Quantum internet using code division multiple access

A crucial open problem inS large-scale quantum networks is how to efficiently transmit quantum data among many pairs of users via a common data-transmission medium. We propose a solution by developing a quantum code division multiple access (q-CDMA) approach in which quantum information is chaotically encoded to spread its spectral content, and then decoded via chaos synchronization to separate different sender-receiver pairs. In comparison to other existing approaches, such as frequency division multiple access (FDMA), the proposed q-CDMA can greatly increase the information rates per channel used, especially for very noisy quantum channels.


Q
2][3] .Recent achievements in quantum information 4-10 have brought quantum networking much closer to realization.Quantum networks exhibit advantages when transmitting classical and quantum information with proper encoding into and decoding from quantum states [11][12][13][14][15][16][17] .However, the efficient transfer of quantum information among many nodes has remained as a problem yet to be solved [18][19][20][21][22][23][24] , which becomes more crucial for the limited-resource scenarios in large-scale networks.Multiple access, which allows simultaneous transmission of multiple quantum data streams in a shared channel, may provide a remedy to this problem.
Popular multiple-access methods in classical communication networks include time-division multiple-access (TDMA), frequency-division multiple-access (FDMA), and code-division multipleaccess (CDMA).See Fig. 1 for an illustration of different multiple-access methods.In TDMA, different users share the same frequency but transmit on different time slots, but timing synchronization and delays become serious problems in large-scale networks.In FDMA, different users share the same time slots but operate on different frequency bands.However, only a narrow band of the data transmission line has a low leakage rate and the bands assigned to different users should be sufficiently separated to suppress interference.Unlike TDMA and FDMA, CDMA utilizes the entire spectrum and time slots to encode the information for all users, while distinguishes different users with their own unique codes.Therefore, CDMA is adopted as the key technology of the currently-used third generation mobile communication systems, and can accommodate more bits per channel use 25 compared with TDMA and FDMA.It has achieved great success in commercial applications of classical communications.
Although FDMA has already been used in quantum key distribution networks [26][27][28][29][30] , to the best of our knowledge, CDMA has not yet been applied in quantum networks and internet 1 .A q-CDMA network would require that the states sent by each transmitting node of the quantum network are encoded into their coherent superposition before being sent to the common channel, and the quantum information for each of the intended receiving node is coherently and faithfully extracted by proper decoding at the end of the common channel.This, however, is not a trivial task but rather a difficult one.
In this paper, we propose a q-CDMA method via chaotic encoding and chaos synchronization among senders and receivers, which require a quantum channel to transmit quantum superposition states and N classical channels for chaos synchronization to decode the quantum signals at the receiver nodes.It can be seen that the proposed q-CDMA provides higher transmission rates for both classical and quantum information, especially in very noisy channels.

Results
To present the underlying principle of our method, we consider the simplest case, where two pairs of sender and receiver nodes communicate quantum information, encoded into quantized electromagnetic fields with the same frequencies, via a single quantum channel [see Fig. 2 The actions of the chaotic devices CPS i=1,2,3,4 induce the phase shifts exp [−iθ i (t)], where θ i (t) = t 0 δ i (τ ) dτ .Thus, to achieve faithful transmission between the senders and the receivers, the effects of δ 1 (t) and δ 2 (t) on the quantum signals should be minimized in the fields received by the nodes 3 and 4. Intuitively, this could be done by simply adjusting the system parameters such that δ 1 (t) = δ 3 (t) and δ 2 (t) = δ 4 (t).However, such an approach is impractical, because any small deviation in the system parameters can be greatly amplified by the chaotic motion, making it impossible to keep two chaotic circuits with the same exact parameters and initial conditions.
Instead, auxiliary classical channels between senders and the intended receivers can be used to synchronize the chaotic circuit as shown in Fig. 2(b).This classical chaotic synchronization helps to reduce the parameter differences between the chaotic phase shifters and to extract the quantum information faithfully.
Modelling of quantum CDMA network.Hereafter, for the sake of simplicity, we assume that CPS 1 (CPS 2 ) and CPS 3 (CPS 4 ) have been synchronized before the start of the transmission of quantum information, i.e., θ The whole information transmission process in this quantum network can be described by the input-output relationship where a † LA and a BS are the creation and annihilation operators of the auxiliary vacuum fields entering the linear amplifier LA and the second beamsplitter BS 2 .For the pseudo-noise chaotic phase-shift θ i (t), we should take an average over this broadband random signal, which leads to In Eq. (2), S δ i (ω) is the power spectrum density of the signal δ i (t), and ω li and ω ui are the lower and upper bounds of the frequency band of δ i (t), respectively.Equation (1) can then be reduced to For a chaotic signal with broadband frequency spectrum, the factor M i is extremely small, and can be neglected in Eq. (3).This leads to a 3 ≈ a 1 and a 4 ≈ a 2 , implying efficient and faithful transmission of quantum information from nodes 1 and 2 to nodes 3 and 4, respectively.
In our q-CDMA network, the information-bearing fields a 1 and a 2 , having the same fre-quency ω c , are modulated by two different pseudo-noise signals, which not only broaden them in the frequency domain but also change the shape of their wavepackets [see Fig. 2(b)].Thus, the energies of the fields a 1 and a 2 are distributed over a very broad frequency span, in which the contribution of ω c is extremely small and impossible to extract without coherent sharpening of the ω c components.This, on the other hand, is possible only via chaos synchronization which effectively eliminates the pseudo-noises in the fields and enables the recovery of a 1 (a 2 ) at the output a 3 (a 4 ) with almost no disturbance from a 2 (a 1 ).This is similar to the classical CDMA.Thus, we name our protocol as q-CDMA.

Quantum state transmission.
Let us further study the transmission of qubit states over the proposed q-CDMA network using a concrete model.The qubit states and , to be transmitted are encoded in the dark states of two Λ-type three-level atoms; i.e., atom 1 in cavity 1 and atom 2, in cavity 2, as shown in Fig. 3(a).The qubit states are transferred to the cavities by Raman transitions and are transmitted over the q-CDMA network.At the receiver nodes, the quantum states are transferred and stored in two Λ-type atoms; i.e., atom 3 in cavity 3, and atom 4 in cavity 4. We assume that the four coupled atom-cavity systems have the same parameters.Let |g i , |e i , and |r i be the three energy levels of atom i.As shown in Fig. 3(a), the |g i → |r i and |e i → |r i transitions are coupled with a classical control field and a quantized cavity field with coupling strengths Ω i (t) and g i (t).By adiabatically eliminating the highest energy level |r i , the Hamiltonian of the atom-cavity system can be expressed as where c i is the annihilation operator of the i-th cavity mode; g i (t) = gΩ i (t) /∆ is the coupling strength which can be tuned by the classical control field Ω i (t); and ∆ is the atom-cavity detuning.
The cavity fields c i are related to the travelling fields a i by where κ is the decay rate of the cavity field; and a 1,in , a 2,in (both in vacuum states) and a 3,out , a 4,out are the input and output fields of the whole system, respectively.
The chaotic phase shifters CPS i=1,2,3,4 are realized by coupling the optical fields to four driven Duffing oscillators, with damping rates γ, described by the Hamiltonian where x i and p i are the normalized position and momentum of the nonlinear Duffing oscillators, ω 0 /2π is the frequency of the fundamental mode, µ is a nonlinear constant, and is the driving force.The interaction between the field a i and the i-th Duffing oscillator is given by the Hamiltonian where g f−o is the coupling strength between the field and the oscillator.Under the semiclassical approximation for the degrees of freedom of the oscillator, the interaction Hamiltonian H i leads to a phase factor exp −i t 0 g f−o x i (τ ) dτ for the field a i .To simplify the discussion, we assume that all of the four Duffing oscillators have the same ω 0 , µ, f d , and ω d , but different initial states.Finally, the chaotic synchronization between CPS 1 (CPS 2 ) and CPS 3 (CPS 4 ) is achieved by coupling two Duffing oscillators by a harmonic potential The nonlinear coupling between the optical fields and the Duffing oscillators and the chaos synchronization to achieve the chaotic encoding and decoding could be realized using different physical platforms.For example, in optomechanical systems, the interaction Hamiltonian (7) can be realized by coupling the optical field via the radiation pressure to a moving mirror connected to a nonlinear spring (see Fig. 3(b)).Chaotic mechanical resonators can provide a frequencyspreading of several hundreds of MHz for a quantum signal, and this is broad enough, compared to the final recovered quantum signal, to realize the q-CDMA and noise suppression.Chaos synchronization between different nonlinear mechanical oscillators can be realized by coupling the two oscillators via a linear spring.This kind of synchronization of mechanical oscillators have been realized in experiments 32 , but it is not suitable or practical for long-distance quantum communication.Chaos synchronization with a mediating optical field, similar to that used to synchronize chaotic semiconductor lasers for high speed secure communication 33 , would be the method of choice for long-distance quantum communication.The main difficulty in this method, however, will be the coupling between the classical chaotic light and the information-bearing quantum light.
This, on the other hand, can be achieved via Kerr interactions.There is a recent report 34 that proposes to use Kerr nonlinearity in whispering gallery mode resonators to solve this problem.
Another approach for chaotic encoding and chaos synchronization between distant nodes of the network could be the use of electro-optic modulators (EOMs).See, e.g., Fig. 3(c).In this case, the input information-bearing quantum signal is modulated by the EOM driven by a chaotic electrical signal 35 .The EOM can prepare the needed broadband signal, and there have been various proven techniques of chaotic signal generation and synchronization in electrical circuits.Indeed, recently experimental demonstration of chaos synchronization in a four-node optoelectronic network was reported 36 .
To show the efficiency of state transmission in q-CDMA, let us calculate the fidelities states to be transmitted.By designing the control parameters g i (t), using the Raman transition technique 18 , we find for the particular chosen quantum states that the fidelities F 1 and F 2 can be approximated as When the Duffing oscillator enters the chaotic regime, we have M ≈ 0, leading to fidelities F 1 , F 2 ≈ 1, which means that the qubit states can be faithfully transmitted over the q-CDMA network.

Information transmission rates.
Next we consider the maximum transmission rates of classical and quantum information over the proposed q-CDMA network, and compare them, under certain energy constraints, with the achievable bounds of transmission rates in a q-FDMA network and in quantum networks without any multiple access method (i.e., single user-pair network).Here the classical information transmission rates are calculated in terms of the Holevo information 37,38 and the quantum information transmission rates are defined by the coherent information [39][40][41] .We assume that the frequencies allocated to different user pairs in the FDMA network are equally spaced such that the number of users is maximized and cross-talks between adjacent channels are suppressed.Moreover, we restrict our discussion to Gaussian channels and Bosonic channels, respectively for the transmissions of quantum and classical information.
We briefly summarize the main results here and in Fig. 5(a)-(c).(i) For lossless channels (i.e., η = 1 where η denotes the transmissivity of the central frequency of the information-bearing field), upper bounds of classical and the quantum information transmission rates for the proposed q-CDMA network are higher than those of the quantum FDMA and the single user-pair networks if the crosstalk in the q-CDMA is suppressed by setting M ≪ 1. (ii) With the increasing number N of user-pairs in the networks, q-CDMA increasingly performs better than the q-FDMA for classical and quantum information.(iii) Information transmission rates for the q-CDMA is more robust to noise.For fixed N, quantum information transmission rates of the q-FDMA and the single userpair networks degrades very fast to zero as the loss 1 − η increases from zero (ideal channel) to 1/2, whereas the q-CDMA network retains its non-zero rate even for very noisy channels.For the classical information transmission, the situation is similar except that the transmission rates of q-FDMA and the single user-pair network drops to zero when η = 0 which corresponds to a completely lossy channel.
The robustness of the proposed q-CDMA network for noisy channel can be explained as follows.The chaotic phase shifters in the q-CDMA network spread the information-bearing field across a broad spectral band.Thus, the energy distributed in a particular mode is almost negligible, and thus the photon loss is also almost negligible.Therefore, increasing η has very small effect on the transmission rates.In Fig. 5(b)-(c), we consider the noise to be broadband, and shows that the transmission rates of classical and quantum information over the q-CDMA network change only slightly.

Discussion
We have introduced a q-CDMA network based on chaotic synchronization where quantum information can be faithfully transmitted with fidelities as high as 0.99 between multiple pairs of nodes sharing a single quantum channel.The proposed quantum multiple-access network is robust against channel noises, and attains higher transmission rates for both classical and quantum information when compared to other approaches.A q-CDMA network based on our proposal requires the realization of two important issues.First, quantum interference of signals from different chaotic sources.This has recently been demonstrated by Nevet et.al. 42 .Second is the implementation of chaotic phase shifters and their synchronization.These could be implemented in various systems, including but not limited to optomechanical, optoelectrical 35 , and all-optical systems 33 .
In particular, whispering-gallery-mode (WGM) optical resonators are possible platforms as chaotic behavior in a WGM microtoroid resonator has been reported in Ref. 43 .Although synchronization of self-sustaining oscillations in directly coupled microring resonators have been demonstrated 44 , and mechanical mode synchronization in two distant resonators coupled via waveguides has been proposed 45 , demonstration of chaos synchronization in such optomechanical resonators are yet to be demonstrated.Although the tasks to be fulfilled are not trivial, we believe that we are not far away from such realizations due to the rapid pace of experimental and theoretical developments we have seen in the field in the past few years.We think that our proposal will pave the way for long distance q-CDMA networks, and will give new perspectives for the optimization of quantum networks.

Methods
Averaging over the chaotic phase shift.A chaotic signal δ i (t) can be expressed as a combination of many high-frequency components, i.e., where A iα , ω iα , φ iα are the amplitude, frequency, and phase of each component, respectively.Then the phase of the signal at any given time t can be written as Using the Fourier-Bessel series identity 31 : with J n (x) as the n-th Bessel function of the first kind, we can write If we take average over the "random" phase θ i (t), the components related to the frequencies ω iα should appear as fast-oscillating terms and thus can be averaged out.This treatment corresponds to averaging out the components that are far off-resonance with the information-bearing field, and keeping only the near-resonance components.Hence, only the lowest-frequency terms, with n α = 0, dominate the dynamical evolution.Thus, we have Since the chaotic signal δ i (t) is mainly distributed in the high-frequency regime, we have A iα ≪ ω iα .Using the expressions J 0 (x) ≈ 1 − x 2 /4, log (1 + x) ≈ x for x ≪ 1, it is easy to show that where Consequently, from Eqs. ( 9) and ( 10), we obtain the equation Input-output relationship of the quantum CDMA network.Here we calculate the input-output relationship of the quantum CDMA network shown in Fig. 6, we can express the input-output relationships of the chaotic phase shifters CPS i=1,2,3,4 as and those of the two beam splitters BS 1 and BS 2 and the linear quantum amplifier "LA", respectively, as and Then, using Eqs.(12-15), we obtain the total input-output relationship of the quantum network as where θ 1 and θ 2 are independent chaotic "noises" as we have not considered chaos synchronization yet.
1. Kimble, H. J.      Upper bounds of classical (quantum) information transmission rates of different methods for noisy channel with 0 < η < 1.The correction factor in the q-CDMA network is M = 0.01.FDMA is constrained by the frequency bandwidth δω/ω = 0.2.All the methods are constrained with the total energy ǫ/ω = 1.C η c(q),CDMA(FDMA) denote the classical (c) and quantum (q) information transmission rates in q-CDMA and q-FDMA networks with transmissivity η.The rates for the single user-pair channel are C η c,sig and C η q,sig .
(a)].The schematic diagram of our strategy is shown in Fig. 2(b).The quantum information sent by the nodes 1 and 2 is first encoded by two chaotic phase shifters CPS 1 and CPS 2 , whose operation can be modelled by the effective Hamiltonian δ i (t) a † i a i , with δ i (t) being time dependent classical chaotic signals and i = 1, 2. This encoding spreads the spectral content of the quantum information across the entire spectrum.The two beams are then combined at the 50 : 50 beamsplitter BS 1 and transmitted via a common channel to the receivers.At the end of the channel, the quantum signal is first amplified by a phase-insensitive linear amplifier (LA), then divided into two branches by a second 50 : 50 beamsplitter BS 2 , and finally sent to nodes 3 and 4 through two chaotic phase shifters CPS 3 and CPS 4 , which are introduced to decode the information by applying the effective Hamiltonian −δ j (t) a † j a j , with j = 3, 4. Amplifier gain is set as G = 4 to compensate the losses induced by the beamsplitters.

Fig. 4
Fig.4(a), it is seen that there are three distinct regions representing how the chaotic motion affects

Figure 1 :
Figure 1: Illustration for different multiple-access methods.(a) TDMA: the users share

Figure 2 :
Figure 2: Diagrams of the quantum multiple access networks.(a) Quantum information

Figure 3 :
Figure 3: Quantum state transmission over q-CDMA network.(a) The broom-shaped or

Figure 4 :
Figure 4: Fidelities of quantum state transmission.(a) Fidelities F 1 and F 2 versus the

Figure 5 :
Figure 5: Quantum information transmission rates.(a) Fidelities F 1 and F 2 versus the

Figure 6 :
Figure 6: Input-output structure of quantum CDMA netwrok.The black dashed lines