Cascaded Multi-Level Linear-Optical Quantum Router

Quantum information processing is an interdisciplinary of physics and information science [1, 2]. Using quantum laws to process information has great advantage, which the traditional classical methods do not have [3]. It has attracted much attention and reached remarkable achievements. Quantum communication protocols, such as quantum teleportation [4], quantum key distribution (QKD) [5, 6, 7, 8, 9, 10, 11, 12, 13, 14], quantum secure direct communication [15, 16, 17, 18, 19, 20, 21, 22, 23] and some other protocols have been widely discussed in the past ten years [24, 25, 26, 27]. On the other hand, quantum network is an indispensable technology allowing people to transmit quantum information [28, 29].

It is known that the important building block of classical information networks are the router devices, which are used to direct the information from the source to its intended destination. Similar to classical router, quantum router as a quantum node coherently connects different quantum channels and different quantum networks. Another function of the quantum router is the path selection of quantum communication, but cannot influence the quantum states. Quantum router will have the similar function with classical router in the future, such as selecting a smooth and rapid path, improving the communication speed, reducing the network communication load, saving network resources, and improving the network system flow rate. Quantum router is an important part of the quantum network composition which represents the rapidly developing research area of the quantum information processing. Recently, many theoretical proposals and experimental demonstrations of quantum router have been carried out in various systems [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]. For example, in 2009, Aoki and Parkins realized a photon router using single cesium atoms coupled to a microtoroidal cavity in the over coupled regime [39]. In 2013, Lemr and Černoch proposed a programmable quantum router based on linear-optical element [40], and their protocol is developed subsequently [41].

On the other hand, the multi-port quantum router can be more resilient to the future complex quantum network, compared with the two output-port quantum router. Quantum router with multi-port can send signals to more destinations and realize more complex functionality. In 2013, Zhou and Yang proposed a quantum routing of single photons with cyclic three-level system [42]. In 2014, Lu and Zhou proposed a single-photon router which is a coherent control of multichannel scattering for single photons with quantum interferences using the coupled-resonator waveguide as the quantum channel [43]. In 2014, Yan and Fan proposed a scheme to achieve the multi-channel quantum routing of the single photons in a waveguide-emitter system [44]. Recently, Shomroni and Rosenblum demonstrated the experimental realization which was an all-optical coherent routing of single photons by single photons, with no need for any additional control fields [45].

It is known that the linear optics which takes advantage of easy manipulation, has been widely discussed in quantum information processing. In Re. [41], Lemr et al. described the two-port quantum router based on the linear optics. However, they do not discuss the multi-level and multi-port quantum router. In this paper, we will discuss the multi-level quantum router, where the basic elements are based on the Re. [41]. It is shown that the signals in the router output ports of the ith level can be regarded as the input signals of the (i + 1)th level. In this way, the cascaded multi-level quantum router can be constructed. We can obtain a K level quantum router with 2 ^K output ports. We also show that with the help of tunable c-phase gate, the success probability of the quantum router can be increased. Moreover, by exploiting the quantum nondemolition (QND) measurement, the control qubits can be reused to decrease the resource of the router.

The paper is organized as follows: in Section 2, we describe the principle of the multiple-port quantum router, give the expression of the output states, and calculate the success probability of each output port. In Section 3, we use the tunable c-phase gate in our multiple-port quantum router to increase the success probability. In Section 4, we will briefly discuss the quantum router with quantum nondemolition (QND) measurement. In Section 5, we will provide a discussion and conclusion.

As shown in Fig. 1, a signal qubit enters the router through the signal input port while the control qubit enters the router using the port of control input [41]. In their protocol, they exploit the generalized c-phase gate [46, 47], quantum nondemolition detector and programmable-phase gate to complete the quantum router. The generalized c-phase gate is composed by a polarization dependent beam splitter(PDBS) with intensity transmissivities \(T_{V}=\frac {1}{3}\) and T _H = 1 for vertical and horizontal polarization respectively. In 2012, Miková et al. accomplished the linear optical implementation of the programmable phase gate(PPG) with high success probability. The gate implementation is based on fiber optics components. Qubits are encoded into spatial modes of single photons. The signal from the feed-forward detector is led directly to a phase modulator using only a passive voltage divider [48]. Assuming that the signal qubit takes the form of a general quantum polarization state |Ψ _s 〉 = α|H〉 + β|V〉, where |H〉 and |V〉 denote the states of horizontal and vertical linear polarizations and |α|² + |β|² = 1. The state of the control qubit is

$$ |\phi_{c}\rangle=\cos\theta|H\rangle+e^{iv}\sin\theta|V\rangle. $$

(1)

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The superscript “(1)” means that it is the first level of the quantum router. \(P_{s_{1}}\), \(P_{s_{2}}\) and \(P_{succ}^{(1)}\) as a function of θ are shown in Fig. 2. Now we extend the protocol described in Re. [41] to explain the cascaded quantum router. As shown in Fig. 3, we design a two-level quantum router with four outports. In the two-level quantum router, we require three control qubits, which can be written as

$$\begin{array}{@{}rcl@{}} &&|\phi_{c}\rangle_{1,1}=\cos\theta_{1,1}|H\rangle+e^{iv_{1,1}}\sin\theta_{1,1}|V\rangle,\\ &&|\phi_{c}\rangle_{2,1}=\cos\theta_{2,1}|H\rangle+e^{iv_{2,1}}\sin\theta_{2,1}|V\rangle,\\ &&|\phi_{c}\rangle_{2,2}=\cos\theta_{2,2}|H\rangle+e^{iv_{2,2}}\sin\theta_{2,2}|V\rangle. \end{array} $$

(5)

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In Fig. 4, we show the success probability of all the four ports and the total success probability.

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Here, i = 1, 2, ⋯ and j = 1, 2, ⋯2 ⁱ⁻¹. The index of i means the ith level quantum router and the index of j means the jth quantum router.

If the control qubits of all the routers are the same, we can get the success probability of the third level \(P_{succ}^{(3)}=\left (P_{succ}^{(1)}\right )^{3}=(|A^{\prime }|^{2}+|A^{\prime \prime }|^{2})^{3}\), and the success probability of the mth level \(P_{succ}^{(m)}=\left (P^{(1)}_{succ}\right )^{m}=(|A^{\prime }|^{2}+|A^{\prime \prime }|^{2})^{m}\). The success probability of the cascaded quantum router is described in Fig. 5. Here we let m = 1, 2, ⋯ , 5, respectively.

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Further more, if the control qubit of each router and all the c-phase gate are all the same, the success probability of the state in the third level can be simplified as \(P_{succ}^{(3)}=\left (P_{succ}^{(1)}\right )^{3}=\left (|A^{\prime }|^{2}+|A^{\prime \prime }|^{2}\right )^{3}\), and the success probability of the mth level \(P_{succ}^{(m)}=\left (P_{succ}^{(1)}\right )^{m}=\left (|A^{\prime }|^{2}+|A^{\prime \prime }|^{2}\right )^{m}\).

So far, we have fully described our cascaded multi-level quantum router. From above description, each signals in the output ports can be regarded as the initial signal, which can be used to implement the cascaded multi-level quantum router, by adding another control qubits. Suppose that the level of the router is K, we can obtain 2 ^K output ports, which is analogy with the classical router in current network.

It is known that the success probability is an important index for the quantum router. In Figs. 2 and 4, we calculated the total success probability for the quantum router. Figure 2 shows the success probability described in Re. [41]. Figure 4 describes the success probability for the two-level quantum router in our protocol. Interestingly, from Figs. 2 and 4, if \(\theta =\frac {\pi }{3}\), we can find that the signals in all the output ports have the same probability. For the Kth level, we can also obtain the same results, that all 2 ^K output ports have the same success probability in the point of \(\theta =\frac {\pi }{3}\). It can be explained as follows. In the first level, if the success probability of the two output ports are the same, it essentially means that the two input signals in the second level are the same. By using the same control qubits, they can also obtain the same states in four output ports. In this way, the success probability of signals in all the output ports of the second level must be the same. Certainly, the total success probability decreases with the level of the router. In Fig. 5, we show the success probability of the top five levels. It is shown that, in the first level, the success probability is ∼ 10⁻¹, while it is ∼ 10⁻⁵ in the fifth level. In Re. [41], it is shown that by using the tunable c-phase gate, one can increase the total success probability of the router. The tunable c-phase gate is also suitable for the cascaded multi-level quantum router. In Fig. 6, we show that the success probability can also be increased. From Fig. 6, curve A is the success probability altered with θ. By using the tunable c-phase gate, we can obtain a high success probability with different φ in the approximate region θ ∈ (0, 1). We let \(\varphi =\frac {\pi }{6}\), \(\frac {\pi }{4}\) and \(\frac {\pi }{3}\), respectively, and change θ from 0 to \(\frac {\pi }{2}\). m is the level of the quantum router.

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Finally, let us briefly discuss the basic elements in our protocol. In this protocol, we essentially exploit the basic block of quantum router to obtain the cascaded quantum router. The basic block of the quantum router is shown in Fig. 1, which is first discussed in Re. [41]. As discussed in Re. [41], the basic elements such as the generalized c-phase gate, PPG, and QND are all based on the linear optics. Certainly, we should point out that to implement the QND with linear optics, they should require the auxiliary entangled photon pair. Moreover, the total success probability cannot reach 100 %. Usually, the QND can be implemented with the nonlinear optics [53, 54]. For example, the cross-Kerr nonlinearity is a good candidate to construct the QND, which has been widely discussed and used in current quantum information processing [55, 56, 57, 58, 59, 60].

In conclusion, we have described an approach to implement the cascaded multi-level quantum router. We can obtain a K level quantum router with 2 ^K output ports. The signals in the output port of the ith level can be regarded as the signals of the input ports of the (i + 1)th level. We calculated all the states and each success probability in the output ports. On the other hand, it is shown that, with the help of tunable c-phase gate, we can also improve the total success probability. Moreover, with the help of the QND, we can improve this protocol by reusing the control qubits, which can save the initial resources. We hope that this protocol is useful in future quantum network.