Download 271, 31 (2000) .

19 June 2000
Physics Letters A 271 Ž2000. 31–34
www.elsevier.nlrlocaterpla
Quantum clone and states estimation for n-state system
Chuan-Wei Zhang, Chuan-Feng Li ) , Guang-Can Guo
Laboratory of Quantum Communication and Quantum, Computation and Department of Physics, UniÕersity of Science and Technology of
China, Hefei 230026, PR China
Received 1 February 2000; received in revised form 3 May 2000; accepted 5 May 2000
Communicated by V.M. Agranovich
Abstract
We derive a lower bound for the optimal fidelity for deterministic cloning a set of n pure states. In connection with states
estimation, we obtain a lower bound about average maximum correct states estimation probability. q 2000 Published by
Elsevier Science B.V.
PACS: 03.67.-a; 03.65.Bz; 89.70.q c
Quantum no-cloning theorem w1,2x has prohibited
cloning and estimating an arbitrary quantum state
exactly by any physical means in a consequence of
linearity of quantum theory. The unitarity of quantum theory does not allow to clone Židentify. no-orthogonal states though orthogonal states can be
cloned Židentified. perfectly w3,4x. However, clone
and estimation of quantum states with a limited
degree of success are always possible. Universal
quantum cloning machine ŽUQCM. w5–13x acts on
any unknown quantum state and produce optimal
approximate copies. This machine is called universal
because it produces copies that are state-independent.
State-dependent quantum cloning machines is designed to clone states belonging to a finite set and
)
Corresponding author.
E-mail addresses: [email protected] ŽC.-F. Li.,
[email protected] ŽG.-C. Guo..
may be divided into two main categories: deterministic w14,15x, probabilistic w16–19x and hybrid w20x.
Deterministic state-dependent cloning machine generates approximate clones with probability 1. Deterministic exact clone violates the no-cloning theorem,
thus perfectly clone must be probabilistic. Probabilistic quantum cloning machines can clone states perfectly, though the success probability cannot be unit
all the time. It is shown that a set of non-orthogonal
states can be probabilistically cloned if and only if
the states are linearly independent. Hybrid clone
interpolates between deterministic and probabilistic
ones, that is, the copies Žnot exact. are better than
those in deterministic clone, but the success probability Žless than 1. is greater than probabilistic exact
clone. Universal quantum states estimation were considered in Ref. w21,22x, given M independent realizations. What’s more, we w23x have discussed general
states discrimination strategies for state-dependent
system.
0375-9601r00r$ - see front matter q 2000 Published by Elsevier Science B.V.
PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 3 5 2 - 2
C.-W. Zhang et al.r Physics Letters A 271 (2000) 31–34
32
Optimal results for two-state deterministic clone
have been obtained in Refs. w14,15,20x. In this Letter
we consider deterministic clone for a set of n pure
states c i :,i s 1,2, . . . ,n4 . When c i : are non-orthogonal, they cannot be cloned perfectly. What we
require is that the final states should be most similar
as the target states, that is, the fidelity between the
final and target states should be optimal. We derive a
lower bound for the optimal fidelity of the cloning
machine. Applying it to states estimation, we obtain
the lower bound about average maximum correct
identification probability in deterministic states estimation.
A quantum state-dependent cloning device is a
quantum machine which performs a prescribed unitary transformation on an extended input which contains M original states in system A and N y M
blank states in system B with N output copies. The
unitary evolution transfers states as follows
U c i M :A S NyM :B s a i :,
Ž 1.
where c i M :A s c i :1 m . . . m c i :M are the M original states, S Ny M :B are the blank states and a i :
are the output cloned states. The n = n inter-innerproducts of Eq. Ž1. yield the matrix equation1
X Ž M . s X˜ ,
Ž 2.
where n = n matrices X˜ s
² ai < a j: ,
X
ŽM .
M
² ci < cj : .
We require a figure of merit to characterize how
closely our copies a i : resemble exact copies c i N :.
Denoting the priori probability of the state c i M : by
hi , one interesting measure of the final states is the
global fidelity introduced by Bruß et al. w14,15x,
which is defined formally as
s
n
Fs
2
Ý hi ² a i ci N : .
Ž 3.
is1
As a criterion for optimality of the state-dependent
cloner, the unitary evolution U should maximize the
global fidelity F of n final states a i : with respect
to the perfect cloned states c i N :. We focus here on
the global fidelity since it has an important interpretation in connection with states estimation w20x.
Now the remained problem is to find the maximum value of the fidelity F, which means optimal
clone. It is equivalent to the problem of maximizing
F under the condition of Eq. Ž2.. This problem is a
nonlinear programming and fairly difficult to solve.
Nevertheless a lower bound of the optimal fidelity
could still be derived by adopting an auxiliary function F X , which is defined as
n
FX s
Ý hi ² ci N N a i : .
Ž 4.
is1
Such function also describes how closely our output
copies resemble exact copies. There exists a bound
between F and F X Žsee below, inequality Ž9.., therefore a bound for F may be obtained by optimizing
FX.
We find that the optimal output states a i : must
lie in the subspace spanned by the exact clones
c i N :. This conclusion may be easily come to with
the method of Lagrange Multipliers Žplease refer to
w14,15x, where n s 2. and here we omit the proof.
If a set of states a˜ i : fulfil Eq. Ž2., that is,
XŽ M . s X˜ s ² a˜ i < a˜ j : , there must exist a unitary
transformation V satisfies V a˜ i :s a i :, thus we can
vary V to optimize F X with chosen states a˜ i :.
Suppose ² c i N N a i : s l i² c i N N a i : with l i g "1 4
in the optimal situation Žthe determination of l i will
be shown in later part., the optimal F X is
n
X
Fopt
s max V F X s max V
Ý hi l i² ci N V a˜ i : .
Ž 5.
is1
Choose n orthogonal states x i : which span a
space H and the space spanned by c i N : is a
subspace of H 2 . Set a˜ i :s Ý njs1 a i j x j :, c i N :s
1
We notice the preserving inner-product property of unitary
transformation, that is, if two sets of states f 1 :, f 2 :, . . . , fn :4
and f˜ 1 :, f˜ 2 :, . . . , f˜ n : satisfy the condition ² f i N f j : s ² f˜ i N
f˜ j :, there exists a unitary operate U to make U fi :s f˜ i :
Ž is1,2, . . . ,n..
4
2
We consider space c i N :,is1,2, . . . ,n 4 may be a subspace
of H since c i : may be linear-dependent and cannot span a
n-dimensional Hilbert space.
C.-W. Zhang et al.r Physics Letters A 271 (2000) 31–34
Ý njs1 bi j x j : on the orthogonal bases
1,2, . . . ,n, we get
X
Fopt
s max V
x i :, i s
tr Ž hl BVAq . smax V tr Ž VO .
s tr'Oq O ,
Ž 6.
where A s w a i j x , B s bi j , h s diagŽh1 ,h 2 , . . . ,
hn ., l s diagŽ l1 , l2 , . . . , l n ., O s Aq hl B. We have
used the freedom in V to make the inequality as tight
as possible. To do this we have recalled w24,25x that
max V tr Ž VO . s tr'Oq O , where O is any operator
and the maximum is achieved only by those V such
that
VO s e i n'Oq O ,
Ž 7.
where n is arbitrary. Generally, we choose n s 0.
As we require above, l i should satisfy
l i² c i N V a˜ i :G 0. This condition can be represented
as ² x i l BVAq x i :G 0, which means the diagonal
elements of matrix l BVAq should be positive. Since
l i g "1 4 , a simple method to determine l i is to
enumerate the 2 n possible results of l s
diagŽ l1 , l2 , . . . , l n . and verify which one fulfils
above inequality. With a chosen basis x i :, matrix
A, B can be given by equations Aq A s XŽ M . and
Bq B s XŽ N . respectively, V can be represented with
parameters l i , thus above postcalculation method
can determine matrix l and then give the maximum
X
Fopt
. According to Eq. Ž6., we obtain a tight upper
bound for the function F X ,
F X F tr Bq lh XŽ M . hl B .
(
Ž 8.
The fidelity F of the cloning machine is constrained by the following inequality
n
Fs
ž
žÝ
Ý hi ² a i
ciN :
2
is1
is1
/ž /
is1
2
n
G
n
Ý hi
hi ² a i c i N :
/
2
s Ž FX . ,
Ž 9.
where the equation is met if and only if ² a i c i N :
are constant. Obviously F is not always optimal
even if F X is optimal. However optimal F should be
2
X
greater than or equal to Ž Fopt
. . When n s 2 and
h1 s h 2 , equation in Ineq. Ž9. is satisfied and gives
the optimal results, which has been provided in Refs.
w14,15,20x.
33
™
State-dependent clone has a close connection with
states estimation in the limit as N `. Given infinite copies of n non-orthogonal states, we can discriminate them exactly with probability 1. On the
other hand, if we can discriminate n states, we can
obtain infinite copies. There are two ways in which
an attempt to discriminate between non-orthogonal
states; it can give either an erroneous or an inconclusive result w23x. In the following we will consider a
strategy without inconclusive results using above
results in the limit as N `. In fact, since the
optimal output states a i : lie in the subspace spanned
by the exact clones c i N :, Eq. Ž1. may be rewritten
as
™
n
U c i M : S NyM :s
Ý ci j
c j N :,
Ž 10 .
js1
™
where c i j s ² c j N N a i :. If N
`,
c j N :,
j s 1,2, . . . ,n4 are orthogonal. After the evolution,
the cloning system is measured and if c j` : is obtained, the original state is estimated as c j M :. The
2
states estimation is correct with probability c i i
when j s i. If j / i, errors occur with probability
2
Ý j/ i c i j . The inter-inner products of Eq. Ž10. give
the matrix equation in the limit N `,
½
™
X Ž M . y EEqs 0,
Ž 11 .
where E s w c i j x . The diagonal elements is corresponding to the probabilities of correct states estimation while non-diagonal elements to those of error.
This equation describes the bound between the maximum probabilities of correct discrimination and those
of incorrect one. In fact, this result is a special case
of that we have derived in w23x. In Ref. w23x, we have
consider two possible ways in which an attempt to
discriminate between non-orthogonal states can fail,
by giving either an erroneous or an inconclusive
result. Above strategy just gives an erroneous result
with some probability. Our principal result in Ref.
w23x is the matrix inequality which prescribes the
bound among the probabilities of correct, error and
inconclusive discrimination results. Such bound may
have intriguing implications for quantum communication theory and cryptography w26x since it offers a
potential eavesdropper increased flexibility by a
C.-W. Zhang et al.r Physics Letters A 271 (2000) 31–34
34
compromise between inconclusive and erroneous results.
An important optimality criterion of the states
estimation is the average maximum correct probabil2
ity, that is, P s Ý ihi c i i s F in the limit N `3.
N
In this situation c j : are orthogonal, thus matrix
B s In . Applying Eqs. Ž8. and Ž9., we obtain
™
2
ž(
P s Ý hi c i i G tr lh X Ž M . hl
i
Acknowledgements
This work was supported by the National Natural
Science Foundation of China.
2
/.
Ž 12 .
References
Such F is not always optimal bound of the average
maximum probability of correct states estimation,
however, the optimal one is always greater than
2
tr lh X Ž M .hl .
We note that above bound about F and P have
the meaning in average. They describe the optimality
approach to the final states we can reach in average
of the n initial states and does not mean the best for
each initial state. However, since we do not know
which one the initial state is in the clone or estimation process, such average may be the most important value to describe the efficiencies of cloning
Žestimating. machines.
In summary, we have derived a lower bound for
the optimal fidelity for the state-dependent quantum
clone. In connection with states estimation, we obtained the matrix inequality which describes the
bound between the maximum probabilities of correct
discrimination and those of incorrect one. A lower
bound about average maximum probability of correct
identification has also been presented. Our results
give some bounds which the optimal cloner and
states estimation can be better than in average, however, we have not found a limit which optimal cloner
ž(
can reach at most. It is still an open question needed
to be explored.
/
w1x
w2x
w3x
w4x
w5x
w6x
w7x
w8x
w9x
w10x
w11x
w12x
w13x
w14x
w15x
w16x
w17x
w18x
w19x
w20x
w21x
w22x
w23x
w24x
w25x
3
It is the reason why we choose the definition of F as that in
Eq. Ž3..
w26x
W.K. Wootters, W.H. Zurek, Nature 299 Ž1982. 802.
D. Dieks, Phys. Lett. A 92 Ž1982. 271.
H.P. Yuen, Phys. Lett. A 113 Ž1986. 405.
G.M. D’Ariano, H.P. Yuen, Phys. Rev. Lett. 76 Ž1996. 2832.
V. Buzek,
˘ M. Hillery, Phys. Rev. A 54 Ž1996. 1844.
V. Buzek,
˘ S.L. Braunstein, M. Hillery, D. Bruß, Phys. Rev.
A 56 Ž1997. 3446.
N. Gisin, S. Massar, Phys. Rev. Lett. 79 Ž1997. 2153.
D. Bruß, A. Ekert, C. Macchiavello, Phys. Rev. Lett. 81
Ž1998. 2598.
R.F. Werner, Phys. Rev. A 58 Ž1998. 1827.
M. Keyl, R.F. Werner, J. Math. Phys. 40 Ž1999. 3283.
V. Buzek,
˘ M. Hillery, Phys. Rev. Lett. 81 Ž1998. 5003.
C.-S. Niu, R.B. Griffiths, Phys. Rev. A 60 Ž1999. 2764.
N.J. Cerf, J. Mod. Opt. 47 Ž2000. 187.
M. Hillery, V. Buzek,
˘ Phys. Rev. A 56 Ž1997. 1212.
D. Bruß, D.P. DiVincenzo, A. Ekert, C.A. Fuchs, C. Macchiavello, J.A. Smolin, Phys. Rev. A 57 Ž1998. 2368.
L.-M. Duan, G.-C. Guo, Phys. Rev. Lett. 80 Ž1998. 4999.
L.-M. Duan, G.-C. Guo, Phys. Lett. A 243 Ž1998. 261.
C.-W. Zhang, Z.-Y. Wang, C.-F. Li, G.-C. Guo, Phys. Rev.
A 61 Ž2000. 062310.
A.K. Pati, Phys. Rev. Lett. 83 Ž1999. 2849.
A. Chefles, S.M. Barnett, Phys. Rev. A 60 Ž1999. 136.
S. Massar, S. Popescu, Phys. Rev. Lett. 74 Ž1995. 1259.
R. Derka, V. Buzek,
˘ A. Ekert, Phys. Rev. Lett. 80 Ž1998.
1571.
C.-W. Zhang, C.-F. Li, G.-C. Guo, Phys. Lett. A 261 Ž1999.
25.
R. Jozsa, J. Mod. Opt. 41 Ž1994. 2315.
R. Schatten, Norm Ideals of Completely Continuous Operators, Springer, Berlin, 1960.
C.H. Bennett, Phys. Rev. Lett. 68 Ž1997. 3121.