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PHYSICAL REVIEW A, VOLUME 61, 062310
Realizing probabilistic identification and cloning of quantum states
via universal quantum logic gates
Chuan-Wei Zhang, Zi-Yang Wang, Chuan-Feng Li,* and Guang-Can Guo†
Laboratory of Quantum Communication and Quantum Computation and Department of Physics,
University of Science and Technology of China, Hefei 230026, People’s Republic of China
共Received 18 June 1999; revised manuscript received 7 September 1999; published 16 May 2000兲
Probabilistic quantum cloning and identifying machines can be constructed via unitary-reduction processes
关Duan and Guo, Phys. Rev. Lett. 80, 4999 共1998兲兴. Given the cloning 共identifying兲 probabilities, we derive an
explicit representation of the unitary evolution and corresponding Hamiltonian to realize probabilistic cloning
共identification兲. The logic networks are obtained by decomposing the unitary representation into universal
quantum logic operations. The robustness of the networks is also discussed. Our method is suitable for a
k-partite system, such as quantum computer, and may be generalized to general state-dependent cloning and
identification.
PACS number共s兲: 03.67.⫺a, 03.65.Bz, 89.70.⫹c
I. INTRODUCTION
Quantum no-cloning theorem 关1兴, which asserts that unknown pure states cannot be reproduced exactly by any
physical means, is one of the most astonishing features of
quantum mechanics. Wootters and Zurek 关1兴 have shown
that the cloning machine violates the quantum superposition
principle. Yuen and D’Ariano 关2,3兴 showed that a violation
of unitarity makes the cloning of two nonorthogonal states
impossible. Barnum et al. 关4兴 have extended such results to
the case of mixed states and shown that two noncommuting
mixed states cannot be broadcast. Furthermore, Koashi and
Imoto 关5兴 generalized the standard no-cloning theorem to the
entangled states. The similar problem exists in the situation
of identifying an arbitrary unknown state 关6兴. Since perfect
quantum cloning and identification are impossible, the inaccurate cloning and identification of quantum states have attracted much attention with the development of quantum information theory 关7兴.
The inaccurate cloning and identification may be divided
into two main categories: deterministic and probabilistic.
The deterministic quantum cloning machine generates approximate copies and further we get two subcategories: universal and state-dependent. Universal quantum cloning machines, first addressed by Bužek and Hillery 关8兴, act on any
unknown quantum state and produce approximate copies
equally well. The Bužek-Hillery cloning machine has been
optimized and generalized in Refs. 关9–13兴. Massar and
Popescu 关14兴 and Derka et al. 关15兴 have also considered the
problem of universal states estimation, given M independent
realizations. The deterministic state-dependent cloning machine, proposed originally by Hillery and Bužek 关16兴, is designed to generate approximate clones of states belonging to
a finite set. Optimal results for two-state cloning have been
obtained by Bruß et al. 关10兴 and Chelfes and Barnett 关17兴.
Deterministic exact cloning violates the no-cloning theorem,
thus faithful cloning must be probabilistic. The probabilistic
*Electronic address: [email protected]
†
Electronic address: [email protected]
1050-2947/2000/61共6兲/062310共9兲/$15.00
cloning machine was first considered by Duan and Guo
关18,19兴 using a general unitary-reduction operation with a
postselection of the measurement results. They showed that a
set of nonorthogonal but linear-independent pure states can
be faithfully cloned with optimal success probability. Recently, Chelfes and Barnett 关17兴 presented the idea of hybrid
cloning, which interpolates between deterministic and probabilistic cloning of a two-state system. In addition, we 关20兴
have provided general identifying strategies for statedependent system.
Clearly, it is important to obtain a physical means to carry
out this cloning and identification. Quantum networks for
universal cloning have been proposed by Bužek et al. 关21兴.
Chelfes and Barnett 关17兴 have constructed the cloning machine in a two-state system.
In this paper we provide a method to realize probabilistic
identification and cloning for an n-state system. The method
is also applicable to general cloning and identification of
state-dependent systems. As any unitary evolution can be
accomplished via universal quantum logic gates 关22,23兴, the
key to realizing probabilistic identification and cloning is to
obtain the unitary representation or the Hamiltonian of the
evolution in the machines. We derive the explicit unitary
representation and the Hamiltonian which are determined by
the probabilities of cloning or identification. Furthermore, we
obtain the logic networks of probabilistic cloning and identification by decomposing the unitary representation into universal quantum logic operations. The robustness of the networks is also discussed.
The plan of the paper is the following. In Sec. II we
derive the unitary representation matrix and Hamiltonian for
quantum identification provided with one copy and generalize this method to M →N quantum cloning and identification
with M initial copies. For the special case of a quantum
computer, we should be concerned with the system which
includes k partites, each of them being an arbitrary two-state
quantum system 共qubit兲. The identification and cloning in
such k-partite quantum systems have more prospective applications, which include normal qubits and multipartite entangled states. In Sec. III, we provide the networks of probabilistic cloning and identification of k-partite systems and
discuss their stability properties.
61 062310-1
©2000 The American Physical Society
ZHANG, WANG, LI, AND GUO
PHYSICAL REVIEW A 61 062310
II. UNITARY EVOLUTIONS AND HAMILTONIANS
FOR IDENTIFICATION AND CLONE
Any operation in quantum mechanics can be represented
by a unitary evolution together with a measurement. Considering the states secretly chosen from the set S⫽ 兵 兩 ␺ i 典 ,i
⫽1,2, . . . ,n 其 which span an n-dimensional Hilbert space,
Duan and Guo 关19兴 have shown that these states can be
probabilistically cloned by a general unitary-reduction operation if and only if 兩 ␺ 1 典 , 兩 ␺ 2 典 , . . . , 兩 ␺ n 典 are linearindependent. By introducing a probe P in an n P -dimensional
Hilbert space, where n P ⭓n⫹1, the unitary evolution Û in
the M →N probabilistic cloning machine can be written as
follows:
Û 兩 ␺ i 典 丢 M 兩 ␸ 1 典 丢 (N⫺M ) 兩 P 0 典 ⫽ 冑␥ i 兩 ␺ i 典 丢 N 兩 P i 典
⫹
兺j C i j 兩 ␣ j 典 兩 ␸ 1 典
丢 (N⫺M )
兩 P 0典 ,
fication and cloning. When the probe states 兩 P i 典 are orthogonal to each other, we can identify and clone the input states
simultaneously. When 兩 P i 典 are the same for all the to-becloned states, we obtain no information about the input states
and the probabilities of successful clone approach the maximum. For a normal situation interpolating between the cloning and identification, where states 兩 P i 典 ⫽ 兩 P j 典 exist, we can
identify them with no-zero probability and get some information about the input, which means the cloning probabilities must decrease.
Now that the existence of probabilistic cloning and identifying machines has been demonstrated, the next step is to
determine the representations of the unitary evolution Û for
the cloning and identifying machines with the given probability matrix ⌫.
To simplify the deduction, we start with probabilistic
identification of one initial copy. A unitary evolution Û is
utilized to identify 兩 ␺ i 典 ,
共2.1兲
where 兩 P 0 典 and 兩 P i 典 are normalized states of the probe system 共not generally orthogonal, but each of 兩 P i 典 is orthogonal
to 兩 P 0 典 ), and 兩 ␺ i 典 丢 M ⫽ 兩 ␺ i 典 1 兩 ␺ i 典 2 ••• 兩 ␺ i 典 M ( 兩 ␺ i 典 k is the kth
copy of state 兩 ␺ i 典 ). The n-dimensional Hilbert spaces
spanned by state sets 兵 兩 ␺ i 典 其 , 兵 兩 ␺ i 典 丢 M 其 , or 兵 兩 ␺ i 典 丢 N 兩 P i 典 其 are
denoted by H, H M , and H N , respectively, and 兵 兩 ␸ i 典 其 ,
兵 兩 ␣ i 典 其 , and 兵 兩 ␤ i 典 其 are the orthogonal bases of each space.
The probe P is measured after the evolution. With probability ␥ i , the cloning attempt succeeds and the output state is
兩 ␺ i 典 丢 N if and only if the measurement result of the probe is
兩 P i 典 . The n⫻n inter-inner products of Eq. 共2.1兲 yield the
matrix equation
X
(M )
†
⫽ 冑⌫X (N)
P 冑⌫⫹CC ,
共2.2兲
where the n⫻n matrices are C⫽ 关 C i j 兴 , X (M ) ⫽ 关 具 ␺ i 兩 ␺ j 典 M 兴 ,
N
and X (N)
P ⫽ 关 具 ␺ i 兩 ␺ j 典 具 P i 兩 P j 典 兴 . The diagonal efficiency matrix ⌫ is defined as ⌫⫽diag( ␥ 1 , ␥ 2 , . . . , ␥ n ). Since CC †
⭓0 (CC † is positive semidefinite兲, Eq. 共2.2兲 yields
X (M ) ⫺ 冑
⌫X (N)
P
冑⌫⭓0.
Û 兩 ␺ i 典 兩 P 0 典 ⫽ 冑␥ i 兩 ␸ i 典 兩 P 1 典 ⫹
共2.6兲
X⫽⌫⫹CC † .
Denoting matrix A⫽ 关 具 ␸ i 兩 ␺ j 典 兴 n⫻n , we get
共2.7兲
X⫽A † A.
Obviously
A
is
reversible.
Since
兩 ␺ i典兩 P 0典
n
⫽ 兺 m⫽1
兩 ␸ m 典 兩 P 0 典具 ␸ m 兩 ␺ i 典 , Eq. 共2.5兲 can be rewritten as
Û 共 兩 ␸ 1 典 兩 P 0 典 , . . . , 兩 ␸ n 典 兩 P 0 典 )
⫽ 共 兩 ␸ 1 典 兩 P 1 典 , . . . , 兩 ␸ n 典 兩 P 1 典 ) 冑⌫A ⫺1
⫹ 共 兩 ␸ 1 典 兩 P 0 典 , . . . , 兩 ␸ n 典 兩 P 0 典 )C † A ⫺1 .
On the orthogonal bases 兵 兩 ␸ i 典 兩 P j 典 ,i⫽1,2, . . . ,n, j⫽0,1其 in
Hilbert space HA P ⫽H 丢 H P , Û can be represented as
␥ 1⫹ ␥ 2
1⫺ 兩 具 ␺ 1 兩 ␺ 2 典 兩 M
1⫺ 兩 具 ␺ 1 兩 ␺ 2 典 兩 M
⭐ max
⫽
.
N
2
1⫺ 兩 具 ␺ 1 兩 ␺ 2 典 兩 N
兵 兩 P i 典 其 1⫺ 兩 具 ␺ 1 兩 ␺ 2 典 兩 兩 具 P 1 兩 P 2 典 兩
共2.4兲
In the limit as N→⬁, the M →N probabilistic clone has a
close connection with the problem of identification of a set
of states. That is, Eq. 共2.1兲 is applicable to describe the
probabilistic identification evolution, since 兵 兩 ␺ i 典 丢 ⬁ , i
⫽1,2, . . . ,n 其 are the orthogonal bases of n -dimension Hilbert space. Inequality 共2.3兲 turns into X (M ) ⫺⌫⭓0 , and inequality 共2.4兲 results in ( ␥ 1 ⫹ ␥ 2 ) /2⭐1⫺円具 ␺ 1 兩 ␺ 2 典 円M , which
is the maximum identification probability when n⫽2, with
M initial copies. In fact, there is a trade-off between identi-
共2.5兲
where 兩 P 0 典 and 兩 P 1 典 are the orthogonal bases of the probe
system. If a postselective measurement of probe P results in
兩 P 0 典 , the identification fails. Otherwise we make a further
measurement of the to-be-identified system and if 兩 ␸ k 典 is
detected, the input state should be identified as 兩 ␺ k 典 . The
inter-inner products of Eq. 共2.5兲 yield the matrix equation
共2.3兲
This inequality determines the optimal cloning efficiencies.
For example, when n⫽2, we get 关17兴
兺j C i j 兩 ␸ j 典 兩 P 0 典 ,
U⫽
冉冑
C † A ⫺1
M
⌫A ⫺1
N
冊
,
共2.8兲
where M ,N are n⫻n matrices. In Appendix A, we derive the
expressions of the four submatrices in Eq. 共2.8兲 and get
共2.9兲
U⫽ṼSṼ † ,
where Ṽ⫽diag(V,V),
062310-2
S⫽
冉
F
⫺E
E
F
冊
REALIZING PROBABILISTIC IDENTIFICATION AND . . .
PHYSICAL REVIEW A 61 062310
with
E⫽diag( 冑m 1 , . . . , 冑m n ),
and
F
⫽diag( 冑1⫺m 1 , . . . , 冑1⫺m n ). V and m i are determined by
I n ⫺C † X ⫺1 C⫽V diag共 m 1 , . . . ,m n 兲 V † .
共2.16兲
共2.11兲
The unitary representation U in Eq. 共2.9兲 can be diagnolized
by interchanging the columns and rows of the matrix 共refer
to Appendix B兲 as
U⫽O diag共 e i ␪ 1 ,e ⫺i ␪ 1 , . . . ,e i ␪ n ,e ⫺i ␪ n 兲 O † ,
共2.12兲
where O is a unitary matrix and ␪ j , j⫽1, . . . ,n are determined by
冉
e i ␪ j ⫽ 冑1⫺m j ⫹i 冑m j 0⬍ ␪ j ⭐
冊
␲
.
2
共2.13兲
Comparing Eq. 共2.12兲 with Eq. 共2.11兲, the eigenvalues E ⫾k
of the Hamiltonians should be
E ⫾k ⫽⫿
␪ k ប 2 ␲ N ⫾k ប
⫹
,
⌬t
⌬t
共2.14兲
where N ⫾k are arbitrary integers. H can be represented as
H⫽O diag共 E 1 ,E ⫺1 , . . . ,E n ,E ⫺n 兲 O † .
兺j C i j 兩 ␣ j 典 兩 P 0 典 ,
共2.10兲
Since the coefficient matrix C can be deduced from Eq.
共2.6兲, the parameters V and m i ,i⫽1, . . . ,n are determined
by the probabilities ␥ i ,i⫽1, . . . ,n. Hence, the representation U is obtained from the given probabilities. The expressions of E and F require 0⭐m i ⭐1, i⫽1,2, . . . ,n. In Appendix A we show a more strict limitation 0⬍m i ⭐1.
Equation 共2.9兲 is fundamental in obtaining the Hamiltonian and realizing a quantum probabilistic identifying machine. Based on this representation, we use the following
method to derive the corresponding Hamiltonian. We adopt
the approach in the quantum computation literature of assuming that a constant Hamiltonian H acts during a short
time interval ⌬t. Here we only consider evolution from t to
t⫹⌬t. The time interval is then related to the strength of
couplings in H, which are of the order ប/䉭t. Under this
condition we deduce H with
U⫽e ⫺iH⌬t/ប .
Û 兩 ␺ i 典 丢 M 兩 P 0 典 ⫽ 冑␥ i 兩 ˜␸ i 典 兩 P 1 典 ⫹
共2.15兲
Now we have successfully derived the diagonalized representation and Hamiltonian of the evolution described by
Eq. 共2.5兲, which are essential to realizing the identification
via universal quantum logic gates. We will extend the result
to M-initial-copy identification and M →N cloning in a similar way. In the situation of probabilistic identification with M
initial copies, we generalize Eq. 共2.5兲 to
where 兵 兩 ˜␸ i 典 ,i⫽1,2, . . . ,n 其 is a set of orthogonal states in
n M -dimensional Hilbert space H 丢 M . With the method mentioned above, we can prove that U has the same representation as that in Eq. 共2.9兲 on different orthogonal bases
兵兵 兩 ␣ i 典 兩 P 0 典 其 , 兵 兩 ˜␸ j 典 兩 P 1 典 其 ,i, j⫽1,2, . . . ,n 其 , where the definitions of V, m i , E, and F are also the same as that of Eq.
共2.9兲. However, they are different in fact because the determining condition Eq. 共2.10兲 turns into
I n ⫺C † 共 X (M ) 兲 ⫺1 C⫽V diag共 m 1 , . . . ,m n 兲 V † . 共2.17兲
As to M →N probabilistic cloning, the unitary evolution
equation is Eq. 共2.1兲. Under the same condition of Eq. 共2.17兲
but different orthogonal bases 兵兵 兩 ␣ i 典 兩 ␸ 1 典 丢 (N⫺M ) 兩 P 0 典 其 ,
兵 兩 ␤ i 典 其 ,i⫽1,2, . . . ,n 其 , U may still be represented as that in
Eq. 共2.9兲.
We notice that in different situations for probabilistic
identification and cloning, the unitary representation and
Hamiltonian are of the same form. However, since the determining conditions are different, the values of V, m i , ␪ i , and
E ⫾k are different as well. The unitary representations and
Hamiltonians of different identifications and clones are based
on different bases. All these show that these Û or Ĥ are
actually different.
In this section, we choose appropriate orthogonal bases
and represent the 2n N -dimensional unitary evolution as Eq.
共2.9兲 in a 2n-dimensional subspace. In the subspace orthogonal to such 2n-dimensional subspace, U⫽I.
III. NETWORKS OF PROBABILISTIC CLONING
AND IDENTIFICATION IN A k-PARTITE SYSTEM
So far we have derived the explicit representation of the
unitary evolutions for quantum probabilistic cloning and
identification. The next problem is how to realize these cloning and identifying transformations by physical means. The
fundamental unit of quantum information transmission is the
quantum bit 共qubit兲, i.e., a two-state quantum system, which
is capable of existing in a superposition of Boolean states
and of being entangled with one another. Just as classical bit
strings can represent the discrete states of arbitrary finite dimensionality, a string of k qubits can be used to represent
quantum states in any 2 k -dimensional Hilbert space. Obviously there exist 2 k linear-independent states in such a
k-partite system. In this section we apply the method provided in Sec. II to this special system and realize probabilistic cloning and identification of an arbitrary state secretly
chosen from a linear-independent state set via universal logic
gates. This solution may be essential to the realization of a
quantum computer.
062310-3
ZHANG, WANG, LI, AND GUO
PHYSICAL REVIEW A 61 062310
A. Some basic ideas and notations
Quantum logic gates have the same number of input and
output qubits and a k -qubit gate carries out a unitary operation of the group U(2 k ), i.e., a generalized rotation in a
2 k -dimensional Hilbert space. The formalism we use for
quantum computation, which is called a quantum gate array,
was introduced by Deutsch 关22兴, who showed that a simple
generalization of the Toffoli gate is sufficient as a universal
gate for quantum computation. We introduce this gate as
follows.
For any 2⫻2 unitary matrix
U⫽
冉
u 00
u 01
u 10 u 11
冊
and m苸 兵 0,1,2, . . . 其 , the matrix corresponding to the (m
⫹1) -bit operation is ⌳ m (U)⫽diag(I 2 m ,U), where the bases
are lexicographically ordered, i.e., 兩000典,兩001典, . . . ,兩111典. For
a general U, ⌳ m (U) can be regarded as a generalization of
the Toffoli gate, which, on the m⫹1 input bits, applies U to
the (m⫹1)th bit if and only if the other m bits are all on
state 兩1典. Barenco et al. 关23兴 have further demonstrated that
arbitrary ⌳ m (U) can be executed by the combination of a set
of one-bit quantum gates 关 U(2) 兴 and two-bit ControlledNOT 共C-NOT兲 gate 关that maps Boolean values (x,y) to
(x,x 丣 y)].
We first introduce a lemma which shows how to decompose a general unitary matrix to the product of the matrices
⌳ m (U).
Lemma 1. Any unitary matrix U⫽ 关 u i j 兴 n⫻n can be decomposed into
冉兿
n⫺1
U⫽
n
冊冉 兿 冊
n
兿
t⫽1 l⫽t⫹1
A tl
k⫽1
共3.1兲
Bk ,
†
†
where A tl ⫽ 关 a (tl)
i j 兴 n⫻n ⫽ P t,n⫺1 P l,n Â(û tl ) P l,n P t,n⫺1 , B k
⫽ P k,n B̂„exp(i␣k)…P k,n , Â(û tl )⫽diag(1,1, . . . ,1,û tl ), û tl is a
2⫻2 unitary matrix, B̂„exp(i␣k)…⫽diag„1,1, . . . ,1,exp(i␣k)…,
P i j left-multiplying a matrix interchanges the ith and jth row
of the matrix, and similarly P †i j right-multiplying a matrix
interchanges columns. On the lexicographically ordered orthogonal bases, the representations of operators P i j and P †i j
are identical. When n⫽2 m⫹1 , obviously Â(û tl )⫽⌳ m (û tl ),
and B̂„exp(i␣k)…⫽⌳ m (diag„1, exp(i␣k)…).
T̃⫽
冉
The meaning of this decomposition in mathematics is that
some unitary matrices, namely A †tl , left-multiply U to transfer it to a upper triangular matrix. Since U is unitary, it
should be diagonal and can be decomposed into the product
of matrices B k . Thus unitary matrix U is decomposed into
the form of Eq. 共3.1兲.
We show how to transfer the operation P i j to operation
⌳ m ( ␴ x ) via C-NOT operations. In fact P i j ⫽ 兩 兵 x ik 其 典具 兵 x kj 其
兩 ⫹ 兩 兵 x kj 其 典具 兵 x ik 其 兩 ⫹ 兺 l⫽i, j 兩 兵 x lk 其 典具 兵 x lk 其 兩 ,
where
兩 兵 x tk 其 典
t
t
t
t
⫽ 兩 x 1 , x 2 , . . . , x m⫹1 典 with x k 苸 兵 0,1其 , k⫽1,2, . . . ,m⫹1.
These C-NOT operations transfer the subspace spanned by
兩 兵 x ik 其 典 and 兩 兵 x kj 其 典 to the subspace spanned by 兩11¯11典 and
兩 11•••10典 . For i⫽ j, there must exist k satisfying x ik ⫽x kj .
Denote the minimum value of k by k 0 and assume x ki ⫽1,
0
x kj ⫽0. For an integer s, k 0 ⬍s⭐n, if x si ⫽x sj , we execute
0
C-NOT operation 共the k 0 th bit controls the sth bit兲. Then for
1⭐h⭐n, x ih ⫽x hj ⫽0 , we execute ␴ hx on the hth bit. At last
we interchange the input sequence of the k 0 th bit and the
(m⫹1)th bit.
With the lemma above, a unitary evolution can be expressed as the product of some controlled unitary operations.
The representations of the input states are another important problem. As to two linear-independent states 兩 ␺ 1 典 , 兩 ␺ 2 典
of one qubit system, we set them symmetric,
兩 ␺ 1,2共 ␪ 兲 典 ⫽ 兩 ␺ ⫾ 共 ␪ 兲 典 ⫽cos ␪ 兩 0 典 A ⫾sin ␪ 兩 1 典 A ,
where 0⭐ ␪ ⭐ ␲ /4 and A represents the system for identification and cloning. 关This simplification is reasonable because
arbitrary states 兩 ␺ 1 典 , 兩 ␺ 2 典 can be transformed to Eq. 共3.2兲 via
unitary rotation.兴
In the case of a two-partite system, the orthogonal bases
are 兵 兩 ␾ i 典 1,2其 ⫽ 兵 兩 00典 1,2 , 兩 01典 1,2 , 兩 10典 1,2 , 兩 11典 1,2其 and the input
states are 兵 兩 ␺ i 典 1,2 , i⫽1,2,3,4其 , each of which may be expressed as 兩 ␺ i 典 1,2⫽ 兺 4j⫽1 t i j 兩 ␾ j 典 1,2 with 兺 4j⫽1 兩 t i j 兩 2 ⫽1. However, they cannot be converted to symmetric forms as those
in Eq. 共3.2兲. Define T⫽ 关 t i j 兴 4⫻4 ; the determinant of T should
be nonzero since 兩 ␺ i 典 1,2 are linear-independent.
Lemma 2. For any four states 兵 兩 ␺ i 典 1,2 , i⫽1,2,3,4其 in Hilbert space H 丢 2 共two-partite system兲, there exists a unitary
operator U 0 ,
U 0 共 兩 ␺ 1 典 1,2 , 兩 ␺ 2 典 1,2 , 兩 ␺ 3 典 1,2 , 兩 ␺ 4 典 1,2)
⫽ 共 兩 ␾ 1 典 1,2 , 兩 ␾ 2 典 1,2 , 兩 ␾ 3 典 1,2 , 兩 ␾ 4 典 1,2)T̃,
where
1
e i ␮ 2 cos ␪ (1)
2
(1)
(2)
e i ␮ 3 cos ␪ (1)
3 cos ␪ 3
0
sin ␪ (1)
2
0
0
0
0
共3.2兲
(1)
(2)
(3)
e i ␮ 4 cos ␪ (1)
4 cos ␪ 4 cos ␪ 4
(1)
(2)
e i ␮ 3 cos ␪ (1)
3 sin ␪ 3
(2)
(2)
(3)
e i ␮ 4 cos ␪ (1)
4 cos ␪ 4 sin ␪ 4
sin ␪ (1)
3
(2)
e i ␮ 4 cos ␪ (1)
4 sin ␪ 4
0
sin ␪ (1)
4
with 0⭐ ␪ 共i 1 兲 ⬍ ␲ , 0⭐ ␪ (i j) ⬍2 ␲ , 0⭐ ␮ 共i j 兲 ⬍2 ␲ .
062310-4
(2)
(3)
冊
共3.3兲
REALIZING PROBABILISTIC IDENTIFICATION AND . . .
PHYSICAL REVIEW A 61 062310
eters in matrix T̃ in lemma 2. We generalize the D-gate to
two-partite system, which acts as
FIG. 1. The networks of of D gate 关17兴. 䊉 and 丣 denote the
controller and target bit of a C-NOT operation, respectively.
If 兵 兩 ␺ i 典 1,2其 are linear-independent, then ␪ (1)
i ⬎0. The unitarity of U 0 yields
T † T⫽T̃ † T̃.
共3.4兲
Lemma 2 can be generalized to a k-partite system. According to this lemma, we may concentrate on probabilistic
cloning and identification of states 兩 ˜␺ i 典 1,2⫽U 0 兩 ␺ i 典 1,2 , i
⫽1,2,3,4.
All unitary representations have physical meaning only
when the orthogonal bases have been set. To represent the
bases 兵 兩 ␣ i 典 其 and 兵 兩 ␤ i 典 其 , we adopt the distinguishability
transfer gate (D-gate兲 operation introduced in Ref. 关17兴 and
generalize it to a k-partite system. This operation compresses
all the information of the M input copies into one qubit and
acts as follows:
D 共 ␪ 1 , ␪ 2 兲兩 ␺ ⫾共 ␪ 1 兲 典 兩 ␺ ⫾共 ␪ 2 兲 典 ⫽ 兩 ␺ ⫾共 ␪ 3 兲 典 兩 0 典 .
共3.5兲
The unitarity of operation D( ␪ 1 , ␪ 2 ) requires
cos 2 ␪ 3 ⫽cos 2 ␪ 1 cos 2 ␪ 2 .
共3.6兲
This condition, together with 0⭐ ␪ j ⭐ ␲ /4, suffices to determine ␪ 3 uniquely. Since D( ␪ 1 , ␪ 2 ) is Hermitian 关17兴, it can
also transfer state 兩 ␺ ⫾ ( ␪ 3 ) 典 兩 0 典 back to 兩 ␺ ⫾ ( ␪ 1 ) 典 兩 ␺ ⫾ ( ␪ 2 ) 典 .
This accomplishes the process of information decompression. Both the compression and decompression will be useful
in implementing the cloning and identification.
The D-gate operation can be used as an element in a network for M →N
cloning. Define D K ⫽D 1 ( ␪ 1 ,
␪ K⫺1 )D 2 ( ␪ 1 , ␪ K⫺2 )•••D K⫺1 ( ␪ 1 , ␪ 1 ), where the operation
D j ( ␪ 1 , ␪ K⫺ j ) compresses the information of partites j, j⫹1
to partite j, and angles ␪ j are uniquely determined by
cos 2␪ j⫹1⫽cos 2␪1cos 2␪ j (0⭐ ␪ j ⭐ ␲ /4). D K acts as
D K 兩 ␺ ⫾ 共 ␪ 1 兲 典 丢 K ⫽ 兩 ␺ ⫾ 共 ␪ K 兲 典 1 兩 0 典 丢 (K⫺1) .
共3.7兲
The operations D K , by pairwise interactions, compress all
the information to partite 1. D( ␪ 1 , ␪ 2 ) may be decomposed
into universal operations 关17兴, i.e., local unitary 共LU兲 operations on a single qubit and C-NOT operations. Here we directly use the results obtained by Chelfes and Barnett 关17兴
and illustrate the D gate in Fig. 1.
Operation D K that is suitable for a one-partite system can
be generalized to a k-partite system. In the previous part of
this subsection we have discussed the special representations
of input states in a k-partite system and we will adopt them
below. Consider a two-partite system. Define ˜␪
⫽ 兵 ␪ (i j) , ␮ (i j) , 2⭐i⭐4, 1⭐ j⭐i⫺1 其 to represent the param-
D 共 ˜␪ ,˜␰ 兲 兩 ˜␺ i 共 ˜␪ 兲 典 A 兩 ˜␺ i 共˜␰ 兲 典 B ⫽ 兩 ˜␺ i 共 ˜␩ 兲 典 A 兩 00典 B ,
共3.8兲
where ˜␰ and ˜␩ have a definition similar to ˜␪ . The unitarity of
operation D(˜␪ ,˜␰ ) yields
X 共 ˜␪ ,˜␰ 兲 ⫽T̃ † 共 ˜␩ 兲 T̃ 共 ˜␩ 兲 ,
共3.9兲
where X(˜␪ ,˜␰ )⫽ 关 A 具 ˜␺ i (˜␪ ) 兩 ˜␺ j (˜␪ ) 典 AB 具 ˜␺ i (˜␰ ) 兩 ˜␺ j (˜␰ ) 典 B 兴 4⫻4 . The
upper triangular representation of T̃( ˜␩ ) determines ˜␩
uniquely through Eq. 共3.9兲.
To obtain an explicit expression for the operation D(˜␪ ,˜␰ ),
we must specify how it transforms states in the subspace
orthogonal to that spanned by 兩 ˜␺ i (˜␪ ) 典 A 兩 ˜␺ i (˜␰ ) 典 B . Equation
共3.8兲 may be rewritten as
D ⫺1 共 ˜␪ ,˜␰ 兲 兵 ␾ i 典 A 兩 00典 B , i⫽1,2,3,4其 ⫽ 兵 兩 ␾ i 典 A 兩 ␾ j 典 B ,
i, j⫽1,2,3,4其 GT̃ ⫺1 共 ˜␩ 兲 ,
共3.10兲
where G 16⫻4 is the matrix representation of states
兵 兩 ˜␺ i (˜␪ ) 典 A 兩 ˜␺ i (˜␰ ) 典 B 其 on the bases 兵 兩 ␾ i 典 A 兩 ␾ j 典 B 其 , which are
lexicographically ordered, i.e., 兩0000典,兩0001典, . . . ,兩1111典 in
Hilbert space H 丢 2 丢 H 丢 2 . We denote GT̃ ⫺1 ( ˜␩ )
⫽( ␻ 1 , ␻ 5 , ␻ 9 , ␻ 13). States 兩 ␻ i 典 are orthogonal in the space
Denote
G̃ ⫺1
spanned
by
兵 兩 ˜␺ i (˜␪ ) 典 A 兩 ˜␺ i (˜␰ ) 典 B 其 .
⫽( ␻ 1 , ␻ 2 , . . . , ␻ 16),
where
states
兵 兩 ␻ j 典 , 1⭐ j
⭐16, j苸 兵 1,5,9,13其其 are selected in the subspace orthogonal
to that spanned by 兵 兩 ␻ i 典 , i⫽1,5,9,13其 . With Eq. 共3.10兲, we
let
D ⫺1 共 ˜␪ ,˜␰ 兲 兵 兩 ␾ i 典 A 兩 ␾ j 典 B 其 ⫽ 兵 兩 ␾ i 典 A 兩 ␾ j 典 B 其 G̃ ⫺1 .
共3.11兲
Thus we represent D(˜␪ ,˜␰ ) as matrix G̃ on the orthogonal
bases 兵 兩 ␾ i 典 A 兩 ␾ j 典 B 其 . Similar to Eq. 共3.7兲, define D K
⫽D 1 (˜␪ ,˜␰ K⫺1 )D 2 (˜␪ ,˜␰ K⫺2 )•••D K⫺1 (˜␪ ,˜␪ ) (˜␪ ⫽˜␰ 1 ), where
D j (˜␪ ,˜␰ ) compresses the information of partite systems
A j ,A j⫹1 to A j , and ˜␰ j⫹1 is uniquely determined by
X(˜␪ ,˜␰ j )⫽T̃ † (˜␰ j⫹1 )T̃(˜␰ j⫹1 ). D K acts as follows:
D K 兩 ␺ i 共 ˜␪ 兲 典 丢 K ⫽ 兩 ␺ i 共˜␰ K 兲 典 A 1 兩 00典 丢 (K⫺1) .
共3.12兲
We can also define the similar operation D K (˜␪ ,˜␰ ) that compresses the information of K input copies into one for a
k-partite system, where ˜␪ ⫽( ␪ (i j) , ␮ (i j) ,i⫽2,3, . . . ,2k ; j
⫽1,2, . . . ,i⫺1). With lemma 1 we can realize D K via universal logic gates.
For operation D K , we may introduce a new gate called
the Controlled-D K gate, which can transfer the complicated
orthogonal bases to lexicographically ordered ones of a multipartite system. In the information compression process, we
perform a Controlled-D K gate on the controlled partites with
062310-5
ZHANG, WANG, LI, AND GUO
PHYSICAL REVIEW A 61 062310
P as the controller. In the information decompression process, a Controlled-D K⫹ gate is needed. With all these operations and controlled operations, we can express the orthogonal bases and transfer them to those suitable for the
realization of quantum cloning and identification via universal quantum logic gates.
B. Representation of unitary evolution and realization
via universal gates
Suppose that ⍀ k ⫽ 兵 兩 ⍀ i 典 ,i⫽1,2, . . . ,2k 其 are the bases
which are lexicographically ordered in Hilbert space H 丢 k .
For the given probability matrix ⌫, with a D K gate, we can
represent the orthogonal bases 兵兵 兩 ␣ i 典 兩 P 0 典 其 , 兵 兩 ˜␸ j 典 兩 P 1 典 其 , i, j
⫽1,2, . . . ,2k 其 关of Eq. 共2.16兲 for probabilistic identification兴
and 兵兵 兩 ␣ i 典 兩 ␸ 1 典 丢 (N⫺M ) 兩 P 0 典 其 , 兵 兩 ␤ j 典 其 ,i, j⫽1,2, . . . ,2k 其 关of Eq.
共2.1兲 for probabilistic cloning兴 as
丢 (M ⫺1)
兩 P 0 典 其 , 兵 兩 ⍀ j 典 兩 ⍀ 1 典 丢 (M ⫺1) 兩 P 1 典 其 ,
兵兵 D ⫺1
M 兩 ⍀ i典兩 ⍀ 1典
i, j⫽1,2, . . . ,2k 其 ,
FIG. 2. The networks of probabilistic identification for a onepartite system. 兩 ␺ ⫾ ( ␪ ) 典 A i are to-be-identified states and 兩 P 0 典 is the
probe.
as S⫽diag(K 1 ,K 2 , . . . ,K 2 k ), where
K i⫽
共3.13兲
兵兵 兩 ⍀ i 典 A 1 兩 P 0 典 其 , 兵 兩 ⍀ j 典 A 1 兩 P 1 典 其 ,i, j⫽1,2, . . . ,2k 其
共3.14兲
where K⫽M is for identification and K⫽N is for cloning.
On these new orthogonal bases, the evolution Û is a unitary
controlled operation on a composite system of A 1 and probe
P with the composite system of subsystem A 2 ,A 3 , . . . ,A K
as the controller. If the controller is in state
兩 ⍀ 1 典 A丢 (K⫺1)
, we perform operation Û on the composite
2 ,A 3 , . . . ,A K
system of A 1 P. Otherwise we make no operation. Denote
兩 P 0 典 ⫽ 兩 0 典 P , 兩 P 1 典 ⫽ 兩 1 典 P , on the bases 兵兵 兩 ⍀ i 典 A 1 兩 0 典 P 其 ,
兵 兩 ⍀ j 典 A 1 兩 1 典 P 其 , i, j⫽1,2, . . . ,2 k 其 ; U can be represented as
Ŝ⫽
兿
i⫽1
A P
P 2i,2k⫹1 P 2i⫺1,2k⫹1 ⫺1 ⌳ k 1 共 K i 兲
共3.16兲
where K⫽M is for identification and K⫽N is for cloning.
We have shown in lemma 1 that the unitary operations U 0 ,
P 2i⫺1,2k⫹1 ⫺1 , P 2i,2k⫹1 , and V̂ A 1 can be decomposed into the
product of basis operations such as C-NOT and ⌳ k (û). The
decomposition of ⌳ k (û) has been completed by Barenco
et al. 关23兴. Thus we complete the decomposition of the unitary evolution via universal quantum logic gates, so as to
realize probabilistic cloning and identification of a k -partite
system.
In the following we will give some examples. First we
shall be concerned with quantum probabilistic identification
of a one-partite system, provided with M initial copies. With
the given maximum probability ␥ 1 ⫽ ␥ 2 ⫽1⫺cosM2␪, we obtain
U⫽ṼSṼ † 关Eq. 共2.9兲兴. Ṽ corresponds to the operation
Ṽ⫽
⫽ 兩 ⍀ i 1 典 兩 ⍀ i 2 典 ••• 兩 ⍀ i K⫺1 典 , K⫽M is for identification, and K
⫽N for cloning. The matrix corresponding to the operation
V̂ A 1 on the bases 兵 兩 ⍀ i 典 A 1 其 is V. Î P represents unit operation
of a probe system. On the new orthogonal bases
兵兵 兩 ⍀ i 典 A 1 兩 P 0 典 , 兩 ⍀ i 典 A 1 兩 P 1 典 其 , i⫽1,2, . . . ,2k 其 , we express
S⫽
冉
E
F
冊
冑2
共 I A ⫹i ␴ Ay 兲 I P ⫽R Ay
m 2⫽
j
⫺E
1
共3.15兲
where Ĵ⫽ 兺 兩 兵 ⍀ i 其 典 ⫽ 兩 ⍀ 1 典 丢 (K⫺1) Î A 1 P 兩 兵 ⍀ i j 其 典具 兵 ⍀ i j 其 兩 with 兩 兵 ⍀ i j 其 典
F
.
⫻ P 2i⫺1,2k⫹1 ⫺1 P 2i,2k⫹1 兩 ⍀ 1 典 丢 (K⫺1) 丢 (K⫺1) 具 ⍀ 1 兩 ⫹Ĵ,
where the first expression is for identification and the second
is for cloning. With a controlled-D M gate and a controlledD N gate, we can transfer these orthogonal bases into
V̂ A 1 Î P 兩 ⍀ 1 典 丢 (K⫺1) 丢 (K⫺1) 具 ⍀ 1 兩 ⫹Ĵ,
冊
2k
i, j⫽1,2, . . . ,2k 其 ,
丢 (K⫺1)
冑1⫺m i ⫺ 冑m i
冑m i 冑1⫺m i
So we obtain
丢 (N⫺1)
兩 P 0 典 其 , 兵 D N⫺1 兩 ⍀ j 典 兩 ⍀ 1 典 丢 (N⫺1) 兩 P 1 典 其 ,
兵兵 D ⫺1
M 兩 ⍀ i典兩 ⍀ 1典
丢 兩 ⍀ 1 典 A ,A , . . . ,A ,
2 3
K
冉
K 2⫽
where
062310-6
冉
1⫺cosM 2 ␪
1⫹cos 2 ␪
M
冑
冑
,
2cosM 2 ␪
1⫹cosM 2 ␪
1⫺cosM 2 ␪
1⫹cosM 2 ␪
冉冊
␲ P
I ,
2
K 1⫽
冉
冑
冑
⫺
m 1 ⫽1,
0
⫺1
1
0
冊
,
1⫺cosM 2 ␪
1⫹cosM 2 ␪
2cosM 2 ␪
1⫹cosM 2 ␪
冊
,
REALIZING PROBABILISTIC IDENTIFICATION AND . . .
PHYSICAL REVIEW A 61 062310
FIG. 3. The networks of an S gate for a one-partite system.
R y共 ␹ 兲⫽
冉
␹
cos
2
␹
sin
2
␹
⫺sin
2
␹
cos
2
冊
FIG. 5. The networks of an S gate for a two-partite system.
.
The network of quantum probabilistic identification for a
one-partite system via universal logic gates is shown in Fig.
2 (M ⫽2).
The S gate in Fig. 2 is illustrated in Fig. 3.
For a two-partite system, with the given maximum probability matrix ⌫ which satisfies the inequality X (M ) ⫺⌫⭓0,
we obtain
The network of quantum probabilistic identification for a
two-partite system is shown in Fig. 4 (M ⫽2).
The S gate in Fig. 4 is illustrated in Fig. 5
As to probabilistic cloning, we also begin with a onepartite system. With inequality 共2.4兲, we give the maximum
probability ␥ max⫽(1⫺cosM2␪)/(1⫺cosN2␪). Then
Ṽ⫽
1
冑2
A
共 I A 1 ⫹i ␴ y 1 兲 兩 0 典 A 2 A 2 具 0 兩 I P ⫽R Ay
Ŝ⫽ ␴ 1x ␴ 2x ⌳ 2 共 K 1 兲 ␴ 2x ␴ 1x ␴ 1x ⌳ 2 共 K 2 兲 ␴ 1x ␴ 2x ⌳ 2 共 K 3 兲
K 1⫽
⫻ ␴ 2x ⌳ 2 共 K 4 兲 兩 00典 丢 (M ⫺1) 丢 (M ⫺1) 具 00兩 ⫹Ĵ.
K 2⫽
冉
冑
冑
2 共 cosM 2 ␪ ⫺cosN 2 ␪ 兲
共 1⫺cosN 2 ␪ 兲共 1⫹cosM 2 ␪ 兲
冑
冑
⫺
共 1⫹cosN 2 ␪ 兲共 1⫺cosM 2 ␪ 兲
共 1⫺cosN 2 ␪ 兲共 1⫹cosM 2 ␪ 兲
The network of quantum probabilistic clone for a one-partite
system is shown in Fig. 6 (M ⫽2, N⫽3).
For a two-partite system, with the given maximum probability matrix ⌫ satisfying X (M ) ⫺ 冑⌫X (N) 冑⌫⭓0, we obtain
Ŝ⫽ ␴ 1x ␴ 2x ⌳ 2 共 K 1 兲 ␴ 2x ␴ 1x ␴ 1x ⌳ 2 共 K 2 兲 ␴ 1x ␴ 2x ⌳ 2 共 K 3 兲
冉
0
⫺1
1
0,
共 1⫹cosN 2 ␪ 兲共 1⫺cosM 2 ␪ 兲
共 1⫺cosN 2 ␪ 兲共 1⫹cosM 2 ␪ 兲
2 共 cosM 2 ␪ ⫺cosN 2 ␪ 兲
共 1⫺cosN 2 ␪ 兲共 1⫹cosM 2 ␪ 兲
冊
冊
冉冊
␲
兩 0 典 A2A2具 0 兩 I P,
2
,
.
So far we have realized quantum probabilistic identification and cloning in a k -partite system via universal quantum
logic gates, which have important applications in quantum
cryptography 关24,25兴, quantum programming 关26兴, and quantum state preparation 关27兴.
C. Robustness of the quantum networks
⫻ ␴ 2x ⌳ 2 共 K 4 兲 兩 00典 丢 (N⫺1) 丢 (N⫺1) 具 00兩 ⫹Ĵ.
The network of quantum probabilistic cloning for a twopartite system is shown in Fig. 7 共where M ⫽2, N⫽3).
FIG. 4. The networks of probabilistic identification for a twopartite system. 兩 ˜␺ i 典 A j are to-be-identified states.
The robustness properties of the cloning and identifying
machines may prove to be crucial in practice. In this subsection, we show whether any errors occur in the input target
systems A M ⫹1 , A M ⫹2 , . . . ,A N , we can detect them without
FIG. 6. The networks of probabilistic cloning of a one-partite
system. The S gate has been illustrated in Fig. 3. 兩 ␺ ⫾ ( ␪ ) 典 A i are
to-be-cloned states and 兩 0 典 A 3 is the input target state.
062310-7
ZHANG, WANG, LI, AND GUO
PHYSICAL REVIEW A 61 062310
If the errors are caused by state preparation, after the evolution of the system, the output target system corresponding
to 兩 0 典 P is the superposition of two different terms. We measure the output target states, and if they result in 兩 ⍀ 1 典 丢 (N⫺M )
, the clone really fails. Otherwise, the errors work and the
to-be-cloned state remains undestroyed.
To the two error situations mentioned above, we can reinput the to-be-cloned system to the cloning machines at the
location immediately behind the Controlled-D K gate 共the
first operation of the cloning machine兲 and clone again.
FIG. 7. The networks of probabilistic cloning for a twopartite
system. The S gate has been illustrated in Fig. 5. 兩 ˜␺ i 典 A j are to-becloned states and 兩 00典 A 3 is the input target state.
destroying the to-be-cloned states in systems A 1 ,
A 2 , . . . ,A M , and the to-be-cloned states can be recycled.
The input target state with errors may be generally expressed as
␳ A M ⫹1 ,A M ⫹2 , . . . ,A N
冉
2k
⫽ 共 1⫺ ␦ 1 兲 兩 ⍀ 1 典具 ⍀ 1 兩 ⫹ ␦ 1
兺 ⑀ i兩 ⍀ i 典具 ⍀ i兩
i⫽2
冊
丢 (N⫺M )
,
共3.17兲
IV. CONCLUSIONS
In summary, we have considered the realization of quantum probabilistic identifying and cloning machines by physical means. We showed that the unitary representation and the
Hamiltonian of probabilistic cloning and identifying machines are determined by the probabilities of success. The
logic networks have been obtained by decomposing the unitary representation into universal quantum logic operations.
We have discussed the robustness of the networks and found
that if error occurs in the input target system, we can detect
it and the to-be-cloned states can be recycled. Our method is
suitable for a k-partite system, such as a quantum computer,
and may be generalized to general state-dependent cloning
and identification.
k
2
兩 ⑀ i 兩 ⫽1 and ␦ 1 is the error rate, or
where 兺 i⫽2
兩 ␾ 典 A M ⫹1 ,A M ⫹2 , . . . ,A N
⫽
冉
2k
冑1⫺ 兩 ␦ 2 兩 2 兩 ⍀ 1 典 ⫹ ␦ 2 兺 ␶ i 兩 ⍀ i 典
i⫽2
ACKNOWLEDGMENTS
冊
We thank Dr. L.-M. Duan for helpful discussion. This
work was supported by the National Natural Science Foundation of China.
丢 (N⫺M )
,
APPENDIX A
共3.18兲
In this appendix, we determine M and N and derive the
representation of U. U is a unitary matrix, that is,
k
2
兩 ␶ i 兩 2 ⫽1 and 兩 ␦ 2 兩 2 is the error rate. Equation
where 兺 i⫽2
共3.17兲 expresses the errors caused by the decoherence due to
the environment. Equation 共3.18兲 represents the errors in
state preparation. The errors occur in the (N⫺M ) input target systems for cloning with the approximate rate (N
⫺M ) ␦ 1 关 (N⫺M ) 兩 ␦ 2 兩 2 兴 , which cannot be omitted in practice when N is relatively large.
After the cloning process, if measurement of probe P results in 兩 0 典 P , the cloning attempt should be regarded as a
failure in a normal sense. However, it may be caused by
errors.
If errors caused by the decoherence occur in any input
target systems, at least one system occupies state 兩 ⍀ i 典 , i
⫽1. According to Eqs. 共3.15兲 and 共3.16兲, the controlled operations V̂ A 1 , Ŝ, and Controlled-D K⫹ gate in the information
decompression, function as unit evolutions, in other words,
only Controlled-D K gate in the information compression
works. Thus the to-be-cloned state remains undestroyed. According to Eq. 共2.1兲 and the above discussion, the input target states remain unchanged if probe P is in 兩 0 典 P , whenever
the clone fails or errors occur. These two cases can be
checked out by measuring the output target states.
UU † ⫽U † U⫽I 2n .
共A1兲
Equation 共A1兲 can be proved equivalent to the following
two equations:
N⫽⫺ 共 冑⌫ 兲 ⫺1 CM ,
共A2兲
M M † ⫽I n ⫺C † X ⫺1 C.
共A3兲
It is obvious that I n ⫺C † X ⫺1 C is a symmetric matrix.
According to Eq. 共2.6兲, we yield
I n ⫺C † X ⫺1 C⫽ 共 I n ⫹C † ⌫ ⫺1 C 兲 ⫺1 .
共A4兲
For ⌫ positive definite, C † ⌫ ⫺1 C is semipositive definite.
Thus I n ⫹C † ⌫ ⫺1 C is positive definite and its reversed matrix
I n ⫺C † X ⫺1 C is also positive definite.
I n ⫺C † X ⫺1 C can be represented as the following:
I n ⫺C † X ⫺1 C⫽V diag共 m 1 , . . . ,m n 兲 V † ,
共A5兲
where V is unitary. Together with Eq. 共A3兲, M is determined
by
062310-8
REALIZING PROBABILISTIC IDENTIFICATION AND . . .
M ⫽⫺V diag共 冑m 1 , . . . , 冑m n 兲 V † .
PHYSICAL REVIEW A 61 062310
APPENDIX B
共A6兲
Here we diagonalize Û.
S can be rewritten as
Furthermore, we can also prove several useful conclusions to replace the submatrices of U in Eq. 共2.8兲,
C † A ⫺1 ⫽V diag共 冑1⫺m 1 , . . . , 冑1⫺m n 兲 V † ,
共A7兲
冑⌫A ⫺1 ⫽V diag共 冑m 1 , . . . , 冑m n 兲 V † ,
共A8兲
N⫽⫺ 共 冑⌫ 兲 ⫺1 CM ⫽V 共 冑1⫺m 1 , . . . , 冑1⫺m n 兲 V † .
Hence, we get
U⫽
冉 冊冉
V
0
F
⫺E
0
V
E
F
冊冉
V†
0
0
V†
冊
where K⫽diag(K 1 ,K 2 , . . . ,K n ),
K i⫽
共A9兲
,
共A10兲
冊
,
1
冑2
冉
1
⫺i
⫺i
1
冊
,
j⫽1,2, . . . ,n, L̃⫽diag(L 1 ,L 2 , . . . ,L n ), we have
K⫽L̃ diag共 e i ␪ 1 ,e ⫺i ␪ 1 , . . . ,e i ␪ n ,e ⫺i ␪ n 兲 L̃ † ,
where ␪ j , and j⫽1,2, . . . ,n is determined by
i⫽1,2, . . . ,n.
e i ␪ j ⫽ 冑1⫺m j ⫹i 冑m j ,
C † X ⫺1 C⫽V diag共 1⫺m 1 , . . . ,1⫺m n 兲 V † .
For X positive definite, C † X ⫺1 C is semipositive definite.
So 1⫺m i ⭓0, that is, m i ⭐1, i⫽1, . . . ,n. Combining the
results above, we get the range of m i as
i⫽1,2, . . . ,n.
冑1⫺m i ⫺ 冑m i
冑m i
冑1⫺m i
L j⫽
On the other hand, Eq. 共A5兲 can be rewritten as
0⬍m i ⭐1,
冉
T is a unitary matrix which interchanges the rows of K, and
T † interchanges the columns. Denoting
where E⫽diag( 冑m 1 , . . . , 冑m n ), F⫽diag( 冑1⫺m 1 , . . . ,
冑1⫺m n ).
According to Eq. 共A5兲 and I n ⫺C † X ⫺1 C⬎0, we yield
m i ⬎0,
共B1兲
S⫽TKT † ,
共A11兲
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冉
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共B2兲
共B3兲
According to Eqs. 共2.9兲, 共B1兲, and 共B2兲, U is completely
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U⫽O diag共 e i ␪ 1 ,e ⫺i ␪ 1 , . . . ,e i ␪ n ,e ⫺i ␪ n 兲 O † ,
共B4兲
where O⫽ṼTL̃.
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