Optimal cloning of two known nonorthogonal quantum states

PHYSICAL REVIEW A 86, 022322 (2012)

Optimal cloning of two known nonorthogonal quantum states Wen-Hai Zhang,1,* Long-Bao Yu,2 Zhuo-Liang Cao,2 and Liu Ye3 1

2

Department of Physics, Huainan Normal University, Huainan 232001, People’s Republic of China School of Electronic and Information Engineering, Hefei Normal University, Hefei 230061, People’s Republic of China 3 School of Physics and Material Science, Anhui University, Hefei 230039, People’s Republic of China (Received 2 April 2012; published 22 August 2012) Two known nonorthogonal quantum states can be cloned either deterministically or probabilistically. In this paper, we investigate quantum cloning by combining these two extreme cases, i.e., a trade-off between the copy fidelity and the success probability. For a special set of two known nonorthogonal quantum states, we start with an explicit unitary transformation and derive the copy fidelity as a function of the success probability. The result shows that the higher is the copy fidelity, the less is the success probability. This quantum cloning may have important applications in quantum cryptography. DOI: 10.1103/PhysRevA.86.022322

PACS number(s): 03.67.−a, 03.65.−w

I. INTRODUCTION

The no-cloning theorem states that an arbitrary quantum state cannot be cloned [1], and this provides a fundamental support to the absolute security of quantum cryptography [2]. Though unknown quantum states are forbidden in perfect duplication, some information about them can be obtained by suitable physical processes, such as unitary transformations. In other words, after quantum cloning transformations, the copies at the outputs should ensemble to the most extent in the input states to be cloned. Therefore, the upper bound of the copy fidelity definitely is one of the essential issues in the quantum cloning process [3]. Quantum cloning has important applications in quantum information science [4–7]. Buˇzek and Hillery [8] first investigated optimal 1 → 2 universal quantum cloning (UQC) in two dimensions. Since then, quantum cloning has attracted a vast amount of research and has become an interesting theory [3] in quantum information science, and UQC has been further studied [9–13]. There are other quantum cloners, such as phase-covariant cloning [14–19], entanglement state cloning [20,21], and orthogonal qubit cloning [22–24]. Some kinds of quantum cloning machines of the discrete variables have been implemented in experiment [25–32]; continuous-variable cloning has been investigated both theoretically [33–38] and experimentally [39–42]. In comparison with the many fruitful contributions on quantum cloners, there is little literature on the cloning of two known nonorthogonal states. To clone two known nonorthogonal states, existing cloners are different and separate, i.e., either deterministic [43] or probabilistic [44]. The first cloner is statedependent cloning (SDC) [43], which can deterministically produce copies with a copy fidelity of less than a unit, and the second cloner is probabilistic quantum cloning (PQC) [44], which can create faithful copies, but sometimes may fail. From these two different cloners, it is natural to question whether there exists a uniform cloning, which can include SDC and PQC as two extreme cases, and may illuminate some relation between the copy fidelity and the success probability. In this paper, starting with a special set of two known nonorthogonal

*

[email protected]

1050-2947/2012/86(2)/022322(6)

states, we investigate optimal 1 → 2 uniform cloning of a set. We present the explicit transformation, and derive the expression of the copy fidelity as a function of the success probability. Our result shows that the higher is the fidelity, the less is the success probability. The most important practical use for SDC may be in the B92 protocol [45], where two known nonorthoganal quantum states are explored as quantum keys in quantum cryptography. An eavesdropper intercepts the keys sent by a sender, and exploits SDC to obtain two identical copies such that one copy is held by herself and the other is sent to a receiver. The eavesdropper may consider storing the clone and delaying the actual measurement until any further public communication between the sender and the receiver takes place. In order to detect the security of the communication channel, the receiver may measure the disturbance to judge the presence of the eavesdropper. This eavesdropping strategy was discussed in Ref. [46], and the disturbance as a criterion for eavesdropping in quantum cryptography is considered as the conventional method [47]. Using our cloner, the eavesdropper can sacrifice success probability to obtain two clones with a higher fidelity, and so the receiver cannot detect eavesdropping if only the disturbance criterion is used. Therefore, an important criterion for eavesdropping should involve both the disturbance and the detect probability of the keys. The paper is organized as follows. In Sec. II, we briefly review some previous contributions, including UQC, SDC, and PQC. In Ref. [43], the authors derived the explicit expression of the optimal fidelity and did not present the cloning coefficients; in Ref. [44], the authors presented the cloning formulation of PQC. So in Sec. III, we give the explicit transformations of optimal 1 → 2 SDC and PQC, as well as an illustration of our scheme. In Sec. IV, we design an explicit transformation and derive the expression of the copy fidelity as a function of the success probability, where the explicit transformation includes optimal SDC and optimal PQC as the two extreme cases. The paper ends with a summary. II. PREVIOUS CONTRIBUTIONS

We want to clone an unknown quantum state in the form |ψ = α|0 + β|1, with α and β being the complex numbers and satisfying |α|2 + |β|2 = 1. The unitary transformation of 022322-1

©2012 American Physical Society

WEN-HAI ZHANG, LONG-BAO YU, ZHUO-LIANG CAO, AND LIU YE

optimal 1 → 2 QUC in two dimensions is defined as [8] 2 1 |01 |02 |0a + (|01 + |10)12 |1a |01 |02 |0a → 3 6 = |(0) 12a , 2 1 |11 |12 |1a + (|10 + |01)12 |0a |11 |02 |0a → 3 6 (1) = |(1) 12a , where particle 1 is the input state to be cloned, particle 2 is the blank copy, and particle a is the ancillary system. After the input system |(in) 12a = |ψ|02 |0a is acted on by the (0) (1) evolution, the output is |(out) 12a = α|12a + β|12a . Tracing out particles 2 and a gives a reduced density matrix of clone 1 in the form ρ1 = Tr2a [| (out) 12a |]. Due to the symmetry of the transformation (1), we have ρ1 = ρ2 , and then the fidelity of each clone reads F (UQC) = ψ|ρ1(2) |ψ = 5/6.

The equality holds if and only if γ1(PQC) = γ2(PQC) = γ = 1/(1 + s). Obviously, the copies are faithful, i.e., F (PQC) = 1. Recently, optimal 1 → 2 PQC has been realized in experiment [49]. The authors presented a simple formulation [49] |χi |0|0p → γ (PQC) |χi |χi |0p + 1 − γ (PQC) | |1p , i = 1,2, (8) where the state | = − 1/(1 + tan4 θ/2)|00 − −4 1/(1 + tan θ/2)|11 is the normalized state of the composite system. Generally, the states given by Eq. (3) can be cloned either by SDC or by PQC. It is natural to ask the following question: Is there a uniform transformation which can include both SDC and PQC as two extreme cases? In the next section, we present explicit 1 → 2 transformations of optimal SDC and optimal PQC to clone the states given by Eq. (3). (PQC)

(2)

The fidelity has no relation to α and β, so the cloning machine is universal. UQC is used to produce clones deterministically, which can be regarded where the success probability is a unit. Suppose we are given with equal probability one quantum state from a set of two known nonorthogonal quantum states in the form

III. OPTIMAL 1 → 2 TRANSFORMATIONS OF SDC AND PQC

In Refs. [43,48], the authors derived the expression of the fidelity (5), and did not present the cloning coefficients. As in Ref. [43], the transformation of optimal 1 → 2 SDC takes the following form: |00 → a|00 + b(|01 + |10) + c|11, |10 → a|11 + b(|10 + |01) + c|00,

|χ1 = cos θ |0 + sin θ |1, |χ2 = sin θ |0 + cos θ |1, (3) where θ ∈ [0,π/4], with their scale product s = χ1 |χ2 = sin 2θ.

(4)

The fidelity of optimal 1 → 2 SDC is expressed as [43,48] √ 2 1 (SDC) F = + (1 + s)(3 − 3s + 1 − 2s + 9s 2 ) 2 32s × 3s 2 + 2s − 1 + (1 − s) 1 − 2s + 9s 2 . (5) To date, cloning coefficients are needed, and in the next section we will determine them. The states given by Eq. (3) can also be cloned by PQC [44]. The formulation of 1 → 2 PQC is given as [44] U (|χi | |P0 ) n (j ) = γi(PQC) |χi |χi |P0 + cij AB Pj , i = 1,2, (6) j =1

(9)

where the cloning coefficients a, b, and c are the real numbers. Due to the symmetry of the transformation given by Eq. (9), we get copy fidelities F1(SDC) (|χ1 ) = F2(SDC) (|χ1 ) and F1(SDC) (|χ2 ) = F2(SDC) (|χ2 ). The states given by Eq. (3) are also symmetric, and with a little calculation we can obtain F1(SDC) (|χ1 ) = F1(SDC) (|χ2 ) = f (SDC) . Thus the fidelity is f (SDC) = 14 {3a 2 + 2ab + 4b2 + 2bc + c2 + (a − 2b − c)(a + c) cos 4θ + 4(a + b)(b + c) sin 2θ }.

(10)

We next determine the cloning coefficients of the optimal fidelity given by Eq. (5). We want to derive the optimal fidelity from Eq. (10), which should be maximized under some constraints. For Eq. (9), the normalization condition is a 2 + 2b2 + c2 = 1,

(11)

and the inner product is given as

γi(PQC)

where is the success probability to obtain two perfect (j ) clones, and | AB are normalized states of the composite system AB. If the probe state |P0 is detected, the cloning process succeeds with a success probability γi(PQC) . Otherwise, the cloning process fails. In the case of two input states, the success probabilities are satisfied with γ1(PQC) + γ2(PQC) 1 . 2 1+s


(7)

2ac + 2b2 = 0.

(12)

Using the method of Lagrange multipliers and after some algebra, we determine the cloning coefficients as 1 b = (1 − csc 2θ + csc 2θ 1 − 2 sin 2θ + 9 sin2 2θ ), 8 (13) 1 b b 1 1− −1 . 1− +1 , c = l a= 2 2 2 2

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OPTIMAL CLONING OF TWO KNOWN NONORTHOGONAL . . .


So the optimal fidelity has an expression F (SDC)

√ 1 1+s s − 1 + 9s 2 − 2s + 1 2 2 [3 − 3s + 9s − 2s + 1] 1 − = + , 2 8 4s

(14)

which is an equivalent expression to Eq. (5). One character of optimal SDC can be thought of as F (SDC) < 1 and γ (SDC) = 1. For optimal 1 → 2 PQC, a simpler formulation may be given: |χi |0|0p → γ (PQC) |χi |χi |0p + 1 − γ (PQC) |00|1p , i = 1,2. (15) In order to derive the explicit transformation, we can construct the following transformation: |0|0|0p → γ (PQC) (a1 |00 + a2 |01 + a3 |10 + a4 |11)|0p + b 1 − γ (PQC) |00|1p , |1|0|0p → γ (PQC) (c1 |00 + c2 |01 + c3 |10 + c4 |11)|0p + d 1 − γ (PQC) |00|1p ,

(16)

where γ (PQC) = 1/(1 + s). We assume that all the parameters are real numbers. The normalized condition and the unitarity conditions are written down, respectively, as γ

(PQC)

4 i=1

ai2

+ (1 − γ

(PQC)

)b = 1, 2

γ

(PQC)

4

ci2

+ (1 − γ

(PQC)

)d = 1, 2

γ

i=1

(PQC)

4

ai ci + (1 − γ (PQC) )bd = 0.

After the transformation acts on |χ1 and |χ2 , we have |χ1 |0|0p → γ (PQC) [(cos θ a1 + sin θ c1 )|00 + (cos θ a2 + sin θ c2 )|01 + (cos θ a3 + sin θ c3 )|10 + (cos θ a4 + sin θ c4 )|11]|0p + |χ2 |0|0p → γ (PQC) [(sin θ a1 + cos θ c1 )|00 + (sin θ a2 + cos θ c2 )|01 + (sin θ a3 + cos θ c3 )|10 + (sin θ a4 + cos θ c4 )|11]|0p +

1 − γ (PQC) (cos θ b + sin θ d)|00|1p , 1 − γ (PQC) (sin θ b + cos θ d)|00|1p .

The output system should be satisfied with our definition given by Eq. (15), i.e., |χ1 |0|0p → γ (PQC) |χ1 ⊗2 |0p + 1 − γ (PQC) |00|1p = γ (PQC) [cos2 θ |00 + cos θ sin θ (|01 + |10) + sin2 θ |11]|0p + 1 − γ (PQC) |00|1p , |χ2 |0|0p → γ (PQC) |χ2 ⊗2 |0p + 1 − γ (PQC) |00|1p = γ (PQC) [sin2 θ |00 + cos θ sin θ (|01 + |10) + cos2 θ |11]|0p + 1 − γ (PQC) |00|1p .

By comparing with the two equations above, the coefficients of the computational basis must be equal, and thus we have a1 = c4 =

(17)

i=1

1 + cos θ sin θ cos θ sin θ , c1 = a4 = − , cos θ + sin θ cos θ + sin θ

(20) cos θ sin θ 1 a2 = c2 = a3 = c3 = , b=d= . cos θ + sin θ cos θ + sin θ Note that we may choose b = sin θ and d = cos θ . We wish Eq. (16) to be symmetric, and so we make the choice of b = d. Therefore, the explicit transformation of optimal 1 → 2 PQC may be defined as |0|0|0p → [x|00 + y(|01 + |10) − y|11]|0p √ + γ (PQC) s|00|1p , |1|0|0p → [x|11 + y(|10 + |01) − y|00]|0p √ + γ (PQC) s|00|1p , (21)

(18)

(19)

where x = (1 + γ (PQC) )/2 and y = (1 − γ (PQC) )/2. One characteristic of optimal PQC may be thought as F (PQC) = 1 and γ (PQC) < 1. Comparing Eq. (9) with Eq. (21), one can observe that the distributions of the computational basis of two clones have an equivalent symmetry. Therefore, we want to find a uniform transformation that includes the two cloners. In the next section, our task aims at finding a trade-off between the copy fidelity and the success probability. IV. UNIFORM TRANSFORMATION

We assume that a formulation of the uniform transformation can be satisfied with √ |χi |0|0p → γ |Xi |0p + 1 − γ |00|1p , i = 1,2, (22) where the composite state |Xi is a normalized state containing the two clones. If the probe state |0p is detected, we can obtain two clones with a success probability γ , and if |0p is detected,

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WEN-HAI ZHANG, LONG-BAO YU, ZHUO-LIANG CAO, AND LIU YE

uniform cloning fails. For this uniform cloning, the success probability γ = 1 means SDC, and γ = γ (PQC) is PQC. From Eq. (21), we assume the transformation as |0 → [A|00 + B(|01 + |10) + C|11]|0p + D|00|1p , |1 → [A|11 + B(|01 + |10) + C|00]|0p + D|00|1p , (23) where the cloning coefficients A, B, C, and D are the real members. The transformation acting on the states {|χ1 ,|χ2 } yields + B(cos θ + sin θ )(|01 + |10) + (C cos θ + A sin θ )|11]|0p + D(cos θ + sin θ)|00|1p √ = γ |X1 |0p + 1 − γ |00|1p ,

A2 + 2B 2 + C 2 + D 2 = 1,

|χ2 (out) = [(A sin θ + C cos θ )|00 + B(sin θ + cos θ )(|01 + |10) + (C sin θ + A cos θ )|11|0p ] (24)

The two-clone state |Xi (i = 1,2) should be the normalized state, and so the normalization factor is Ni2 = (A2 + 2B 2 + C 2 ) + sin 2θ (2AC + 2B 2 ) = γ .

(25)

From Eq. (24), the cloning coefficients of the state |00|1p should be equal, so we have 1−γ , (26) D(sin θ + cos θ ) = 1 − γ ⇒ D 2 = 1+s

F =

which implies that the parameter D is known. We should point out that the clone states |X1 and |X2 have different expressions, but the clone fidelities are the same, which has been explained below Eq. (9). The fidelity is ready to obtain 1 {3A2 + 2AB + 4B 2 + 2BC + C 2 f = 4γ + (A − 2B − C)(A + C) cos 4θ + 4(A + B)(B + C) sin 2θ }. (27) Note that Eq. (27) can be reduced to Eq. (10) when taking A = a, B = b, C = c, and γ = γ (SDC) = 1. From Eq. (23), the normalization condition and the unitary condition are written as

|χ1 (out) = [(A cos θ + c sin θ )|00

+ D(sin θ + cos θ)|00|1p √ = γ |X2 |0p + 1 − γ |00|1p .


2AC + 2B 2 + D 2 = 0.

(28)

Restricted by Eq. (28), our task is to maximize the fidelity given by Eq. (27). The cloning coefficients and the fidelity should be a function of the success probability. Since the parameter D is given by Eq. (26), we only repeat the procedure of the derivation of Eq. (14), and determine the cloning coefficients s 2 − 1 + 1 + 9s 4 + 16s 2 γ (1 + s) − 10s 2 B= , 8s(1 + s) 1−γ D= , 1+s (29) 1 2 2 A = ( 1 − 4B − 2D + 1), 2 1 C = ( 1 − 4B 2 − 2D 2 − 1), 2 and the fidelity

1 1 γ + [3 − 3s 2 + 1 + 9s 4 + 16γ s 2 (1 + s) − 10s 2 ] 2γ 4 2 s − 1 + 1 + 9s 4 + 16γ s 2 (1 + s) − 10s 2 2 s − 1 − 2γ . − × 1+s 4s

(30)

The explicit optimization procedure is reported in the Appendix. From Eq. (30), it can be easily testified that F = F (SDC) when γ = 1, and F = 1 when γ (PQC) = 1/(1 + s). In Fig. 1, we show the relation of the fidelity and the inner products of the input states for some particular parameters γ = 0.7,0.8,0.9,1. When the input state is are in the range ,1], the fidelity is as shown in Fig. 1, and when of s ∈ ( 1−γ γ s ∈ [0, 1−γ ], which can only be cloned by PQC, the fidelity γ reaches a unit. To clone two known nonorthogonal states defined by Eq. (3), we have derived the relation between the fidelity and the success probability. The relation suggests that the fidelity and the success probability should be considered together when gaining information from a set of two known nonorthogonal quantum states.

FIG. 1. (Color online) The relation of the fidelity and the inner products of the input state as γ = 0.7,0.8,0.9,1.

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OPTIMAL CLONING OF TWO KNOWN NONORTHOGONAL . . .

SDC can be regarded as the state estimation [7,50] and PQC as the state discrimination [51], therefore, in the limit M → ∞, the 1 → M uniform transformation may reveal some relation between the state estimation and the state discrimination, which may be an interesting issue both in quantum information and in quantum mechanics. As discussed in Refs. [43,46,48], SDC may be regarded as an efficient eavesdropping strategy for the B92 protocol [45]. The disturbance produced by SDC is defined as D (SDC) = (SDC) 1− 0.0102, where the equality holds when s = √F 1/ 2. The legitimate party may measure the disturbance to judge the presence of the eavesdropper. Using our cloner, the eavesdropper can sacrifice the success probability in order to obtain two clones with a higher fidelity, and thus the receiver cannot detect eavesdropping if he only √ uses the disturbance criterion. For example, when s = 1/ 2, the success probability γ = 0.9 gives the disturbance D = 1 − F = 0.007. Therefore, a reasonable criterion for eavesdropping should involve both the disturbance and the detect probability of the keys. V. SUMMARY

In this paper, we investigate a uniform transformation by combining SDC and PQC. Starting with a special set of two known nonorthogonal quantum states, we design the uniform transformation and derive the explicit expression of the copy fidelity as a function of the success probability, together with the explicit unitary transformation. This quantum cloning may have important applications in quantum cryptography. ACKNOWLEDGMENTS

The present authors are grateful to anonymous referees for helpful comments. This work was funded by the National Science Foundation of China under Grants No. 11074002, No. 61073048, and No. 11104057, the Natural Science Foundation of the Education Department of Anhui Province of China under Grants No. KJ2010ZD08 and No. KJ2012A245, and the Postgraduate Program of Huainan Normal University.

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PHYSICAL REVIEW A 86, 022322 (2012) APPENDIX: OPTIMIZATION FOR THE UNIFORM TRANSFORMATION

We use MATHEMATIC 5.0 for the derivation of the optimal fidelity. We rewrite Eqs. (27) and (28):

f =

A2 + 2B 2 + C 2 + D 2 = 1,

(A1)

2AC + 2B 2 + D 2 = 0,

(A2)

1 {3A2 + 2AB + 4B 2 + 2BC + C 2 4γ + (A − 2B − C)(A + C) cos 4θ + 4(A + B)(B + C) sin 2θ }.

(A3)

We want to maximize the value of the fidelity (A3) under the conditions (A1) and (A2). Subtracting Eq. (A2) from Eq. (A1), we obtain (A − C)2 = 1, and, by contrasting with Eq. (13), the reasonable solution should be A − C = 1.

(A4)

After solving Eqs. (A2) and (A4), we obtain two groups of the solutions of the parameters A and C, as functions of the parameter B. The reasonable solution is √ A = 12 (1 + 1 − 4B 2 − 2D 2 ), (A5) √ C = 12 (−1 + 1 − 4B 2 − 2D 2 ). Note that D 2 = 1−γ is given by Eq. (26). Inserting the 1+s parameters A, C, and D into Eq. (A3), the maximum of the fidelity, only as a function f (B) of the parameter B, can be easily derived by using the method of Lagrange multipliers. (B) Solving ∂f∂B = 0 will give two explicit values of the parameter B, and the reasonable value reads s 2 − 1 + 1 + 9s 4 + 16s 2 γ (1 + s) − 10s 2 B= . (A6) 8s(1 + s) Thus, the parameters A and C are determined. By inserting the four parameters into Eq. (A3), the optimal fidelity is expressed as Eq. (30).

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