons of random polarization. T his leads to
a ® delity
f rand ˆ
N ‡1
n 1
p …n† ‡ ;
2n
X
n ˆ1
…9 †
Even in this case there is still a bias towards the right photons, because there is one
such photon coming in.
Figure 2 shows the time evolution of these three ® delities, of the mean number
of photons, and of the mean number of right photons for the case N ˆ 6. T he
curves for smaller atom numbers are similar. One observes a kind of Rabi
oscillation, i.e. an imperfect emission± reabsorption cycle. T he curves show that
there is a regime where our scheme produces a reasonable number of clones with a
® delity that is almost optimal and substantially better than the one achieved by
random photon production. But one could actually do even better, at least in
principle. Figure 3 shows that the short-time regime is optimal for cloning. For
early times, all three ® delities tend towards 1, because then there is only the
incoming photon present. T heref ore we consider the ratio
Optimal quantum cloning and universal NOT
237
Figure 2.
T ime dependence of (a) the ® delities f opt , f clones , and f rand and (b) the mean
number of all photons N all and of `right’ photons N right for the case of N ˆ 6 atoms.
T ime is measured in units of ® t. One sees that almost optimal cloning together with
a reasonable number of clones is achieved, consider, e.g. ® t ˆ 0 :4, where one has
f opt ˆ 0 :837, f rand ˆ 0:755, f clones ˆ 0:825, N all ˆ 2:557, and N right ˆ 2 :013.
Figure 3.
For early times the probability f clones , i.e. the ® delity of the three-level cloning
procedure, is very close to optimum, not only absolutely but also compared to
random photon production. (a ) T ime dependence of the ratio Q for N ˆ 6 atoms. It
appears to be tending versus in® nity for ® t ˆ 0. T his indicates that optimal cloning
is achieved in the short-time limit. (b) Q increases with N for ® xed times, so that in
principle there is no obstacle against increasing the number of atoms.
238
C. S imon et al.
Q ˆ
f clones ¡ f rand
f opt ¡ f clones
…10†
to assess the quality of the cloning procedure. Figure 3 (a) shows that Q seems to
tend towards in® nity for early times, which indicates that optimum cloning is
achieved in this limit. It is clear that in this regime a large number of atoms is
necessary in order to produce a substantial number of clones. Figure 3 (b) shows
that Q increases with N for ® xed times, which means that in principle there is no
obstacle against increasing the number of atoms.
T he practical realization of this scheme probably requires a cavity in order to
achieve the strong interaction of one mode of the radiation ® eld with several (or
even many ) atoms. As discussed above, the interaction does not have to be very
strong in the sense that ®t is not required to be of order one, if there is a su cient
number of atoms involved. But it should still be possible to neglect other ® eld
modes, in particular those corresponding to di€ erent directions of propagation,
otherwise our idealized treatment will lose its validity. Cloning could still be
realizable under such more general conditions. T rapping several atoms in a cavity
could be possible. T he atoms could also ¯ y through the cavity, as in experiments
performed in Paris and Pasadena [14]. We believe that the system deserves further
investigation.
3.
Clon in g w ith param e tric d ow n -c on v e rsion
T he case of weak interaction combined with a large number of atoms is very
similar to the scheme that we will discuss next, namely parametric downconversion (PDC ) driven by a classical pump. T he role of the inverted medium
is now played by the pump beam, in the sense that it is the reservoir out of which
the clones are produced. T his indicates that the down-conversion case might be
even more advantageous for cloning. We will now show that optimal cloning can
indeed be realized in this way. In PDC a strong light beam is sent through a
crystal. T here is a certain (very low ) probability for a photon from the beam to
decay into two photons such that energy and crystal momentum are conserved. In
type-II PDC the two photons that are created have di€ erent polarization. T hey are
denoted as signal and idler.
Figure 4 shows the set-up that we have in mind. We consider pulsed type-II
frequency- degenerate PDC. It is possible to choose two conjugate directions for
the signal and idler beams such that photon pairs, created along these two
directions, are entangled in polarization [15]. Moreover, one can achieve that the
two photons are created in a singlet state, which will make our cloner universal.
T his source of polarization entanglement has been used in many experiments [16].
We consider the quasi-collinear case (i.e. the two directions almost coincide ) , so
that the transverse motion of the photons in the crystal is not important. It is
interesting to note that in the scheme of ® gure 4 one is actually cloning a photon
that is part of an entangled pair. T herefore, in the ® nal state, all three outputs are
entangled.
For stimulated emission to work optimally, there has to be maximum overlap
of the amplitudes of the incoming photon and all the photons that are produced in
the second crystal. T his can be achieved by using a pulsed scheme together with
® ltering of the photons before detection [21]. T he pump pulse can be seen as an
Optimal quantum cloning and universal NOT
239
Figure 4.
Setup for cloning by parametric down-conversion [17± 19]. T he pump pulse
is split at the beamsplitter BS . T he smaller part of the pump pulse hits the ® rst
crystal C 1 , where photon pairs are created at a certain rate. Consider the case where
exactly one pair is created. T he photon created in the lower mode is used as a
trigger. T he upper photon is the system to be cloned. T his photon is directed
towards the second crystal C 2 , as is the rest of the pump pulse, where it stimulates
emission of photons of the same polarization along the same direction. T he path
lengths are adjusted in such a way that the dc-photon and the pump pulse reach C 2
simultaneously. T he photons in mode 1 can be considered as clones of the incoming
photon. Mode 2 is a negative image of mode 1 (apart from the incoming photon)
and can be seen as realization of an optimal universal NOT gate. T his particular
set-up is chosen in order to make the principle transparent. In practice one would
probably use a di€ erent set-up, where the pump pulse is not split into two parts,
but re¯ ected so that it propagates through the crystal twice [20], and also the downconversion photons that are to be cloned are re¯ ected in the same way.
active volume that moves through the crystal. If the photons are ® ltered so much
that the smallest possible size of their wavepackets is substantially bigger than the
pump pulse, then there is maximum overlap between di€ erent pairs created in the
same pulse. Of course, ® ltering limits the achievable count rates. T he group
velocities of pump pulse, signal (V ) and idler (H ) photons are not all identical.
T his leads to separations (of the order of a few hundred femtoseconds per
millimetre in Beta-Barium Borate (BBO)), which have to be kept small compared
to the size of the dc-photon wave packets. T here is a trade-o€ between ® ltering and
crystal length, i.e. one can choose narrower ® lters to make a longer crystal (i.e.
longer interaction times ) possible.
If the above-mentioned conditions are ful® lled, then a single spatial mode (i.e.
one mode for the signal and one for the idler photons ) description is valid. T he
PDC process can then be described in the limit of a large classical pump pulse, in
the interaction picture, by a Hamiltonian
y ¡ a y a y † h:c:;
H ˆ ® …a Vy 1 a H
2
H1 V 2 ‡
…11†
where aVy 1 is the creation operator for a photon with polarization V propagating
along direction 1 etc. T he coupling constant and the intensity of the classical pump
pulse are contained in ® .
T he Hamiltonian H is invariant under general common S U …2 † transf ormations
y † for modes 1 and 2, while a phase transf ormaof the polarization vectors …a Vy ; a H
tion will only change the phase of ® . T his makes our cloner universal, i.e. its
performance is polarization independent. T he argument is the same as in the atom
case. Consider a photon of arbitrary polarization coming in. T he Hamiltonian
looks exactly the same in the new polarization basis de® ned by this photon, so the
® nal state will also look the same. T heref ore it is su cient to analyse the `cloning’
process in one basis, e.g. for an incoming one- photon state jÁi i ˆ a Vy 1 j0 i.
240
C. S imon et al.
In the Heisenberg picture one ® nds the following equations for the time
dependent operators:
y 0 ;
a V 1 …t† ˆ cosh …® t†a V 1 …0 † ¡ i sinh …® t†a H
2… †
aH 2 …t† ˆ cosh …® t†a H 2 …0† ¡ i sinh …® t†aVy 1 …0†;
…12†
and analogous equations for H 1 and V 2. T herefore, the mean numbers of photons
in the output state are
N V 1 …t† ˆ hÁi ja Vy 1 …t †aV 1 …t†jÁi i ˆ 2 sinh 2 ® t ‡ 1 ;
N H2 …t† ˆ 2 sinh 2 ® t ;
2
N H1 …t† ˆ N V 2 …t † ˆ sinh ® t:
…13†
As a ® rst estimate for the performance of this cloning procedure, one can calculate
the ratio of mean values
N V 1 …t†
1
1
2
2
:
ˆ
ˆ
N V 1 …t† ‡ N H 1 …t † 3 ‡ 3 …3 sinh 2 ® t ‡ 1 † 3 ‡ 3…N V 1 …t† ‡ N H 1 …t ††
…14†
T his is identical to the expression for the ® delity of an optimal cloning machine
that produces N V 1 …t† ‡ N H1 …t† clones, which is already an encouraging result.
T o analyse the performance of the down-conversion cloner in more detail, we
change to the SchroÈdinger picture. T he time development operator exp …¡iHt†
clearly factorizes into a V 1 ¡ H 2 and an H 1 ¡ V 2 part. Consider cloning starting
N
1=2
from N identical photons, i.e. an initial state jÁi i ˆ ‰…a Vy 1 † =…N !† Šj0 i. Making
use of the disentangling theorem [22] one ® nds that (cf. [18])
1
¡ ¢
X
£
Xi
jÁf i ˆ exp …¡iHt†jÁi i ˆ K
k ˆ0
1
l ˆ0
k
…¡iG†
l
…iG† jl
k ‡N
N
H 1 jl V 2
i
1=2
jk ‡ N iV 1 jk iH2
;
…15†
where G ˆ tanh ® t and K is a normalizing factor.
T he component of this state that has a ® xed number M of photons in mode 1, is
proportional to
M ¡N
¡ ¢
X
l ˆ0
l
…¡1 †
M ¡l
N
1=2
jM ¡ l iV 1 jl iH1 jl iV 2 jM ¡ N ¡ l iH2 :
…16†
T his is identical to the state produced by the unitary transformation written down
in [23] which can be seen as a special version of the Gisin± Massar cloners [6] which
implements optimal universal cloning and the optimal universal NOT gate at the
same time. T he M photons in mode 1 are the clones, while the M ¡ N photons in
mode 2 are the output of the universal NOT gate, the `anti-clones’ . T his means
that the set-up of ® gure 4 works as an ensemble of optimal universal cloning (and
universal NOT ) machines, producing di€ erent numbers of clones and anti-clones
with certain probabilities. Note that each of the modes can be used as a trigger for
the other one so that cloning or anti-cloning with a ® xed number of output systems
can be realized by post- selection.
Optimal quantum cloning and universal NOT
241
In principle our demonstration that optimality is achieved could stop with the
above remark that the state jÁf i is identical to the one of [23]. T o see it more
explicitly, consider again the subspace of all possible output states that has a ® xed
number of photons M in mode 1, and let us calculate the relative frequency of
`right’ (V -polarized ) photons in this mode. From the expression for the state (16) ,
one sees readily that it is given by
¡ ¡¢ ¢
¡¢ ¡ ¢
M ¡N
X
X
X
pN
right …M † ˆ
Using
it follows that
M ¡l
…M ¡ l †
N
l ˆ0
M
M ¡N
l ˆ0
M
k ˆN
k
N
pN
right …M † ˆ
ˆ
M¡l
:
N
…17†
M ‡1
N ‡1
NM ‡ N ‡ M
;
M …N ‡ 2†
…18†
which is exactly the optimum ® delity for an N to M quantum cloner [7].
T he relative frequency of right photons in the ® nal state, summed over all M, is
given by
1 NM N M M 1 2M
‡ ‡
‡ G
M
N
2
N
… ‡ †
‡1
M ˆN
pN
right ˆ
1 M ‡ 1 2M
G
N ‡1
M ˆN
¡
¢
¡ ¢
X
X
G2
G4
:
…19†
‡
N ‡ 1 …N ‡ 1†…N ‡ 2 †
Note that it tends towards …N ‡ 1 †=…N ‡ 2 † for G tending towards 1, i.e. for large
photon numbers.
In order to make explicit the optimality of the universal NOT realized in
mode 2, calculate the mean relative frequency of H photons in mode 2, for
the subspace where there are M ¡ N photons in mode 2, again ref erring to
equation (16) :
M ¡N
M¡l
…M ¡ N ¡ l †
N
l ˆ0
N
:
F …M † ˆ
…20†
M ¡N
M¡l
M
N
l ˆ0
ˆ1¡
¡ ¡¢ ¢
X
X
Using the same combinatorial identities as above one easily ® nds that
F N …M † ˆ …N ‡ 1 †=…N ‡ 2†, which is exactly the optimum ® delity for the universal
NOT operation [23]. Note that it is identical to the optimum state- estimation
® delity and does not depend on the number of `anti-clones’ produced.
T he set-up of ® gure 4 clones photons of all polarizations (all complex linear
combinations of V and H ) with the same ® delity. If universality is only demanded
for linear polarizations (real linear combinations of V and H ) , a di€ erent
construction becomes possible. By a simple polarization transf ormation in one
242
C. S imon et al.
mode, e.g. taking H 2 into V 2 and V 2 into ¡H 2, the down-conversion Hamiltonian can be brought to the form
y a y † h:c:;
H ˆ ® …a Vy 1 a Vy 2 ‡ a H1
H2 ‡
…21†
which is no longer invariant under simultaneous general unitary transformations of
the polarization vector …V ; H † in both modes, but is still invariant under orthogonal (unitary and real ) transformations. T he loss of universality is accompanied
by a gain in performance in the sense that now in mode 2 there is also a majority of
V (`right’ ) photons, i.e. the photons in mode 2 can also be considered as clones.
T his only works for real combinations of V and H . For a circularly polarized
y =21=2 j0 ) , the photons in mode 2
photon coming in (e.g. the state ‰…a Vy ‡ ia H
†
Š i
remain anti-clones, because circular polarization is invariant, apart from a phase,
under the above transformation (H !V ; V !¡H ) . T he ® delity of the additional
clones in mode 2 is only …N ‡ 1 †=…N ‡ 2 †, as we know from the calculation for the
universal NOT . T he ® delity of the clones in mode 1 is, of course, unchanged with
respect to the previous situation, because the relative frequency of V photons in
mode 1 is not a€ ected by a polarization transf ormation in mode 2. Considering all
the photons in both modes together as clones, and again considering a subspace
with M photons in mode 1 and M ¡ N photons in mode 2, the overall ® delity is
theref ore given by
N 1
MF opt …M ; N † ‡…M ¡ N † ‡
N ‡2 ;
F ˆ
…22†
2M ¡ N
which is equal to F opt …2M ¡ N ; N †, the optimal ® delity for the N to 2M ¡ N
cloner. T his means that the ® delity of the optimal universal cloner is achieved by
this non- universal cloner which does not produce any anti-clones. But note that
this is not the optimum result that is in principle achievable in this case [24].
We have shown a method of realizing optimal quantum cloning machines. Now
we want to discuss whether it is experimentally feasible with current technology.
In our group, pair production probabilities of the order of 4 £10¡3 have been
achieved with a 76 MHz pulsed laser system (UV-power about 0.3 W ) and a 1 mm
BBO crystal, for a 5 nm ® lter bandwidth. Past experiments show that good overlap
of photons originating from di€ erent pairs is achieved under these conditions.
With detection e ciencies around 10%, this leads to a rate of two-pair detections of
the order of one per a few seconds. A new 300 kHz laser system is currently being
set up in our laboratory. An improvement of at least the order of 76=0 :3 in the
average rate of pairs per pulse is to be expected, for identical pump power. T his
will also make several-pair events far more likely. T his means that production of a
few clones with a reasonable rate should be possible.
4.
Clon in g an d sign allin g
In the introduction we mentioned Gisin’ s [8] result that the bound on the
® delity of the 1 to 2 cloner can be derived from the no-signalling condition.
Whether the bounds for N to M cloning can also be derived in this way, is, to our
knowledge, an open problem. It has been shown that quantum mechanics does not
allow superluminal signalling [25]. We know of no reason to believe that this result
does not hold for relativistic quantum ® eld theory. T herefore, we do not expect
Optimal quantum cloning and universal NOT
243
our down-conversion cloner (or in general, any optimal cloning machine ) to allow
superluminal signalling. See also the argument following equation (23). On the
other hand, at ® rst sight, there seems to be some hope for superluminal communication using the set-up of ® gure 4. T herefore it is quite instructive to see how the
most obvious scheme one can think of fails.
Following Herbert’ s original idea, one could think of implementing a superluminal communication channel in the following way. T he two parties that want to
communicate (Alice and Bob) have to share a pair of entangled particles, e.g.
photons in the state
1
jÁi ˆ 1 =2 …jV H i¡ jHV i†:
…23†
2
Alice can measure the polarization of her particle either in the basis fjV i; jH ig or
=
=
in the basis fjP i; jM ig, where jP i ˆ 2¡1 2 …jV i‡ jH i† and jM i ˆ 2¡1 2 …jV i¡ jH i†.
I f Alice measures in the V =H basis and ® nds jV i (jH i), Bob’ s photon is reduced
to jH i (jV i) , while if she measures in the P =M basis (results jP i and jM i)
Bob’ s photon is reduced to the corresponding states in that basis (jM i and jP i
respectively ).
One way of explaining why Alice cannot signal to Bob is the following. Bob
does not know which result Alice got, so his photon is described by a density
matrix 12 …jV ihV j ‡ jH ihH j†, if she measured in the V =H basis, and
1
=
2 …j P ihP j ‡ j M ihM j†, if she measured in the P M basis. Of course, these two
density matrices are identical, so there is no way for him to tell what she did. T his
also shows that a cloner does not help, as long as it is describable by quantum
mechanics, because any quantum device will, fed by identical input density
matrices, produce identical outputs.
A di€ erent point of view is the following. Although the states on his side are
di€ erent depending on Alice’ s choice of basis, this does not allow Bob to know her
choice because he cannot discriminate jV i from jP i by a single measurement
(having only a single copy) . If he had a perfect cloning machine, he could produce
an arbitrary number of copies of his photon and in that way determine its state. Of
course, a perfect cloning machine does not exist. But could not non- perfect cloning
also be su cient? Consider the set-up of ® gure 4. If a photon with polarization V
is fed in, the cloner will produce a state with a clear excess of V -polarized photons
in mode 1. Equation (13) shows that the mean di€ erence between the numbers of
V and H photons in this case is sinh 2 ® t ‡ 1. In the same way, an H -photon will
result in an excess of H -polarized photons. On the other hand, P and M photons
coming in will result in states where V - and H -polarized photons occur with equal
probability. T his is clear from symmetry considerations and can also easily be
veri® ed directly, as discussed below. S hould it not be possible to detect this
di€ erence and in this way infer the basis Alice used? Bob would just have to
perform a polarization analysis in the V =H basis of all photons coming out from
his cloner. If he ® nds a clear excess of one polarization, this would be an indication
that Alice measured in the V =H basis, if he ® nds similar numbers of V - and H polarized photons, it would mean that she measured in the P =M basis.
T o see why this scheme does not work, de® ne ¢N ˆ N V ¡ N H , which denotes
the di€ erence in number between V and H polarized photons in the output of the
cloner. T he ® nal state (15) is a superposition of states with di€ erent values of ¢ N .
For an incoming V and an incoming H photon ¢ N has probability distributions
244
C. S imon et al.
p V …¢N † and pH …¢N † respectively, where p V is peaked around a large positive
value (namely sinh2 ®t ‡ 1 ), and pH …¢N † ˆ p V …¡ ¢N †. T he probability distributions for ¢N in the case of P or M polarized photons coming in are
p P …¢N † ˆ pM …¢N † ˆ 12…pV …¢ N † ‡ p H …¢N ††:
…24†
T his can be seen in the following way. All the terms in the expansion of the ® nal
state jÁf i for an initial V photon are of the form jk ‡ 1 iV 1 jk iH2 jl iH1 jl iV 2 . All these
states consist of V 1 ¡ H 2 and H 1 ¡ V 2 pairs plus the initial V photon in mode 1.
T he terms occurring in the expansion for jÁf i for an initial H state consist of pairs
plus an additional H 1 photon and are theref ore orthogonal to the terms for an
initial V -polarized photon. T his means that there is no interference and one
simply has to add the probabilities.
Now imagine that Bob chooses some threshold value ¢ N th . If he ® nds
¢
j N j > ¢N th , he assumes that Alice measured in the V =H basis. How big is the
error he makes? T he probability to ® nd j¢N j > ¢ N th if she chose the V =H basis is
given by
1
¢ > ¢N th † ‡ 12 P H …j¢N j > ¢N th †;
…25†
2 P V …j N j
where
P V …H †…j¢N j > ¢ N th † ˆ
p V …H †…¢ N †:
…26†
X
j¢N j>¢N th
T he probability to ® nd j¢N j > ¢N th if Alice chose the P =M basis is
1
2 PP
1
…j¢N j > ¢N th † ‡ 2 PM …j¢N j > ¢N th †;
…27†
which is exactly identical, because of (24) . T hus, somewhat counter-intuit ively,
events with a high asymmetry in photon numbers are exactly as likely when Alice
measures in the P =M basis as when she measures in the V =H basis.
T his means that, following this procedure, Bob makes an error in exactly half
of the cases, which is the value he would obtain by random guessing ; he does not
gain any information whatsoever about Alice’ s basis. T he same argument applies,
of course, to the attempt to infer from small values of ¢N that Alice was measuring
in the P =M basis.
5.
Con c lu sion
We have presented possible ways of realizing quantum cloning by stimulated
emission. We have discussed procedures based on three- level systems that could
allow the production of large numbers of clones, and could be easier to realize than
comparable schemes using quantum gates. We also have shown a scheme for
realizing optimal universal cloning based on parametric down-conversion, and at
the same time an implementation of the optimal universal NOT operation. We
have argued that this scheme should be realizable with current technology. We
have discussed how cloning and the question of superluminal communication are
related, and analysed the situation for our down-conversion cloner in some detail.
We ® nd it interesting that quantum operations that have so far been mainly
discussed in terms of quantum gates can also be realized in a di€ erent way. It is
conceivable that for other basic quantum inf ormation procedures there could also
be physical processes that allow a gateless realization. While this may not be the
Optimal quantum cloning and universal NOT
245
direction to go if one wants to construct a universal quantum computer, we still
consider it as a promising and fascinating ® eld for further investigation.
Ac kn ow le d gm e n ts
We would like to thank CÏ . Brukner, V. BuzÏek, J. I. Cirac, M. Hillery,
T . Jennewein, J. W. Pan, and H. Weinf urter for helpful comments and discussions. T his work has been supported by the Austrian Science Foundation (FWF,
Project No. S 6502).
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