Probabilistically cloning two single-photon states using weak cross-Kerr nonlinearities

By using quantum nondemolition detectors (QNDs) based on weak cross-Kerr nonlinearities, we propose an experimental scheme for achieving 1 → 2 ?> probabilistic quantum cloning (PQC) of a single-photon state, secretly choosing from a two-state set. In our scheme, after a QND is performed on the to-be-cloned photon and the assistant photon, a single-photon projection measurement is performed by a polarization beam splitter (PBS) and two single-photon trigger detectors (SPTDs). The measurement is to judge whether the PQC should be continued. If the cloning fails, a cutoff is carried out and some operations are omitted. This makes our scheme economical. If the PQC is continued according to the measurement result, two more QNDs and some unitary operations are performed on the to-be-cloned photon and the cloning photon to achieve the PQC in a nearly deterministic way. Our experimental scheme for PQC is feasible for future technology. Furthermore, the quantum logic network of our PQC scheme is presented. In comparison with similar networks, our PQC network is simpler and more economical.


Introduction
The quantum no-cloning theorem, initially recognized by Wootters and Zurek [1] in 1982, is one of the most distinguishing features of quantum systems due to the linearity of quantum mechanics. It guarantees the absolute security of quantum cryptography [2]. The famous theorem has two different kinds of statements: one [1] indicates that the perfect cloning of an arbitrary unknown state in a deterministic way is impossible; the other [3][4][5] asserts that nonorthogonal states cannot be deterministically cloned with unit fidelity. However, the nocloning theorem does not preclude the approximate quantum cloning of an arbitrary state or the probabilistic quantum cloning (PQC) of nonorthogonal states. In 1996, Bužek and Hillery [6] first designed a universal quantum cloning (UQC) process for optimal cloning of arbitrary twodimension states. By their UQC process, any two-dimension state can be approximately cloned in a deterministic way. Because of the important applications in quantum information science [7][8][9], much attention has been paid to UQC, including phase-covariant cloning [10,11], realstate cloning [11][12][13], and economical phase-covariant cloning [14][15][16][17][18][19]. Some kinds of UQCs have also been experimentally achieved based on optical systems [20][21][22][23] and nuclear magnetic resonance (NMR) systems [24][25][26].
PQC of nonorthogonal states was first studied by Duan and Guo [27] in 1998. They showed that states randomly chosen from a known set of states can be probabilistically cloned with unit fidelity iff the states in the set are linearly independent. It has been shown that PQC can improve the performance of some quantum computation tasks [7]. Moreover, PQC has important applications in quantum cryptography. Therefore, since Duan and Guoʼs initial work, PQC has attracted a great deal of attention. Novel PQC machines, which produce linear superposition of multiple copies of the input state [28,29], and probabilistic cloning with supplementary information [30] have been studied. Analytical solutions have been investigated for the PQC of two qubit states [31], three symmetric qutrit states [32], and equidistant qudit states [33]. In 2011, Chen et al [34] experimentally implemented the PQC of a single-qubit state randomly chosen from two nonorthogonal qubit states set in an NMR system. In 2012, Araneda et al [35] presented a feasible experimental setup for implementing the PQC of two singlephoton polarization states based on linear optics together with a pair of entangled twin photons.
Recently quantum nondemolition detectors (QNDs) with the help of weak cross-Kerr nonlinearity have attracted a great deal of attention. They have been investigated for realizing controlled NOT (CNOT) gates [36,37], generating entangled states [38,39], implementing entanglement purification and concentration [40][41][42][43][44][45][46][47][48][49], etc. In this paper, by using QNDs based on weak cross-Kerr nonlinearity, we put forward an experimental scheme for realizing the economical PQC of two single-photon polarization states. The realization scheme is feasible in the future. We also give the quantum logic network of our PQC scheme. From the network, it can be seen that if the PQC fails, many unnecessary quantum resources can be omitted; and in this sense our scheme is an economical one. Moreover, if the PQC succeeds, our network needs fewer and simpler quantum logic gates than the existing PQC networks in [34,50,51].
The rest of this paper is planned as follows. In section 2, we briefly review some previous contributions, including PQC and existing PQC networks. In section 3, we first introduce QNDs based on weak cross-Kerr nonlinearity. Then we propose an experimental scheme for the PQC of two single-photon polarization states by using QNDs. After that, we present the quantum logic network for our PQC scheme. In section 4, we briefly discuss and make comparisons involving our experimental scheme and network. Finally, a summary is given in section 5.
( We now present Chen et alʼs PQC machine and network for cloning two nonorthogonal qubit states. [34] mentions that Chen et alʼs network is simpler than those in [50,51]. The to-becloned state is prepared in where 0 | 〉 and 1 | 〉 are two orthogonal bases of a single qubit. Chen et alʼs simple PQC machine is expressed as where  is the unitary evolution of qubits x, y, and z; 1 (1 cos 2 ) Υ ξ = + is the optimal success probability of PQC; and xy Φ | 〉 is the normalized state of qubits x and y. After the unitary operation , a projection measurement is performed on the assistant qubit z under bases 0 | 〉 and 1 | 〉. From equation (4) one can see that, if the MR is 0 | 〉 with the probability Υ, the original state φ | 〉 ± is successfully cloned. On the other hand, with the probability 1 Υ − one can obtain the MR 1 | 〉. In this case, qubits x and y collapse to

Experimental scheme for economical PQC of two single-qubit states via QND
In this section, we first introduce weak cross-Kerr nonlinearity and QND. Then we propose our experimental scheme for economical PQC of two single-photon polarization states by using QNDs based on weak cross-Kerr nonlinearity. Finally we present the quantum network for our economical PQC scheme.

Weak cross-Kerr nonlinearity and QND
The Hamiltonian of cross-Kerr nonlinearity is expressed as [49,[52][53][54], H a a a a , where a † and a are respectively the creation and annihilation operators, the subscripts P and S denote the probe beam and the signal beam respectively, and κ is the coupling strength of the cross-Kerr nonlinearity. Suppose the input state of the signal mode is the Fock state n S |˜〉 , whereas the probe state is initially in a coherent state P α | 〉 . The cross-Kerr nonlinearity then causes the combined system of P and S to evolve as follows:  [34]. Qubits x, y, and z are the to-be-cloned qubit, cloning qubit, and assistant qubit, respectively. ○ and • respectively mean that the control states are 0 | 〉 and 1 | 〉.
= , with t being the interaction time. By the way, for the weak cross-Kerr nonlinearity the assumption 1 θ ≪ is always satisfied. The coherent beam picks up a phase shift θ directly proportional to the number of the photons in the Fock state n S |˜〉 . Specifically, if n = 0 or 1, the evolution is respectively expressed as From equation (6) one can see that as soon as the phase shift is measured, we can infer the number of photons in the signal mode. Now we present the QND based on weak cross-Kerr nonlinearity. See figure 2 for the demonstration. Suppose the probe beam is initially in the coherent state P α | 〉 and the two signal beams are in the following Fock state: S S S S S S S S 00 11 01 10 First, the probe beam and the signal beam S 1 simultaneously pass through the first cross-Kerr media (CKM), which induces phase shift θ + of the probe beam if the signal beam is in the state 1 |˜〉. After that, the probe beam and the signal beam S 2 simultaneously interact with the second CKM, which leads to θ − for the probe beam if the signal beam S 2 has only one photon. After the two interactions, from equation (6)   state of the two signal modes is 10 S S  [53]. After the X quadrature homodyne measurement, the state of modes S 1 and S 2 is ( , cos ) 01 10 , π . In equation (9), f X ( , ) α and f X ( , cos ) α θ are two Gaussian curves with the midpoint between the peaks located at X (1 cos ) , and the peaks are separated by a distance X 2 (1 cos ) If the distance is large enough, the overlap between the two Gaussian curves is small. When X X 0 > or X X 0 < , we respectively have the following two outputs: e e 00 11 , or 01 10 , where ∼ means that there is a low probability of error in distinguishing the two states in equation (10) (7)], from equation (6) one can re-express equation (10) The QND using weak cross-Kerr nonlinearities has now been introduced. It is the kernel of the following experimental scheme for economical PQC.

Experimental scheme for economical PQC of two single-qubit states
In this subsection, we introduce a scheme for achieving 1 2 → PQC based on QNDs combined with linear optics elements. The demonstration of our scheme is shown in figure 3. Photon x is the to-be-cloned photon. The state of x is randomly chosen from the set { , and H | 〉 and V | 〉 denote the horizontal and vertical polarization modes of the photon respectively. Note that the value of ξ is given to us, but we do not know whether the to-be- = is satisfied. This representation is reasonable because any pair of arbitrary qubit states 1 φ | 〉 and 2 φ | 〉 can be transformed into the form of equation (12) by a unitary rotation [34]. To probabilistically clone the state with unit fidelity, an assistant photon z is needed. The initial state of photon z is x where the subscripts 0, 1, 2, and 3 denote the different paths of the photons. After that, photons in paths 1 and 2 are sent to a QND (QND1 in figure 3). It should be mentioned that the photons in paths 1 and 2 correspond to beams S 1 and S 2 in figure 2. Subsequently, the photon in path 0 is reflected by the one-way mirror (OWM), whereas the photon in path 1 passes through the OWM. The QND has two possible outputs. In subsection 3.1, we demonstrated that the output of a QND corresponds to the X quadrature homodyne measurement result of the probe beam.
Corresponding to the MR X X 0 > or X X 0 < , from equation (10) The state in equation (16) can evolve to that in equation (15) by modulating an additional phase shift X 2 ( ) ϑ of the photon in path 2 and a σ x operation on photon z. The operations are performed by a classical feed-forward. Incidentally, the feed-forward technique plays an important role in quantum information and computation in that future operations or measurements depend on earlier measurement results. Using current technologies and customized fast electro-optical modulators, we were able to achieve high fidelity ( 99% > ) for detected photons [56]. Before the feed-forward, photons x and z in the state of equations (15) or (16) are delayed in optical single-mode fibers with lengths of 30 m (150 ns). After the classical feed-forward, the state of x and z evolves to equation (15). Then we use a PBS, whose optical axis is placed at the angle where Υ is the same as in equation (4). From equation (18), one can see that the upper SPTD in figure 3 captures a photon with the probability 1 Υ − . With this capture, one knows that the MR is u z | 〉 ⊥ , and consequently, photon x collapses to the state H x 3 | 〉 . It is obvious that the cloning process fails and that further operations are unnecessary. So this capture means the cutoff of the scheme. On the other hand, the right SPTD is triggered, which takes place with the probability Υ. It means that the MR is u z | 〉 . It follows that the collapse state of photon x is In the following, we show that the 1 2 → cloning of the state φ | 〉 ± can be deterministically realized from the state φ | ′ 〉 ± . In our scheme, the right SPTD is the switch of the components in the dot-line rectangle (see figure 3). Unless the right SPTD is triggered, these components do not work. As soon as it is triggered, a single-photon source emits the cloning photon y and an HWP is used for generating the state After that, the QND2, an OWM, and the x σ operation and phase shift X 2 ( ) ϑ corresponding to the outcome of the QND2 are utilized. Here the photons in paths 5 and 2 correspond to beams S 1 and S 2 , respectively, in figure 2. From figure 3, it can be easily seen that these operations are exactly the same as the preceding part. One can use calculations similar to those in equations (14)- (16) to get the following state: . At the same time, the photon in path 2 is reflected by a PBS, whereas the photon in path 3 transits the PBS. Then, by intensive calculations, one can see that the state in equation (22) After that, photon y is split into paths 6 and 7 and x is split into 8 and 9 by using a PBS and a 50:50 BS, respectively. Subsequently, the photon in path 9 passes through an HWP For the collapse of equation (25), the classical feed-forward does nothing. On the other hand, if the collapse state is equation (26), the classical feed-forward performs the additional phase shift X 2 ( ) ϑ of the photon in path 7 and carries out the x σ operation on photon x. Obviously, after the classical feed-forward, the state of photons x and y is that in equation (25) This transformation shows that the state of qubit x is cloned to qubit y. The quantum network for 1 2 → PQC has now been presented in detail.

Discussion and comparisons
In this section, we discuss and make some comparisons involving cross-Kerr nonlinearities and our experimental scheme and network.
Cross-Kerr nonlinearity has been studied extensively with a view to carrying out a number of quantum information processes [36,37,[40][41][42][43][44][45][46][47][48][49]57]. It should be mentioned that clean cross-Kerr nonlinearity is a rather controversial assumption given current technology. Relative to the atom-cavity system [58] and the Rydberg atomic ensemble [59], natural cross-Kerr nonlinearities are extremely weak [60]. Therefore, the phase shift induced by cross-Kerr nonlinearities is rather small. In 2006, Shapiro [61] pointed out that single-photon Kerr nonlinearities do not help quantum computation. However, given an idealized single-mode coherent state α | 〉 and a single-photon Fock state 1 | 〉, one can perform the transformation (here θ is a rather small angle) by utilizing weak cross-Kerr nonlinearity [36,53]. In 2011, He et al [62] investigated the cross-Kerr nonlinearity between the coherent state and single photons. The work of He et al is a significant contribution toward making the treatment of coherent state and single-photon progress interactions more realistic. With the help of weak measurement, Feizpour et al [63] showed that the cross-Kerr phase shift can be amplified to an observable value that is much larger than the intrinsic magnitude of singlephoton-level nonlinearity. Zhu and Huang studied the linear and nonlinear propagation of probe and signal pulses that are coupled in a double-quantum-well structure with a four-level double-Λ-type configuration. They showed that giant cross-Kerr nonlinearities can be obtained with nearly vanishing optical absorption [64]. Now we discuss our experimental scheme for PQC. (i) Our scheme is economical. From figure 3, one can see that, after the first QND and classical feed-forward, a projection measurement is performed by a PBS and two SPTDs. If the right SPTD captures a photon with the probability Υ, the latter two QNDs and unitary operations are activated to fulfill the entire cloning process. On the other hand, if the upper SPTD is triggered, cloning is suspended. Therefore, many resources and operations can be omitted if cloning fails. In this sense, our scheme is economical. (ii) The single-photon sources needed in our scheme can be realized. In our scheme, single-photon sources are necessary. Single-photon sources have been recently established in many experiments [65][66][67][68]. For example, in 1986, a single-photon state was generated via spontaneous parametric down-conversion and photoelectric detection [65]. In 1999, a single photon source based on controlled single molecule fluorescence was established [66]. In 2000, a train of single-photon pulses was generated by a single-photon turnstile device based on the pulsed laser excitation of a single quantum dot [67]. In 2001, a polarization photon was isolated by utilizing the electrostatic interactions between a laser pulse and a single quantum dot [68]. Therefore, the single-photon sources needed in our PQC scheme can be realized with current technology. (iii) The efficiency of the QND used in our scheme is almost unit. In our PQC scheme, the discrimination between the states in equation (10) is implemented by the X quadrature homodyne measurement. It is the kernel of our experimental scheme for PQC. It has been mentioned that there is a low probability of error with respect to distinguishing the states in equation (10) from each other. In [36], Nemoto et al have shown that in the regime of weak cross-Kerr nonlinearities (θ π ≪ ), the probability of this error occurring is less than 10 −5 when the distance X 9 d 2 αθ ∼ > , In our experimental scheme for PQC, three QNDs are necessary. Therefore, the total efficiency of the QND is (1 10 ) 3 5 3 η > − − . It is obvious that the efficiency is almost unit.
The following is a comparison of our PQC network with some existing networks. To the best of our knowledge, three quantum logic networks for PQC of the states in equation (3) have been reported on extensively [34,50,51]. The success probabilities of these three networks and our network are exactly the same. The networks in [50,51] require a two-qubit S gate, a two-qubit D gate, and a single-qubit projection measurement. From figures 1 and 3 in [51], it can be seen that the S gate is equal to two single-qubit unitary operations and two controlled-U gates, and the D gate is equal to seven single-qubit unitary operations and three CNOT gates. Therefore, in this network four CNOT gates, two controlled-U gates, nine single-qubit unitary operations, and a single-qubit projection measurement are necessary. The network in [34] was reviewed in subsection 2.1. From figure 1(a), one can see that two CNOT gates, two controlled-U gates, a single-qubit unitary operation, and a single-qubit projection measurement are used. Obviously, the network in [34] is simpler than those in [50,51], and it is indeed the simplest one by far. In figure 1(b), we have decomposed each controlled-U gate in figure 1(a)  Therefore, four CNOT gates, five single-qubit unitary operations, and a single-qubit projection measurement are necessary in this PQC network. In our network, if the PQC is successful, three CNOT gates, a single-qubit unitary operation, and a single-qubit projection measurement are enough (see figure 4). It is obvious that our network uses the smallest number of quantum logic gates and unitary operations. More importantly, if the PQC fails, only a CNOT gate and a projection measurement are consumed (i.e., the other two CNOT gates and the unitary operation can be omitted). This characteristic has not been mentioned before. Therefore, our network is an economical one. Incidentally, using this network, the 1 2 → PQC can be achieved in other physical systems, such as the cavity QED system and the ion trap system. We will report them elsewhere.

Summary
To summarize, in this paper we propose a PQC scheme using QNDs with the help of CKMs and linear optics elements. This scheme is economical and is feasible for use in future technology. The quantum network of our scheme is presented, and it is simpler and more economical than existing networks for PQC of two single-qubit states.