Probabilistic quantum cloning of real states

Probabilistic quantum cloning of real states

Optics Communications 283 (2010) 1956–1960 Contents lists available at ScienceDirect Optics Communications journal homepage: www.elsevier.com/locate...

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Optics Communications 283 (2010) 1956–1960

Contents lists available at ScienceDirect

Optics Communications journal homepage: www.elsevier.com/locate/optcom

Probabilistic quantum cloning of real states Wen-Hai Zhang a,b,c,*, Jie-Lin Dai b, Zhuo-Liang Cao b,c, Ming Yang c a

Department of Physics, Huainan Normal University, Huainan 232001, PR China Department of Physics and Electronic Engineering, Hefei Teachers College, Hefei 230061, PR China c School of Physics and Material Science, Anhui University, Hefei 230039, PR China b

a r t i c l e

i n f o

Article history: Received 10 August 2009 Received in revised form 6 December 2009 Accepted 11 December 2009

Keywords: Quantum cloning Probabilistic quantum cloning

a b s t r a c t We construct the explicit formulation of the probabilistically perfect quantum cloning machine that perfectly duplicates the input states chosen from the special set fjui igdi¼1 consisting of the linearly independent and nonorthogonal quantum states with hui juji = r 2 (0, 1)(i – j). The success probabilities of cloning the input states are equal and maximal. As two examples, we present the explicit transformations of the optimal 1 ? 2 probabilistically perfect quantum cloning of the real states in 2 and 3 dimensions. The success probabilities of each of two cloning machines are equal and maximal. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction As well known, the no-cloning theorem [1] states that an arbitrary quantum state can not be cloned perfectly. This theorem guarantees the unconditional security of quantum key distribution [2]. Though for the constraint of the no-cloning theorem, it does not forbid obtaining the limited information about an input state via quantum cloning. In general, quantum cloning can be divided into two categories. If the copies of the input states are required to be obtained deterministically, they must be imperfect (or approximate). Perfect copies are possible only in the case of a discrete set of linearly independent and nonorthogonal states, but this cloning process must be probabilistic, i.e., allowing a failure to occur. The former cloning is called the deterministically approximate quantum cloning (DAQC) (see Ref. [3] for a review) and the latter the probabilistically perfect quantum cloning (PPQC) [4,5]. Since a seminal paper proposed the optimal universal quantum cloning [6], the DAQC has been extensively investigated. Many categories of the DAQC [7–11] have been designed for quantum information tasks [12–17]. In experiment, the realizations of the DAQC have been reported [18–23]. Comparing with the fruitful contributions on the DAQC, however, there were very few studies on the PPQC. In Ref. [4,5], the authors discussed the possibility of obtaining perfect clones [4] and derived the best efficiency of the 1 ? M PPQC [5]. However, their formulas of the cloning operations are not explicit and can not instruct to find the explicit transformation of cloning different

* Corresponding author. Address: Department of Physics, Huainan Normal University, Huainan 232001, PR China. E-mail address: [email protected] (W.-H. Zhang). 0030-4018/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.12.023

input sets. Moreover, the dimensionality of the probe is larger than that of the input states to be cloned. For instance, if an input state is qubits (2 dimensions), the dimensionality of the probe should at least be 3-dimensional. For the higher dimensions in practical physical systems, the present experimental techniques can hardly reach. This will prevent from the realization of the PPQC in experiment. These two disadvantages [4,5] are all along an obstacle to the research of the PPQC. In this paper, we study the PPQC to clone the input states in the d special set fjui igi¼1 . The input states we select are linearly independent and nonorthogonal and their inner products are assumed in the narrow range of huijuji = r 2 (0, 1)(i – j). We derive the explicit formulation of the PPQC with the optimal success probabilities being equal, i.e., ci = cj = c 2 (0, 1), which is covered by the previous contributions [4,5]. The dimensionality of the probe is only 2dimensional, independent to the dimensionality of the input states to be cloned. With the help of the explicit formulation, we can easily obtain the explicit transformation of cloning different input sets. As one of two examples, we select one useful set

S ¼ fjs1 i ¼ cos hj0i þ sin hj1i; js2 i ¼ sin hj0i þ cos hj1ig:

ð1Þ

  We take h 2 0; p2 and then have hs1js2i = sin2h = s 2 (0, 1). Making use of our explicit formulation, we derive the explicit transformation of the optimal 1 ? 2 PPQC in 2 dimensions. The input set of the second example we select will be in Section 4. The paper is constructed as follows. In Section 2, we briefly review some previous contributions of two categories of quantum cloning, including the DAQC and the PPQC. In Section 3, we derive the explicit formulation of the PPQC to clone the special set fjui igdi¼1 we define. While in Section 4, we present two explicit

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transformations of the optimal 1 ? 2 PPQC of the real states in 2 and 3 dimensions. The paper ends with a summary.

ðMÞ cðMÞ þ cj i

2. Some previous contributions

where the equality holds if ci ¼ cj ði; j ¼ 1; 2; . . . ; dÞ. Obviously, Eq. (6) covers the Duan’s result. From mathematical point of view, it is indeed enough to determine the relation of the optimal success probabilities given by Eq. (6). However, by Eqs. (3) and (5) one can not construct the explicit transformation of the PPQC due to the ambiguity of the failure states. Furthermore, even if all the failure states are determined, the dimensionality of the probe is larger than d. For the higher dimensions in practical physical systems, it is very much difficult to control. In the following, we will construct the explicit formulation of the PPQC of the special set fjui igdi¼1 . Our task is twofold: finding the simplest form of the failures states and their relation, and determining the least dimensionality of the probe.

rffiffiffi rffiffiffi 2 1 ðj01i þ j10iÞab j1iA ; j000iabA þ j0ia j0ib j0iA ! 3 6 rffiffiffi rffiffiffi 2 1 ðj10i þ j01iÞab j0iA ; j111iabA þ j1ia j0ib j0iA ! 3 6

ð2Þ

where the particle a is an arbitrary quantum state to be cloned, b the blank state as the blank paper in a classical computer, and A the ancillary system. The unitary transformation acting on the input ðoutÞ system j/iaj0ibj0iA produces the output as Uj/ia j0ib j0iA ¼ j/iabA . The optimal 1 ? 2 universal quantum cloning is qualified by the ðoutÞ fidelity F (or the distance) defined by F aðbÞ ¼a h/jqaðbÞ j/ia , where ðoutÞ qaðbÞ denotes the reduced density operator of the clone a (b) by taking the partial trace over the particles b and A (a and A). For examðoutÞ ðoutÞ ðoutÞ ple, qb ¼ Tr aA ½qabA  ¼ TraA ½j/iabA h/j. The fidelity of the two clones is Fa = Fb = 5/6 < 1. Obviously, the two clones are imperfect. For all the unitary transformations acting on the input system j/iaj0ibj0iA, the unitary transformation defined by Eq. (2) is unique and can make the clone fidelity reach maximal. This unitary transformation can act an arbitrary unknown quantum state in 2 dimensions and can deterministically produce two clones with the equal maximal fidelity. The cloning machine is then defined as the optimal symmetric 1 ? 2 universal quantum cloning in 2 dimensions. This transformation is so important that it triggered the investigations of quantum cloning theory [3]. In Refs. [4,5], the authors discussed another kind of quantum cloning, i.e., the PPQC. Given a set fjwi igdi¼1 consisting of the linearly independent and nonorthogonal quantum states jwii, we require the two clones of any input state jwii to be perfect. This task can be completed by the optimal 1 ? 2 PPQC in d dimensions, given by Ref. [4]

pffiffiffiffi

ci jwi i2 jp0 i þ

d X j¼1

 E  ð jÞ   cij UAB pj ði ¼ 1; 2; . . . ; dÞ; ð3Þ

where jp0i, jp1i,. . . and jpdi are (d + 1) orthonormal states of the ðjÞ probe P, the state jUAB i is the normalized state of the composite system AB. After the evolution defined by Eq. (3), the success probability of obtaining two perfect clones jwii2 is ci if the probe jp0i is detected. If jpji is measured, the cloning process then fails. So this quantum cloning machine is probabilistic, and the success probability is satisfied ci 2 (0, 1). In the case of 2 dimensions, the best success probabilities of c1 and c2 are satisfied the relation [4]

c1 þ c2 2

1  jhw1 jw2 ij

6

1  jhw1 jw2 ij

ð4Þ

: 2

The equality holds if c1 = c2. Afterwards, Duan’s machine was generalized to the superposition case, i.e., the superposition of the 1 ? N (N = 2, 3, . . . , M) PPQC in d dimensions, defined as [5] N c pffiffiffi M qffiffiffiffiffiffiffi X X n ðMnÞ fl jwl iAB jpl i; Uðjwi ijRijpiÞ ¼ cðnÞ jpn1 i þ i jwi i j0i n¼2

1  jhwi jwj ij 1  jhwi jwj ijM

ð6Þ

; ðMÞ

3. Optimal probabilistically perfect quantum cloning In this section, we will prove that for a special set to be cloned with the best success probabilities being equal, the formulation of the PPQC can be taken the simplest form. We select the simplest set fjui igdi¼1 consisting of the linearly independent and nonorthogonal quantum states with huijuji = r 2 (0, 1)(i – j) and intend to construct the explicit formulation to perfectly clone any input state juii probabilistically. We suppose that the explicit formulation of the optimal 1 ? M (M P 2) PPQC in d dimensions takes the following form:

jui ij0iðM1Þ j0ip !

pffiffiffiffi

ci jui iM j0ip þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi 1  ci jUi ij1ip ;

ð7Þ

where the failure state jUii must be the normalized failure state of the composite system containing M particles under the constraint of the unitarity of Eq. (7). In Eq. (7), we conjecture that the failure state is only one, but not the superposition of d failure states as in Eq. (3), and that the dimensionality of the probe is only 2-dimensional, but not at least (d + 1)-dimensional as in Eq. (3). In the following, we intend to derive the relation of the optimal success probabilities of ci, cj 2 (0, 1), and to determine the overlap of the failure states of hUi jUji. The inner products of Eq. (7) directly yield

E pffiffiffiffiffiffiffiffi EM rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi   þ ð1  ci Þ 1  cj hUi Uj : hui juj ¼ ci cj hui juj

ð8Þ

In order to make Eq. (8) hold, the overlap of hUijUji should be a real number in the range of hUijUji 2 [1, 1] because of huijuji, ci, cj 2 R. Since the overlap of huiju ji is fixed in the given set fjui igdi¼1 , our task is then to estimate the relation of the maximal success probabilities, obviously dependent to the value of hUijUji 2 [1, 1] under the two natural constraints

0 < hui juj

EM

E < hui juj < 1;

ð9:1Þ

ci ; cj 2 ð0; 1Þ:

ð9:2Þ

In order to discuss the maximum of the success probabilities, we divide the interval of hUijUji 2 [1, 1] into three subintervals, that is, [1, huijujiM], (huijujiM, hu ijuji], and (huijuji, 1]. In the case of hUijUji 2 [1, huijujiM], by Eq. (8), we get

E pffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi

EM EM ci cj þ ð1  ci Þ 1  cj hui juj 6 hui juj ; hui juj 6

l¼M

ð5Þ ðnÞ

6

ðMÞ

In this section, we briefly review the DAQC. The simplest and most important case is to clone any completely unknown quantum state j/ia = aj0ia + bj1ia, where a and b are complex numbers and satisfy the normalization condition jaj2 + jbj2 = 1. If we intent to clone the state j/ia and deterministically obtain two copies, our task can be done by the following unitary transformation [6]

U ðjwi ijRijp0 iÞ ¼

2

where ci is the optimal success probability of obtaining n perfect clones of the input state jwii. For the optimal 1 ? M PPQC, the optiðMÞ ðMÞ mal success probabilities ci and cj are satisfied the relation [5]

ð10Þ M

where the equalities hold when hUijUji = huijuji and then ci ¼ cj ¼ 12. To the last inequality of Eq. (10), we use the relation of rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi pffiffiffiffiffiffiffiffi ð1c Þþ 1c ci cj þ ð1  ci Þ 1  cj 6 ci þ2 cj þ i 2ð j Þ ¼ 1 for ci, cj 2 (0, 1)

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pffiffiffiffiffi by exploring the inequality xy 6 xþy ðx; y P 0Þ. Obviously, Eq. (10) 2 is incompatible to Eq. 9.1. By using the inequality pffiffiffiffiffi xþy xy 6 2 ðx; y P 0Þ, Eq. (8) can be rewritten as

E   ci þ cj hUi Uj  hui juj 6 EM ;   2 hUi Uj  hui juj

ð11Þ

  where the equality holds iif ci = cj. In the case of hUi Uj 2 EM E

hUi jUj ihui juj i hui juj ; hui juj , the value of M 6 0 (the equality hUi jUj ihui juj i c þc

holds when hUijUj i = huijuj i) means i 2 j 6 0. Therefore, Eq. (11) is contradictive to Eq. (9.2). So, there is only left the interval of hUijUji 2 (huijuj i,1] on which we will determine the relation of the optimal success probabilities of ci and cj. Eq. (11) can be directly rewritten as

E E EM   hUi Uj  hui juj hui juj  hui juj 6 EM ¼ 1  EM :     hUi Uj  hui juj hUi Uj  hui juj

ci þ cj 2

ð13Þ

E 1  hu i ju j 6 EM : 1  hui juj

2

ð14Þ

This inequality is covered by Eq. (6), strongly suggesting that the explicit formulation defined by Eq. (7) is the optimal PPQC to clone the special set fjui igdi¼1 . By Eq. (13), we take jUii = jUji = jUi for convenience. When cloning the special set fjui igdi¼1 , we define the explicit formulation of the optimal 1 ? M PPQC as ðM1Þ

jui ij0i

j0ip !

pffiffiffiffi

ci jui i

M

pffiffiffiffiffiffiffiffiffiffiffiffiffi j0ip þ 1  ci jUij1ip :

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi ð1  ci Þ 1  cj ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi r ¼ ci ck r M þ ð1  ci Þð1  ck Þ; r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffi 1  cj ð1  ck Þ: r ¼ cj ck r M þ

where f(ci) is a function of ci, which is easily obtained by using some mathematic computational software and takes a very complex and tedious explicit expression having no help for our proof. Inserting the relation of cj = ck into Eq. (16.3), we immediately obtain

cj ¼ ck ¼

1r : 1  rM

ð18Þ

Obviously, Eqs. (17) and (18) are inconsistent. So our assumption is incorrect. Thus we complete our proof. One can observe that Eqs. (16.1)–(16.3) can be simultaneously held if and only if 1r ci ¼ cj ¼ ck ¼ 1r M . Therefore, our result in this section is as follows. When cloning the special set fjui igdi¼1 with huijuji = r 2 (0, 1)(i – j), the relation of the failure states hU1jU2i = 1 always holds as d = 2, and hUijUji = 1 can only hold under the condition of 1r ci ¼ 1r M ði ¼ 1; 2; . . . ; dÞ as d P 3. In other cases, Eq. (15) is invalid.

To show the validity of our formulation, we take two examples to illustrate the optimal 1 ? 2 PPQC in 2 and 3 dimensions. We first select the explicit set S given by Eq. (1) and require the optimal success probabilities to be equal, i.e., c = c1 = c2 = 1/(1 + s). The two mutually nonorthogonal states are without the phase factors, so we call them the real states. We conjecture the transformation of the optimal 1 ? 2 PPQC in 2 dimensions as

pffiffiffi cða1 j00i þ a2 j01i þ a3 j10i þ a4 j11iÞAB j0ip pffiffiffiffiffiffiffiffiffiffiffiffi þ 1  cðb1 j00i þ b2 j01i þ b3 j10i þ b4 j11iÞAB j1ip ; pffiffiffi j1iA j0iB j0ip ! cðc1 j00i þ c2 j01i þ c3 j10i þ c4 j11iÞAB j0ip pffiffiffiffiffiffiffiffiffiffiffiffi þ 1  cðd1 j00i þ d2 j01i þ d3 j10i þ d4 j11iÞAB j1ip :

j0iA j0iB j0ip !

ð19Þ We assume that all the parameters are real numbers. The normalized and the unitarity conditions are written down, respectively,

c

ð16:1Þ ð16:2Þ ð16:3Þ

 1r   1r  1r We suppose ci 2 0; 1r [ 1rM ; 1 a given number, i.e., ci – 1r M M . By Eqs. (16.1) and (16.2), we get the values of cj and ck as a function of ci as

4 X i¼1

ð15Þ

Here, the failure state jUi is very important to determine the explicit cloning transformation and is closely dependent the explicit form of different input sets to be cloned. The dimensionality of the probe is only 2-dimensional, independent to different input sets. We here should point out that Eq. (15) is merely valid to the special set fjui igdi¼1 with huijuji = r 2 (0, 1)(i – j) under the restric1r tion of ci ¼ c ¼ 1r M . In the case of d = 2, Eq. (15) is always held as Duan’s result [4]. That is, we can choose c1 – c2. While in other cases, Eq. (15) does not work. Let us present a brief proof by contradiction when d P 3. If Eq. (15) holds forci – cj when cloning any input state in the special set fjui igdi¼1 , by Eq. (8), then the following equations should be held:

pffiffiffiffiffiffiffiffi r ¼ ci cj r M þ

ð17Þ

4. Optimal probabilistically perfect quantum cloning of real states

Eq. (13) is very important to determining the explicit formulation of cloning the special set fjui igdi¼1 . This will be observed in our paper. Inserting this relation into Eq. (11), the optimal success probabilities of c iand cj are satisfied

ci þ cj

1r ; 1  rM

ð12Þ

Since huijuji > huijujiM, from the last equation of Eq. (12), one can easily observe that the higher the value of hUijUj i 2 (huijuji,1] is, c þc the larger the value of i 2 j 2 ð0; 1Þ. Thus, we get the relation of the failure states as

  hUi Uj ¼ 1:

cj ¼ ck ¼ f ðci Þ–

c

4 X i¼1

a2i þ ð1  cÞ

4 X

2

bi ¼ 1;

i¼1

ai ci þ ð1  cÞ

4 X

c

4 X i¼1

bi di ¼ 0:

c2i þ ð1  cÞ

4 X

2

di ¼ 1;

ð20:1Þ

i¼1

ð20:2Þ

i¼1

After the transformation defined by Eq. (19) acts on the input states js1i and js2i, we have

  U js1 iA j0iB j0ip pffiffiffi ¼ c½ða1 cos h þ c1 sin hÞj00i þ ða2 cos h þ c2 sin hÞj01i þða3 cos h þ c3 sin hÞj10i þ ða4 cos h þ c4 sin hÞj11iAB j0ip pffiffiffiffiffiffiffiffiffiffiffiffi þ 1  c½ðb1 cos h þ d1 sin hÞj00i þ ðb2 cos h þ d2 sin hÞj01iþðb3 cos h þ d3 sin hÞj10i þ ðb4 cos h þ d4 sin hÞj11iAB j1ip ; ð21:1Þ   U js2 iA j0iB j0ip pffiffiffi ¼ c½ða1 sin h þ c1 cos hÞj00i þ ða2 sin h þ c2 cos hÞj01i þða3 sin h þ c3 cos hÞj10i þ ða4 sin h þ c4 cos hÞj11iAB j0ip pffiffiffiffiffiffiffiffiffiffiffiffi þ 1  c½ðb1 sin h þ d1 cos hÞj00i þ ðb2 sin h þ d2 cos hÞj01iþðb3 sin h þ d3 cos hÞj10i ð21:2Þ þ ðb4 sin h þ d4 cos hÞj11iAB j1ip :

The output system should be satisfied with our definition given by Eq. (15), i.e.,

W.-H. Zhang et al. / Optics Communications 283 (2010) 1956–1960

pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi js1 ij0ij0ip ! cjs1 i2 j0ip þ 1  cjU1 ij1ip i pffiffiffih 2 ¼ c cos2 hj00i þ coshsinhðj01i þ j10iÞ þ sin hj11i j0ip AB pffiffiffiffiffiffiffiffiffiffiffi þ 1  cjU1 ij1ip ; ð22:1Þ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffi js2 ij0ij0ip ! cjs2 i2 j0ip þ 1  cjU2 ij1ip i pffiffiffih 2 ¼ c sin hj00i þ coshsinhðj01i þ j10iÞ þ cos2 hj11i j0ip AB pffiffiffiffiffiffiffiffiffiffiffi þ 1  cjU2 ij1ip : ð22:2Þ By comparing with the four equations above, the coefficients of the computational basis of the success states must be equal. We thus get a series of equations 2

a1 cos h þ c1 sin h ¼ cos h;

2

a1 sin h þ c1 cos h ¼ sin h;

a2 cos h þ c2 sin h ¼ cos h sin h; a3 cos h þ c3 sin h ¼ cos h sin h; 2

a4 cos h þ c4 sin h ¼ sin h;

a2 sin h þ c2 cos h ¼ sin h cos h; a3 sin h þ c3 cos h ¼ sin h cos h;

a4 sin h þ c4 cos h ¼ cos2 h: ð23Þ

Eq. (23) is only a system of the liner equations and has one unique solution. The solution is then given as

1 þ cos h sin h cos h sin h ; c1 ¼ a4 ¼  ; cos h þ sin h cos h þ sin h cos h sin h a2 ¼ c2 ¼ a3 ¼ c3 ¼ : cos h þ sin h

a1 ¼ c4 ¼

ð24Þ

We have determined the coefficients of the computational basis of the success state of the transformation. This is not enough to determine the explicit transformation defined by Eq. (19). By Eqs. (22.1) and (22.2) and using an important relation given by Eq. (13), we write the relation of the failure states at the output as,

hU1 jU2 i ¼ ðb1 sin h þ d1 cos hÞðb1 cos h þ d1 sin hÞ þ ðb2 sin h þ d2 cos hÞðb2 cos h þ d2 sin hÞ þ ðb3 sin h þ d3 cos hÞðb3 cos h þ d3 sin hÞ þ ðb4 sin h þ d4 cos hÞðb4 cos h þ d4 sin hÞ ¼ 1:

ð25Þ

Combining Eqs. (20.2) and (25), we obtain the coefficients of the computational basis of the failure states

b1 ¼ d1 ¼

pffiffiffi

c; bi ¼ di ¼ 0;

ð26Þ

where i = 2, 3, 4. Therefore, the explicit transformation defined by Eq. (19) is determined as the following explicit form:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ c2  cðxj00i þ yj01i þ yj10i  yj11iÞj0ip qffiffiffiffiffiffiffiffiffiffiffiffiffi þ c  c2 j00ij1ip ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j1ij0ij0ip ! 1 þ c2  cðyj00i þ yj10i þ yj01i þ xj11iÞj0ip qffiffiffiffiffiffiffiffiffiffiffiffiffi þ c  c2 j00ij1ip ; j0ij0ij0ip !

ð27Þ 

 c c 1 ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi . This transformation where x ¼ p1þ c ¼ 1þs and y ¼ p1 2 2 2

c cþ1

2

c cþ1

acting any one state chosen from the special set S = {js1i, js2i} can yield the following outputs:

pffiffiffi

pffiffiffiffiffiffiffiffiffiffiffiffi

cjs1 i2 j0ip þ 1  cj00ij1ip ; pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 2 js2 ij0ij0ip ! cjs2 i j0ip þ 1  cj00ij1ip : js1 ij0ij0ip !

ð28Þ

Here, we complete the derivation of the explicit transformation of the optimal 1 ? 2 PPQC in 2 dimensions to clone the special set

1959

{js1i, js2i}. The cloning machine is symmetric, i.e., c1 = c2 = c. In 1 as Duan’s result the asymmetric case of c1 – c2 satisfying c1 þ2 c2 6 1þs [4], by using our method above, one can also derive the explicit transformation. The expression of the transformation is only little complex. In the case of d P 3, the relation of hUijUji = 1 is satisfied if ci = c(i = 1, 2, . . . , d). The other set we select takes the following form:

1 jai ¼ pffiffiffi ðj0i þ j1iÞ; 2

1 jbi ¼ pffiffiffi ðj0i þ j2iÞ; 2

1 jci ¼ pffiffiffi ðj1i þ j2iÞ: 2 ð29Þ

The overlaps of the states are the same as hX jYi ¼ 12 ðX; Y ¼ a; b; c:X–YÞ. We design the optimal 1 ? 2 PPQC to clone three input states with equal success probabilities, i.e., c ¼ ci ¼ 1þh1X jYi ¼ 2 3

ði ¼ 1; 2; 3Þ. To cloning the input states given by Eq. (29), the optimal 1 ? 2 PPQC in 3 dimensions is defined as

rffiffiffiffiffiffi rffiffiffi 1 1 ð2j00iþ jð01Þiþ jð02Þi jð12ÞiÞj0ip þ j00ij1ip ; j0ij0ij0ip ! 12 6 rffiffiffiffiffiffi rffiffiffi 1 1 ðjð01Þiþ2j11i jð02Þiþ jð12ÞiÞj0ip þ j00ij1ip ; j1ij0ij0ip ! 12 6 rffiffiffiffiffiffi rffiffiffi 1 1 ðjð01Þiþ jð02Þiþ jð12Þiþ2j22iÞj0ip þ j00ij1ip ; j2ij0ij0ip ! 12 6 ð30Þ where the state j(ij)i is denoted as j(ij)i = jiji + jjii (i,j = 1, 2, 3, i – j). It is very easily to verify that this transformation acting on the input state jXi(X = a, b, c) outputs

rffiffiffi rffiffiffi 2 2 1 jX i j0ip þ j00ij1ip : jX ij0ij0ip ! 3 3

ð31Þ

One can observe that Eq. (31) is satisfied with all the constraints that we have imposed in Section 3. In the case of ci – 1þh1X jYi ¼ 2 ði ¼ 1; 2; 3Þ, the optimal 1 ? 2 probabilistically perfect cloning 3 should be expressed in another transformation, which is a further study. We should point out that our formulation of the optimal PPQC is nothing but a replacement of the previous results, where we assume that the failure state is only one and that the dimensionality of the probe is only 2-dimensional. From the mathematical point of view in determining the relation of the best success probability, the replacement is trivial. To clone the special set fjui igdi¼1 with huijuji = r 2 (0, 1)(i – j) and to require the optimal success probabil1r ities being equal as c ¼ ci ¼ 1r M , this change leads the result of hUijUji = 1, i.e., all the failure states are the same. The relation of the failure states helps us in determining the explicit transformation, as shown in our examples. Furthermore, one can observe that the structure of the transformation given by Eq. (2) has no essential difference to that of our transformation given by Eq. (27) in the practical physical realizations. Both of the two transformations are involved three particles of qubits. From the physical point of view, the optimal PPQC can be testified as the optimal DAQC [18–23] in experiment. Since the emergence of the optimal 1 ? 2 universal quantum cloning in 2 dimensions [6] given by Eq. (2), the optimal DAQC [3] has been found to have many applications to quantum information science [12–17]. Therefore, we are convinced that the optimal PPQC will find its applications to quantum information tasks. 5. Summary In this paper, we have derived the explicit formulation of the optimal PPQC to clone the special set with the equal success probabilities. With the help of the explicit formulation and by using our

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method presented, the explicit transformation of the optimal PPQC can be determined. As two examples, we present the explicit transformations of the optimal 1 ? 2 PPQC in 2 and 3 dimensions. Acknowledgments This work was funded by the National Science Foundation of China under Grant No. 10704001. References [1] W.K. Wootters, W.H. Zurek, Nature (London) 299 (1982) 802. [2] N. Gisin, G. Ribordy, W. Tittel, H. Zbinden, Rev. Mod. Phys. 74 (2002) 145; V. Scarani, H. Bechmann-Pasquinucci, N.J. Cerf, et al., Rev. Mod. Phys. 81 (2009) 1301. [3] V. Scarani, S. Iblisdir, N. Gisin, A. Acı´n, Rev. Mod. Phys. 77 (2005) 1225. [4] L.M. Duan, G.C. Guo, Phys. Rev. Lett. 80 (1998) 4999.

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