Temperature effects on quantum cloning of states and entanglement
S. Baghbanzadeh1,2 and A. T. Rezakhani3, 4
1Department of Physics, Sharif University of Technology, P. O. Box 11155-9161, Tehran, Iran
2Department of Physics, Iran University of Science and Technology, Narmak, P. O. Box 16765-163, Tehran, Iran
3Center for Quantum Information Science and Technology, and Departments of Chemistry and Physics,
University of Southern California, Los Angeles, CA 90089, USA
4Institute for Quantum Information Science, University of Calgary, Alberta T2N 1N4, Canada
Performances of the symmetric universal and phase-covariant cloning transformations and entanglement clon-
ers — qubit case — are investigated when the initial state of the hardware or the original state to be cloned is
weakly coupled to a thermal environment. Different behaviors of each of these transformations are analyzed
and contrasted with the ideal cases.
PACS numbers: 03.67.-a, 03.67.Mn, 03.65.Ud
Introduction.— It has been known that based on the prin-
ciples of quantum mechanics, accurate cloning of arbitrary
quantum state is impossible [1]. Nevertheless, on the account
of the significant role of copying in quantum computation and
quantum communication, a variety of approximate quantum
cloning transformations have been proposed, e.g., universal
cloning (UC) machines producing identical copies from ar-
bitrary input qudits [2, 3, 4, 5, 6], phase-covariant cloning
(PCC) machines of a class of partially known qudits [7, 8],
and optimal entanglement cloning machines [9, 10]. For re-
cent reviews see Refs. [11].
A question of practical relevance is how uncontrollable en-
vironmentally induced decoherence or dissipation can affect
performance of quantum cloning machines. In closed sys-
tems, an initially pure state evolves to another pure state. In
practice, however, preparation of pure states and/or keeping
them pure are not generally easy tasks. In general, interac-
tion with an environment degrades purity of quantum systems
and makes their states mixed. A usual effect that a thermal
environment can cause is thermalization (as a kind of dissipa-
tion) [12, 13]. That is, because of interaction with the environ-
mental degrees of freedom which are in thermal equilibrium,
the quantum system will also be driven toward equilibrium.
It should be noted that a generic isolated quantum many-body
system does also relax to a state well described by the standard
statistical-mechanical prescription [14]. In this paper, our aim
is to investigate temperature effects on the performance of the
cloning machines. It has been known that decoherence can put
a limitation on the number of clones that a quantum cloner can
generate [15]. There is also a model in which the robustness
of the cloner increases with the number of qubits [16].
Through a thermalizing process, the density matrix of the
system ̺ in long time will approach the Boltzmann state
̺th = e
−βH/Z , where kBβ is inverse temperature (kB is
the Boltzmann constant), H is the Hamiltonian of the open
quantum system, and Z = Tr(e−βH) is the partition func-
tion. Energy and phase relaxation processes, with the time-
scales T1 and T2, respectively, are common processes present
when approaching an induced equilibrium state. For a more
precise and elaborate discussion of dissipation, thermaliza-
tion, and decoherence see Ref. [12]. Some (phenomenolog-
ical) models for the underlying dynamics of the thermaliza-
tion have already been proposed [13]. We assume that the
time-scale in which typical correlation functions of the en-
vironment decay, tenv., is much smaller than all other time-
scales, i.e., tenv. ≪ τc, Tdiss. = min{T1, T2, TO}, where τc is
the time-scale of the cloning process and TO is the time-scale
dictated by all other relaxation mechanisms. This assumption
is important for the Markovian analysis of the dynamics of the
thermalization [17, 18]. This implies that during the cloning
process, a negligible amount of information flows from the
environment to the system (or vice versa). Here, we also as-
sume that τc . Tdiss.. This extra condition allows us to ig-
nore dynamical effects of the thermalization, hence consider
a simple static (toy) model — explained below — to bring
temperature into play. Despite these simplifying assumptions,
we will argue that the result is still reliable enough to give a
hint about how temperature effects can change performance
of different cloning machines such as the universal cloners,
phase-covariant cloners, and entanglement cloners. Indeed,
such investigation has an immediate importance in attempts
to realize quantum cloning in systems where (due to thermal
and perhaps other noise effects) the preparation of pure states,
whether initial state of the system to be cloned or the quantum
hardware, is difficult, such as in NMR systems [19, 20]. For
another study using a different approach, see Refs. [21, 22].
For the purpose of illustration, we only consider the case of
symmetric 1 → 2 qubit cloners. Extension to M → N qudits
is straightforward as well.
Optimal universal and phase-covariant cloning transfor-
mations.— In the universal cloning transformation, it is usu-
ally assumed that the qubit state to be cloned is a pure state,
|Ψ〉a = cos θ2 |0〉 + e
iφ sin θ
|1〉, and the blank copy (b) and
the quantum cloning machine (also called ancillary system,
c) are each in a known pure state, say |0〉 [2, 3, 4, 23].
The symmetric cloning transformation, then, acts in this
way: U (|Ψ〉a|0〉b|0〉c) = |Υ〉abc, where Trbc(|Υ〉abc〈Υ|) =
Trac(|Υ〉abc〈Υ|). The latter condition guarantees that the fi-
nal clones both have the same states, ̺outa = ̺
b . A mea-
sure to quantify performance of a cloning machine is the fi-
delity between the original and the output states, F (ρ, σ) =
ρ1/2σρ1/2
. Optimization of the fidelity over all input
states on the Bloch sphere results in the qubit optimal univer-
sal cloner, in which F = 5/6 [2, 3]. For orbital states, where
θ is an a priori known constant and φ ∈ [0, 2π), a class of
http://arxiv.org/abs/0704.1177v3
θ0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
PSfrag replacements (a) (b)
0 00 0
FIG. 1: Fidelity (F ) of UC vs θ for some values of η: (a) ǫ = 5/11
and (b) ǫ = 2/3.
phase-covariant cloning machines has been suggested [8]. Af-
ter the cloning process, in the computational basis {|0〉, |1〉}
(the eigenvectors of σz = diag(1,−1)) each of the clones can
be identified by the density operator: ̺out00 = µ
2̺in00 + ν
2 and
̺out01 = 2µν̺
01, where µ
2 + 2ν2 = 1, and ν2 = 1/6 for UC
and ν2 = (1 − 1√
1+2 tan4 θ
)/4 for PCC. Most of this descrip-
tion is also valid when the original quantum system is initially
mixed.
Our main assumption is that preparation of the initial pure
state |Ψ〉 is diluted by a thermal bath in the following special
and simple form:
̺in = (1− ǫ)|Ψ〉〈Ψ|+ ǫ̺th, 0 6 ǫ < 1. (1)
The parameter ǫ, which measures how thermally perturbed the
preparation is, may in general be time-dependent. Nonethe-
less, based on our earlier assumptions, it would be a fairly
slow-varying time-dependent function so that with a good ap-
proximation we can take it a relatively small constant of the
order of τc/Tdiss.. This state does not seem to arise natu-
rally from a typical thermalization dynamics. Nevertheless,
in Ref. [18] it has been illustrated that general behaviors ob-
tained from such a simple preparation assumption (in the con-
text of the geometric phases) have general features similar to
those obtained from the Lindblad equation for the dynamics.
It is worth mentioning that in the limit of infinite tempera-
ture, the thermalized density matrix ρth is equivalent to pure
noise [24]. In that case, ǫ represents the degree of pure noise
existing during the process (for example, in the case of NMR
systems, due to fluctuations of the external magnetic fields and
similar reasons). A more general analysis of quantum cloning
in the presence of a thermalization mechanism is yet lacking,
but our simple analysis may also shed some light before hav-
ing a more complete analysis at hand.
First, we consider the effect of the thermal term only on the
state of the cloner, that is, the quantum cloning hardware is
thermally diluted as in Eq. (1). In this case, the initial state
of the machine is mixed. Considering the fact that in the op-
timal UC and PCC, the initial state of the cloning machine
can be any pure state [4, 7, 23], one can conclude the opti-
mal fidelity here is achieved by the existing optimal cloning
transformations. By a similar analysis, it appears that for the
case of diluted joint blank and ancillary systems, one can con-
sider the joint state as a new blank copy and attach some new
reservoir to the whole Hilbert space of the input states (i.e.,
the information qubit, the blank copy, and the ancilla state) as
a new ancillary system and then define a new transformation
θ0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
PSfrag replacements (a) (b)
0 00 0
FIG. 2: Variation in the fidelity of PCC with θ for: (a) ǫ = 5/11 and
(b) ǫ = 2/3, and different values of η.
for cloning [23]. This would in fact be the existing optimal
cloning transformation, now acting on a larger Hilbert space,
and hence one obtains the same optimal fidelity again. How-
ever, from an experimental point of view, thermalization ef-
fects are likely to occur during the cloning process rather than
at the initial preparation level — for instance in NMR sys-
tems [19, 20]. Therefore, to be more precise, thermal effects
during the preparation should also be taken into account.
Now, we consider the case in which the input state |Ψ〉 is
thermally diluted as in Eq. (1). Our aim now is to compare
the similarity between the clones and the input state of our in-
terest, i.e., |Ψ〉. Indeed, here we assume that the model of the
cloning machine consists of two parts: the first is the dilution
of the input pure state which models the imperfect feature of
the machine, and the second is some known UC or PCC trans-
formation which is applied to this diluted state. The Hamilto-
nian of the qubit system is taken to be H = ω0σz/2 (ω0 > 0),
whence, Z = 2 cosh η, where η = ω0β/2. More general
cloning transformations in spin networks with more compli-
cated Hamiltonians can be found, for example, in Ref. [21].
The fidelity of the output state and the unperturbed initial state
can be calculated as follows:
F (θ, ǫ, η) =µ2[1− ǫ+ ǫ(e−η cos2 θ
+ eη sin2
+ (µν − µ2/2)(1− ǫ) sin2 θ + ν2.
Figure 1 illustrates how the fidelity in the UC behaves in terms
of θ (orbit of the state) in thermally diluted states, for two
different values of ǫ (the degree of thermalization) and η (∝
1/T ). It can be seen that when
ǫ < cosh η /(e−η sin2
+ eη cos2
), (2)
the fidelity of the UC is higher than the classical value 1/2.
This threshold is the fidelity of a classical-like 1 → M uni-
versal cloning in which with a given probability, an unknown
input state is sent to one of the M parties and a completely
randomized state is transmitted to any of the other ones, of
course, in the limit of large M [25]. In the literature, however,
“classical cloner” has been attributed to some other cloning
transformations as well — see [3, 26]. In other words, in some
cases thermal noise (even in the simple form of Eq. (1)) can
result in a lower performance than a classical machine. For
θ > π/2, the condition (2) implies that for all 0 6 ǫ < 1, the
fidelity of the output of the UC is always greater than that of
PSfrag replacements
FIG. 3: Fidelity vs ǫ and θ in low temperature limit (η → ∞): UC
(left) and PCC (right).
the classical cloner (if ω0 was negative, this would occur for
θ 6 π/2). Equation (2) can also be interpreted as a condition
on temperature for a given θ and ǫ in order to outperform a
classical cloner. Figure 2 shows the variation of the fidelity
of the outputs of the PCC machines in terms of θ, for some
fixed values of ǫ and η. As is clear from this figure, in the case
of equatorial qubits, similar to the case of the UC, the fidelity
of the outputs does not vary with temperature — according to
Eq. (2), this feature is due to the symmetry property of such
states. Low temperature limits of the fidelity for both UC and
PCC have been depicted in Fig. 3. In the case of the UC, for
all θ in [0, π), the fidelity is a decreasing function of ǫ. The
corresponding graph for the PCC also shows a decrease in the
fidelity for different values of θ ∈ [0, π/2) with the pertur-
bation factor ǫ. However, a closer inspection shows that here
there are also some θs (& 2.52 and less than π rad) in which
the fidelity of the PCC is an increasing function of ǫ. At high
temperature limit, the fidelity of both UC and PCC, for all θs,
is a decreasing function of ǫ. Another important point that can
be concluded from the figures is that in some cases, the quality
of the clones at the output of the UC can be better than that of
the PCC — see for example those regions of Fig. 3 in which
ǫ and θ are large and small, respectively. This is indeed con-
trary to what happens when the cloning is performed perfectly
without any external noise.
Entanglement cloning.— Quantum cloning can be used to
clone or broadcast entanglement as well [5, 9, 10, 27, 28].
Let us assume that we have an initial state in the form of
|Ψ−α 〉ab = α|01〉ab −
1− α2|10〉ab, where α is real and
|α| 6 1. As in the cases of the UC and the PCC, sup-
pose that because of a thermal environment, the initializa-
tion is diluted as in Eq. (1). Let us take our system to be
two spin-1/2 particles interacting via the XX Hamiltonian:
H = J(σaxσ
x + σ
y), where σx and σy are Pauli matri-
TABLE I: Inseparability conditions of the output states in the three
different scenarios of cloning.
γ α ǫ
γ > γc
|α2 − 1/2| < αc
0 < α < 1 0 6 ǫ < 1
−1 < α < 0 0 6 ǫ < ǫ1 or ǫ2 < ǫ < 1
|α2 − 1/2| > αc ǫ2 < ǫ < 1
0 < γ 6 γc |α2 − 1/2| < αc
0 < α < 1 0 6 ǫ < ǫ2
−1 < α < 0 0 6 ǫ < ǫ1
ces. Now, we want to compare performances of the following
schemes of entanglement broadcasting between two parties in
the presence of thermal noise: (i) Local cloning by means of
two optimal UC machines copying each qubit separately [27].
In this scenario, after the cloning process and discarding the
ancillas, we will have the overall state ̺aa′bb′ whose two first
(last) qubits are the copies of a (b). (ii) Non-local cloning of
the two-qubit state as a whole with the UC machine of 4-level
quantum states [5]. (iii) Cloning by an optimal entanglement
cloner [9].
After some algebra, it can be seen that the density matrices
of the clones in cases (ii) and (iii), and ̺a′b (also ̺ab′ , ̺ab,
and ̺a′b′ ) — nonlocal copies — in case (i), read as follows:
̺out = (Mǫ
+ 1−M
)(|00〉〈00|+ |11〉〈11|)
+[M(1−ǫ
+ ǫ cosh γ
) + 1−M
+ L(1− ǫ)(2α2 − 1)]|01〉〈01|
+[M(1−ǫ
+ ǫ cosh γ
) + 1−M
− L(1− ǫ)(2α2 − 1)]|10〉〈10|
−M [(1− ǫ)α
1− α2 + ǫ
sinh γ](|01〉〈10|+ |10〉〈01|), (3)
in which γ = 2βJ, Z = 2(1 + cosh γ), L = 3(1 + 2M +√
1 + 4M − 9M2)/26, Mi = (2/3)2, Mii = 3/5, Miii =
6A2 + 4AC, A =
(1/2 + 1/
13)/3, and C = A(
3)/2. Note that, the output states of case (ii) for all values of
ǫ, α, and γ, the nonlocal copies of case (i) ̺a′b, and the output
states of case (iii) for ǫ = 1 and ∀γ or α = ±1/
2 (for all
ǫ and γ) all can be written in the following compact form:
̺out = M̺in + (1 − M)I/4, where I is the 4 × 4 identity
matrix.
To determine the regions in which the output states are
separable or inseparable, we use the well-known Peres-
Horodecki positive partial transposition criterion [29]. Ac-
cording to this criterion, in the case of 2 × 2 and 2 × 3 sys-
tems, a density matrix ̺AB is inseparable (i.e., entangled) iff
(̺AB)
TA (TA: partial transposition with respect to system
A) is not positive. Tables I and II show the results for anti-
ferromagnetic case (J > 0). The parameters in the tables are
TABLE II: Inseparability conditions of the output states in the three
different scenarios of cloning, at low and high temperature limits.
γ ǫ, α
γ → ∞
C1 and 0 6 ǫ 6 1−M
and |α2 − 1/2| < α∞
C1 and 1−M
< ǫ < 1 and α ∈ C1
C2 and 0 6 ǫ < 3M−1
and |α2 − 1/2| < α∞
C2 and 1−M
< ǫ 6 M+1
and |α2 − 1/2| > α∞
C2 and M+1
< ǫ < 1 and α ∈ C2
γ → 0 0 6 ǫ < (1 − 1
) and |α2 − 1/2| < α0
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
PSfrag replacements
(ii) (iii)
(input)
FIG. 4: Entanglement phase diagrams of input and output
states (achieved from three different schemes of entanglement
cloning/broadcasting introduced in the text), when α = 1/
2. The
regions labeled by 1 are the regions in which entanglement exists,
whilst the regions labeled by 2 indicate no-entanglement regions.
This figure shows that for γ < γc (T > Tc), depending on the value
of ǫ, we may or may not have entanglement. γc is a decreasing func-
tion of M . In other words, the area of region 1 increases when M
increases, as well. This may imply the advantage of the entanglement
cloner Miii over the other entanglement broadcasting schemes.
as follows:
α∞1 =
(3M − 1)(M + 1− 4Mǫ)
4M(1− ǫ)
α∞2 =
(M + 1)(3M − 1− 4Mǫ)
4M(1− ǫ)
3M2 + 2M − 1
γc = ln(
M + 1 + 2
M2 +M
3M − 1
ǫ1(2) =
(M − 1∓ 4Mδ)(1 + cosh γ)
2M [1± sinh γ ∓ 2δ(1 + cosh γ)] ,
(3M(1− ǫ)− 1) (M(1− ǫ) + 1)
4M(1− ǫ)
where δ = α
1− α2, C1 ≡ 0 < α 6 1, and C2 ≡
−1 6 α 6 0. When γ → ∞ and M = Miii, since
(3M − 1)/4M > (1 −M)/2M , there exists an overlap be-
tween the ǫ-inequalities in the third and fourth sub-rows of Ta-
ble II. In this case, one should notice that for (1−M)/2M <
ǫ < (3M − 1)/4M , clones are entangled if |α2 − 1/2| < α∞2
or |α2 − 1/2| > α∞1 . This removes the ambiguity in such
cases.
Tables I and II imply that in most temperature regions, the
inseparability inequalities are not symmetric with respect to
α → −α. In other words — unlike the case of ǫ = 0 — de-
pending on the sign of α, the parameter regions over which
the cloned pairs are entangled may be different. Another im-
portant point (see the second row of Table I) is the existence
of a critical temperature Tc (∝ 1/γc) beyond which the cloned
pairs for some α regions, |α2 − 1/2| > αc, for all ǫs are not
entangled.
Overall, by taking into account the behaviors of the upper
and lower bounds of the inseparability inequalities we can find
that in some temperature regions, in Table I (Table II), there
exist intervals of α2 (ǫ) in which the cloned pairs are sepa-
rable. The length of these intervals decreases when M in-
creases (recall that Miii > Mii > Mi). Furthermore, for a
given α2 (ǫ) at intermediate (two limits of) temperatures, the
range of ǫ (α2) in which the clones are entangled increases
when M increases as well. Indeed, for some temperature re-
gions, in Table I (Table II) there exist some α2 (ǫ) in which
clones for all ǫ (α in C1 or C2) and all three Ms are entan-
gled — e.g., see first sub-row of Table I or second and fifth
sub-rows of Table II. These facts together with the entan-
glement phase diagrams in Fig. 4, whose regions show ex-
istence of entanglement or its non-existence for α = 1/
indicate advantage of entanglement cloner Miii, over the other
cloning schemes. That is, the optimal entanglement cloner has
an advantage over other mentioned schemes of entanglement
broadcasting in the sense of robustness against thermal noise.
Conclusion.— We have studied the role of thermal noise
in some quantum cloning schemes through a simple model of
temperature effect on spin states at the input of the cloning
machines. The performance of the cloning machines depends
generally on the values of the thermal perturbation coefficient,
the orbit of the original state on the Bloch sphere, as well as on
the temperature. In addition, three scenarios of entanglement
cloning of thermally diluted two-qubit states have been inves-
tigated. Our analysis shows that the clones generated from
non-local transformations, in particular those out of the opti-
mal entanglement cloner, remain entangled for wider regions
of parameters. I.e., the optimal entanglement cloner shows
a relatively larger region of entanglement in the parameter
space. This can be considered as an advantage of optimal en-
tanglement cloner over the other scenarios in the sense of ro-
bustness against thermal perturbations. This statement, how-
ever, is subject to the thermalization model we have used; so
for a general conclusion a more detailed study is still needed.
Our results may be of importance in practical implementations
of quantum cloning in systems in which thermal effects are
unavoidable, e.g., nuclear spin systems [19, 20]. Indeed, the
large ǫ regime of our approach — when τc is of the same or-
der of magnitude as Tdiss. — has already been experimentally
realized in a different context [19]. This can be considered as
a non-economic cloning process [30].
Acknowledgments.— Supports by the Center of Excellence
in Complex Systems and Condensed Matter (CSCM) at Sharif
University of Technology, iCORE, MITACS, and PIMS are
gratefully acknowledged.
[1] W. K. Wootters and W. H. Zurek, Nature 299 (1982) 802. [2] V. Bužek and M. Hillery, Phys. Rev. A 54 (1996) 1844.
[3] N. Gisin and S. Massar, Phys. Rev. Lett. 79 (1997) 2153.
[4] D. Bruß, D. P. DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchi-
avello, and J. A. Smolin, Phys. Rev. A 57 (1998) 2368.
[5] V. Bužek and M. Hillery, Phys. Rev. Lett. 81 (1998) 5003:
in Quantum Computing and Quantum Communications, Lec-
ture Notes in Computer Science 1509, edited by C. P. Williams
(Springer-Verlag, Berlin, 1999), p. 235.
[6] R. F. Werner, Phys. Rev. A 58 (1998) 1827; P. Zanardi, ibid 58
(1998) 3484; H. Fan, K. Matsumoto, and M. Wadati, ibid 64
(2001) 064301.
[7] D. Bruß, M. Cinchetti, G. M. D’Ariano, and C. Macchiavello,
Phys. Rev. A 62 (2000) 012302.
[8] H. Fan, K. Matsumoto, X. -B. Wang, and M. Wadati, Phys. Rev.
A 65 (2002) 012304; V. Karimipour and A. T. Rezakhani, ibid
66 (2002) 052111; H. Fan, K. Matsumoto, X. -B. Wang, and H.
Imai, J. Phys. A: Math. Gen. 35 (2002) 7415.
[9] L. P. Lamoureux, P. Navez, J. Fiurášek, and N. J. Cerf, Phys.
Rev. A 69 (2004) 040301; E. Karpov, P. Navez, and N. J. Cerf,
ibid 72 (2005) 042314.
[10] A. Kay and M. Ericsson, Phys. Rev. A 73 (2006) 012343.
[11] V. Scarani, S. Iblisdir, N. Gisin, and A. Acı́n, Rev. Mod. Phys.
77 (2005) 1225; N. J. Cerf and J. Fiurášek, Progress in Optics
Vol. 49, edited by E. Wolf (Elsevier, Amsterdam, 2006), p. 455.
[12] M. Merkli, I. M. Sigal, and G. P. Berman, Phys. Rev. Lett. 98
(2007) 130401.
[13] R. Alicki and K. Lendi, Quantum Dynamical Semigroup and
Applications (Springer, Berlin, 1987); V. Scarani, M. Ziman,
P. Štelmachovič, N. Gisin, and V. Bužek, Phys. Rev. Lett. 88
(2002) 097905.
[14] M. Rigol, V. Dunjko, and M. Olshanii, Nature 452 (2008) 854.
[15] K. Maruyama and P. L. Knight, Phys. Rev. A 67 (2003) 032303.
[16] S. -B. Zheng and G. -C. Guo, Phys. Rev. A 72 (2005) 064303.
[17] R. S. Whitney, Y. Makhlin, A. Shnirman, and Y. Gefen, Phys.
Rev. Lett. 94 (2005) 070407; K. M. Fonseca Romero, A. C.
Aguiar Pinto, and M. T. Thomaz, Physica A 307 (2002) 142.
[18] A. T. Rezakhani and P. Zanardi, Phys. Rev. A 73 (2006) 052117.
[19] H. K. Cummins, C. Jones, A. Furze, N. F. Soffe, M. Mosca, J.
M. Peach, and J. A. Jones, Phys. Rev. Lett. 88 (2002) 187901.
[20] T. Durt and J. -F. Du, Phys. Rev. A 69 (2004) 062316; J. -F. Du,
T. Durt, P. Zou, H. Li, L. C. Kwek, C. H. Lai, C. H. Oh, and A.
Ekert, Phys. Rev. Lett. 94 (2005) 040505; T. Durt, J. Fiurášek,
and N. J. Cerf, Phys. Rev. A 72 (2005) 052322.
[21] G. De Chiara, R. Fazio, C. Macchiavello, S. Montangero, and
G. M. Palma, Phys. Rev. A 70 (2004) 062308; ibid 72 (2005)
012328.
[22] A. Ferraro and M. G. A. Paris, J. Opt. B 7 (2005) 532.
[23] A. Roy, A. Sen(De), and U. Sen, Phys. Lett. A 286 (2001) 1.
[24] N. Linden and S. Popescu, Phys. Rev. Lett. 87 (2001) 047901.
[25] R. Filip, Phys. Rev. A 69 (2004) 052301.
[26] N. J. Cerf, O. Krüger, P. Navez, R. F. Werner, and M. M. Wolf,
Phys. Rev. Lett. 95 (2005) 070501; S. Koike, H. Takahashi, H.
Yonezawa, N. Takei, S. L. Braunstein, T. Aoki, and A. Furu-
sawa ibid 96 (2006) 060504.
[27] V. Bužek, V. Vedral, M. B. Plenio, P. L. Knight, and M. Hillery,
Phys. Rev. A 55 (1997) 3327.
[28] S. Bandyopadhyay and G. Kar, Phys. Rev. A 60 (1999) 3296.
[29] A. Peres, Phys. Rev. Lett. 77 (1996) 1413; M. Horodecki, P.
Horodecki, and R. Horodecki, Phys. Lett. A 223 (1996) 1.
[30] Other economic cloning transformations with small ǫ have also
been analyzed in Ref. [20].

  Performances of the symmetric universal and phase-covariant cloning
transformations and entanglement cloners -- qubit case -- are investigated when
the initial state of the hardware or the original state to be cloned is weakly
coupled to a thermal environment. Different behaviors of each of these
transformations are analyzed and contrasted with the ideal cases.

Introduction.— It has been known that based on the prin-
ciples of quantum mechanics, accurate cloning of arbitrary
quantum state is impossible [1]. Nevertheless, on the account
of the significant role of copying in quantum computation and
quantum communication, a variety of approximate quantum
cloning transformations have been proposed, e.g., universal
cloning (UC) machines producing identical copies from ar-
bitrary input qudits [2, 3, 4, 5, 6], phase-covariant cloning
(PCC) machines of a class of partially known qudits [7, 8],
and optimal entanglement cloning machines [9, 10]. For re-
cent reviews see Refs. [11].
A question of practical relevance is how uncontrollable en-
vironmentally induced decoherence or dissipation can affect
performance of quantum cloning machines. In closed sys-
tems, an initially pure state evolves to another pure state. In
practice, however, preparation of pure states and/or keeping
them pure are not generally easy tasks. In general, interac-
tion with an environment degrades purity of quantum systems
and makes their states mixed. A usual effect that a thermal
environment can cause is thermalization (as a kind of dissipa-
tion) [12, 13]. That is, because of interaction with the environ-
mental degrees of freedom which are in thermal equilibrium,
the quantum system will also be driven toward equilibrium.
It should be noted that a generic isolated quantum many-body
system does also relax to a state well described by the standard
statistical-mechanical prescription [14]. In this paper, our aim
is to investigate temperature effects on the performance of the
cloning machines. It has been known that decoherence can put
a limitation on the number of clones that a quantum cloner can
generate [15]. There is also a model in which the robustness
of the cloner increases with the number of qubits [16].
Through a thermalizing process, the density matrix of the
system ̺ in long time will approach the Boltzmann state
̺th = e
−βH/Z , where kBβ is inverse temperature (kB is
the Boltzmann constant), H is the Hamiltonian of the open
quantum system, and Z = Tr(e−βH) is the partition func-
tion. Energy and phase relaxation processes, with the time-
scales T1 and T2, respectively, are common processes present
when approaching an induced equilibrium state. For a more
precise and elaborate discussion of dissipation, thermaliza-
tion, and decoherence see Ref. [12]. Some (phenomenolog-
ical) models for the underlying dynamics of the thermaliza-
tion have already been proposed [13]. We assume that the
time-scale in which typical correlation functions of the en-
vironment decay, tenv., is much smaller than all other time-
scales, i.e., tenv. ≪ τc, Tdiss. = min{T1, T2, TO}, where τc is
the time-scale of the cloning process and TO is the time-scale
dictated by all other relaxation mechanisms. This assumption
is important for the Markovian analysis of the dynamics of the
thermalization [17, 18]. This implies that during the cloning
process, a negligible amount of information flows from the
environment to the system (or vice versa). Here, we also as-
sume that τc . Tdiss.. This extra condition allows us to ig-
nore dynamical effects of the thermalization, hence consider
a simple static (toy) model — explained below — to bring
temperature into play. Despite these simplifying assumptions,
we will argue that the result is still reliable enough to give a
hint about how temperature effects can change performance
of different cloning machines such as the universal cloners,
phase-covariant cloners, and entanglement cloners. Indeed,
such investigation has an immediate importance in attempts
to realize quantum cloning in systems where (due to thermal
and perhaps other noise effects) the preparation of pure states,
whether initial state of the system to be cloned or the quantum
hardware, is difficult, such as in NMR systems [19, 20]. For
another study using a different approach, see Refs. [21, 22].
For the purpose of illustration, we only consider the case of
symmetric 1 → 2 qubit cloners. Extension to M → N qudits
is straightforward as well.
Optimal universal and phase-covariant cloning transfor-
mations.— In the universal cloning transformation, it is usu-
ally assumed that the qubit state to be cloned is a pure state,
|Ψ〉a = cos θ2 |0〉 + e
iφ sin θ
|1〉, and the blank copy (b) and
the quantum cloning machine (also called ancillary system,
c) are each in a known pure state, say |0〉 [2, 3, 4, 23].
The symmetric cloning transformation, then, acts in this
way: U (|Ψ〉a|0〉b|0〉c) = |Υ〉abc, where Trbc(|Υ〉abc〈Υ|) =
Trac(|Υ〉abc〈Υ|). The latter condition guarantees that the fi-
nal clones both have the same states, ̺outa = ̺
b . A mea-
sure to quantify performance of a cloning machine is the fi-
delity between the original and the output states, F (ρ, σ) =
ρ1/2σρ1/2
. Optimization of the fidelity over all input
states on the Bloch sphere results in the qubit optimal univer-
sal cloner, in which F = 5/6 [2, 3]. For orbital states, where
θ is an a priori known constant and φ ∈ [0, 2π), a class of
http://arxiv.org/abs/0704.1177v3
θ0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
PSfrag replacements (a) (b)
0 00 0
FIG. 1: Fidelity (F ) of UC vs θ for some values of η: (a) ǫ = 5/11
and (b) ǫ = 2/3.
phase-covariant cloning machines has been suggested [8]. Af-
ter the cloning process, in the computational basis {|0〉, |1〉}
(the eigenvectors of σz = diag(1,−1)) each of the clones can
be identified by the density operator: ̺out00 = µ
2̺in00 + ν
2 and
̺out01 = 2µν̺
01, where µ
2 + 2ν2 = 1, and ν2 = 1/6 for UC
and ν2 = (1 − 1√
1+2 tan4 θ
)/4 for PCC. Most of this descrip-
tion is also valid when the original quantum system is initially
mixed.
Our main assumption is that preparation of the initial pure
state |Ψ〉 is diluted by a thermal bath in the following special
and simple form:
̺in = (1− ǫ)|Ψ〉〈Ψ|+ ǫ̺th, 0 6 ǫ < 1. (1)
The parameter ǫ, which measures how thermally perturbed the
preparation is, may in general be time-dependent. Nonethe-
less, based on our earlier assumptions, it would be a fairly
slow-varying time-dependent function so that with a good ap-
proximation we can take it a relatively small constant of the
order of τc/Tdiss.. This state does not seem to arise natu-
rally from a typical thermalization dynamics. Nevertheless,
in Ref. [18] it has been illustrated that general behaviors ob-
tained from such a simple preparation assumption (in the con-
text of the geometric phases) have general features similar to
those obtained from the Lindblad equation for the dynamics.
It is worth mentioning that in the limit of infinite tempera-
ture, the thermalized density matrix ρth is equivalent to pure
noise [24]. In that case, ǫ represents the degree of pure noise
existing during the process (for example, in the case of NMR
systems, due to fluctuations of the external magnetic fields and
similar reasons). A more general analysis of quantum cloning
in the presence of a thermalization mechanism is yet lacking,
but our simple analysis may also shed some light before hav-
ing a more complete analysis at hand.
First, we consider the effect of the thermal term only on the
state of the cloner, that is, the quantum cloning hardware is
thermally diluted as in Eq. (1). In this case, the initial state
of the machine is mixed. Considering the fact that in the op-
timal UC and PCC, the initial state of the cloning machine
can be any pure state [4, 7, 23], one can conclude the opti-
mal fidelity here is achieved by the existing optimal cloning
transformations. By a similar analysis, it appears that for the
case of diluted joint blank and ancillary systems, one can con-
sider the joint state as a new blank copy and attach some new
reservoir to the whole Hilbert space of the input states (i.e.,
the information qubit, the blank copy, and the ancilla state) as
a new ancillary system and then define a new transformation
θ0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3
PSfrag replacements (a) (b)
0 00 0
FIG. 2: Variation in the fidelity of PCC with θ for: (a) ǫ = 5/11 and
(b) ǫ = 2/3, and different values of η.
for cloning [23]. This would in fact be the existing optimal
cloning transformation, now acting on a larger Hilbert space,
and hence one obtains the same optimal fidelity again. How-
ever, from an experimental point of view, thermalization ef-
fects are likely to occur during the cloning process rather than
at the initial preparation level — for instance in NMR sys-
tems [19, 20]. Therefore, to be more precise, thermal effects
during the preparation should also be taken into account.
Now, we consider the case in which the input state |Ψ〉 is
thermally diluted as in Eq. (1). Our aim now is to compare
the similarity between the clones and the input state of our in-
terest, i.e., |Ψ〉. Indeed, here we assume that the model of the
cloning machine consists of two parts: the first is the dilution
of the input pure state which models the imperfect feature of
the machine, and the second is some known UC or PCC trans-
formation which is applied to this diluted state. The Hamilto-
nian of the qubit system is taken to be H = ω0σz/2 (ω0 > 0),
whence, Z = 2 cosh η, where η = ω0β/2. More general
cloning transformations in spin networks with more compli-
cated Hamiltonians can be found, for example, in Ref. [21].
The fidelity of the output state and the unperturbed initial state
can be calculated as follows:
F (θ, ǫ, η) =µ2[1− ǫ+ ǫ(e−η cos2 θ
+ eη sin2
+ (µν − µ2/2)(1− ǫ) sin2 θ + ν2.
Figure 1 illustrates how the fidelity in the UC behaves in terms
of θ (orbit of the state) in thermally diluted states, for two
different values of ǫ (the degree of thermalization) and η (∝
1/T ). It can be seen that when
ǫ < cosh η /(e−η sin2
+ eη cos2
), (2)
the fidelity of the UC is higher than the classical value 1/2.
This threshold is the fidelity of a classical-like 1 → M uni-
versal cloning in which with a given probability, an unknown
input state is sent to one of the M parties and a completely
randomized state is transmitted to any of the other ones, of
course, in the limit of large M [25]. In the literature, however,
“classical cloner” has been attributed to some other cloning
transformations as well — see [3, 26]. In other words, in some
cases thermal noise (even in the simple form of Eq. (1)) can
result in a lower performance than a classical machine. For
θ > π/2, the condition (2) implies that for all 0 6 ǫ < 1, the
fidelity of the output of the UC is always greater than that of
PSfrag replacements
FIG. 3: Fidelity vs ǫ and θ in low temperature limit (η → ∞): UC
(left) and PCC (right).
the classical cloner (if ω0 was negative, this would occur for
θ 6 π/2). Equation (2) can also be interpreted as a condition
on temperature for a given θ and ǫ in order to outperform a
classical cloner. Figure 2 shows the variation of the fidelity
of the outputs of the PCC machines in terms of θ, for some
fixed values of ǫ and η. As is clear from this figure, in the case
of equatorial qubits, similar to the case of the UC, the fidelity
of the outputs does not vary with temperature — according to
Eq. (2), this feature is due to the symmetry property of such
states. Low temperature limits of the fidelity for both UC and
PCC have been depicted in Fig. 3. In the case of the UC, for
all θ in [0, π), the fidelity is a decreasing function of ǫ. The
corresponding graph for the PCC also shows a decrease in the
fidelity for different values of θ ∈ [0, π/2) with the pertur-
bation factor ǫ. However, a closer inspection shows that here
there are also some θs (& 2.52 and less than π rad) in which
the fidelity of the PCC is an increasing function of ǫ. At high
temperature limit, the fidelity of both UC and PCC, for all θs,
is a decreasing function of ǫ. Another important point that can
be concluded from the figures is that in some cases, the quality
of the clones at the output of the UC can be better than that of
the PCC — see for example those regions of Fig. 3 in which
ǫ and θ are large and small, respectively. This is indeed con-
trary to what happens when the cloning is performed perfectly
without any external noise.
Entanglement cloning.— Quantum cloning can be used to
clone or broadcast entanglement as well [5, 9, 10, 27, 28].
Let us assume that we have an initial state in the form of
|Ψ−α 〉ab = α|01〉ab −
1− α2|10〉ab, where α is real and
|α| 6 1. As in the cases of the UC and the PCC, sup-
pose that because of a thermal environment, the initializa-
tion is diluted as in Eq. (1). Let us take our system to be
two spin-1/2 particles interacting via the XX Hamiltonian:
H = J(σaxσ
x + σ
y), where σx and σy are Pauli matri-
TABLE I: Inseparability conditions of the output states in the three
different scenarios of cloning.
γ α ǫ
γ > γc
|α2 − 1/2| < αc
0 < α < 1 0 6 ǫ < 1
−1 < α < 0 0 6 ǫ < ǫ1 or ǫ2 < ǫ < 1
|α2 − 1/2| > αc ǫ2 < ǫ < 1
0 < γ 6 γc |α2 − 1/2| < αc
0 < α < 1 0 6 ǫ < ǫ2
−1 < α < 0 0 6 ǫ < ǫ1
ces. Now, we want to compare performances of the following
schemes of entanglement broadcasting between two parties in
the presence of thermal noise: (i) Local cloning by means of
two optimal UC machines copying each qubit separately [27].
In this scenario, after the cloning process and discarding the
ancillas, we will have the overall state ̺aa′bb′ whose two first
(last) qubits are the copies of a (b). (ii) Non-local cloning of
the two-qubit state as a whole with the UC machine of 4-level
quantum states [5]. (iii) Cloning by an optimal entanglement
cloner [9].
After some algebra, it can be seen that the density matrices
of the clones in cases (ii) and (iii), and ̺a′b (also ̺ab′ , ̺ab,
and ̺a′b′ ) — nonlocal copies — in case (i), read as follows:
̺out = (Mǫ
+ 1−M
)(|00〉〈00|+ |11〉〈11|)
+[M(1−ǫ
+ ǫ cosh γ
) + 1−M
+ L(1− ǫ)(2α2 − 1)]|01〉〈01|
+[M(1−ǫ
+ ǫ cosh γ
) + 1−M
− L(1− ǫ)(2α2 − 1)]|10〉〈10|
−M [(1− ǫ)α
1− α2 + ǫ
sinh γ](|01〉〈10|+ |10〉〈01|), (3)
in which γ = 2βJ, Z = 2(1 + cosh γ), L = 3(1 + 2M +√
1 + 4M − 9M2)/26, Mi = (2/3)2, Mii = 3/5, Miii =
6A2 + 4AC, A =
(1/2 + 1/
13)/3, and C = A(
3)/2. Note that, the output states of case (ii) for all values of
ǫ, α, and γ, the nonlocal copies of case (i) ̺a′b, and the output
states of case (iii) for ǫ = 1 and ∀γ or α = ±1/
2 (for all
ǫ and γ) all can be written in the following compact form:
̺out = M̺in + (1 − M)I/4, where I is the 4 × 4 identity
matrix.
To determine the regions in which the output states are
separable or inseparable, we use the well-known Peres-
Horodecki positive partial transposition criterion [29]. Ac-
cording to this criterion, in the case of 2 × 2 and 2 × 3 sys-
tems, a density matrix ̺AB is inseparable (i.e., entangled) iff
(̺AB)
TA (TA: partial transposition with respect to system
A) is not positive. Tables I and II show the results for anti-
ferromagnetic case (J > 0). The parameters in the tables are
TABLE II: Inseparability conditions of the output states in the three
different scenarios of cloning, at low and high temperature limits.
γ ǫ, α
γ → ∞
C1 and 0 6 ǫ 6 1−M
and |α2 − 1/2| < α∞
C1 and 1−M
< ǫ < 1 and α ∈ C1
C2 and 0 6 ǫ < 3M−1
and |α2 − 1/2| < α∞
C2 and 1−M
< ǫ 6 M+1
and |α2 − 1/2| > α∞
C2 and M+1
< ǫ < 1 and α ∈ C2
γ → 0 0 6 ǫ < (1 − 1
) and |α2 − 1/2| < α0
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
1 2 3 4 5
PSfrag replacements
(ii) (iii)
(input)
FIG. 4: Entanglement phase diagrams of input and output
states (achieved from three different schemes of entanglement
cloning/broadcasting introduced in the text), when α = 1/
2. The
regions labeled by 1 are the regions in which entanglement exists,
whilst the regions labeled by 2 indicate no-entanglement regions.
This figure shows that for γ < γc (T > Tc), depending on the value
of ǫ, we may or may not have entanglement. γc is a decreasing func-
tion of M . In other words, the area of region 1 increases when M
increases, as well. This may imply the advantage of the entanglement
cloner Miii over the other entanglement broadcasting schemes.
as follows:
α∞1 =
(3M − 1)(M + 1− 4Mǫ)
4M(1− ǫ)
α∞2 =
(M + 1)(3M − 1− 4Mǫ)
4M(1− ǫ)
3M2 + 2M − 1
γc = ln(
M + 1 + 2
M2 +M
3M − 1
ǫ1(2) =
(M − 1∓ 4Mδ)(1 + cosh γ)
2M [1± sinh γ ∓ 2δ(1 + cosh γ)] ,
(3M(1− ǫ)− 1) (M(1− ǫ) + 1)
4M(1− ǫ)
where δ = α
1− α2, C1 ≡ 0 < α 6 1, and C2 ≡
−1 6 α 6 0. When γ → ∞ and M = Miii, since
(3M − 1)/4M > (1 −M)/2M , there exists an overlap be-
tween the ǫ-inequalities in the third and fourth sub-rows of Ta-
ble II. In this case, one should notice that for (1−M)/2M <
ǫ < (3M − 1)/4M , clones are entangled if |α2 − 1/2| < α∞2
or |α2 − 1/2| > α∞1 . This removes the ambiguity in such
cases.
Tables I and II imply that in most temperature regions, the
inseparability inequalities are not symmetric with respect to
α → −α. In other words — unlike the case of ǫ = 0 — de-
pending on the sign of α, the parameter regions over which
the cloned pairs are entangled may be different. Another im-
portant point (see the second row of Table I) is the existence
of a critical temperature Tc (∝ 1/γc) beyond which the cloned
pairs for some α regions, |α2 − 1/2| > αc, for all ǫs are not
entangled.
Overall, by taking into account the behaviors of the upper
and lower bounds of the inseparability inequalities we can find
that in some temperature regions, in Table I (Table II), there
exist intervals of α2 (ǫ) in which the cloned pairs are sepa-
rable. The length of these intervals decreases when M in-
creases (recall that Miii > Mii > Mi). Furthermore, for a
given α2 (ǫ) at intermediate (two limits of) temperatures, the
range of ǫ (α2) in which the clones are entangled increases
when M increases as well. Indeed, for some temperature re-
gions, in Table I (Table II) there exist some α2 (ǫ) in which
clones for all ǫ (α in C1 or C2) and all three Ms are entan-
gled — e.g., see first sub-row of Table I or second and fifth
sub-rows of Table II. These facts together with the entan-
glement phase diagrams in Fig. 4, whose regions show ex-
istence of entanglement or its non-existence for α = 1/
indicate advantage of entanglement cloner Miii, over the other
cloning schemes. That is, the optimal entanglement cloner has
an advantage over other mentioned schemes of entanglement
broadcasting in the sense of robustness against thermal noise.
Conclusion.— We have studied the role of thermal noise
in some quantum cloning schemes through a simple model of
temperature effect on spin states at the input of the cloning
machines. The performance of the cloning machines depends
generally on the values of the thermal perturbation coefficient,
the orbit of the original state on the Bloch sphere, as well as on
the temperature. In addition, three scenarios of entanglement
cloning of thermally diluted two-qubit states have been inves-
tigated. Our analysis shows that the clones generated from
non-local transformations, in particular those out of the opti-
mal entanglement cloner, remain entangled for wider regions
of parameters. I.e., the optimal entanglement cloner shows
a relatively larger region of entanglement in the parameter
space. This can be considered as an advantage of optimal en-
tanglement cloner over the other scenarios in the sense of ro-
bustness against thermal perturbations. This statement, how-
ever, is subject to the thermalization model we have used; so
for a general conclusion a more detailed study is still needed.
Our results may be of importance in practical implementations
of quantum cloning in systems in which thermal effects are
unavoidable, e.g., nuclear spin systems [19, 20]. Indeed, the
large ǫ regime of our approach — when τc is of the same or-
der of magnitude as Tdiss. — has already been experimentally
realized in a different context [19]. This can be considered as
a non-economic cloning process [30].
Acknowledgments.— Supports by the Center of Excellence
in Complex Systems and Condensed Matter (CSCM) at Sharif
University of Technology, iCORE, MITACS, and PIMS are
gratefully acknowledged.
[1] W. K. Wootters and W. H. Zurek, Nature 299 (1982) 802. [2] V. Bužek and M. Hillery, Phys. Rev. A 54 (1996) 1844.
[3] N. Gisin and S. Massar, Phys. Rev. Lett. 79 (1997) 2153.
[4] D. Bruß, D. P. DiVincenzo, A. Ekert, C. A. Fuchs, C. Macchi-
avello, and J. A. Smolin, Phys. Rev. A 57 (1998) 2368.
[5] V. Bužek and M. Hillery, Phys. Rev. Lett. 81 (1998) 5003:
in Quantum Computing and Quantum Communications, Lec-
ture Notes in Computer Science 1509, edited by C. P. Williams
(Springer-Verlag, Berlin, 1999), p. 235.
[6] R. F. Werner, Phys. Rev. A 58 (1998) 1827; P. Zanardi, ibid 58
(1998) 3484; H. Fan, K. Matsumoto, and M. Wadati, ibid 64
(2001) 064301.
[7] D. Bruß, M. Cinchetti, G. M. D’Ariano, and C. Macchiavello,
Phys. Rev. A 62 (2000) 012302.
[8] H. Fan, K. Matsumoto, X. -B. Wang, and M. Wadati, Phys. Rev.
A 65 (2002) 012304; V. Karimipour and A. T. Rezakhani, ibid
66 (2002) 052111; H. Fan, K. Matsumoto, X. -B. Wang, and H.
Imai, J. Phys. A: Math. Gen. 35 (2002) 7415.
[9] L. P. Lamoureux, P. Navez, J. Fiurášek, and N. J. Cerf, Phys.
Rev. A 69 (2004) 040301; E. Karpov, P. Navez, and N. J. Cerf,
ibid 72 (2005) 042314.
[10] A. Kay and M. Ericsson, Phys. Rev. A 73 (2006) 012343.
[11] V. Scarani, S. Iblisdir, N. Gisin, and A. Acı́n, Rev. Mod. Phys.
77 (2005) 1225; N. J. Cerf and J. Fiurášek, Progress in Optics
Vol. 49, edited by E. Wolf (Elsevier, Amsterdam, 2006), p. 455.
[12] M. Merkli, I. M. Sigal, and G. P. Berman, Phys. Rev. Lett. 98
(2007) 130401.
[13] R. Alicki and K. Lendi, Quantum Dynamical Semigroup and
Applications (Springer, Berlin, 1987); V. Scarani, M. Ziman,
P. Štelmachovič, N. Gisin, and V. Bužek, Phys. Rev. Lett. 88
(2002) 097905.
[14] M. Rigol, V. Dunjko, and M. Olshanii, Nature 452 (2008) 854.
[15] K. Maruyama and P. L. Knight, Phys. Rev. A 67 (2003) 032303.
[16] S. -B. Zheng and G. -C. Guo, Phys. Rev. A 72 (2005) 064303.
[17] R. S. Whitney, Y. Makhlin, A. Shnirman, and Y. Gefen, Phys.
Rev. Lett. 94 (2005) 070407; K. M. Fonseca Romero, A. C.
Aguiar Pinto, and M. T. Thomaz, Physica A 307 (2002) 142.
[18] A. T. Rezakhani and P. Zanardi, Phys. Rev. A 73 (2006) 052117.
[19] H. K. Cummins, C. Jones, A. Furze, N. F. Soffe, M. Mosca, J.
M. Peach, and J. A. Jones, Phys. Rev. Lett. 88 (2002) 187901.
[20] T. Durt and J. -F. Du, Phys. Rev. A 69 (2004) 062316; J. -F. Du,
T. Durt, P. Zou, H. Li, L. C. Kwek, C. H. Lai, C. H. Oh, and A.
Ekert, Phys. Rev. Lett. 94 (2005) 040505; T. Durt, J. Fiurášek,
and N. J. Cerf, Phys. Rev. A 72 (2005) 052322.
[21] G. De Chiara, R. Fazio, C. Macchiavello, S. Montangero, and
G. M. Palma, Phys. Rev. A 70 (2004) 062308; ibid 72 (2005)
012328.
[22] A. Ferraro and M. G. A. Paris, J. Opt. B 7 (2005) 532.
[23] A. Roy, A. Sen(De), and U. Sen, Phys. Lett. A 286 (2001) 1.
[24] N. Linden and S. Popescu, Phys. Rev. Lett. 87 (2001) 047901.
[25] R. Filip, Phys. Rev. A 69 (2004) 052301.
[26] N. J. Cerf, O. Krüger, P. Navez, R. F. Werner, and M. M. Wolf,
Phys. Rev. Lett. 95 (2005) 070501; S. Koike, H. Takahashi, H.
Yonezawa, N. Takei, S. L. Braunstein, T. Aoki, and A. Furu-
sawa ibid 96 (2006) 060504.
[27] V. Bužek, V. Vedral, M. B. Plenio, P. L. Knight, and M. Hillery,
Phys. Rev. A 55 (1997) 3327.
[28] S. Bandyopadhyay and G. Kar, Phys. Rev. A 60 (1999) 3296.
[29] A. Peres, Phys. Rev. Lett. 77 (1996) 1413; M. Horodecki, P.
Horodecki, and R. Horodecki, Phys. Lett. A 223 (1996) 1.
[30] Other economic cloning transformations with small ǫ have also
been analyzed in Ref. [20].
