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Strategies for state-dependent quantum deleting
Physics Letters A 327 (2004) 123–128 www.elsevier.com/locate/pla
Strategies for state-dependent quantum deleting Wei Song ∗ , Ming Yang, Zhuo-Liang Cao School of Physics & Material Science, Anhui University, Hefei 230039, PR China Received 16 March 2004; received in revised form 14 May 2004; accepted 17 May 2004 Available online 28 May 2004 Communicated by P.R. Holland
Abstract A quantum state-dependent quantum deleting machine is constructed. We obtain a upper bound of the global fidelity on N-to-M quantum deleting from a set of K non-orthogonal states. Quantum networks are constructed for the above statedependent quantum deleting machine when K = 2. Our deleting protocol only involves a unitary interaction among the initial copies, with no ancilla. We also present some analogies between quantum cloning and deleting. 2004 Elsevier B.V. All rights reserved. PACS: 03.67.-a; 03.65.-w Keywords: Quantum cloning; Quantum deleting
1. Introduction Manipulation and extraction of quantum information are important tasks in building quantum computer. As is known, the copying and deleting of information in a classical computer are inevitable operations whereas similar operations cannot be realized perfectly in quantum computers. Linearity of quantum mechanics unveils that we cannot duplicate an unknown quantum state accurately [1]. This has been proven by Wootters and Zurek [1] and Dieks [2] which called the quantum no-cloning theorem. Though exact cloning is not possible, in the literature various * Corresponding author.
E-mail addresses: [email protected] (W. Song), [email protected] (M. Yang), [email protected] (Z.-L. Cao). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.05.036
cloning machines have been proposed [3–11] which operate either in a deterministic or probabilistic way. Corresponding to the quantum no-cloning theorem, Pati and Braunstein [12] demonstrated that the linearity of quantum mechanics also forbids one to delete one unknown state ideally against a copy [12], which is called the quantum no-deleting principle and complements the quantum no-cloning theorem in spirit. If quantum deleting could be done, then one would create a standard blank state onto which one could copy an unknown state approximately, by deterministic cloning or exactly, by probabilistic cloning process. When memory in a quantum computer is scare, quantum deleting may play an important role, and one could store new information in an already computed state by deleting the old information. At first glance it seems that quantum deleting is just the reverse of quantum cloning, actually it is not so. In Ref. [13],
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W. Song et al. / Physics Letters A 327 (2004) 123–128
through the analysis of a approximate 2 → 1 deleting machine, the author shows the fidelity of the two modes are different for the deleting operation, whereas in the cloning operation the reduced density matrix of both the modes are same. As indicated emphatically in Refs. [12,14], discussing probabilistic and approximate quantum deleting will not only contribute to processing quantum information, but further understand some connections between quantum cloning and quantum deleting. Recently some probabilistic deleting machines have been established [15,16]. The purpose of this Letter is to investigate how well one can deleting quantum states. Here, we discuss the problem of approximate N -to-M quantum deleting from a set of K non-orthogonal states, and our deleting protocol only involves a unitary interaction among the initial copies, with no ancilla. In Section 2, we prove the impossibility of deleting a copy from two copies of K non-orthogonal states, then the analytic solution for N -to-M quantum deleting from a set of two nonorthogonal states is given, quantum networks are also constructed for this quantum deleting machine. In Section 3, we give an upper bound of the global fidelity on N -to-M quantum deleting from a set of K nonorthogonal states, and some analogies between quantum cloning and deleting are presented. A summary is given in Section 4.
containing K non-orthogonal states. We show a copy from two copies of K non-orthogonal states cannot be perfectly deleted.
2. State-dependent deleting machine
U ΨiN = |Φi |Σ⊗(N−M) ,
Firstly let us review the definition for N -to-M quantum deleting introduced by Pati and Braunstein [13]. In general the quantum deleting operation is defined for N unknown states |Ψ ⊗N such that the linear operator acts on the combined Hilbert space and deletes N − M copies and keeps M copies intact. It is defined by
where |ΨiN are the N -fold tensor product states |ΨiN = |Ψi 1 ⊗ · · · ⊗ |Ψi N which are prepared in the same state, and |Ψi is chosen from a set of K non-orthogonal states, |Φi is the output state after the machine deleting |Ψi ⊗(N−M) . The ideal output state after the machine deleting |Ψi ⊗(N−M) is |Ψi ⊗M . To characterize how closely the output state after the machine deleting |Ψi ⊗(N−M) resembles the ideal state|Ψi ⊗M , we will utilize the global fidelity introduced by Bruss et al. [5], which is defined as
|Ψ ⊗N |A → |Ψ ⊗M |Σ⊗N−M |Aψ ,
(2.1)
where |Σ is the blank state of a qubit, |A is the initial and |AΨ is the final state of the ancilla. Pati and Braunstein has proven the impossibility of deleting an unknown copy of a quantum state in any finitedimensional Hilbert space with linearity. We will consider the following situations [13], if K qubits need not be in orthogonal states nor in completely arbitrary states but they would be chosen secretly from a set
Proof. We restrict ourselves to the K non-orthogonal states |Ψi (i = 1, 2, . . . , K), for simplicity we will not consider the ancilla. If the two input states are identical then machine deletes a copy and if they are different then it allows them to pass through without any change. The deleting machine is a unitary operator which acts on the combined Hilbert space of K qubits and would create the following transformation: U |Ψi |Ψi = |Ψi |Σ, U |Ψi |Ψj = |Ψi |Ψj (i, j = 1, 2, . . . , K, i = j ), using the unitary condition we have Ψi |Ψj 2 = Ψi |Ψj , Ψi |Ψj = Σ|Ψj , 1 = Σ|Ψj , these equation can be satisfied only if |Ψi = |Ψj = |Σ, which contradict with the assumption that the K states are non-orthogonal. Thus, each of a copy from two copies of K non-orthogonal states cannot be deleted by a unitary machine. Within similar approach, we could prove the impossibility of N -to-M quantum deleting from a set of K non-orthogonal states. Since it is impossible to realize perfect N -to-M quantum deleting from a set of K non-orthogonal states, we may discuss how well can we construct an approximately imperfect deleting machine. If the ancilla is not considered, a quantum approximate deleting device is a quantum machine described by the following unitarity transformation
K 2 F= ηi ΨiM Φi ,
(2.2)
(2.3)
i=1
where ηi denotes a priori probability of the state |ΨiN . Noticing the unitary condition of Eq. (2.2), we have
W. Song et al. / Physics Letters A 327 (2004) 123–128
equation Ψi |Ψj N = Φi |Φj .
(2.4)
Now our purpose is to maximizing F under the condition of Eq. (2.4). In the following we will give the solution to this optimization problem when K = 2. For simplicity and clarity we replace the index i by the binary notation ±. We could prove that the optimal state |Φ± lie in the subspace spanned by ideal state |Ψ±M (see Appendix A). Let the states |α and |β be an orthonormal basis for the subspace spanned by |Ψ±M , then the ideal output state after the machine deleting |Ψ± ⊗(N−M) can be parametrized as M Ψ = cos ϕ|α ± sin ϕ|β, (2.5) ± where ϕ ∈ [0, π/4]. The states to be optimized can be expressed as |Φ± = cos φ± |α + sin φ± |β.
(2.6)
Under the condition of K = 2, Eq. (2.4) becomes Ψ+ |Ψ− N = Φ+ |Φ− which implies Φ+ |Φ− is a constant, substituting of Eq. (2.6) to Φ+ |Φ− we can get φ+ − φ− is a constant. Using the definition of Eq. (2.3), F can be written as F = η+ cos (ϕ − φ+ ) + η− cos (ϕ + φ− ). 2
2
(2.7)
We can now use the method of Lagrange multipliers for the constraint φ+ − φ− = C, which gives these equations ∂F + λ = 0, ∂φ+ ∂F − λ = 0. ∂φ−
(2.8a) (2.8b)
For these two equations to be satisfied we must have cos2 (φ+ + φ− ) =
cos2 (2ϕ − φ+ + φ− ) 1 − 4η+ η− sin2 (2ϕ − φ+ + φ− )
(2.9)
using the constraint η+ + η− = 1, F can also be expressed as F=
1 1 + cos(φ+ + φ− ) cos(2ϕ − φ+ + φ− ) 2 + (η+ − η− ) sin(φ+ + φ− ) sin(2ϕ − φ+ + φ− ) . (2.10)
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Without loss of generality, we can take φ+ and φ− to lie in the first and fourth quadrants, then the optimal global fidelity can be obtained by substituting the positive root of cos2 (φ+ + φ− ) and sin2 (φ+ + φ− ) into Eq. (2.10), with the constraint η+ + η− = 1, after little algebra we can write the optimal global fidelity as 1/2 1 1 + 1 − 4η+ η− sin2 (2ϕ − φ+ + φ− ) . 2 (2.11) The value of optimal global fidelity can reach unity only when one of the priori probabilities is zero. We should point out that our discussion only apply to the restricted case when the ancilla is not involved, and the value of optimal global fidelity for N -to-M quantum deleting from a set of two non-orthogonal states has the same form as N -to-M quantum cloning, but the meaning of the parameter is different. ϕ in Eq. (2.11) measures the distinctness of the ideal output state after the machine deleting |Ψi ⊗(N−M) . The ideal output state after the machine deleting |Ψi ⊗(N−M) can be written in the basis {|Ψ+M , |Ψ−M } as |Φ± = γ± Ψ+M + δ± Ψ−M , (2.12) Fopt =
sin(ϕ−φ± ) ±) where γ± = sin(ϕ+φ sin 2θ , δ± = sin 2θ . In the following we would show how to construct the quantum networks to realize this process, we adopt the distinguishability transfer gate (D-gate) operation introduced in Ref. [9]. This operation compresses all the information of the N input copies into one qubit and acts as follows D(θ1 , θ2 )Ψ± (θ1 ) Ψ± (θ2 ) = Ψ± (θ3 ) |Σ. (2.13)
The unitarity condition requires cos 2θ3 = cos 2θ1 × cos 2θ2 , this condition, together with 0 θj π/4, suffices to determine θ3 uniquely. D(θ1 , θ2 ) may be decomposed into universal operations which have been illustrated in Ref. [9]. Since D(θ1 , θ2 ) is Hermitian [9], it can also transfer state |Ψ± (θ3 )|Σ back to |Ψ± (θ1 )|Ψ± (θ2 ). This accomplishes the process of information decompression. We show the D-gate operation can be used as an element in a network for N -to-M quantum deleting. Define DN = D1 (θ1 , θN−1 )D2 (θ1 , θN−2 ) · · · DN−1 (θ1 , θ1 ), where the operation Dj (θ1 , θN−j ) compresses the information of particles j, j + 1 to particle j , and angles θj are uniquely determined by cos 2θj +1 = cos 2θ1 cos 2θj
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Fig. 1. Network for optimum approximate deleting. Here N = 6 and M = 2. The distinguishability of six initial states is transferred to a single qubit. Then a unitary operation T is performed on this qubit, finally a further distinguishability transfer gates decompose the distinguishability from state 1 to states 1, 2 which completes the deleting of the four states.
(0 θj π/4). DN acts as ⊗N DN Ψ± (θ1 ) = Ψ± (θN ) 1 |Σ⊗(N−1) .
(2.14)
This is the first stage of the deleting operation which moves the distinguishability from particles 2, . . . , N to particle 1. Noting the optimal output |Ξ opt = (γ± |Ψ+M + δ± |Ψ−M )|Σ⊗(N−M) , we should then perform a unitary operator T on |Ψ± (θN ) acts as T Ψ± (θN ) = γ± Ψ+ (θM ) + δ± Ψ− (θM ) . (2.15) The unitarity of T have been discussed in Ref. [9]. The final stage of the deleting operation is obvious, we should perform D-gate operation which decompose the distinguishability from particle 1 to particle 2, . . . , M, DM is given by DM = D1 (θ1 , θM−1 ) × D2 (θ1 , θM−2 ) · · · DM−1 (θ1 , θ1 ), DM completes the deleting procedure is given by the transformation DM γ± Ψ+ (θM ) 1
+ δ± Ψ− (θM ) 1 |Σ2 |Σ3 · · · |ΣN
= γ± Ψ+M + δ± Ψ−M |Σ⊗(N−M) . (2.16) Thus the desired optimal state |Ξ opt = (γ± |Ψ+M + δ± |Ψ−M )|Σ⊗(N−M) have been obtained by applying the above procedure. The whole process is illustrated in Fig. 1, and we have taken time to advance from left to right. We may find that the above network structurally resembles the reverse of the optimum approximate cloning network in Ref. [9]. Are these networks timereverse of each other? It is well known that we
could always find D −1 and T −1 to transform the output state to the original input state. Actually since the distinguishability transfer gate is both Hermitian and unitary, then it is its own inverse, so we have D −1 = D. However from the definition in Eq. (2.15), we know the T gate is not necessary Hermitian, so we conclude that the state-dependent deleting network and the state-dependent cloning network in Ref. [9] are time-reverse of each other only when T −1 = T .
3. Upper bound of state-dependent deleting when the set contains N states In this section, we will investigate the state-dependent quantum deleting when the state set contains more than two states. Similar to the above discussion, the ancilla is not involved. It is still difficult to solve this problem completely, we can only give some bounds for this problem. Now our purpose is to maximizing F under the condition of Eq. (2.4). In Ref. [10], the author have obtained the upper bound of F which is given by 1 1 F + 2 2n(n − 1)
n
cos(aj,k − aj,k )
(3.1)
k=j =1
in the quantum deleting process cos aj,k and cos aj,k = should be defined as cos aj,k = ΨjM |ΨkM , cos aj,k ΨjN |ΨkN , we can still used the method proposed in Ref. [10] to get a more stringent bound.
W. Song et al. / Physics Letters A 327 (2004) 123–128
We can find the above deleting machine may be considered as a converse process for which this cloning machine performs. Next we shall give some explanation of the connection and difference between quantum cloning and quantum deleting. As shown in Ref. [4], a quantum cloning machine can be considered as a universal device transforming quantum information into classical information, we may also think quantum deleting machine as a device transforming quantum information into classical information, actually a perfect quantum deleting can be realized by performing a single-particle measurement on each state which to be deleted, but as it is well known that measurement is not unitary process, so we conclude that quantum cloning and quantum deleting can both be regarded as a device which perform a unitary operation to distill classical information from quantum information. This is not to say quantum deleting is just the reverse of quantum cloning, there are still some difference between them which we will generalize as follows, firstly, a quantum cloning process could be think as a swapping operation between the blank qubit and the cloning machine state, but a swapping operation between the cloning state and the deleting machine state could not be think as a successful deleting process. Secondly, the fidelity of the each mode is different for the deleting operation whereas it has the same value in the cloning process for symmetric case.
4. Conclusions In summary, we have constructed the state-dependent deleting machines which may be formally thought of the converse device of the corresponding cloning machines to a certain extent. We also analyze the connection and difference between quantum cloning and quantum deleting. Our results may have potential applications in information processing because it provides strategies for state-dependent quantum deleting in a quantum computer. It tells us how to control the deleting efficiencies from a set of K non-orthogonal states when the ancilla is not considered. On the other hand, one may store new information in an already computed state by deleting copies of the old states. Moreover, we expect our results to play a fundamental role in future understanding of quantum information theory.
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Acknowledgements We are grateful to the referee for valuable comments and helpful suggestions. This work is supported by Anhui Provincial Natural Science Foundation under Grant No. 03042401, the Key Program of the Education Department of Anhui Province under Grant No. 2004kj005zd, and the Talent Foundation of Anhui University.
Appendix A. Optimization for state-dependent deleting when the state set contains two states Let us assume that Φ+ and Φ− have some contribution that does not lie in the space spanned by Ψ+M and Ψ−M . Then the form of Φ+ and Φ− can be written as |Φ+ = a0 Ψ+M + b0 Ψ_M + c0 |C0 , (A.1) M M |Φ− = a1 Ψ+ + b1 Ψ_ + c1 |C1 , (A.2) where vectors C0 and C1 are normalized and lie in the subspace orthogonal to the space spanned by Ψ+M and Ψ−M . The unitarity of Eq. (A.1) and Eq. (A.2) gives the constraints
ϑ1 = Re a0∗ a1 + b0∗ b1 + S 2 a0∗ b1 + b0∗ a1 + c0∗ c1 C0 |C1 − S = 0, (A.3)
∗ ϑ2 = Im a0 a1 + b0∗ b1 + S 2 a0∗ b1 + b0∗ a1 + c0∗ c1 C0 |C1 = 0, (A.4) ϑ3 = |a0 |2 + |b0 |2 + 2S 2 Re a0∗ b0 + |c0 |2 − 1 = 0,
(A.5)
ϑ4 = |a1 | + |b1 | + 2S 2 Re a1∗ b1 + |c1 |2 − 1 = 0,
(A.6)
where S = Ψ+N |Ψ−N . The global fidelity is given by 2 2 F = η+ a0 + b0 S 2 + η− b1 + a1 S 2 .
(A.7)
2
2
Inserting constraints ϑ3 and ϑ4 into Eq. (A.7) yields
F = (η+ + η− ) − 1 − S 4 η+ |b0 |2 + η− |a1 |2
− η+ |c0 |2 + η− |c1 |2 . (A.8)
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Using the method of Langrange multipliers for the remaining two constraints, we get equations ∂ϑi ∂F + = 0, λi ∂|c0 | ∂|c0 | 2
last equation into the other then we could find c0 = |c1 | = 0. Thus, we conclude Φ+ and Φ− lie in the subspace spanned by Ψ+M and Ψ−M .
(A.9)
i=1
References
∂ϑi ∂F + = 0, λi ∂|c1 | ∂|c1 |
(A.10)
∂F ∂ϑi + = 0. λi ∂C0 |C1 ∂C0 |C1
(A.11)
2
i=1
2
i=1
Without loss of generality, we can consider c0 and C0 |C1 real, while c1 should be a general complex (c0 = |c1 |eiδ ), then the above three equations can be simplified as −c0 + λ1 Re |c1 |eiδ C0 |C1 + λ2 Im |c1 |eiδ C0 |C1 = 0, (A.12) iδ −|c|1 + λ1 Re c0 e C0 |C1 + λ2 Im c0 eiδ C0 |C1 = 0, (A.13) λ1 Re c0 |c1 |eiδ + λ2 Im c0 |c1 |eiδ = 0. (A.14) We can multiplying Eq. (A.12) by c0 , Eq. (A.13) by |c1 | and Eq. (A.14) by C0 |C1 and inserting the
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
W.K. Wootters, W.H. Zurek, Nature 299 (1982) 802. D. Dieks, Phys. Lett. A 92 (1982) 271. V. Buzek, M. Hillery, Phys. Rev. A 54 (1996) 1844. N. Gisin, S. Massar, Phys. Rev. Lett. 79 (1997) 2153. D. Bruß, D. DiVincenzo, A. Ekert, C. Fuchs, C. Macchiavello, J. Smolin, Phys. Rev. A 57 (1998) 2368. R.F. Werner, Phys. Rev. A 58 (1998) 1827. L.M. Duan, G.C. Guo, Phys. Rev. Lett. 80 (1998) 4999. H. Fan, K. Matsumoto, X.B. Wang, M. Wadati, Phys. Rev. A 65 (2002) 012304. A. Chefles, S.M. Barnett, Phys. Rev. A 60 (1999) 136. Y.J. Han, Y.S. Zhang, G.C. Guo, Phys. Rev. A 66 (2002) 052301. A.K. Pati, Phys. Rev. Lett. 83 (1999) 2849. A.K. Pati, S.L. Braunstein, Nature 404 (2000) 164. A.K. Pati, S.L. Braunstein, quant-ph/0007121. W.H. Zurek, Nature 404 (2000) 131. J. Feng, Y.-F. Gao, J.-W. Cao, J.-S. Wang, M.-S. Zhan, Phys. Lett. A 292 (2001) 12. D.-W. Qiu, Phys. Rev. A 65 (2002) 052329.
Strategies for state-dependent quantum deleting Wei Song ∗ , Ming Yang, Zhuo-Liang Cao School of Physics & Material Science, Anhui University, Hefei 230039, PR China Received 16 March 2004; received in revised form 14 May 2004; accepted 17 May 2004 Available online 28 May 2004 Communicated by P.R. Holland
Abstract A quantum state-dependent quantum deleting machine is constructed. We obtain a upper bound of the global fidelity on N-to-M quantum deleting from a set of K non-orthogonal states. Quantum networks are constructed for the above statedependent quantum deleting machine when K = 2. Our deleting protocol only involves a unitary interaction among the initial copies, with no ancilla. We also present some analogies between quantum cloning and deleting. 2004 Elsevier B.V. All rights reserved. PACS: 03.67.-a; 03.65.-w Keywords: Quantum cloning; Quantum deleting
1. Introduction Manipulation and extraction of quantum information are important tasks in building quantum computer. As is known, the copying and deleting of information in a classical computer are inevitable operations whereas similar operations cannot be realized perfectly in quantum computers. Linearity of quantum mechanics unveils that we cannot duplicate an unknown quantum state accurately [1]. This has been proven by Wootters and Zurek [1] and Dieks [2] which called the quantum no-cloning theorem. Though exact cloning is not possible, in the literature various * Corresponding author.
E-mail addresses: [email protected] (W. Song), [email protected] (M. Yang), [email protected] (Z.-L. Cao). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.05.036
cloning machines have been proposed [3–11] which operate either in a deterministic or probabilistic way. Corresponding to the quantum no-cloning theorem, Pati and Braunstein [12] demonstrated that the linearity of quantum mechanics also forbids one to delete one unknown state ideally against a copy [12], which is called the quantum no-deleting principle and complements the quantum no-cloning theorem in spirit. If quantum deleting could be done, then one would create a standard blank state onto which one could copy an unknown state approximately, by deterministic cloning or exactly, by probabilistic cloning process. When memory in a quantum computer is scare, quantum deleting may play an important role, and one could store new information in an already computed state by deleting the old information. At first glance it seems that quantum deleting is just the reverse of quantum cloning, actually it is not so. In Ref. [13],
124
W. Song et al. / Physics Letters A 327 (2004) 123–128
through the analysis of a approximate 2 → 1 deleting machine, the author shows the fidelity of the two modes are different for the deleting operation, whereas in the cloning operation the reduced density matrix of both the modes are same. As indicated emphatically in Refs. [12,14], discussing probabilistic and approximate quantum deleting will not only contribute to processing quantum information, but further understand some connections between quantum cloning and quantum deleting. Recently some probabilistic deleting machines have been established [15,16]. The purpose of this Letter is to investigate how well one can deleting quantum states. Here, we discuss the problem of approximate N -to-M quantum deleting from a set of K non-orthogonal states, and our deleting protocol only involves a unitary interaction among the initial copies, with no ancilla. In Section 2, we prove the impossibility of deleting a copy from two copies of K non-orthogonal states, then the analytic solution for N -to-M quantum deleting from a set of two nonorthogonal states is given, quantum networks are also constructed for this quantum deleting machine. In Section 3, we give an upper bound of the global fidelity on N -to-M quantum deleting from a set of K nonorthogonal states, and some analogies between quantum cloning and deleting are presented. A summary is given in Section 4.
containing K non-orthogonal states. We show a copy from two copies of K non-orthogonal states cannot be perfectly deleted.
2. State-dependent deleting machine
U ΨiN = |Φi |Σ⊗(N−M) ,
Firstly let us review the definition for N -to-M quantum deleting introduced by Pati and Braunstein [13]. In general the quantum deleting operation is defined for N unknown states |Ψ ⊗N such that the linear operator acts on the combined Hilbert space and deletes N − M copies and keeps M copies intact. It is defined by
where |ΨiN are the N -fold tensor product states |ΨiN = |Ψi 1 ⊗ · · · ⊗ |Ψi N which are prepared in the same state, and |Ψi is chosen from a set of K non-orthogonal states, |Φi is the output state after the machine deleting |Ψi ⊗(N−M) . The ideal output state after the machine deleting |Ψi ⊗(N−M) is |Ψi ⊗M . To characterize how closely the output state after the machine deleting |Ψi ⊗(N−M) resembles the ideal state|Ψi ⊗M , we will utilize the global fidelity introduced by Bruss et al. [5], which is defined as
|Ψ ⊗N |A → |Ψ ⊗M |Σ⊗N−M |Aψ ,
(2.1)
where |Σ is the blank state of a qubit, |A is the initial and |AΨ is the final state of the ancilla. Pati and Braunstein has proven the impossibility of deleting an unknown copy of a quantum state in any finitedimensional Hilbert space with linearity. We will consider the following situations [13], if K qubits need not be in orthogonal states nor in completely arbitrary states but they would be chosen secretly from a set
Proof. We restrict ourselves to the K non-orthogonal states |Ψi (i = 1, 2, . . . , K), for simplicity we will not consider the ancilla. If the two input states are identical then machine deletes a copy and if they are different then it allows them to pass through without any change. The deleting machine is a unitary operator which acts on the combined Hilbert space of K qubits and would create the following transformation: U |Ψi |Ψi = |Ψi |Σ, U |Ψi |Ψj = |Ψi |Ψj (i, j = 1, 2, . . . , K, i = j ), using the unitary condition we have Ψi |Ψj 2 = Ψi |Ψj , Ψi |Ψj = Σ|Ψj , 1 = Σ|Ψj , these equation can be satisfied only if |Ψi = |Ψj = |Σ, which contradict with the assumption that the K states are non-orthogonal. Thus, each of a copy from two copies of K non-orthogonal states cannot be deleted by a unitary machine. Within similar approach, we could prove the impossibility of N -to-M quantum deleting from a set of K non-orthogonal states. Since it is impossible to realize perfect N -to-M quantum deleting from a set of K non-orthogonal states, we may discuss how well can we construct an approximately imperfect deleting machine. If the ancilla is not considered, a quantum approximate deleting device is a quantum machine described by the following unitarity transformation
K 2 F= ηi ΨiM Φi ,
(2.2)
(2.3)
i=1
where ηi denotes a priori probability of the state |ΨiN . Noticing the unitary condition of Eq. (2.2), we have
W. Song et al. / Physics Letters A 327 (2004) 123–128
equation Ψi |Ψj N = Φi |Φj .
(2.4)
Now our purpose is to maximizing F under the condition of Eq. (2.4). In the following we will give the solution to this optimization problem when K = 2. For simplicity and clarity we replace the index i by the binary notation ±. We could prove that the optimal state |Φ± lie in the subspace spanned by ideal state |Ψ±M (see Appendix A). Let the states |α and |β be an orthonormal basis for the subspace spanned by |Ψ±M , then the ideal output state after the machine deleting |Ψ± ⊗(N−M) can be parametrized as M Ψ = cos ϕ|α ± sin ϕ|β, (2.5) ± where ϕ ∈ [0, π/4]. The states to be optimized can be expressed as |Φ± = cos φ± |α + sin φ± |β.
(2.6)
Under the condition of K = 2, Eq. (2.4) becomes Ψ+ |Ψ− N = Φ+ |Φ− which implies Φ+ |Φ− is a constant, substituting of Eq. (2.6) to Φ+ |Φ− we can get φ+ − φ− is a constant. Using the definition of Eq. (2.3), F can be written as F = η+ cos (ϕ − φ+ ) + η− cos (ϕ + φ− ). 2
2
(2.7)
We can now use the method of Lagrange multipliers for the constraint φ+ − φ− = C, which gives these equations ∂F + λ = 0, ∂φ+ ∂F − λ = 0. ∂φ−
(2.8a) (2.8b)
For these two equations to be satisfied we must have cos2 (φ+ + φ− ) =
cos2 (2ϕ − φ+ + φ− ) 1 − 4η+ η− sin2 (2ϕ − φ+ + φ− )
(2.9)
using the constraint η+ + η− = 1, F can also be expressed as F=
1 1 + cos(φ+ + φ− ) cos(2ϕ − φ+ + φ− ) 2 + (η+ − η− ) sin(φ+ + φ− ) sin(2ϕ − φ+ + φ− ) . (2.10)
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Without loss of generality, we can take φ+ and φ− to lie in the first and fourth quadrants, then the optimal global fidelity can be obtained by substituting the positive root of cos2 (φ+ + φ− ) and sin2 (φ+ + φ− ) into Eq. (2.10), with the constraint η+ + η− = 1, after little algebra we can write the optimal global fidelity as 1/2 1 1 + 1 − 4η+ η− sin2 (2ϕ − φ+ + φ− ) . 2 (2.11) The value of optimal global fidelity can reach unity only when one of the priori probabilities is zero. We should point out that our discussion only apply to the restricted case when the ancilla is not involved, and the value of optimal global fidelity for N -to-M quantum deleting from a set of two non-orthogonal states has the same form as N -to-M quantum cloning, but the meaning of the parameter is different. ϕ in Eq. (2.11) measures the distinctness of the ideal output state after the machine deleting |Ψi ⊗(N−M) . The ideal output state after the machine deleting |Ψi ⊗(N−M) can be written in the basis {|Ψ+M , |Ψ−M } as |Φ± = γ± Ψ+M + δ± Ψ−M , (2.12) Fopt =
sin(ϕ−φ± ) ±) where γ± = sin(ϕ+φ sin 2θ , δ± = sin 2θ . In the following we would show how to construct the quantum networks to realize this process, we adopt the distinguishability transfer gate (D-gate) operation introduced in Ref. [9]. This operation compresses all the information of the N input copies into one qubit and acts as follows D(θ1 , θ2 )Ψ± (θ1 ) Ψ± (θ2 ) = Ψ± (θ3 ) |Σ. (2.13)
The unitarity condition requires cos 2θ3 = cos 2θ1 × cos 2θ2 , this condition, together with 0 θj π/4, suffices to determine θ3 uniquely. D(θ1 , θ2 ) may be decomposed into universal operations which have been illustrated in Ref. [9]. Since D(θ1 , θ2 ) is Hermitian [9], it can also transfer state |Ψ± (θ3 )|Σ back to |Ψ± (θ1 )|Ψ± (θ2 ). This accomplishes the process of information decompression. We show the D-gate operation can be used as an element in a network for N -to-M quantum deleting. Define DN = D1 (θ1 , θN−1 )D2 (θ1 , θN−2 ) · · · DN−1 (θ1 , θ1 ), where the operation Dj (θ1 , θN−j ) compresses the information of particles j, j + 1 to particle j , and angles θj are uniquely determined by cos 2θj +1 = cos 2θ1 cos 2θj
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Fig. 1. Network for optimum approximate deleting. Here N = 6 and M = 2. The distinguishability of six initial states is transferred to a single qubit. Then a unitary operation T is performed on this qubit, finally a further distinguishability transfer gates decompose the distinguishability from state 1 to states 1, 2 which completes the deleting of the four states.
(0 θj π/4). DN acts as ⊗N DN Ψ± (θ1 ) = Ψ± (θN ) 1 |Σ⊗(N−1) .
(2.14)
This is the first stage of the deleting operation which moves the distinguishability from particles 2, . . . , N to particle 1. Noting the optimal output |Ξ opt = (γ± |Ψ+M + δ± |Ψ−M )|Σ⊗(N−M) , we should then perform a unitary operator T on |Ψ± (θN ) acts as T Ψ± (θN ) = γ± Ψ+ (θM ) + δ± Ψ− (θM ) . (2.15) The unitarity of T have been discussed in Ref. [9]. The final stage of the deleting operation is obvious, we should perform D-gate operation which decompose the distinguishability from particle 1 to particle 2, . . . , M, DM is given by DM = D1 (θ1 , θM−1 ) × D2 (θ1 , θM−2 ) · · · DM−1 (θ1 , θ1 ), DM completes the deleting procedure is given by the transformation DM γ± Ψ+ (θM ) 1
+ δ± Ψ− (θM ) 1 |Σ2 |Σ3 · · · |ΣN
= γ± Ψ+M + δ± Ψ−M |Σ⊗(N−M) . (2.16) Thus the desired optimal state |Ξ opt = (γ± |Ψ+M + δ± |Ψ−M )|Σ⊗(N−M) have been obtained by applying the above procedure. The whole process is illustrated in Fig. 1, and we have taken time to advance from left to right. We may find that the above network structurally resembles the reverse of the optimum approximate cloning network in Ref. [9]. Are these networks timereverse of each other? It is well known that we
could always find D −1 and T −1 to transform the output state to the original input state. Actually since the distinguishability transfer gate is both Hermitian and unitary, then it is its own inverse, so we have D −1 = D. However from the definition in Eq. (2.15), we know the T gate is not necessary Hermitian, so we conclude that the state-dependent deleting network and the state-dependent cloning network in Ref. [9] are time-reverse of each other only when T −1 = T .
3. Upper bound of state-dependent deleting when the set contains N states In this section, we will investigate the state-dependent quantum deleting when the state set contains more than two states. Similar to the above discussion, the ancilla is not involved. It is still difficult to solve this problem completely, we can only give some bounds for this problem. Now our purpose is to maximizing F under the condition of Eq. (2.4). In Ref. [10], the author have obtained the upper bound of F which is given by 1 1 F + 2 2n(n − 1)
n
cos(aj,k − aj,k )
(3.1)
k=j =1
in the quantum deleting process cos aj,k and cos aj,k = should be defined as cos aj,k = ΨjM |ΨkM , cos aj,k ΨjN |ΨkN , we can still used the method proposed in Ref. [10] to get a more stringent bound.
W. Song et al. / Physics Letters A 327 (2004) 123–128
We can find the above deleting machine may be considered as a converse process for which this cloning machine performs. Next we shall give some explanation of the connection and difference between quantum cloning and quantum deleting. As shown in Ref. [4], a quantum cloning machine can be considered as a universal device transforming quantum information into classical information, we may also think quantum deleting machine as a device transforming quantum information into classical information, actually a perfect quantum deleting can be realized by performing a single-particle measurement on each state which to be deleted, but as it is well known that measurement is not unitary process, so we conclude that quantum cloning and quantum deleting can both be regarded as a device which perform a unitary operation to distill classical information from quantum information. This is not to say quantum deleting is just the reverse of quantum cloning, there are still some difference between them which we will generalize as follows, firstly, a quantum cloning process could be think as a swapping operation between the blank qubit and the cloning machine state, but a swapping operation between the cloning state and the deleting machine state could not be think as a successful deleting process. Secondly, the fidelity of the each mode is different for the deleting operation whereas it has the same value in the cloning process for symmetric case.
4. Conclusions In summary, we have constructed the state-dependent deleting machines which may be formally thought of the converse device of the corresponding cloning machines to a certain extent. We also analyze the connection and difference between quantum cloning and quantum deleting. Our results may have potential applications in information processing because it provides strategies for state-dependent quantum deleting in a quantum computer. It tells us how to control the deleting efficiencies from a set of K non-orthogonal states when the ancilla is not considered. On the other hand, one may store new information in an already computed state by deleting copies of the old states. Moreover, we expect our results to play a fundamental role in future understanding of quantum information theory.
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Acknowledgements We are grateful to the referee for valuable comments and helpful suggestions. This work is supported by Anhui Provincial Natural Science Foundation under Grant No. 03042401, the Key Program of the Education Department of Anhui Province under Grant No. 2004kj005zd, and the Talent Foundation of Anhui University.
Appendix A. Optimization for state-dependent deleting when the state set contains two states Let us assume that Φ+ and Φ− have some contribution that does not lie in the space spanned by Ψ+M and Ψ−M . Then the form of Φ+ and Φ− can be written as |Φ+ = a0 Ψ+M + b0 Ψ_M + c0 |C0 , (A.1) M M |Φ− = a1 Ψ+ + b1 Ψ_ + c1 |C1 , (A.2) where vectors C0 and C1 are normalized and lie in the subspace orthogonal to the space spanned by Ψ+M and Ψ−M . The unitarity of Eq. (A.1) and Eq. (A.2) gives the constraints
ϑ1 = Re a0∗ a1 + b0∗ b1 + S 2 a0∗ b1 + b0∗ a1 + c0∗ c1 C0 |C1 − S = 0, (A.3)
∗ ϑ2 = Im a0 a1 + b0∗ b1 + S 2 a0∗ b1 + b0∗ a1 + c0∗ c1 C0 |C1 = 0, (A.4) ϑ3 = |a0 |2 + |b0 |2 + 2S 2 Re a0∗ b0 + |c0 |2 − 1 = 0,
(A.5)
ϑ4 = |a1 | + |b1 | + 2S 2 Re a1∗ b1 + |c1 |2 − 1 = 0,
(A.6)
where S = Ψ+N |Ψ−N . The global fidelity is given by 2 2 F = η+ a0 + b0 S 2 + η− b1 + a1 S 2 .
(A.7)
2
2
Inserting constraints ϑ3 and ϑ4 into Eq. (A.7) yields
F = (η+ + η− ) − 1 − S 4 η+ |b0 |2 + η− |a1 |2
− η+ |c0 |2 + η− |c1 |2 . (A.8)
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Using the method of Langrange multipliers for the remaining two constraints, we get equations ∂ϑi ∂F + = 0, λi ∂|c0 | ∂|c0 | 2
last equation into the other then we could find c0 = |c1 | = 0. Thus, we conclude Φ+ and Φ− lie in the subspace spanned by Ψ+M and Ψ−M .
(A.9)
i=1
References
∂ϑi ∂F + = 0, λi ∂|c1 | ∂|c1 |
(A.10)
∂F ∂ϑi + = 0. λi ∂C0 |C1 ∂C0 |C1
(A.11)
2
i=1
2
i=1
Without loss of generality, we can consider c0 and C0 |C1 real, while c1 should be a general complex (c0 = |c1 |eiδ ), then the above three equations can be simplified as −c0 + λ1 Re |c1 |eiδ C0 |C1 + λ2 Im |c1 |eiδ C0 |C1 = 0, (A.12) iδ −|c|1 + λ1 Re c0 e C0 |C1 + λ2 Im c0 eiδ C0 |C1 = 0, (A.13) λ1 Re c0 |c1 |eiδ + λ2 Im c0 |c1 |eiδ = 0. (A.14) We can multiplying Eq. (A.12) by c0 , Eq. (A.13) by |c1 | and Eq. (A.14) by C0 |C1 and inserting the
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
W.K. Wootters, W.H. Zurek, Nature 299 (1982) 802. D. Dieks, Phys. Lett. A 92 (1982) 271. V. Buzek, M. Hillery, Phys. Rev. A 54 (1996) 1844. N. Gisin, S. Massar, Phys. Rev. Lett. 79 (1997) 2153. D. Bruß, D. DiVincenzo, A. Ekert, C. Fuchs, C. Macchiavello, J. Smolin, Phys. Rev. A 57 (1998) 2368. R.F. Werner, Phys. Rev. A 58 (1998) 1827. L.M. Duan, G.C. Guo, Phys. Rev. Lett. 80 (1998) 4999. H. Fan, K. Matsumoto, X.B. Wang, M. Wadati, Phys. Rev. A 65 (2002) 012304. A. Chefles, S.M. Barnett, Phys. Rev. A 60 (1999) 136. Y.J. Han, Y.S. Zhang, G.C. Guo, Phys. Rev. A 66 (2002) 052301. A.K. Pati, Phys. Rev. Lett. 83 (1999) 2849. A.K. Pati, S.L. Braunstein, Nature 404 (2000) 164. A.K. Pati, S.L. Braunstein, quant-ph/0007121. W.H. Zurek, Nature 404 (2000) 131. J. Feng, Y.-F. Gao, J.-W. Cao, J.-S. Wang, M.-S. Zhan, Phys. Lett. A 292 (2001) 12. D.-W. Qiu, Phys. Rev. A 65 (2002) 052329.