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Some general probabilistic quantum cloning and deleting machines
Physics Letters A 308 (2003) 335–342 www.elsevier.com/locate/pla
Some general probabilistic quantum cloning and deleting machines Daowen Qiu a,b a Department of Computer Science, Zhongshan University, Guangzhou 510275, People’s Republic of China b State Key Laboratory of Intelligent Technology and Systems, Department of Computer Science and Technology, Tsinghua University,
Beijing 100084, People’s Republic of China Received 28 September 2002; received in revised form 15 January 2003; accepted 15 January 2003 Communicated by J.P. Vigier
Abstract Through systematically analyzing some impossible operations in quantum information, we then construct two general probabilistic quantum machines to process these limitations. In particular, we analyze that they unify various probabilistic devices, such as the standard cloning, the anti-cloning, a phase-conjugated inputs cloning, an orthogonal qubits inputs cloning, the standard deleting, the anti-deleting, the phase-conjugated inputs deleting and the spin-flipping. The efficiencies of success for each considered machine are given. 2003 Elsevier Science B.V. All rights reserved. PACS: 03.67.-a; 03.65.Ta; 89.70.+c Keywords: Quantum cloning; Quantum deleting
Both entanglement property between subsystems and superposition principle are the distinguishing features of quantum mechanic different from classical physics, which lies at the heart of many applications in the emerging field of quantum information theory [1–3] and, has revealed many fascinating and intrinsic characteristics, for example, one can gain more information from two anti-parallel spins than from two parallel ones [4], whereas in the classical situation what we obtain are exactly equal; another intriguing instance is that there is more information capable of being encoded in a pair of phase-conjugated pure states |ψ|ψ ∗ than that in two identical replicas
E-mail addresses: [email protected], [email protected] (D. Qiu).
|ψ|ψ [5]. Lately, the authors of Ref. [6] have proved that the cloning machines with phase-conjugated or anti-parallel input qubits yield better fidelities than standard N → M cloning machines [7]. More excitingly, quantum computing can efficiently solve some intractable problems on classical computers such as factorization [8]. Nevertheless, in quantum information processing, there also do exist some restrictions and limitations due to the linearity and unitarity of quantum physics. A well-known fact is the so-called quantum nocloning theorem recognized initially by Wootters and Zurek and Dieks [9], which asserts that an unknown quantum state cannot be perfectly cloned as a consequence of linearity, while the unitarity prohibits one from copying two non-orthogonal states [10]. Another difference from classical case is the quantum
0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00086-0
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D. Qiu / Physics Letters A 308 (2003) 335–342
no-deleting principle proposed recently by Pati and Braunstein [11], which seems to be the reverse of no-cloning theorem, but in general it is not so [12]. This conclusion states that the linearity forbids one to delete exactly an unknown quantum state against a copy. Also, similar to Ref. [10] it is easy to know that the unitarity does not allow one to delete two nonorthogonal states [13]. Notably, Zurek [14] further verified the existence of limitations on cloning and deleting completely an unknown state and pointed out the importance of studying approximate and probabilistic deletion corresponding to cloners [7,15,16]. Besides the no-cloning theorem and the no-deleting principle, there are also two important limitations called, respectively, the quantum no-complementing principle [4,17,18] which indeed is the quantum noanti-cloning property that we shall give, and the impossibility of spin-flipping [4] which was also considered as one cannot complement arbitrary qubits. In other words, one cannot flip a spin of unknown polarization, because the flip operator V defined as V |Ψ = Ψ ⊥ (1) is not unitary but anti-unitary, where if qubit |Ψ = α|0 + β|1 belongs to two-dimensional Hilbert space H2 , then its opposite direction is defined as |Ψ ⊥ = β ∗ |0 − α ∗ |1. But there is no physical operation capable of implementing such a transformation. As for the quantum no-anti-cloning property, formally, it says that given an arbitrary state |Ψ ∈ H2 of an unknown qubit and a blank state |Σ ∈ H2 , there does not exist an isometric operator U such that U |Ψ |Σ = |Ψ Ψ ⊥ (2) holds [18]. As indicated above, a particularly interesting effect discovered recently [4] is that one can measure the spin direction |Ψ with better fidelity when two qubits are in anti-parallel |Ψ, Ψ ⊥ than in parallel |Ψ, Ψ . So Gisin and Popescu [4] emphasized the importance of considering quantum spin-flip machine and anti-cloning device, while Bu˘zek, Hillery and Werner in [17] also clearly pointed out the existence of the above limits implied by Eqs. (1) and (2). Motivated by these and by approximate cloner [15] as well as probabilistic one [16], we naturally hope to establish approximate or probabilistic anticloning machine and spin-flipping one. Actually, Gisin and Popescu [4] defined an optimal universal quantum
spin-flip machine by a unitary transformation |Ψ → ρ(Ψ )
(3)
such that ρ(Ψ ) is as close as possible to |Ψ ⊥ in terms of the usual fidelity F = dΨ Ψ ⊥ |ρ(Ψ )|Ψ ⊥ and they pointed out the optimal fidelity of this machine is of 23 . It is worth pointing out that Caves and Nemoto et al. [19] also studied this transformation. A probabilistic quantum anti-cloning machine analogous to Duan and Guo’s cloning one [16] was also considered in Ref. [20] and described by U |Ψi |Σ|P0 =
n ⊥ f |Ψi Ψi |P0 + aik Φ (j ) |Pj .
(4)
k=1
In Ref. [18], a general limit was demonstrated, which is the non-existence of the isometric map |Ψ |Σ → |Ψ F (Ψ ) (5) where |F (Ψ ) = T |Ψ for any given unitary or antiunitary operator T . From the fundamental viewpoint of functional analysis it directly follows that its reverse transformation |Ψ F (Ψ ) → |Ψ |Σ (6) required to be isometric does not exist either. Consequently, we have presented another general limitation in quantum information processing, wherefore it follows the no-anti-deleting property (when |F (Ψ ) = |Ψ ⊥ ). So far we have known that there are several important limits, that is, the quantum no-cloning theorem, the quantum no-deleting principle, the quantum no-complementing principle and the no-anti-deleting property. Also it has been known that reversing any spin 12 state |Ψ → |Ψ ⊥ is not possible, neither is conjugating an unknown qubit, i.e., |Ψ → Ψ ∗ , (7) since it is an anti-unitary transformation, where |Ψ ∗ = α ∗ |0 + β ∗ |1 if |Ψ = α|0 + β|1. Similarly, from Eq. (5) it follows that the transformation |Ψ |Σ → |Ψ Ψ ∗ (8) required to be unitary is impossible either [18].
D. Qiu / Physics Letters A 308 (2003) 335–342
As mentioned above, in Ref. [6] Cerf and Fiurá˘sek et al. constructed the approximate cloning machines with phase-conjugated and anti-parallel (i.e., orthogonal qubits) inputs, respectively, which may approximately implement the following transformations |Ψ Ψ ∗ → |Ψ ⊗M (9) and
|Ψ Ψ ⊥ → |Ψ ⊗M ,
(10)
and they verified that these devices have better cloning fidelities than the standard ones [7,15]. Gisin and Popescu [4] suggested one should consider some interesting machines which may be formally described by the following transformations |Ψ |Ψ → Ψ ⊥ |Σ, (11) ⊥ |Ψ |Ψ → |Ψ Ψ , (12) ⊥ |Ψ |Ψ |Σ → |Ψ |Ψ Ψ . (13) Roughly speaking, transformation (11) implies the deleting and spin-flipping process; transformation (13) represents the anti-cloning one with two parallel spins, while transformation (12) may be thought of as flipping the second spin. Basing on the preceding review and exposition we can easily know that the above transformations (11), (12) and (13) are not unitary but anti-unitary, and thus no physical operation could perfectly implement them. Notably, Fiurá˘sek [21] developed a considerably general approximate transformation and applied it to universal-NOT gate, quantum cloning machine [7,15], quantum entangler [22], and qubit ϑ shifter [23]. As we know, the probabilistic cloning machines [16, 24] can produce perfect copies with some non-zero probabilities. However, most of above-indicated those limitations have not been considered to process with probabilistic fashions, so inspired by Refs. [16,21,24], we in this Letter aim to set up some probabilistic quantum devices to execute those operations. In what follows, our purpose is to establish general probabilistic quantum machines. By analyzing their various special cases, we shall see these machines can execute those impossible operations indicated above with respective probabilities. Indeed they also unify most of the existing probabilistic quantum machines, some of which are mentioned above. In other words, those impossible operations known or just presented
337
may be probabilistically realized through our machines. Furthermore, it is worth indicating that the inputs in our devices are those states of n-bits, so, except the anti-unitary operator T satisfying T |Ψ = |Ψ ⊥ is undefined for the case of n-bits with n 3, our probabilistic machines are still valid for above-mentioned those impossible operations in the situation of n-bits. First we define that a transformation T on ndimensional Hilbert space H is called anti-unitary operation, if it satisfies: (i) ψ|φ = φ |ψ , where T |ψ = |ψ , T |φ = , and |φ n−1 ∗ n−1 (ii) T i=0 ci |i = i=0 ci T |i, where {|i} is an orthonormal basis of H. For example, operator T defined as T |Ψ = |Ψ ∗ , i.e., n−1
n−1 T ci |i = ci∗ |i i=0
i=0
is a typical example of anti-unitary operator; another typical anti-unitary operator is defined as T α|0 + β|1 = β ∗ |0 − α ∗ |1, i.e., T |Ψ = |Ψ ⊥ for the situation of qubits. Now these general probabilistic quantum devices may be described by the following theorems and corollaries. Theorem 1. There exists a unitary operator U such that for any unknown state chosen from a set A = {|Ψi } (i = 1, 2, . . . , n) U |Ψi |Σ|Σ|P0 (i) = s1 F1 (Ψi ) |Σ⊗2 |P0 + s2(i) |Ψi F2 (Ψi ) F3 (Ψi ) |P1 +
n
aik Φ (k) |Qk
(14)
k=1 (i)
(i)
for some real numbers s1 , s2 0 and aik 0 with i, k = 1, 2, . . . , n, if and only if |Ψ1 , |Ψ2 , . . . , |Ψn are linearly independent, where |Fl (Ψ ) equals Tl |Ψ for some unitary or anti-unitary operator Tl , l = 1, 2, 3; |P0 , |P1 , |Q1 , |Q2 , . . . , |Qn denoting the probe states are orthonormal; and the states |Φ (k) ’s
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D. Qiu / Physics Letters A 308 (2003) 335–342
of composite system are normalized but unnecessarily orthogonal. In Eq. (14), input states |Ψi from A belongs to a Hilbert space HA having dimension not smaller than n, say dA ; |Fk (Ψi ) (k = 1, 2, 3) represent a copy of |Ψi , or conjugated state |Ψi∗ of |Ψi , or even anti-parallel (orthogonal ) state |Ψi⊥ of |Ψi . sk(i) (k = 1, 2; i = 1, 2, . . . , n) mean the success probabilities and aik the failure ones.
{|Ψi } (i = 1, 2, . . . , n) U |Ψi T0 |Ψi |Σ|P0 (i) = t1 F1 (Ψi ) |Σ⊗2 |P0 (i) + t2 |Ψi F2 (Ψi ) |Σ|P1 +
n
aik Φ (k) |Qk
(17)
k=1
Corollary 1. There exists a unitary operator U such that for any unknown state chosen from a set A = {|Ψi } (i = 1, 2, . . . , n) U |Ψi |Σ|Σ|P0 = s1(i) F1 (Ψi ) |Σ⊗2 |P0 (i) + s2 |Ψi F2 (Ψi ) |Σ|P1 +
n
aik Φ (k) |Qk
(15)
k=1
for some real numbers s1(i) , s2(i) 0 and aik 0 with i, k = 1, 2, . . . , n, if and only if |Ψ1 , |Ψ2 , . . . , |Ψn are linearly independent, where |Fl (Ψ ), |Pi and |Qj are as those in Theorem 1. Theorem 2. There exists a unitary operator U such that for any unknown state chosen from a set A = {|Ψi } (i = 1, 2, . . . , n) U |Ψi T0 |Ψi |Σ|P0 (i) = t1 F1 (Ψi ) |Σ⊗2 |P0 (i) + t2 |Ψi F2 (Ψi ) F3 (Ψi ) |P1 +
n
aik Φ (k) |Qk
(16)
k=1 (i)
(i)
for some real numbers t1 , t2 0 and aik 0 with i, k = 1, 2, . . . , n, if and only if |Ψ1 , |Ψ2 , . . . , |Ψn are linearly independent, where T0 is either a unitary operator or anti-unitary operator; |Fl (Ψ ) equals Tl |Ψ for some unitary or anti-unitary operator Tl , l = 1, 2, 3; the others symbols have the same implications as those in Theorem 1. Corollary 2. There exists a unitary operator U such that for any unknown state chosen from a set A =
(i)
(i)
for some real numbers t1 , t2 0 and aik 0 with i, k = 1, 2, . . . , n, if and only if |Ψ1 , |Ψ2 , . . . , |Ψn are linearly independent, where |Fl (Ψ ) (l = 1, 2) are as those in Theorem 1. The proofs of the above theorems are put in Appendix A. Next let us analyze the two general machines, by considering their various special cases. Actually, each case represents some more especial probabilistic quantum devices. Notably, these states |Φ (k) ’s may differ in different cases. We consider them by two scenarios. Scenario I. By means of Theorem 1 and Corollary 1 we can obtain the following probabilistic machines. (i)
Case 1. In Corollary 1 we take s1 = 0 for any i ∈ {1, 2, . . . , n}, then Eq. (14) reduces essentially to U |Ψi |Σ|P0 n aik Φ (k) |Qk . (18) = s (i) |Ψi F (Ψi ) |P0 + k=1
So, though transformation |Ψ |Σ → |Ψ F (Ψ ) usually cannot be unitary or even isometric, which was studied by Pati in Ref. [18], we may still achieve it with certain probability through the machine described by Eq. (18). Obviously, in the case of |F (Ψi ) = |Ψi for any i = 1, 2, . . . , n, the above machine becomes Duan and Guo’s one [16], while in the case of qubits, if |F (Ψi ) = |Ψi⊥ for any i = 1, 2, . . . , n, then Eq. (18) describes a probabilistic anti-cloning machine proposed first in Ref. [20]. Furthermore, since | F (Ψi )|F (Ψj )| = | Ψi |Ψj |, one can derive a bound of the cloning efficiency for this machine
D. Qiu / Physics Letters A 308 (2003) 335–342
described by Eq. (18) as 1 1 (i) s + s (j ) . 2 1 + | Ψi |Ψj |
(19)
This bound is the same as that of Duan and Guo’s cloning machine [16] and the anti-cloning machine described in [20]. Case 2. Take s1(i) = 0 for any i = 1, 2, . . . , n in Theorem 1. Then Eq. (14) reduces to U |Ψi |Σ⊗2 |P0 = s (i) |Ψi F2 (Ψi ) F3 (Ψi ) |P0 +
n
aik Φ (k) |Qk .
(20)
k=1
This machine exactly generalizes correspondingly the case of 1 → 3 cloning in Pati’s one [25], which can be clearly seen by taking |F2 (Ψi ) = |F3 (Ψi ) = |Ψi for any i = 1, 2, . . . , n. Similar to Case 1 one can obtain a bound of the cloning efficiency for this device described by Eq. (20) as 1 1 (i) s + s (j ) . 2 1 + | Ψi |Ψj | + | Ψi |Ψj |2
(21)
From inequalities (19) and (21) it is showed that the bounds decrease with the number of copies increasing. (i)
Case 3. In case s2 = 0 for any i = 1, 2, . . . , n in Theorem 1, then Eq. (14) reduces intrinsically to U |Ψi |P0 =
n s (i) F1 (Ψi ) |P0 + aik Φ (k) |Qk .
(22)
deleting machine described by U |Ψi T0 |Ψi |P0 =
n t (i) F1 (Ψi ) |Σ|P0 + aik Φ (j ) |Pj . (23) k=1
Based on the no-deleting principle [11], we know that transformation |Ψ F (Ψ ) → |Ψ |Σ (24) required to be isometric is impossible. However, here this deleting process may be executed with certain non-zero probability in terms of Eq. (23) with |F1 (Ψi ) = |Ψi . Particularly, if n = 2, T0 |Ψi = |Ψi (i = 1, 2) and |F1 (Ψi ) = |Ψi , Eq. (23) describes a probabilistic quantum deleting device that was exactly discussed in Ref. [25], while in the case of T0 |Ψi = |Ψi⊥ and |F1 (Ψi ) = |Ψi with qubits |Ψi for any i = 1, 2, . . . , n, Eq. (23) becomes a probabilistic quantum anti-deleting machine described by U |Ψi Ψi⊥ |P0 =
t (i) |Ψi |Σ|P0 +
n
aik Φ (j ) |Pj
(25)
k=1
which seems to be the reverse process of probabilistic anti-cloning machine. Similarly, if T0 |Ψi = |Ψi∗ and |F1 (Ψi ) = |Ψi for any i = 1, 2, . . . , n, we have a phase-conjugated inputs probabilistic deleting machine described by U |Ψi Ψi∗ |P0 =
t (i) |Ψi |Σ|P0 +
n
aik Φ (j ) |Pj .
(26)
k=1
k=1
More especially, if |F1 (Ψi ) = |Ψi⊥ with qubit inputs |Ψi in Eq. (22) for any i = 1, 2, . . . , n, it becomes a probabilistic quantum spin-flip machine which was mentioned in Ref. [20].
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The two probabilistic deleting machines are first given here and, especially, they show the possibility of executing the following transformations with certain probabilities: |Ψ Ψ ⊥ → |Ψ |Σ, (27)
Scenario II. The following probabilistic devices result from Theorem 2 or Corollary 2.
|Ψ |Ψ ∗ → |Ψ |Σ.
Case 1. By taking t2(i) = 0 for any i = 1, 2, . . . , n in Theorem 2, then Eq. (16) becomes essentially the
Denote |F0 (Ψ ) = T0 |Ψ . Then analogous to Eqs. (19) and (21) one can easily obtain a bound on the success probability of the general deleting machine described
(28)
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D. Qiu / Physics Letters A 308 (2003) 335–342
by Eq. (23) as 1 − | Ψi |Ψj || F0 (Ψi )|F0 (Ψi )| 1 (i) s + s (j ) . 2 1 − | Ψi |Ψj | (29) Case 2. If in Theorem 2 t1(i) = 0 for any i = 1, 2, . . . , n, then Eq. (16) reduces to U |Ψi T0 |Ψi |P0 = t (i) |Ψi F2 (Ψi ) F3 (Ψi ) |P0 +
n
aik Φ (k) |Qk .
(30)
k=1
This is indeed a type of generalized probabilistic cloning machine from 2 to 3. As pointed out above, transformation (13): |Ψ |Ψ |Σ → |Ψ |Ψ |Ψ ⊥ , cannot be ideally realized in quantum communication. However, in Eq. (30) with qubit inputs, by taking T0 |Ψi = |F2 (Ψi ) = |Ψi and |F3 (Ψi ) = |Ψi⊥ for any i = 1, 2, . . . , n, we know that it is possible under certain probability. Also, if |F2 (Ψi ) = |F3 (Ψi ) = |Ψi , Eq. (30) describes implicitly two probabilistic quantum cloning machines with phase-conjugated inputs (T0 |Ψi = |Ψi∗ ) and orthogonal qubits inputs (T0 |Ψi = |Ψi⊥ ), respectively, which process probabilistically transformations (9) and (10), i.e., |Ψ |Ψ ∗ → |Ψ ⊗M and |Ψ |Ψ ⊥ → |Ψ ⊗M . We may call them the phase-conjugated inputs cloning machine and the orthogonal qubits inputs cloning machine, respectively, which to certain extent, complement correspondingly the approximate machines investigated in Ref. [6]. Similar to the above various cases, one can easily obtain the bound on the success probability for this device described by Eq. (30) as 1 − | Ψi |Ψj || F0 (Ψi )|F0 (Ψi )| 1 (i) s + s (j ) . 2 1 − | Ψi |Ψj |3 (i)
Case 3. Let t1 = 0 for any i = 1, 2, . . . , n in Corollary 2. Then with Eq. (17) we have U |Ψi T0 |Ψi |Σ|P0 n = t (i) |Ψi F2 (Ψi ) |Σ|P0 + aik Φ (k) |Qk . k=1
(31)
In Eq. (31), if T0 |Ψi = |Ψi and |F2 (Ψi ) = |Ψi⊥ for the case of qubits, we conclude that transformations (12), that is, |Ψ |Ψ → |Ψ |Ψ ⊥ , is also possible in this sense. Case 4. In fact, from the following proof in Appendix A it is easy to see that the machine described by U |Ψi |Ψi |P0 n = t (i) Ψi⊥ |Σ|P0 + (32) aik Φ (k) |Qk k=1
also exists if |Ψ1 , |Ψ2 , . . . , |Ψn are linearly independent for the case of qubits. This shows that transformation (11): |Ψ |Ψ → |Ψ ⊥ |Σ, can be executed with certain non-zero probabilities. Summarily, by presenting some of the existing impossible operations and analyzing the corresponding devices to realize them with approximate or probabilistic fashions in quantum information, we further address others related impossible operations described by Eq. (6) that contains the so-called noanti-deleting property. However, processing them with certain probabilities is always possible. So we have constructed two considerably general probabilistic quantum information processing machines. They unify lots of probabilistic quantum devices such as the probabilistic cloning machine [16], the probabilistic deleting machine [25], the probabilistic anti-cloning machine (i.e., the probabilistic complementing one) [20], and the probabilistic spin-flip one [20], while they also combine various probabilistic machines first proposed, containing the probabilistic anti-deleting (i.e., orthogonal qubits inputs deleting) machine, the phase-conjugated inputs probabilistic deleting machine, a phase-conjugated inputs probabilistic cloning machine, and an orthogonal qubits inputs probabilistic cloning machine. Particularly, they exactly contain some generalized probabilistic quantum devices for processing probabilistically those generally impossible operations proposed in Refs. [4,17]. The efficiencies of success for these machines have also been estimated. As we have known, those processes above are either unable to be perfectly realized, or just anti-unitary thus beyond the physical operations capable of implementing them at present, whereas it is known that classical bits can be exactly copied, deleted
D. Qiu / Physics Letters A 308 (2003) 335–342
and complemented. On the other hand, the quantum cloning and the anti-cloning (i.e., the quantum complementing) seem the reverse of the quantum deleting and the anti-deleting, respectively, but in general it is not so. However there may be some links to be clarified. In a sense, our discussion may partially give answers for them, and particularly the presented probabilistic machines perhaps play an important role for those impossible operations [4,11,17] in quantum information. To conclude, a natural problem inspired is how to construct a general universal quantum approximate machine to realize those limitations for which we have mostly executed them in this Letter with some probabilistic devices? However, basing on the construction of universal quantum deleting machines in Ref. [26], we anticipate this discussion will be considerably complicated, since such a general quantum device includes deleting one. For the details we shall investigate them elsewhere. In a word, we hope that our investigation will provide some useful ideas for quantum information processing in the future.
Acknowledgements I am grateful to the anonymous referee for critical comments and invaluable suggestions that help me improve the presentation of the Letter. This work was supported by the National Key Project for Basic Research, the Natural Science Foundation of Guangdong Province (Grant No. 020146) and the Young Foundation of Zhongshan University of China.
Appendix A Before the demonstration, we need to know two facts: Fact 1. Let T be either a unitary or anti-unitary operator. Then the linear independence of {|Ψi : i = 1, 2, . . . , n} implies that {T |Ψi : i = 1, 2, . . . , n} is also of linear independence. Fact 2. Let T be a unitary or anti-unitary operator. If set {|Ψi : i = 1, 2, . . . , n} is linearly independent, then so is {|Ψi T |Ψi : i = 1, 2, . . . , n}.
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Proof of Theorem 1. First with Case 1 in Scenario I we note the machine described by Eq. (14) generalizes exactly Duan and Guo’s one [16], so, in terms of Theorem 1 in [16], the existence of unitary transformation (14) infers that {|ψi } are linearly independent. Next, we aim to demonstrate the converse derivation. Our main purpose is to show that if {|ψi } are linearly independent, then there exists a unitary operator U such that Eq. (14) holds. With Lemma 1 in Ref. [16] it suffices to demonstrate that the following matrix equation holds, i.e.,
Ψi |Ψj = S2 Ψi |Ψj F2 (Ψi )F2 (Ψj ) × F3 (Ψi )F3 (Ψj ) S2† + S1
n † F1 (Ψi ) F1 (Ψj ) S + Dk , 1
k=1
(A.1) where Sl = diag(sl(1) , sl(2) , . . . , sl(n) ), l = 1, 2, and Dk = [aik aj k ]. We just consider the case of |Fl (Ψi ) = |Σ for any l = 1, 2, 3 and i = 1, 2, . . . , n, because the other cases, i.e., |Fl (Ψi ) = |Σ for some l ∈ {1, 2, 3}, are actually simpler. Since |Fl (Ψi ) = Tl |Ψi , and Tl is either unitary or anti-unitary operator for l = 1, 2, 3, it is easy to deduce that Tl |Ψ1 , Tl |Ψ2 , . . . , Tl |Ψn are linearly independent from the linear independence of {|Ψi : i = 1, 2, . . . , n} (Fact 1). Therefore, with the assumption of the linear independence of |Ψ1 , |Ψ2 , . . . , |Ψn , both the sets {|F1 (Ψi ): i = 1, 2, . . . , n} and {|Ψi |F2 (Ψi )|F3 (Ψi ): i = 1, 2, . . . , n} are linearly independent (Fact 2). As a consequence, one can show that all these matrices [ Ψi |Ψj ], [ F1 (Ψi )|F1 (Ψj )] and [ Ψi |Ψj F2 (Ψi )|F2 (Ψj )
F3 (Ψi )|F3 (Ψj )] are positive definite. Indeed, for any n-dimensional vector c = (c1 , c2 , . . . , cn ), we have c Ψi |Ψj F2 (Ψi )F2 (Ψj ) F3 (Ψi )F3 (Ψj ) c† = ci cj∗ Ψi |Ψj F2 (Ψi )F2 (Ψj ) i,j
× F3 (Ψi )F3 (Ψj ) n 2 = ci |Ψi F2 (Ψi ) F3 (Ψi ) , i=1
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D. Qiu / Physics Letters A 308 (2003) 335–342
which shows that matrix [ Ψi |Ψj F2 (Ψi )|F2 (Ψj )
F3 (Ψi )|F3 (Ψj )] is positive definite, since |Ψi F2 (Ψi ) F3 (Ψi ) : i = 1, 2, . . . , n is linearly independent. Likewise, one can verify the others are also positive definite. Utilizing the analogous method in Refs. [16,24] one can verify that Eq. (A.1) holds with appropriate Dk (k = 1, 2, . . . , n) and small enough S1 and S2 . In reality, for fixed unitary or anti-unitary operators Tl (l = 1, 2, 3) with |Fl (Ψi ) = Tl |Ψi , because matrices [ Ψi |Ψj ], [ F1 (Ψi )|F1 (Ψj )] and [ Ψi |Ψj F2 (Ψi )|F2 (Ψj )
F3 (Ψi )|F3 (Ψj )] are positive definite, and Ψi |Ψj = Fl (Ψi )Fl (Ψj ) , from continuity argument [16,24] it follows that for small enough S1 , S2 and S3 , matrix M(S1 , S2 )
= Ψi |Ψj − S1 F1 (Ψi )F1 (Ψj ) S1† − S2 Ψi |Ψj F2 (Ψi )F2 (Ψj ) × F3 (Ψi )F3 (Ψj ) S2† is also positive definite. Therefore there is unitary matrix V such that V M(S1 , S2 )V † = diag(a1 , a2 , . . . , an ) for some real numbers ai > 0 (i = 1, 2, . . . , n). Now we choose Dk = V † diag d1(k) , d2(k) , . . . , dn(k) V for some real numbers di(k) 0 (i = 1, 2, . . . , n; k = 1, 2, . . . , n) satisfying n
(k)
di
= ai ,
i = 1, 2, . . . , n,
k=1
then M(S1 , S2 ) = V † diag(a1 , a2 , . . . , an )V n (k) (k) = V † diag d1 , d2 , . . . , dn(k) V k=1
=
n k=1
Dk .
So the proof of Theorem 1 has been completed. ✷ Proof of Theorem 2. Note that Ψi |Ψj = F0 (Ψi )F0 (Ψj ) , where |F0 (Ψj ) = T0 |Ψi for any unitary or antiunitary operator T0 . Actually, the procedure of verifying Theorem 2 just repeats that of Theorem 1, and we omit the details. ✷
References [1] M.A. Nielsen, I.L. Chuang, Quantum Computation Quantum Information, Cambridge Univ. Press, Cambridge, 2000. [2] G. Albert, T. Beth, M. Horodecki, et al., in: Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments, in: Springer Tracts in Modern Physics, Vol. 173, Springer, 2001. [3] R.F. Werner, in: G. Alber, et al. (Eds.), Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments, in: Springer Tracts in Modern Physics, Vol. 173, Springer, 2001. [4] N. Gisin, S. Popescu, Phys. Rev. Lett. 83 (1999) 433. [5] N.J. Cerf, S. Iblisdir, quant-ph/0012020. [6] J. Fiurá˘sek, S. Iblisdir, S. Massar, N.J. Cerf, Phys. Rev. A 65 (2002) 040302; N.J. Cerf, S. Iblisdir, quant-ph/0102077. [7] N. Gisin, S. Massar, Phys. Rev. Lett. 79 (1997) 2153. [8] P.W. Shor, SIAM J. Comput. 26 (1997) 1484. [9] W.K. Wootters, W.H. Zurek, Nature 299 (1982) 802; D. Dieks, Phys. Lett. A 92 (1982) 271. [10] H.P. Yuen, Phys. Lett. A 113 (1986) 405. [11] A.K. Pati, S.L. Braunstein, Nature 404 (2000) 164. [12] A.K. Pati, S.L. Braunstein, quant-ph/0007121. [13] D.W. Qiu, Phys. Rev. A 65 (2002) 052303. [14] W.H. Zurek, Nature 404 (2000) 130. [15] V. Bu˘zek, M. Hillery, Phys. Rev. A 54 (1996) 1844. [16] L.M. Duan, G.C. Guo, Phys. Rev. Lett. 80 (1998) 4999; L.M. Duan, G.C. Guo, Phys. Lett. A 243 (1998) 261. [17] V. Bu˘zek, M. Hillery, R.F. Werner, Phys. Rev. A 60 (1999) R2626. [18] A.K. Pati, quant-ph/0111153. [19] P. Rungta, W.J. Munro, et al., quant-ph/0001075. [20] D.D. Song, L. Hardy, quant-ph/0001105. [21] J. Fiurá˘sek, Phys. Rev. A 64 (2001) 062310. [22] V. Bu˘zek, M. Hillery, Phys. Rev. A 62 (2000) 022303. [23] L. Hardy, D.D. Song, Phys. Rev. A 64 (2001) 032301. [24] A.K. Pati, Phys. Rev. Lett. 83 (1999) 2849. [25] J. Feng, Y.F. Gao, J.W. Cao, J.S. Wang, M.-S. Zhan, Phys. Lett. A 292 (2001) 12. [26] D.W. Qiu, Phys. Lett. A 301 (2002) 112.
Some general probabilistic quantum cloning and deleting machines Daowen Qiu a,b a Department of Computer Science, Zhongshan University, Guangzhou 510275, People’s Republic of China b State Key Laboratory of Intelligent Technology and Systems, Department of Computer Science and Technology, Tsinghua University,
Beijing 100084, People’s Republic of China Received 28 September 2002; received in revised form 15 January 2003; accepted 15 January 2003 Communicated by J.P. Vigier
Abstract Through systematically analyzing some impossible operations in quantum information, we then construct two general probabilistic quantum machines to process these limitations. In particular, we analyze that they unify various probabilistic devices, such as the standard cloning, the anti-cloning, a phase-conjugated inputs cloning, an orthogonal qubits inputs cloning, the standard deleting, the anti-deleting, the phase-conjugated inputs deleting and the spin-flipping. The efficiencies of success for each considered machine are given. 2003 Elsevier Science B.V. All rights reserved. PACS: 03.67.-a; 03.65.Ta; 89.70.+c Keywords: Quantum cloning; Quantum deleting
Both entanglement property between subsystems and superposition principle are the distinguishing features of quantum mechanic different from classical physics, which lies at the heart of many applications in the emerging field of quantum information theory [1–3] and, has revealed many fascinating and intrinsic characteristics, for example, one can gain more information from two anti-parallel spins than from two parallel ones [4], whereas in the classical situation what we obtain are exactly equal; another intriguing instance is that there is more information capable of being encoded in a pair of phase-conjugated pure states |ψ|ψ ∗ than that in two identical replicas
E-mail addresses: [email protected], [email protected] (D. Qiu).
|ψ|ψ [5]. Lately, the authors of Ref. [6] have proved that the cloning machines with phase-conjugated or anti-parallel input qubits yield better fidelities than standard N → M cloning machines [7]. More excitingly, quantum computing can efficiently solve some intractable problems on classical computers such as factorization [8]. Nevertheless, in quantum information processing, there also do exist some restrictions and limitations due to the linearity and unitarity of quantum physics. A well-known fact is the so-called quantum nocloning theorem recognized initially by Wootters and Zurek and Dieks [9], which asserts that an unknown quantum state cannot be perfectly cloned as a consequence of linearity, while the unitarity prohibits one from copying two non-orthogonal states [10]. Another difference from classical case is the quantum
0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0375-9601(03)00086-0
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D. Qiu / Physics Letters A 308 (2003) 335–342
no-deleting principle proposed recently by Pati and Braunstein [11], which seems to be the reverse of no-cloning theorem, but in general it is not so [12]. This conclusion states that the linearity forbids one to delete exactly an unknown quantum state against a copy. Also, similar to Ref. [10] it is easy to know that the unitarity does not allow one to delete two nonorthogonal states [13]. Notably, Zurek [14] further verified the existence of limitations on cloning and deleting completely an unknown state and pointed out the importance of studying approximate and probabilistic deletion corresponding to cloners [7,15,16]. Besides the no-cloning theorem and the no-deleting principle, there are also two important limitations called, respectively, the quantum no-complementing principle [4,17,18] which indeed is the quantum noanti-cloning property that we shall give, and the impossibility of spin-flipping [4] which was also considered as one cannot complement arbitrary qubits. In other words, one cannot flip a spin of unknown polarization, because the flip operator V defined as V |Ψ = Ψ ⊥ (1) is not unitary but anti-unitary, where if qubit |Ψ = α|0 + β|1 belongs to two-dimensional Hilbert space H2 , then its opposite direction is defined as |Ψ ⊥ = β ∗ |0 − α ∗ |1. But there is no physical operation capable of implementing such a transformation. As for the quantum no-anti-cloning property, formally, it says that given an arbitrary state |Ψ ∈ H2 of an unknown qubit and a blank state |Σ ∈ H2 , there does not exist an isometric operator U such that U |Ψ |Σ = |Ψ Ψ ⊥ (2) holds [18]. As indicated above, a particularly interesting effect discovered recently [4] is that one can measure the spin direction |Ψ with better fidelity when two qubits are in anti-parallel |Ψ, Ψ ⊥ than in parallel |Ψ, Ψ . So Gisin and Popescu [4] emphasized the importance of considering quantum spin-flip machine and anti-cloning device, while Bu˘zek, Hillery and Werner in [17] also clearly pointed out the existence of the above limits implied by Eqs. (1) and (2). Motivated by these and by approximate cloner [15] as well as probabilistic one [16], we naturally hope to establish approximate or probabilistic anticloning machine and spin-flipping one. Actually, Gisin and Popescu [4] defined an optimal universal quantum
spin-flip machine by a unitary transformation |Ψ → ρ(Ψ )
(3)
such that ρ(Ψ ) is as close as possible to |Ψ ⊥ in terms of the usual fidelity F = dΨ Ψ ⊥ |ρ(Ψ )|Ψ ⊥ and they pointed out the optimal fidelity of this machine is of 23 . It is worth pointing out that Caves and Nemoto et al. [19] also studied this transformation. A probabilistic quantum anti-cloning machine analogous to Duan and Guo’s cloning one [16] was also considered in Ref. [20] and described by U |Ψi |Σ|P0 =
n ⊥ f |Ψi Ψi |P0 + aik Φ (j ) |Pj .
(4)
k=1
In Ref. [18], a general limit was demonstrated, which is the non-existence of the isometric map |Ψ |Σ → |Ψ F (Ψ ) (5) where |F (Ψ ) = T |Ψ for any given unitary or antiunitary operator T . From the fundamental viewpoint of functional analysis it directly follows that its reverse transformation |Ψ F (Ψ ) → |Ψ |Σ (6) required to be isometric does not exist either. Consequently, we have presented another general limitation in quantum information processing, wherefore it follows the no-anti-deleting property (when |F (Ψ ) = |Ψ ⊥ ). So far we have known that there are several important limits, that is, the quantum no-cloning theorem, the quantum no-deleting principle, the quantum no-complementing principle and the no-anti-deleting property. Also it has been known that reversing any spin 12 state |Ψ → |Ψ ⊥ is not possible, neither is conjugating an unknown qubit, i.e., |Ψ → Ψ ∗ , (7) since it is an anti-unitary transformation, where |Ψ ∗ = α ∗ |0 + β ∗ |1 if |Ψ = α|0 + β|1. Similarly, from Eq. (5) it follows that the transformation |Ψ |Σ → |Ψ Ψ ∗ (8) required to be unitary is impossible either [18].
D. Qiu / Physics Letters A 308 (2003) 335–342
As mentioned above, in Ref. [6] Cerf and Fiurá˘sek et al. constructed the approximate cloning machines with phase-conjugated and anti-parallel (i.e., orthogonal qubits) inputs, respectively, which may approximately implement the following transformations |Ψ Ψ ∗ → |Ψ ⊗M (9) and
|Ψ Ψ ⊥ → |Ψ ⊗M ,
(10)
and they verified that these devices have better cloning fidelities than the standard ones [7,15]. Gisin and Popescu [4] suggested one should consider some interesting machines which may be formally described by the following transformations |Ψ |Ψ → Ψ ⊥ |Σ, (11) ⊥ |Ψ |Ψ → |Ψ Ψ , (12) ⊥ |Ψ |Ψ |Σ → |Ψ |Ψ Ψ . (13) Roughly speaking, transformation (11) implies the deleting and spin-flipping process; transformation (13) represents the anti-cloning one with two parallel spins, while transformation (12) may be thought of as flipping the second spin. Basing on the preceding review and exposition we can easily know that the above transformations (11), (12) and (13) are not unitary but anti-unitary, and thus no physical operation could perfectly implement them. Notably, Fiurá˘sek [21] developed a considerably general approximate transformation and applied it to universal-NOT gate, quantum cloning machine [7,15], quantum entangler [22], and qubit ϑ shifter [23]. As we know, the probabilistic cloning machines [16, 24] can produce perfect copies with some non-zero probabilities. However, most of above-indicated those limitations have not been considered to process with probabilistic fashions, so inspired by Refs. [16,21,24], we in this Letter aim to set up some probabilistic quantum devices to execute those operations. In what follows, our purpose is to establish general probabilistic quantum machines. By analyzing their various special cases, we shall see these machines can execute those impossible operations indicated above with respective probabilities. Indeed they also unify most of the existing probabilistic quantum machines, some of which are mentioned above. In other words, those impossible operations known or just presented
337
may be probabilistically realized through our machines. Furthermore, it is worth indicating that the inputs in our devices are those states of n-bits, so, except the anti-unitary operator T satisfying T |Ψ = |Ψ ⊥ is undefined for the case of n-bits with n 3, our probabilistic machines are still valid for above-mentioned those impossible operations in the situation of n-bits. First we define that a transformation T on ndimensional Hilbert space H is called anti-unitary operation, if it satisfies: (i) ψ|φ = φ |ψ , where T |ψ = |ψ , T |φ = , and |φ n−1 ∗ n−1 (ii) T i=0 ci |i = i=0 ci T |i, where {|i} is an orthonormal basis of H. For example, operator T defined as T |Ψ = |Ψ ∗ , i.e., n−1
n−1 T ci |i = ci∗ |i i=0
i=0
is a typical example of anti-unitary operator; another typical anti-unitary operator is defined as T α|0 + β|1 = β ∗ |0 − α ∗ |1, i.e., T |Ψ = |Ψ ⊥ for the situation of qubits. Now these general probabilistic quantum devices may be described by the following theorems and corollaries. Theorem 1. There exists a unitary operator U such that for any unknown state chosen from a set A = {|Ψi } (i = 1, 2, . . . , n) U |Ψi |Σ|Σ|P0 (i) = s1 F1 (Ψi ) |Σ⊗2 |P0 + s2(i) |Ψi F2 (Ψi ) F3 (Ψi ) |P1 +
n
aik Φ (k) |Qk
(14)
k=1 (i)
(i)
for some real numbers s1 , s2 0 and aik 0 with i, k = 1, 2, . . . , n, if and only if |Ψ1 , |Ψ2 , . . . , |Ψn are linearly independent, where |Fl (Ψ ) equals Tl |Ψ for some unitary or anti-unitary operator Tl , l = 1, 2, 3; |P0 , |P1 , |Q1 , |Q2 , . . . , |Qn denoting the probe states are orthonormal; and the states |Φ (k) ’s
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D. Qiu / Physics Letters A 308 (2003) 335–342
of composite system are normalized but unnecessarily orthogonal. In Eq. (14), input states |Ψi from A belongs to a Hilbert space HA having dimension not smaller than n, say dA ; |Fk (Ψi ) (k = 1, 2, 3) represent a copy of |Ψi , or conjugated state |Ψi∗ of |Ψi , or even anti-parallel (orthogonal ) state |Ψi⊥ of |Ψi . sk(i) (k = 1, 2; i = 1, 2, . . . , n) mean the success probabilities and aik the failure ones.
{|Ψi } (i = 1, 2, . . . , n) U |Ψi T0 |Ψi |Σ|P0 (i) = t1 F1 (Ψi ) |Σ⊗2 |P0 (i) + t2 |Ψi F2 (Ψi ) |Σ|P1 +
n
aik Φ (k) |Qk
(17)
k=1
Corollary 1. There exists a unitary operator U such that for any unknown state chosen from a set A = {|Ψi } (i = 1, 2, . . . , n) U |Ψi |Σ|Σ|P0 = s1(i) F1 (Ψi ) |Σ⊗2 |P0 (i) + s2 |Ψi F2 (Ψi ) |Σ|P1 +
n
aik Φ (k) |Qk
(15)
k=1
for some real numbers s1(i) , s2(i) 0 and aik 0 with i, k = 1, 2, . . . , n, if and only if |Ψ1 , |Ψ2 , . . . , |Ψn are linearly independent, where |Fl (Ψ ), |Pi and |Qj are as those in Theorem 1. Theorem 2. There exists a unitary operator U such that for any unknown state chosen from a set A = {|Ψi } (i = 1, 2, . . . , n) U |Ψi T0 |Ψi |Σ|P0 (i) = t1 F1 (Ψi ) |Σ⊗2 |P0 (i) + t2 |Ψi F2 (Ψi ) F3 (Ψi ) |P1 +
n
aik Φ (k) |Qk
(16)
k=1 (i)
(i)
for some real numbers t1 , t2 0 and aik 0 with i, k = 1, 2, . . . , n, if and only if |Ψ1 , |Ψ2 , . . . , |Ψn are linearly independent, where T0 is either a unitary operator or anti-unitary operator; |Fl (Ψ ) equals Tl |Ψ for some unitary or anti-unitary operator Tl , l = 1, 2, 3; the others symbols have the same implications as those in Theorem 1. Corollary 2. There exists a unitary operator U such that for any unknown state chosen from a set A =
(i)
(i)
for some real numbers t1 , t2 0 and aik 0 with i, k = 1, 2, . . . , n, if and only if |Ψ1 , |Ψ2 , . . . , |Ψn are linearly independent, where |Fl (Ψ ) (l = 1, 2) are as those in Theorem 1. The proofs of the above theorems are put in Appendix A. Next let us analyze the two general machines, by considering their various special cases. Actually, each case represents some more especial probabilistic quantum devices. Notably, these states |Φ (k) ’s may differ in different cases. We consider them by two scenarios. Scenario I. By means of Theorem 1 and Corollary 1 we can obtain the following probabilistic machines. (i)
Case 1. In Corollary 1 we take s1 = 0 for any i ∈ {1, 2, . . . , n}, then Eq. (14) reduces essentially to U |Ψi |Σ|P0 n aik Φ (k) |Qk . (18) = s (i) |Ψi F (Ψi ) |P0 + k=1
So, though transformation |Ψ |Σ → |Ψ F (Ψ ) usually cannot be unitary or even isometric, which was studied by Pati in Ref. [18], we may still achieve it with certain probability through the machine described by Eq. (18). Obviously, in the case of |F (Ψi ) = |Ψi for any i = 1, 2, . . . , n, the above machine becomes Duan and Guo’s one [16], while in the case of qubits, if |F (Ψi ) = |Ψi⊥ for any i = 1, 2, . . . , n, then Eq. (18) describes a probabilistic anti-cloning machine proposed first in Ref. [20]. Furthermore, since | F (Ψi )|F (Ψj )| = | Ψi |Ψj |, one can derive a bound of the cloning efficiency for this machine
D. Qiu / Physics Letters A 308 (2003) 335–342
described by Eq. (18) as 1 1 (i) s + s (j ) . 2 1 + | Ψi |Ψj |
(19)
This bound is the same as that of Duan and Guo’s cloning machine [16] and the anti-cloning machine described in [20]. Case 2. Take s1(i) = 0 for any i = 1, 2, . . . , n in Theorem 1. Then Eq. (14) reduces to U |Ψi |Σ⊗2 |P0 = s (i) |Ψi F2 (Ψi ) F3 (Ψi ) |P0 +
n
aik Φ (k) |Qk .
(20)
k=1
This machine exactly generalizes correspondingly the case of 1 → 3 cloning in Pati’s one [25], which can be clearly seen by taking |F2 (Ψi ) = |F3 (Ψi ) = |Ψi for any i = 1, 2, . . . , n. Similar to Case 1 one can obtain a bound of the cloning efficiency for this device described by Eq. (20) as 1 1 (i) s + s (j ) . 2 1 + | Ψi |Ψj | + | Ψi |Ψj |2
(21)
From inequalities (19) and (21) it is showed that the bounds decrease with the number of copies increasing. (i)
Case 3. In case s2 = 0 for any i = 1, 2, . . . , n in Theorem 1, then Eq. (14) reduces intrinsically to U |Ψi |P0 =
n s (i) F1 (Ψi ) |P0 + aik Φ (k) |Qk .
(22)
deleting machine described by U |Ψi T0 |Ψi |P0 =
n t (i) F1 (Ψi ) |Σ|P0 + aik Φ (j ) |Pj . (23) k=1
Based on the no-deleting principle [11], we know that transformation |Ψ F (Ψ ) → |Ψ |Σ (24) required to be isometric is impossible. However, here this deleting process may be executed with certain non-zero probability in terms of Eq. (23) with |F1 (Ψi ) = |Ψi . Particularly, if n = 2, T0 |Ψi = |Ψi (i = 1, 2) and |F1 (Ψi ) = |Ψi , Eq. (23) describes a probabilistic quantum deleting device that was exactly discussed in Ref. [25], while in the case of T0 |Ψi = |Ψi⊥ and |F1 (Ψi ) = |Ψi with qubits |Ψi for any i = 1, 2, . . . , n, Eq. (23) becomes a probabilistic quantum anti-deleting machine described by U |Ψi Ψi⊥ |P0 =
t (i) |Ψi |Σ|P0 +
n
aik Φ (j ) |Pj
(25)
k=1
which seems to be the reverse process of probabilistic anti-cloning machine. Similarly, if T0 |Ψi = |Ψi∗ and |F1 (Ψi ) = |Ψi for any i = 1, 2, . . . , n, we have a phase-conjugated inputs probabilistic deleting machine described by U |Ψi Ψi∗ |P0 =
t (i) |Ψi |Σ|P0 +
n
aik Φ (j ) |Pj .
(26)
k=1
k=1
More especially, if |F1 (Ψi ) = |Ψi⊥ with qubit inputs |Ψi in Eq. (22) for any i = 1, 2, . . . , n, it becomes a probabilistic quantum spin-flip machine which was mentioned in Ref. [20].
339
The two probabilistic deleting machines are first given here and, especially, they show the possibility of executing the following transformations with certain probabilities: |Ψ Ψ ⊥ → |Ψ |Σ, (27)
Scenario II. The following probabilistic devices result from Theorem 2 or Corollary 2.
|Ψ |Ψ ∗ → |Ψ |Σ.
Case 1. By taking t2(i) = 0 for any i = 1, 2, . . . , n in Theorem 2, then Eq. (16) becomes essentially the
Denote |F0 (Ψ ) = T0 |Ψ . Then analogous to Eqs. (19) and (21) one can easily obtain a bound on the success probability of the general deleting machine described
(28)
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D. Qiu / Physics Letters A 308 (2003) 335–342
by Eq. (23) as 1 − | Ψi |Ψj || F0 (Ψi )|F0 (Ψi )| 1 (i) s + s (j ) . 2 1 − | Ψi |Ψj | (29) Case 2. If in Theorem 2 t1(i) = 0 for any i = 1, 2, . . . , n, then Eq. (16) reduces to U |Ψi T0 |Ψi |P0 = t (i) |Ψi F2 (Ψi ) F3 (Ψi ) |P0 +
n
aik Φ (k) |Qk .
(30)
k=1
This is indeed a type of generalized probabilistic cloning machine from 2 to 3. As pointed out above, transformation (13): |Ψ |Ψ |Σ → |Ψ |Ψ |Ψ ⊥ , cannot be ideally realized in quantum communication. However, in Eq. (30) with qubit inputs, by taking T0 |Ψi = |F2 (Ψi ) = |Ψi and |F3 (Ψi ) = |Ψi⊥ for any i = 1, 2, . . . , n, we know that it is possible under certain probability. Also, if |F2 (Ψi ) = |F3 (Ψi ) = |Ψi , Eq. (30) describes implicitly two probabilistic quantum cloning machines with phase-conjugated inputs (T0 |Ψi = |Ψi∗ ) and orthogonal qubits inputs (T0 |Ψi = |Ψi⊥ ), respectively, which process probabilistically transformations (9) and (10), i.e., |Ψ |Ψ ∗ → |Ψ ⊗M and |Ψ |Ψ ⊥ → |Ψ ⊗M . We may call them the phase-conjugated inputs cloning machine and the orthogonal qubits inputs cloning machine, respectively, which to certain extent, complement correspondingly the approximate machines investigated in Ref. [6]. Similar to the above various cases, one can easily obtain the bound on the success probability for this device described by Eq. (30) as 1 − | Ψi |Ψj || F0 (Ψi )|F0 (Ψi )| 1 (i) s + s (j ) . 2 1 − | Ψi |Ψj |3 (i)
Case 3. Let t1 = 0 for any i = 1, 2, . . . , n in Corollary 2. Then with Eq. (17) we have U |Ψi T0 |Ψi |Σ|P0 n = t (i) |Ψi F2 (Ψi ) |Σ|P0 + aik Φ (k) |Qk . k=1
(31)
In Eq. (31), if T0 |Ψi = |Ψi and |F2 (Ψi ) = |Ψi⊥ for the case of qubits, we conclude that transformations (12), that is, |Ψ |Ψ → |Ψ |Ψ ⊥ , is also possible in this sense. Case 4. In fact, from the following proof in Appendix A it is easy to see that the machine described by U |Ψi |Ψi |P0 n = t (i) Ψi⊥ |Σ|P0 + (32) aik Φ (k) |Qk k=1
also exists if |Ψ1 , |Ψ2 , . . . , |Ψn are linearly independent for the case of qubits. This shows that transformation (11): |Ψ |Ψ → |Ψ ⊥ |Σ, can be executed with certain non-zero probabilities. Summarily, by presenting some of the existing impossible operations and analyzing the corresponding devices to realize them with approximate or probabilistic fashions in quantum information, we further address others related impossible operations described by Eq. (6) that contains the so-called noanti-deleting property. However, processing them with certain probabilities is always possible. So we have constructed two considerably general probabilistic quantum information processing machines. They unify lots of probabilistic quantum devices such as the probabilistic cloning machine [16], the probabilistic deleting machine [25], the probabilistic anti-cloning machine (i.e., the probabilistic complementing one) [20], and the probabilistic spin-flip one [20], while they also combine various probabilistic machines first proposed, containing the probabilistic anti-deleting (i.e., orthogonal qubits inputs deleting) machine, the phase-conjugated inputs probabilistic deleting machine, a phase-conjugated inputs probabilistic cloning machine, and an orthogonal qubits inputs probabilistic cloning machine. Particularly, they exactly contain some generalized probabilistic quantum devices for processing probabilistically those generally impossible operations proposed in Refs. [4,17]. The efficiencies of success for these machines have also been estimated. As we have known, those processes above are either unable to be perfectly realized, or just anti-unitary thus beyond the physical operations capable of implementing them at present, whereas it is known that classical bits can be exactly copied, deleted
D. Qiu / Physics Letters A 308 (2003) 335–342
and complemented. On the other hand, the quantum cloning and the anti-cloning (i.e., the quantum complementing) seem the reverse of the quantum deleting and the anti-deleting, respectively, but in general it is not so. However there may be some links to be clarified. In a sense, our discussion may partially give answers for them, and particularly the presented probabilistic machines perhaps play an important role for those impossible operations [4,11,17] in quantum information. To conclude, a natural problem inspired is how to construct a general universal quantum approximate machine to realize those limitations for which we have mostly executed them in this Letter with some probabilistic devices? However, basing on the construction of universal quantum deleting machines in Ref. [26], we anticipate this discussion will be considerably complicated, since such a general quantum device includes deleting one. For the details we shall investigate them elsewhere. In a word, we hope that our investigation will provide some useful ideas for quantum information processing in the future.
Acknowledgements I am grateful to the anonymous referee for critical comments and invaluable suggestions that help me improve the presentation of the Letter. This work was supported by the National Key Project for Basic Research, the Natural Science Foundation of Guangdong Province (Grant No. 020146) and the Young Foundation of Zhongshan University of China.
Appendix A Before the demonstration, we need to know two facts: Fact 1. Let T be either a unitary or anti-unitary operator. Then the linear independence of {|Ψi : i = 1, 2, . . . , n} implies that {T |Ψi : i = 1, 2, . . . , n} is also of linear independence. Fact 2. Let T be a unitary or anti-unitary operator. If set {|Ψi : i = 1, 2, . . . , n} is linearly independent, then so is {|Ψi T |Ψi : i = 1, 2, . . . , n}.
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Proof of Theorem 1. First with Case 1 in Scenario I we note the machine described by Eq. (14) generalizes exactly Duan and Guo’s one [16], so, in terms of Theorem 1 in [16], the existence of unitary transformation (14) infers that {|ψi } are linearly independent. Next, we aim to demonstrate the converse derivation. Our main purpose is to show that if {|ψi } are linearly independent, then there exists a unitary operator U such that Eq. (14) holds. With Lemma 1 in Ref. [16] it suffices to demonstrate that the following matrix equation holds, i.e.,
Ψi |Ψj = S2 Ψi |Ψj F2 (Ψi )F2 (Ψj ) × F3 (Ψi )F3 (Ψj ) S2† + S1
n † F1 (Ψi ) F1 (Ψj ) S + Dk , 1
k=1
(A.1) where Sl = diag(sl(1) , sl(2) , . . . , sl(n) ), l = 1, 2, and Dk = [aik aj k ]. We just consider the case of |Fl (Ψi ) = |Σ for any l = 1, 2, 3 and i = 1, 2, . . . , n, because the other cases, i.e., |Fl (Ψi ) = |Σ for some l ∈ {1, 2, 3}, are actually simpler. Since |Fl (Ψi ) = Tl |Ψi , and Tl is either unitary or anti-unitary operator for l = 1, 2, 3, it is easy to deduce that Tl |Ψ1 , Tl |Ψ2 , . . . , Tl |Ψn are linearly independent from the linear independence of {|Ψi : i = 1, 2, . . . , n} (Fact 1). Therefore, with the assumption of the linear independence of |Ψ1 , |Ψ2 , . . . , |Ψn , both the sets {|F1 (Ψi ): i = 1, 2, . . . , n} and {|Ψi |F2 (Ψi )|F3 (Ψi ): i = 1, 2, . . . , n} are linearly independent (Fact 2). As a consequence, one can show that all these matrices [ Ψi |Ψj ], [ F1 (Ψi )|F1 (Ψj )] and [ Ψi |Ψj F2 (Ψi )|F2 (Ψj )
F3 (Ψi )|F3 (Ψj )] are positive definite. Indeed, for any n-dimensional vector c = (c1 , c2 , . . . , cn ), we have c Ψi |Ψj F2 (Ψi )F2 (Ψj ) F3 (Ψi )F3 (Ψj ) c† = ci cj∗ Ψi |Ψj F2 (Ψi )F2 (Ψj ) i,j
× F3 (Ψi )F3 (Ψj ) n 2 = ci |Ψi F2 (Ψi ) F3 (Ψi ) , i=1
342
D. Qiu / Physics Letters A 308 (2003) 335–342
which shows that matrix [ Ψi |Ψj F2 (Ψi )|F2 (Ψj )
F3 (Ψi )|F3 (Ψj )] is positive definite, since |Ψi F2 (Ψi ) F3 (Ψi ) : i = 1, 2, . . . , n is linearly independent. Likewise, one can verify the others are also positive definite. Utilizing the analogous method in Refs. [16,24] one can verify that Eq. (A.1) holds with appropriate Dk (k = 1, 2, . . . , n) and small enough S1 and S2 . In reality, for fixed unitary or anti-unitary operators Tl (l = 1, 2, 3) with |Fl (Ψi ) = Tl |Ψi , because matrices [ Ψi |Ψj ], [ F1 (Ψi )|F1 (Ψj )] and [ Ψi |Ψj F2 (Ψi )|F2 (Ψj )
F3 (Ψi )|F3 (Ψj )] are positive definite, and Ψi |Ψj = Fl (Ψi )Fl (Ψj ) , from continuity argument [16,24] it follows that for small enough S1 , S2 and S3 , matrix M(S1 , S2 )
= Ψi |Ψj − S1 F1 (Ψi )F1 (Ψj ) S1† − S2 Ψi |Ψj F2 (Ψi )F2 (Ψj ) × F3 (Ψi )F3 (Ψj ) S2† is also positive definite. Therefore there is unitary matrix V such that V M(S1 , S2 )V † = diag(a1 , a2 , . . . , an ) for some real numbers ai > 0 (i = 1, 2, . . . , n). Now we choose Dk = V † diag d1(k) , d2(k) , . . . , dn(k) V for some real numbers di(k) 0 (i = 1, 2, . . . , n; k = 1, 2, . . . , n) satisfying n
(k)
di
= ai ,
i = 1, 2, . . . , n,
k=1
then M(S1 , S2 ) = V † diag(a1 , a2 , . . . , an )V n (k) (k) = V † diag d1 , d2 , . . . , dn(k) V k=1
=
n k=1
Dk .
So the proof of Theorem 1 has been completed. ✷ Proof of Theorem 2. Note that Ψi |Ψj = F0 (Ψi )F0 (Ψj ) , where |F0 (Ψj ) = T0 |Ψi for any unitary or antiunitary operator T0 . Actually, the procedure of verifying Theorem 2 just repeats that of Theorem 1, and we omit the details. ✷
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