A quantum access network^[1–7] has been gaining increasing interest, since it can provide a reliable infrastructure layer for a large number of users. In a quantum access network, multiple pairs of nodes can transmit quantum information with proper encoding into and decoding from the quantum states by using multiple access, which permits simultaneous transmission of multiple quantum data via a common channel. Nowadays, the popular methods of multiple access in optical cryptography include frequency division multiple access^[4,5] and time division multiple access,^[1] while code division multiple access is not applied to enlarge the scale. Although a corresponding keyed code division multiple access (CDMA) in quantum noise scheme^[8,9] is proposed to obtain key expansion in the quantum case, it offers the nonorthogonal set of M-ry states and cannot be used to expand the number of users in the quantum network. Recently, the chaotic phase shifter^[10,11] has been proved to be a decoherence suppressor and then suppress the influence from the environment to the quantum states. Besides, the chaotic shifters are almost orthogonal and broadband due to the quality of chaotic signal, which perfectly satisfy the requirement of the CDMA network. By using Kerr interactions in whispering gallery mode resonators,^[12] we can successfully achieve coupling between the information-bearing light and the chaotic light for chaotic phase shifter modulation. Experimental demonstration of the group synchrony in a system of chaotic optoelectronic oscillators is also achieved in a four-node optoelectronic network.^[13]

In this letter, we propose to implement the CDMA quantum access network by using chaotic phase shifters and synchronization among senders and receivers. We use chaotic phase shifters to modulate the quantum states and then use beamsplitter (BS) multiplexers to get quantum superposition. After a common channel, we use beamsplitter demultiplexers and chaotic synchronization to decode the quantum signals at the receiver. By analysis, we find the proposed CDMA network can faithfully transmit the quantum states in very noisy channels and defend the crosstalk between different nodes.

Motivated by code division multiple access theory, we have shown a schematic diagram of our strategy in Fig. 1.

Fig. 1.

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	Fig. 1. (color online) Quantum access network scheme using code division multiple access.

The quantum information sent by any nodes is first encoded by the chaotic phase shifters with the Hamiltonian , while is a time-dependent classical chaotic signal, . Compared to the information-bearing field a_i, this encoding spreads the spectral content of quantum information across the entire spectrum due to the chaotic signals. After encoding, all of these quantum signals are combined by beamsplitters multiplexer, and then transmitted over the common channel. At the end of the channel, the quantum signal is amplified by a phase-insensitive linear amplifier (LA) to compensate the loss of the multiplexer, demultiplexer and the channel, and then divided to an N beam by a beamsplitters demultiplexer with the inverse infrastructure to the multiplexer. All the decomposed beam can be decoded to get the information using with the effective Hamiltonian .

Each pair of land induces phase shifts and the inverse phase shifts respectively, where and . To maintain the high fidelity of the quantum signals, we should precisely control the process of quantum communication and keep the same parameters and evolutions of the related chaotic phase shifters to guarantee . However, this precise control is impractical for remote participants because any small deviation in the system can greatly affect the evolution of the chaotic process. Therefore, additional classical channels between senders and receivers are assumed to exist to synchronize each pair of and as shown in Fig. 1. is the displacement operator, which is used to generate the coherent state.

We first introduce the structure of the multimultiplexer and demultiplexer. We introduce two kinds of multiplexer, corresponding to two kinds of demultiplexer. To achieve the communication between N pairs of users through a common channel, we should build the multiplexer to combine the N pieces of quantum signals. Considering the structure of the BS in the reference,^[14] we design the same structure to split and inverse structure to combine the N signals.

Figure 2(a) shows the structure of BS with different transmittance. are the annihilation operators of the signal fields. It is easy to get the output of the multiplexer is

Fig. 2.

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	Fig. 2. The multiplexer and demultiplexer by using BS with different transmittance ratio.

Figure 2(b) shows the inverse demultiplexer to help the user to get the same component of the quantum signals from the linear amplifier, where is the annihilation operator of the vacuum fields entering the beamsplitter, and are the annihilation operator of the demultiplexed signal fields. However, it is very difficult to produce the different BS when the number of users is so large. Thus, we give another structure of the multiplexer and demultiplexer.

Figure 3(a) shows the structure of BS with the same transmission. Due to the quality of the binary tree, we require the amount of users to be . Therefore, the output of the multiplexer is

Fig. 3.

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	Fig. 3. The multiplexer and demultiplexer by using BS with the same transmittance ratio.

Figure 3(b) shows the inverse structure of the binary tree, which aims to get the same component from the source signal, for example, , .

Then the effect of quantum channel and linear amplifier can be showed as:

Using the chaotic synchronization, the global quantum transmission process of the CDMA network can be described as

According to the characteristic of the chaotic phase shift ,^[10,11] we can take an average over this broadband random signal. We can get the relationship , and

Considering the broadband frequency spectrum of the chaotic signal, all M_i are extremely small,^[10] which leads to . Therefore, faithful transmission of quantum information is successfully achieved from the node a_i to node . Furthermore, every different pair of users can set a different standard for the chaotic phase shifter, which means high safe-level users can have the smaller factor M_i. In the following discussion, we assume all the .

To analyze the quantum network using code division multiple access, we first consider its quantum fidelities. We calculate every node fidelity , where is the i-th received quantum sates. According to Eq. (6), the fidelity F_i can be approximated as . For the chaotic phase shifters, when the Duffing oscillator enters the chaotic regime,^[10] the parameter , the fidelities , which means the qubit states can be successfully transmitted through a common channel by using code division multiple access.

Next, we analyze the maximum quantum information transmission rate. The quantum information transmission rate is defined by the quantum capacity,^[15–17] and we restrict our discussion to Gaussian channels. The quantum information transmission rate can be shown as^[15]

First of all, we have shown the information rate under the 2, 8, and 16 pairs of users in Fig. 4(a). We can see the information rate keeps almost the same while the transmittance varies in a wide range. When the quantum channel is severely lossy, the CDMA network is also a robust network for information transmission. In Fig. 4(b), we have shown the information rate of 8 pairs of users under the different channel noise . The information transmission rate is almost not influenced by the channel excess noise, which is the feature of the CDMA technology. In Fig. 4(c), we have shown the relationship between the total information rate and the number of user pairs. We can see the information rate increases slower compared to the increase of the user. It is because the linear amplifier vacuum noise term and the channel excess noise term become larger and larger when the number of user pairs increases.

Fig. 4.

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	Fig. 4. (color online) Quantum information transmission rates. (a) Quantum information transmission rate under different channel transmittance in the same noise. (b) Quantum information transmission rate under different channel transmittance in different noise of 8 pairs of users. (c) Total quantum information transmission rate related to the number of user pairs.

Finally, we are especially concerned about the crosstalk between different nodes for the CDMA network. Here, we calculate the mutual information of different nodes to reflect crosstalk. According to Eq. (6), the mutual information of the p-th received node and q-th sent node can be expressed as

Here we use the expression , for .

In Fig. 5, we have shown the mutual information between different nodes. We can see that the mutual information value is extremely small, which shows crosstalk from the different nodes can be efficiently defended by the chaotic phase shifter due to its orthogonality. When the number of users increases, the single pair mutual information decreases, but it will increase the crosstalk in total. Meanwhile, the mutual information of different nodes is almost not influenced by the noise in the channel.

Fig. 5.

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	Fig. 5. (color online) The mutual information between p-th received node and q-th sent node under different pairs of users.

From these results, we have seen the robustness of the CDMA network using a chaotic phase shifter. When the quantum channel is not totally lossy, the information rate of the CDMA network is not obviously influenced by the channel loss. Besides, the excess noise in the channel has almost no effect on the transmission.

Adding the number of user pairs, the total information rate increases while the slope becomes slower. Meanwhile, it can help defend the crosstalk between the different nodes. All of these are mainly due to the character of the code division multiple access. The chaotic phase shifter spreads the information-bearing field across a broad spectral band, which helps defend the loss and excess noise in the channel to a particular mode. At the same time, the non-matched phase shifter will spread other signal as noise and then the crosstalk can be defended, which reveals the orthogonality of different pairs of chaotic phase shifters. However, when the number of user pairs increase, the linear amplifier parameter G should be large enough to compensate the loss of multiplexer and demultiplexer, which will bring a large amount of vacuum noise and decrease the information transmission rate for a single pair, although the chaotic phase shifter also decreases the vacuum noise of the amplifier. If the non-loss multiplexer and demultiplexer get and split the superposition of quantum states, the CDMA network can have a better performance in all aspects, especially in the number of users.

We have introduced the quantum access network using code division multiple access. Based on the chaotic phase shifter and its synchronization, we can build the CDMA network to achieve the faithful quantum transmission with high fidelity. We thoroughly analyze the CDMA network, finding that it is robust against channel loss, excess noises and crosstalk between different users. Meanwhile, we point out the function of the chaotic phase shifters in the CDMA network by its orthogonality and broadband spectra.