General Sequential Quantum Cloning
Gui-Fang Dang and Heng Fan
Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China.
(Dated: November 4, 2018)
Some multipartite quantum states can be generated in a sequential manner which may be im-
plemented by various physical setups like microwave and optical cavity QED, trapped ions, and
quantum dots etc. We analyze the general N to M (N ≤ M) qubits Universal Quantum Cloning
Machine (UQCM) within a sequential generation scheme. We show that the N to M sequential
UQCM is available. The case of d-level quantum states sequential cloning is also presented.
PACS numbers: 03.67.Mn, 03.65.Ud, 52.50.Dv
Quantum entanglement plays a key role in quantum
computation and quantum information [1]. Multipartite
entangled states arise as a resource for quantum infor-
mation processing tasks such as the well known quantum
teleportation[2], quantum communication [3, 4], clock
synchronization [5] etc. In general it is extremely dif-
ficult to generate experimentally multipartite entangled
states through single global unitary operations. In this
sense, the sequential generation of the entangled states
appears to be promising. Actually most of the quantum
computation networks are designed to implement quan-
tum logic gates through a sequential procedure [6]. Re-
cently sequential implementing of quantum information
processing tasks has been attracting much attention. It
is pointed out that photonic multiqubit states can be
generated by letting a source emit photonic qubits in a
sequential manner [7]. The general sequential generation
of entangled multiqubit states in the realm of cavity QED
was systematically studied in Refs.[8, 9]. It is also shown
that the class of sequentially generated states is identical
to the matrix-product-state (MPS) which is very useful
in study of spin chains of condensed matter physics [10].
On the other hand, much progress has already been
made in the past years in studying quantum cloning ma-
chines, for reviews see, for example, Refs.[11, 12, 13].
And various quantum cloning machines have been im-
plemented experimently by polarization of photons [14,
15, 16, 17, 18],nuclear spins in Nuclear Magnetic Reso-
nance [19, 20], etc. However, these experiments are for
1 to 2 (one qubit input and two-qubit output) or 1 to 3
cloning machines. The more general case will be much
difficult. There are some schemes proposed for the gen-
eral quantum cloning machines which are not in a sequen-
tial manner, see for example, [21, 22]. Recently a 1 to M
sequential universal quantum cloning is proposed [23] by
using the cloning transformation presented in Ref.[24].
Since it is in a sequential procedure, potentially it re-
duces the difficult in implementing this quantum cloning
machine. However, as is well known the collective quan-
tum cloning machine (the N identical input states are
cloned collectively to M copies) is better than the quan-
tum cloning machine which can only deal with the in-
dividual input(only one input is copied to several copies
each time). We know that the general N to M cloning
transformation is also available in Refs.[24, 25]. Then a
natural question arise is that whether the general N to
M sequential cloning machine is possible. In this Letter,
we will present the general sequential universal quantum
cloning machine.
The 1 to M cloning transformations used in Ref.[23]
was proposed by Gisin and Massar in Ref.[24]. And the
N toM UQCM was also presented in Ref.[24]. However,
to use the method proposed in Refs.[8, 23] to find the se-
quential cloning machine, the input state |Φ〉⊗N should
be expanded in computational basis {|0〉, |1〉}. The ex-
plicit quantum cloning transformations with this kind of
input were proposed by Fan et al in Ref.[25]. In this Let-
ter, based on the result of Ref.[25], the general sequential
UQCM will be presented.
As presented in Refs.[8, 23], the sequential generation
of a multiqubit state is like the following. Let HA be
a D-dimensional Hilbert space which acts as the ancil-
lary system, and a single qubit (e.g., a time-bin qubit)
is in a two-dimensional Hilbert space HB. In every step
of the sequential generation of a multiqubit state, a uni-
tary time evolution will be acting on the joint system
HA ⊗HB. We assume that each qubit is initially in the
state |0〉 which is like a blank or an empty state and
will not be written out in the formulas. So the unitary
time evolution is written in the form of an isometry V :
HA → HA⊗HB, where V =
i,α,β V
α,β |α, i〉〈β|, each V i
is a D×D matrix, and the isometry condition takes the
i=0 V
i†V i = 1. By applying successively n oper-
ations of V (not necessarily the same) on an initial ancil-
lary state |φI〉 ∈ HA, we obtain |Ψ〉 = V [n]...V [2]V [1]|φI〉.
The generated n qubits are in general an entangled state,
but the last step qubit-ancilla interaction can be chosen
so as to decouple the final multiqubit entangled state
from the auxiliary system, so the sequentially generated
state is
|ψ〉 =
i1...in=0
〈φF |V [n]in ...V [1]i1 |φI〉|in, ..., i1〉, (1)
where |φF 〉 is the final state of the ancilla. This is the
MPS. It was proven that any MPS can be sequentially
generated [8].
http://arxiv.org/abs/0704.0323v2
Suppose there are N identical pure quantum states
|Φ〉⊗N = (x0|0〉+x1|1〉)⊗N need to be cloned toM copies,
where |x0|2 + |x1|2 = 1. We know that the input state
can be represented by a basis in symmetric subspace.
|Φ〉⊗N =
xN−m0 x
CmN |(N −m)0,m1〉, (2)
where |(N − m)0,m1〉 denotes the symmetric and nor-
malized state with (N −m) qubits in the state |0〉 and m
qubits in the state |1〉, and we have CmN = N !/(N−m)!m!
in standard notation. So if we find the quantum cloning
transformations for all states in symmetric subspace, we
can clone N pure states to M copies. The UQCM with
input in symmetric subspace can be written as [25],
|(N −m)0,m1〉 → |ΦmM 〉, (3)
where
|ΦmM 〉 =
βmj |(M −m− j)0, (m+ j)1〉 ⊗Rj ,(4)
βmj =
M−N−j
M−m−jC
(m+j)
/CN+1M+1, (5)
where Rj are the ancillary states of the cloning machine
and are orthogonal with each other for different j. For
a sequential quantum cloning machine in this Letter, we
choose a realization Rj ≡ |(M −N − j)1, j0〉 for the an-
cilla states. This UQCM is optimal in the sense that
the fidelity between single qubit output state reduced
density operator ρoutreduced and the single input |Φ〉 is op-
timal. The optimal fidelity is F = 〈Φ|ρoutreduced|Φ〉 =
(MN +M + N)/M(N + 2), see Refs.[11, 12, 13] for re-
views and the references therein. A realization of this
UQCM with photon stimulated emission can be found in
Ref.[22] which is not in a sequential manner. We next
show that this general N to M UQCM can be generated
through a sequential procedure.
The basic idea is to show that the final state of the
cloning, |ΦmM 〉 in (4), can be expressed in its MPS form.
As shown in Ref.[8], any MPS can be sequentially gen-
erated. We shall follow the method, for example, as in
Refs.[23, 26]. By Schmidt decomposition, we first ex-
press the quantum state |ΦmM 〉 as a bi-partite state across
1 : 2... cut,
|ΦmM 〉 = λ
1 |0〉|φ
[2...(2M−N)]
1 〉+ λ
2 |1〉|φ
[2...(2M−N)]
Γ[1]i1α1 λ
|i1〉|φ[2...(2M−N)]α1 〉, (6)
where Γ
α1 = δα1,1,Γ
α1 = δα1,2, and λ
α1 are eigen-
values of the first qubit reduced density operator, and
we find λ
∑M−m−1
k=−m β
M−1/C
M , λ
∑M−m−1
k=−m β
mk+1C
M−1/C
m+k+1
M . To correspond with
the MPS in (1), we can define V
[1]i1
α1 = Γ
[1]i1
α1 . Suc-
cessively by Schmidt decomposition, the quantum state
|ΦmM 〉 in (4) is divided into a bi-partite state with the first
n qubits as one part, and the rest as another part, where
1 < n ≤M − 1. We find
|ΦmM 〉 =
j+1|(n− j)0, j1〉|φ
[(n+1)...(2M−N)]
j+1 〉, (7)
when 1 < n ≤M−N+m,n′ = n; whenM−N+m< n ≤
M − 1, n′ =M −N +m, λ[n]j+1 are eigenvalues of the first
n qubits reduced density operator of |ΦmM 〉. According to
the results in Eqs.(4,5), we can obtain,
j+1 =
M−m−n
m(j+k)
Cm+kM−n
m+j+k
. (8)
And we also have
|φ[(n+1)...(2M−N)]j+1 〉 =
M−m−n
β2m(j+k) ×
(m+k)
m+j+k
|(M − n−m− k)0, (m+ k)1〉 ⊗Rj+k.
By induction and a concise formula, we have
|Φn...(2M−N)]j+1 〉
αn,in
[n]in
(j+1)αn
λ[n]αn |in〉|φ
[(n+1)...(2M−N)]
[n−1]
|0〉|φ[(n+1)...(2M−N)]j+1 〉
+|1〉|φ[(n+1)...(2M−N)]j+2 〉
, (9)
where we denote
(j+1)αn
= δ(j+1)αn
n−1/(λ
[n−1]
n), (10)
(j+1)αn
= δ(j+2)αn
n−1/(λ
[n−1]
n ). (11)
Still we define that
V [n]inαnαn−1 = Γ
[n]in
αn−1αn
λ[n]αn . (12)
It is thus in the MPS representation. We can further con-
sider other cases including the ancilla state of the cloning
machine represented as Rj (Note it is not the ancilla state
in the MPS representation). We can find that the out-
put state of the general UQCM can be expressed as MPS
as in form (1). So it can be created sequentially. The
explicit results are summarized in the appendix.
We have shown that the output states of the general
UQCM in (4,5) are MPS’s and thus can be generated
sequentially. The sequential matrices V [n] of course de-
pend on the input |(N−m)0,m1〉 which are W-like states
and are generally multiqubit entangled. For later con-
venience, we denote V (m) to express that it depends
on input state for different m. By a straightforward
method, the sequential cloning operation, i.e., the iso-
metrices, depending on different input may take the form
m |(N − m)0,m〉〈(N − m)0,m1| ⊗ V (m). However,
this operation may need a single global unitary opera-
tor which involves N -qubit entangled states except for
m = 0,m = N . This contradicts with our aim that each
operation should be divided into sequential unitary oper-
ators in a quDit (quantum state in D-dimensional space)
times qubit system. Here we can use a scheme like the
following: the ancillary state interacts with each qubit
according to the (N + 1) × D-dimensional isometrices
CmN |0〉〈0|⊗N−m⊗|1〉〈1|⊗m⊗V (m) sequentially,
here a whole normalization factor is omitted. We know
that the operation |0〉〈0|⊗N−m ⊗ |1〉〈1|⊗m acts on each
qubit individually. Thus this scheme reduces the com-
plexity of the operation. This finishes our general se-
quential UQCM for the case of qubit. In case N = 1,
we recover the result of Ref.[23] for 1 to M cloning.
We should remark that similar as the case of sequen-
tial 1 to M UQCM in Ref.[23], for the general sequential
UQCM, the minimal dimension D of the ancillary state
grows linearly at most with M −N/2 + 1 for even N or
M − (N − 1)/2 for odd N .
Next we will consider a more general case that the se-
quential cloning machine is about the quantum state in d-
dimensional Hilbert space. We will use the d-dimensional
UQCM proposed by Fan et al in Ref. [25]. This UQCM
is a generalization of the cloning machine proposed in
Ref.[24] and we can use this UQCM to study its sequen-
tial form for d-dimensional case.
An arbitrary d-dimensional pure state takes the form
|Φ〉 =
i=0 xi|i〉 with
i=0 |xi|2 = 1. N identical pure
states can be expanded in terms of state in symmet-
ric subspace |Φ〉⊗N =
m1!...md!
xm10 ...x
d−1|~m〉,
where |~m〉 ≡ |m1, ...,md〉 is a symmetric state with mi
states of |i − 1〉, and also mi should satisfy a relation
i=1mi = N . The cloning transformations with states
in symmetric subspace can be written as
|~m〉 → |Φ~mM 〉 =
|~m+~j〉 ⊗ |~j〉, (13)
i=1 C
mi+ji
CM−NM+d−1
where ~j should satisfy
i ji = M − N . This cloning
machine is optimal and the corresponding fidelity of a
single quantum state between input and output is F =
(N(d+M) +M −N) /(d+N)M .
As for qubit system, we next show that the output
states for all symmetric states input can be expressed
as the sequential form. We consider the case 1 < n ≤
M − 1, and the state |Φ~mM 〉 is a bipartite state across
1...n : (n+ 1)... cut,
|Φ~mM 〉 =
|~j〉|φ[(n+1)...(M+1)]
〉 (15)
where
~m(~j−~m+~k)
i=1 C
ji+ki
, (16)
|φ[(n+1)...(M+1)]
~m(~j−~m+~k)
i=1 C
ji+ki
|~k〉|~j − ~m+ ~k〉/λ[n]
. (17)
By the same procedure as that of qubit case, we can
obtain the following
|φ[n...(M+1)]
[n]in
λ[n]αn |in〉|φ
(n+1)...(M+1)]
〉. (18)
Then we have
[n]in
= δαn(~j+~ein+1)
jin+1 + 1
[n−1]
. (19)
Still we can define V
[n]in
αnαn−1 = Γ
[n]in
αn−1αnλ
αn , and thus we
can find that each state |Φ~mM 〉 is a MPS and thus can be
sequentially generated. The detailed result of this part
will be presented elsewhere [27].
In conclusion, we show that the generalN toM univer-
sal quantum cloning machine can be implemented by a se-
quential manner. Since the sequential generation of mul-
tipartite state can be implemented in various physical se-
tups such as microwave and optical cavity QED, trapped
ions and quantum dots etc. This general sequential quan-
tum cloning machine may be implemented much easier
than the single global implementation scheme. This re-
duces dramatically the complexity in implementing the
general UQCM. We also show that for d-dimensional
quantum state, the sequential UQCM is also available.
Besides the universal cloning machine, the 1 toM phase-
covariant quantum cloning machine can also be sequen-
tially implemented. It will be interesting to consider sim-
ilarly the generalN toM phase-covariant cloning and the
economic phase-covariant cloning. The sequential asym-
metric quantum cloning machine may also be an inter-
esting topic.
Acknowledgements: HF was supported by ”Bairen”
program, NSFC and ”973” program (2006CB921107).
Appendix.–The explicit form of matrices V are pre-
sented as:
V [n]0αnαn−1 = δαnαn−1 ×
∑M−m−n
k=−m X
m+αn−1−1+k
∑M−m−n+1
k=−m X
M−n+1
m+αn−1−1+k
V [n]0αnαn−1 = δαnαn−1+1 ×
∑M−m−n
k=−m X
m+αn−1+k
∑M−m−n+1
k=−m X
M−n+1
m+αn−1−1+k
where notations X = β2m(αn−1−1+k), X
′ = β2m(αn−1+k)
are used. For case 1 < n ≤ M − N + m,αn−1 =
1, ..., n;αn = 1, ..., (n+1), and for caseM−N+m < n ≤
M − 1, αn−1, αn = 1, ..., (M −N +m+1). We can check
that the above defined V satisfies the isometry condition
V [n]in
V [n]in = 1. Similarly we have
V [M ]0αMαM−1 = δαMαM−1 ×
M−1−1−m)
M−1−1−m)
M−1−1
M−1−m)
V [M ]1αMαM−1 = δαM (αM−1+1) ×
M−1−m)
M−1−1−m)
M−1−1
M−1−m)
where 0 ≤ m ≤ N −m,αM−1, αM = 1, 2, ..., (M − N +
m+ 1).
For case concerning about ancilla state of the UQCM,
assume 1 ≤ l ≤M −N , we have
V [M+l]0αM+lαM+l−1 = δαM+l(αM+l−1−1) ×
αM+l−1 −m− 1
M −N − l + 1
V [M+l]1αM+lαM+l−1 = δαM+lαM+l−1 ×
M −N − l − αM+l−1 +m+ 1
M −N − l + 1
(1) For (m+ 1) ≤ αM+l ≤ (M −N +m− l+ 1),
(m+ 2) ≤ αM+l−1 ≤ (M −N +m− l+ 2),
[M+l]0
αM+lαM+l−1 = δαM+l(αM+l−1−1)
αM+l−1−m−1
M−N−l+1 .
For αM+l = (M −N +m− l + 2), 1 ≤ αM+l−1 ≤
(M −N +m+ 1), V [M+l]0αM+lαM+l−1 = 0. Otherwise
[M+l]0
αM+lαM+l−1 = δαM+lαM+l−1
(2) For (m+ 1) ≤ αM+l, αM+l−1 ≤
(M −N +m− l + 1), V [M+l]1αM+lαM+l−1 =
δαM+lαM+l−1
M−N−l−αM+l−1+m+2
M−N−l+1 . For αM+l =
(M −N +m− l + 2), 1 ≤ αM+l−1 ≤ (M −N +m+ 1),
[M+l]0
αM+lαM+l−1 = 0. Otherwise V
[M+l]0
αM+lαM+l−1 =
δαM+lαM+l−1
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http://arxiv.org/abs/quant-ph/0607105

  Some multipartite quantum states can be generated in a sequential manner
which may be implemented by various physical setups like microwave and optical
cavity QED, trapped ions, and quantum dots etc. We analyze the general N to M
qubits Universal Quantum Cloning Machine (UQCM) within a sequential generation
scheme. We show that the N to M sequential UQCM is available. The case of
d-level quantum states sequential cloning is also presented.

General Sequential Quantum Cloning
Gui-Fang Dang and Heng Fan
Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China.
(Dated: November 4, 2018)
Some multipartite quantum states can be generated in a sequential manner which may be im-
plemented by various physical setups like microwave and optical cavity QED, trapped ions, and
quantum dots etc. We analyze the general N to M (N ≤ M) qubits Universal Quantum Cloning
Machine (UQCM) within a sequential generation scheme. We show that the N to M sequential
UQCM is available. The case of d-level quantum states sequential cloning is also presented.
PACS numbers: 03.67.Mn, 03.65.Ud, 52.50.Dv
Quantum entanglement plays a key role in quantum
computation and quantum information [1]. Multipartite
entangled states arise as a resource for quantum infor-
mation processing tasks such as the well known quantum
teleportation[2], quantum communication [3, 4], clock
synchronization [5] etc. In general it is extremely dif-
ficult to generate experimentally multipartite entangled
states through single global unitary operations. In this
sense, the sequential generation of the entangled states
appears to be promising. Actually most of the quantum
computation networks are designed to implement quan-
tum logic gates through a sequential procedure [6]. Re-
cently sequential implementing of quantum information
processing tasks has been attracting much attention. It
is pointed out that photonic multiqubit states can be
generated by letting a source emit photonic qubits in a
sequential manner [7]. The general sequential generation
of entangled multiqubit states in the realm of cavity QED
was systematically studied in Refs.[8, 9]. It is also shown
that the class of sequentially generated states is identical
to the matrix-product-state (MPS) which is very useful
in study of spin chains of condensed matter physics [10].
On the other hand, much progress has already been
made in the past years in studying quantum cloning ma-
chines, for reviews see, for example, Refs.[11, 12, 13].
And various quantum cloning machines have been im-
plemented experimently by polarization of photons [14,
15, 16, 17, 18],nuclear spins in Nuclear Magnetic Reso-
nance [19, 20], etc. However, these experiments are for
1 to 2 (one qubit input and two-qubit output) or 1 to 3
cloning machines. The more general case will be much
difficult. There are some schemes proposed for the gen-
eral quantum cloning machines which are not in a sequen-
tial manner, see for example, [21, 22]. Recently a 1 to M
sequential universal quantum cloning is proposed [23] by
using the cloning transformation presented in Ref.[24].
Since it is in a sequential procedure, potentially it re-
duces the difficult in implementing this quantum cloning
machine. However, as is well known the collective quan-
tum cloning machine (the N identical input states are
cloned collectively to M copies) is better than the quan-
tum cloning machine which can only deal with the in-
dividual input(only one input is copied to several copies
each time). We know that the general N to M cloning
transformation is also available in Refs.[24, 25]. Then a
natural question arise is that whether the general N to
M sequential cloning machine is possible. In this Letter,
we will present the general sequential universal quantum
cloning machine.
The 1 to M cloning transformations used in Ref.[23]
was proposed by Gisin and Massar in Ref.[24]. And the
N toM UQCM was also presented in Ref.[24]. However,
to use the method proposed in Refs.[8, 23] to find the se-
quential cloning machine, the input state |Φ〉⊗N should
be expanded in computational basis {|0〉, |1〉}. The ex-
plicit quantum cloning transformations with this kind of
input were proposed by Fan et al in Ref.[25]. In this Let-
ter, based on the result of Ref.[25], the general sequential
UQCM will be presented.
As presented in Refs.[8, 23], the sequential generation
of a multiqubit state is like the following. Let HA be
a D-dimensional Hilbert space which acts as the ancil-
lary system, and a single qubit (e.g., a time-bin qubit)
is in a two-dimensional Hilbert space HB. In every step
of the sequential generation of a multiqubit state, a uni-
tary time evolution will be acting on the joint system
HA ⊗HB. We assume that each qubit is initially in the
state |0〉 which is like a blank or an empty state and
will not be written out in the formulas. So the unitary
time evolution is written in the form of an isometry V :
HA → HA⊗HB, where V =
i,α,β V
α,β |α, i〉〈β|, each V i
is a D×D matrix, and the isometry condition takes the
i=0 V
i†V i = 1. By applying successively n oper-
ations of V (not necessarily the same) on an initial ancil-
lary state |φI〉 ∈ HA, we obtain |Ψ〉 = V [n]...V [2]V [1]|φI〉.
The generated n qubits are in general an entangled state,
but the last step qubit-ancilla interaction can be chosen
so as to decouple the final multiqubit entangled state
from the auxiliary system, so the sequentially generated
state is
|ψ〉 =
i1...in=0
〈φF |V [n]in ...V [1]i1 |φI〉|in, ..., i1〉, (1)
where |φF 〉 is the final state of the ancilla. This is the
MPS. It was proven that any MPS can be sequentially
generated [8].
http://arxiv.org/abs/0704.0323v2
Suppose there are N identical pure quantum states
|Φ〉⊗N = (x0|0〉+x1|1〉)⊗N need to be cloned toM copies,
where |x0|2 + |x1|2 = 1. We know that the input state
can be represented by a basis in symmetric subspace.
|Φ〉⊗N =
xN−m0 x
CmN |(N −m)0,m1〉, (2)
where |(N − m)0,m1〉 denotes the symmetric and nor-
malized state with (N −m) qubits in the state |0〉 and m
qubits in the state |1〉, and we have CmN = N !/(N−m)!m!
in standard notation. So if we find the quantum cloning
transformations for all states in symmetric subspace, we
can clone N pure states to M copies. The UQCM with
input in symmetric subspace can be written as [25],
|(N −m)0,m1〉 → |ΦmM 〉, (3)
where
|ΦmM 〉 =
βmj |(M −m− j)0, (m+ j)1〉 ⊗Rj ,(4)
βmj =
M−N−j
M−m−jC
(m+j)
/CN+1M+1, (5)
where Rj are the ancillary states of the cloning machine
and are orthogonal with each other for different j. For
a sequential quantum cloning machine in this Letter, we
choose a realization Rj ≡ |(M −N − j)1, j0〉 for the an-
cilla states. This UQCM is optimal in the sense that
the fidelity between single qubit output state reduced
density operator ρoutreduced and the single input |Φ〉 is op-
timal. The optimal fidelity is F = 〈Φ|ρoutreduced|Φ〉 =
(MN +M + N)/M(N + 2), see Refs.[11, 12, 13] for re-
views and the references therein. A realization of this
UQCM with photon stimulated emission can be found in
Ref.[22] which is not in a sequential manner. We next
show that this general N to M UQCM can be generated
through a sequential procedure.
The basic idea is to show that the final state of the
cloning, |ΦmM 〉 in (4), can be expressed in its MPS form.
As shown in Ref.[8], any MPS can be sequentially gen-
erated. We shall follow the method, for example, as in
Refs.[23, 26]. By Schmidt decomposition, we first ex-
press the quantum state |ΦmM 〉 as a bi-partite state across
1 : 2... cut,
|ΦmM 〉 = λ
1 |0〉|φ
[2...(2M−N)]
1 〉+ λ
2 |1〉|φ
[2...(2M−N)]
Γ[1]i1α1 λ
|i1〉|φ[2...(2M−N)]α1 〉, (6)
where Γ
α1 = δα1,1,Γ
α1 = δα1,2, and λ
α1 are eigen-
values of the first qubit reduced density operator, and
we find λ
∑M−m−1
k=−m β
M−1/C
M , λ
∑M−m−1
k=−m β
mk+1C
M−1/C
m+k+1
M . To correspond with
the MPS in (1), we can define V
[1]i1
α1 = Γ
[1]i1
α1 . Suc-
cessively by Schmidt decomposition, the quantum state
|ΦmM 〉 in (4) is divided into a bi-partite state with the first
n qubits as one part, and the rest as another part, where
1 < n ≤M − 1. We find
|ΦmM 〉 =
j+1|(n− j)0, j1〉|φ
[(n+1)...(2M−N)]
j+1 〉, (7)
when 1 < n ≤M−N+m,n′ = n; whenM−N+m< n ≤
M − 1, n′ =M −N +m, λ[n]j+1 are eigenvalues of the first
n qubits reduced density operator of |ΦmM 〉. According to
the results in Eqs.(4,5), we can obtain,
j+1 =
M−m−n
m(j+k)
Cm+kM−n
m+j+k
. (8)
And we also have
|φ[(n+1)...(2M−N)]j+1 〉 =
M−m−n
β2m(j+k) ×
(m+k)
m+j+k
|(M − n−m− k)0, (m+ k)1〉 ⊗Rj+k.
By induction and a concise formula, we have
|Φn...(2M−N)]j+1 〉
αn,in
[n]in
(j+1)αn
λ[n]αn |in〉|φ
[(n+1)...(2M−N)]
[n−1]
|0〉|φ[(n+1)...(2M−N)]j+1 〉
+|1〉|φ[(n+1)...(2M−N)]j+2 〉
, (9)
where we denote
(j+1)αn
= δ(j+1)αn
n−1/(λ
[n−1]
n), (10)
(j+1)αn
= δ(j+2)αn
n−1/(λ
[n−1]
n ). (11)
Still we define that
V [n]inαnαn−1 = Γ
[n]in
αn−1αn
λ[n]αn . (12)
It is thus in the MPS representation. We can further con-
sider other cases including the ancilla state of the cloning
machine represented as Rj (Note it is not the ancilla state
in the MPS representation). We can find that the out-
put state of the general UQCM can be expressed as MPS
as in form (1). So it can be created sequentially. The
explicit results are summarized in the appendix.
We have shown that the output states of the general
UQCM in (4,5) are MPS’s and thus can be generated
sequentially. The sequential matrices V [n] of course de-
pend on the input |(N−m)0,m1〉 which are W-like states
and are generally multiqubit entangled. For later con-
venience, we denote V (m) to express that it depends
on input state for different m. By a straightforward
method, the sequential cloning operation, i.e., the iso-
metrices, depending on different input may take the form
m |(N − m)0,m〉〈(N − m)0,m1| ⊗ V (m). However,
this operation may need a single global unitary opera-
tor which involves N -qubit entangled states except for
m = 0,m = N . This contradicts with our aim that each
operation should be divided into sequential unitary oper-
ators in a quDit (quantum state in D-dimensional space)
times qubit system. Here we can use a scheme like the
following: the ancillary state interacts with each qubit
according to the (N + 1) × D-dimensional isometrices
CmN |0〉〈0|⊗N−m⊗|1〉〈1|⊗m⊗V (m) sequentially,
here a whole normalization factor is omitted. We know
that the operation |0〉〈0|⊗N−m ⊗ |1〉〈1|⊗m acts on each
qubit individually. Thus this scheme reduces the com-
plexity of the operation. This finishes our general se-
quential UQCM for the case of qubit. In case N = 1,
we recover the result of Ref.[23] for 1 to M cloning.
We should remark that similar as the case of sequen-
tial 1 to M UQCM in Ref.[23], for the general sequential
UQCM, the minimal dimension D of the ancillary state
grows linearly at most with M −N/2 + 1 for even N or
M − (N − 1)/2 for odd N .
Next we will consider a more general case that the se-
quential cloning machine is about the quantum state in d-
dimensional Hilbert space. We will use the d-dimensional
UQCM proposed by Fan et al in Ref. [25]. This UQCM
is a generalization of the cloning machine proposed in
Ref.[24] and we can use this UQCM to study its sequen-
tial form for d-dimensional case.
An arbitrary d-dimensional pure state takes the form
|Φ〉 =
i=0 xi|i〉 with
i=0 |xi|2 = 1. N identical pure
states can be expanded in terms of state in symmet-
ric subspace |Φ〉⊗N =
m1!...md!
xm10 ...x
d−1|~m〉,
where |~m〉 ≡ |m1, ...,md〉 is a symmetric state with mi
states of |i − 1〉, and also mi should satisfy a relation
i=1mi = N . The cloning transformations with states
in symmetric subspace can be written as
|~m〉 → |Φ~mM 〉 =
|~m+~j〉 ⊗ |~j〉, (13)
i=1 C
mi+ji
CM−NM+d−1
where ~j should satisfy
i ji = M − N . This cloning
machine is optimal and the corresponding fidelity of a
single quantum state between input and output is F =
(N(d+M) +M −N) /(d+N)M .
As for qubit system, we next show that the output
states for all symmetric states input can be expressed
as the sequential form. We consider the case 1 < n ≤
M − 1, and the state |Φ~mM 〉 is a bipartite state across
1...n : (n+ 1)... cut,
|Φ~mM 〉 =
|~j〉|φ[(n+1)...(M+1)]
〉 (15)
where
~m(~j−~m+~k)
i=1 C
ji+ki
, (16)
|φ[(n+1)...(M+1)]
~m(~j−~m+~k)
i=1 C
ji+ki
|~k〉|~j − ~m+ ~k〉/λ[n]
. (17)
By the same procedure as that of qubit case, we can
obtain the following
|φ[n...(M+1)]
[n]in
λ[n]αn |in〉|φ
(n+1)...(M+1)]
〉. (18)
Then we have
[n]in
= δαn(~j+~ein+1)
jin+1 + 1
[n−1]
. (19)
Still we can define V
[n]in
αnαn−1 = Γ
[n]in
αn−1αnλ
αn , and thus we
can find that each state |Φ~mM 〉 is a MPS and thus can be
sequentially generated. The detailed result of this part
will be presented elsewhere [27].
In conclusion, we show that the generalN toM univer-
sal quantum cloning machine can be implemented by a se-
quential manner. Since the sequential generation of mul-
tipartite state can be implemented in various physical se-
tups such as microwave and optical cavity QED, trapped
ions and quantum dots etc. This general sequential quan-
tum cloning machine may be implemented much easier
than the single global implementation scheme. This re-
duces dramatically the complexity in implementing the
general UQCM. We also show that for d-dimensional
quantum state, the sequential UQCM is also available.
Besides the universal cloning machine, the 1 toM phase-
covariant quantum cloning machine can also be sequen-
tially implemented. It will be interesting to consider sim-
ilarly the generalN toM phase-covariant cloning and the
economic phase-covariant cloning. The sequential asym-
metric quantum cloning machine may also be an inter-
esting topic.
Acknowledgements: HF was supported by ”Bairen”
program, NSFC and ”973” program (2006CB921107).
Appendix.–The explicit form of matrices V are pre-
sented as:
V [n]0αnαn−1 = δαnαn−1 ×
∑M−m−n
k=−m X
m+αn−1−1+k
∑M−m−n+1
k=−m X
M−n+1
m+αn−1−1+k
V [n]0αnαn−1 = δαnαn−1+1 ×
∑M−m−n
k=−m X
m+αn−1+k
∑M−m−n+1
k=−m X
M−n+1
m+αn−1−1+k
where notations X = β2m(αn−1−1+k), X
′ = β2m(αn−1+k)
are used. For case 1 < n ≤ M − N + m,αn−1 =
1, ..., n;αn = 1, ..., (n+1), and for caseM−N+m < n ≤
M − 1, αn−1, αn = 1, ..., (M −N +m+1). We can check
that the above defined V satisfies the isometry condition
V [n]in
V [n]in = 1. Similarly we have
V [M ]0αMαM−1 = δαMαM−1 ×
M−1−1−m)
M−1−1−m)
M−1−1
M−1−m)
V [M ]1αMαM−1 = δαM (αM−1+1) ×
M−1−m)
M−1−1−m)
M−1−1
M−1−m)
where 0 ≤ m ≤ N −m,αM−1, αM = 1, 2, ..., (M − N +
m+ 1).
For case concerning about ancilla state of the UQCM,
assume 1 ≤ l ≤M −N , we have
V [M+l]0αM+lαM+l−1 = δαM+l(αM+l−1−1) ×
αM+l−1 −m− 1
M −N − l + 1
V [M+l]1αM+lαM+l−1 = δαM+lαM+l−1 ×
M −N − l − αM+l−1 +m+ 1
M −N − l + 1
(1) For (m+ 1) ≤ αM+l ≤ (M −N +m− l+ 1),
(m+ 2) ≤ αM+l−1 ≤ (M −N +m− l+ 2),
[M+l]0
αM+lαM+l−1 = δαM+l(αM+l−1−1)
αM+l−1−m−1
M−N−l+1 .
For αM+l = (M −N +m− l + 2), 1 ≤ αM+l−1 ≤
(M −N +m+ 1), V [M+l]0αM+lαM+l−1 = 0. Otherwise
[M+l]0
αM+lαM+l−1 = δαM+lαM+l−1
(2) For (m+ 1) ≤ αM+l, αM+l−1 ≤
(M −N +m− l + 1), V [M+l]1αM+lαM+l−1 =
δαM+lαM+l−1
M−N−l−αM+l−1+m+2
M−N−l+1 . For αM+l =
(M −N +m− l + 2), 1 ≤ αM+l−1 ≤ (M −N +m+ 1),
[M+l]0
αM+lαM+l−1 = 0. Otherwise V
[M+l]0
αM+lαM+l−1 =
δαM+lαM+l−1
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