Experimental simulation of the quantum secure direct communication using MATLAB and Simulink

Abstract

Quantum secure direct communication is an important mode of quantum communication in which secret messages are securely communicated directly over a quantum channel without the requirement of symmetric key generation at first between the two nodes. Quantum secure direct communication is also a basic cryptographic primitive for constructing other quantum communication tasks, such as quantum authentication and quantum dialog. Here, we report the first detailed experimental simulation of quantum secure direct communication based on the modified version of the DL04 protocol for the phase degree of freedom and equipped with single-photon frequency coding that explicitly demonstrated block transmission. In our experiment simulator, we demonstrated both the noiseless and noisy channel versions of the protocol. In the noisy channel version, we have used 16 different aperiodic frequency channels, equivalent to a nibble of four-bit binary numbers for direct information transmission. We have explicitly demonstrated the transmission of the word ’QNu’ using block frequency at the Bob node and decoding at the Alice node. The experiment simulator firmly demonstrates the feasibility of quantum secure direct communication in the presence of noise and loss.

Appendices

Appendix A: Alice optical components simulation

1.1 1. Laser

Laser is a complex yet self-consistent system that is capable of demonstrating a wide range of dynamic. Semiconductor lasers are most frequently used optical source in optical fibre communication technologies due to their compact size, low power consumption and affordability. For long-haul optical communication DFB laser (Distributed Feedback Laser) are most popular, credit to their superior performances, narrow spectral width and low noise.

The operating characteristics of a DFB semiconductor lasers are well described by a set of rate equations (A1) that governs the interaction of photons and charge carriers inside the active cavity region. These rate equations are used for the simulation of the frequency chirp and output power waveform. For a single-mode laser, the rate equations are given below:

$$\begin{aligned} \frac{dN(t)}{dt}= & {} \frac{I(t)}{e V_a} - \frac{N(t)}{\tau _n} - g_0 \frac{N(t)-N_0}{1+\epsilon _c S(t)}S(t). \nonumber \\ \frac{dS(t)}{dt}= & {} \Big (\Gamma g_0 \frac{N(t)-N_0}{1+\epsilon _c S(t)} - \frac{1}{\tau _p}\Big )S(t) + \frac{\beta \Gamma N(t)}{\tau _{n}}. \nonumber \\ P(t)= & {} \frac{S(t)V_a \eta _0 h \nu }{2 \Gamma \tau _p}. \end{aligned}$$

(A1)

Equation (A1) is coupled nonlinear differential equations between the charge carrier density, N(t) and photon density, S(t). The carrier density N(t) increases due to the injection current I(t) into the active layer volume $V_a$ and decreases due to stimulated and spontaneous emission of photon density S(t). Similarly, the photon density S(t) is increased by stimulated and spontaneous emission S(t) and decreased by internal and mirror losses. The time variations of the output optical power are related to the photon density as shown in Eq. (A1). The electric field output can be described as:

$$\begin{aligned} E(t) = \sqrt{P(t)}e^{i (\omega _c t + \phi (t)))}. \end{aligned}$$

(A2)

Generally, the laser runs in a continuous-wave mode, and external modulation is performed, providing a constant injection current with environmental noise. Additionally, the laser source exhibits two types of noise: (a) relative intensity noise ($\delta P(t)$) and (b) phase noise ($\delta \phi (t)$). In our implementation, RIN is assumed to be white Gaussian and we can evaluate the standard deviation as $\sigma _P = \sqrt{10^{RIN/10}\Delta f} P_{av}$ where RIN is in dB/Hz, $\Delta f$ is the system bandwidth and $P_{av}$ is the average power output. The phase noise is related to the spectral line width ($\Delta \omega$) and inverse of clock frequency ($\tau$); the standard deviation is given by $\sigma \phi = 2 \pi \Delta \omega \tau$. Gaussian noise with corresponding variance is added to the power and the phase to replicate actual system imperfections.

Operating temperature is another critical parameter to consider as variation in temperature changes the central wavelength of the laser. This variation is assumed to be linear and can be expressed by: $\Delta \lambda = \lambda _0 + \delta \lambda \Delta T.$ $\delta \lambda$ is the coefficient representing a change in wavelength per unit change in temperature.

The rate equation model along with emulation of laser noise and temperature dependence demonstrates the response of a monochromatic laser with a reasonable fidelity. The laser sub-module outputs a bus signal with Ex(t) and Ey(t) as its two components. The signal propagation is shown in Fig. 12. The detailed list of the laser parameters is mentioned in Table 4.

Fig. 12

Amplitude of the electric field in the x and y direction coming out of the DFB laser submodule

Table 4 List of the parameters in DFB laser submodule

1.2 2. Inline polariser

An in-line polariser passes linearly polarised light while blocking the orthogonal polarisation from an unpolarised (or randomly polarised) light source. The orthogonally polarised light is attenuated with desired high extinction ratio. We have attenuated the electric field in the y-direction as shown in Fig. 13. The relevant parameters are mentioned in Table 5.

Fig. 13

Amplitude of the electric field coming out of the inline polariser

Table 5 List of the parameters in inline polariser submodule

1.3 3. Intensity modulator

Intensity modulators are used to externally modulate the optical signal from the source to result in a pulsed output. A Mach–Zehnder interferometer structure is used to allow modulation of the optical output power of the device. The devices include two electrical ports: one for the modulation driving signal and one for biasing the modulator.

Consider an interferometric intensity modulator that consists of an input wave-guide split into two branches and then recombined to a single output wave-guide. If the two electrodes are initially biased with voltages, then the initial phases exerted on the light waves would be $\phi _1(t)$ and $\phi _2(t)$ for the two branches. The output field of the light wave carrier can be represented by:

$$\begin{aligned} E_o(t) = \frac{1}{2}E_i\Big (e^{i \phi _1(t)} + e^{i \phi _2(t)}\Big ). \end{aligned}$$

(A3)

For simplicity, the bias voltages applied to the two electrodes are equal and opposite in signs and the final transfer function of field is shown in Fig. 4 and given by:

$$\begin{aligned} E_o(t) = E_i \cos (\frac{\pi V(t)}{V_{\pi }}). \end{aligned}$$

(A4)

where $V_{\pi }$ is the driving voltage such that a $\pi$ phase shift is exerted on the light wave carrier along the optical path of the MZIM. Extinction ratio of the resultant optical signal is given by:

$$\begin{aligned} ER = \frac{\cos ^2(\frac{\pi V_1}{V_{\pi }})}{\cos ^2(\frac{\pi V_0}{V_{\pi }})}. \end{aligned}$$

(A5)

$V_1$ and $V_0$ represent the voltage required max and min of the optical pulse. For a desired extinction ratio, bounds on the electrical signal can be calculated using the above equation. An arbitrary waveform can be selected along with the calculated bounds to achieve the desired extinction ratio.

The spectra of the modulated optical signals would include sets of symmetric side bands arranged around the laser carrier peak at frequency $f_o$. The side bands are displaced from the laser carrier peak frequency at integer multiples of the modulation frequency $f_s =$ ($f_o \pm n f_m \text {with } n = 1, 2, \cdots )$. The relative heights of the side bands are a function of the modulation depth, which is in turn a function of the peak-to-peak value of the RF driving voltage. As a result of these side bands, a multi-model interference appears at the DLI on Bob’s side. The list of the parameters is mentioned in Table 6. The electric field signal at the output of the IM is shown in Fig. 14.

Table 6 List of the parameters in laser module

Fig. 14

Amplitude of the electric field signal at the output of the intensity modulator submodule. Note that the output is pulsed with the duration of 1 ns

1.4 4. Variable optical attenuator (VOA)

VOA is another crucial device for realising quantum-safe communication. The primary function of VOA is to attenuate the signal to the quantum level, reducing the mean photon number to less than unity. The extent of attenuation is dependent upon the length of the transmission line; considering this the attenuation level of VOA can be set to any extent. The list of the parameters is mentioned in Table 7. The electric field signal at the output of the VOA is shown in Fig. 15.

Fig. 15

Electric field signal at the output of the variable optical submodule

Table 7 List of the parameters in variable optical attenuator submodule

1.5 5. Phase modulator

The phase modulator modulates the phase of optical carrier signals by passing it through an optoelectrical wave-guide under an RF voltage. As the refractive index of the optoelectrical medium changes with the voltage applied, any desired phase shift can be provided to the input signal. An important parameter to characterise PM is $V_{\pi }$, the RF bias voltage required to give a phase shift of $\pi$. PM is also referred to as the Mach–Zehnder modulator. Since our QSDC protocol is based on phase encoding of information, PM is an essential device to realise the protocol. The list of the parameters is mentioned in Table 8. The electric field signal at the output of the PM is shown in Fig. 16.

Fig. 16

Electric field at the output of the phase modulator submodule. See that from the amplitude of the electric field, phase cannot be determined

We use Simulink’s Bernoulli random number generator, yielding a random binary string at a rate equal to clock frequency. Phase modulator uses this random string to govern the voltage across the electrodes to ensure a phase shift of $0(\pi )$ for an input logical bit of 0(1).

Appendix B: Optical fibre

Each optical fibre represents a transmission system, which is frequency dependent. A pulse propagation inside this transmission system can be described by the nonlinear Schrodinger equation (NLSE), which is derived from Maxwell equations. From the NLSE equation, we can express effects in optical fibres that can be classified as (a) linear effects, which are wavelength depended, and (b) nonlinear effects, which are intensity (power) dependent. Major impairments of optical signals transmitted via optical fibre are mainly caused by linear effects—the dispersion and the attenuation. The attenuation limits the power of optical signals and represents transmission losses. Another source of linear effects represents the dispersion that causes the broadening of optical pulses in time and phase shifting of signals at the fibre end. The two types of dispersion most prevalent in single-mode fibre (SMF) are:

• Chromatic dispersion (CD)—caused by the different propagation velocities of signal wavelengths from a laser source via optical fibres. Also known as Group Velocity Dispersion (GVD)

• Higher-order dispersion—governed by the slope of the total dispersion curve, considering dispersion over the complete optical spectrum

  Table 8 List of the parameters in phase modulator module

• Polarisation mode dispersion (PMD)—caused by the birefringence effect of nonsymmetrical and imperfect optical fibres.

Nonlinear effects play an important role in the long-haul optical signal transmission. We can classify effects in the following way: Kerr nonlinearities are self-induced effects, where the phase velocity of the pulse depends on the pulse’s own intensity. The Kerr effect describes a change in the fibre refractive index due to electrical perturbations. Due to the Kerr nonlinearities, the nonlinear effects modelled, while propagation in SMF is:

• Self-phase Modulation (SPM)—an effect that changes the refractive index of the transmission medium caused by the intensity of the pulse.

• Cross-phase effect (XPM)—an effect where an optical pulse can change the phase of another pulse with different wavelengths when multiple channels are present.

• Four Wave Mixing (FWM)—an effect that by mixing optical signals from different channels a fourth wave can be arisen and can appear in the same wavelength as one of the mixed waves.

Scattering nonlinearities occur due to a photon inelastic scattering to lower energy photons. The pulse energy is transferred to another wave with a different wavelength. Two effects appear in the optical fibre: Stimulated Brillouin Scattering (SBS) and Stimulated Raman Scattering (SRS)—effects that change the variance of light waves into different waves when the intensity reaches a certain threshold. Both scattering effects shift the frequency of the signal downwards. These effects are considerably more dominant at high pump powers as the intensity of scattered light grows exponentially once a power threshold is exceeded. Since operating power in QSDC is significantly low, it is safe to neglect their effects.

The numerical method used to solve the NLSE is known as the Split-Step Fourier Method (SSFM). It accurately models the fibre nonlinearities within the system. This method only applies to the un-normalised expression of the Schrodinger equation and requires the NLSE to be transformed into:

$$\begin{aligned} \frac{dA}{dz} = A(\hat{D}+\hat{N}). \end{aligned}$$

(B1)

where $\hat{D}$ represents the differential operator that accounts for dispersion and absorption in a linear regime and $\hat{N}$ is a nonlinear operator that governs the effect of fibre nonlinearities.

The resulting non-normalised NLSE in split-step form including the differential group delay (DGD), $\Delta \tau$ for PMD in the orthogonal principle state of polarisations can be represented as

$$\begin{aligned} \hat{D}= & {} -\frac{\alpha }{2}+\frac{\iota }{2}\beta _2\frac{\delta ^2}{\delta t^2}+\frac{1}{6}\beta _3\frac{\delta ^3}{\delta t^3}\pm \frac{\iota \Delta \tau }{2}\frac{\delta }{\delta t}. \nonumber \\ \hat{N}= & {} \iota |A \sqrt{\gamma }|^2. \end{aligned}$$

(B2)

In theory, the nonlinear effects and dispersion work in conjunction with the transmission fibre. As SSFM is only an approximation method, these two factors are broken up and solved individually. The fibre transmission distance is partitioned into a large number of segments of width h. Within a segment, the effect of nonlinearity is included at the mid-plane with linear dispersion effects at both ends. Both dispersion and nonlinearity can be solved through an analytical approach, which involves: (i) Converting from the time domain to the Fourier domain, (ii) Adding a phase shift for the linear effect, (iii) Taking an inverse Fourier transform and back into time domain, (iv) Adding a phase shift for the nonlinear shift. The following effects are included in the model: loss, group velocity dispersion (GVD), third-order dispersion, polarisation mode dispersion (PMD) and self-phase modulation (SPM). The list of the relevant parameters is mentioned in Table 9.

Table 9 List of the parameters in optical fibre module

Appendix C: Bob’s optical components simulation

1.1 1. Phase modulator

The phase modulator on Bob’s side modulates the phase of the incoming optical pulses based on the message that he wants to send and the frequency encoding for the respective 1-ms block. For ex: if Bob wants to send the message ‘0100’, the modulation frequency is 100 kHz. So, in a 1-ms block, there are a total of 100 modulation pulses and $10^6$ optical pulses. Whereever modulation pulse is high, Bob would flip the relative phase of the incoming pulses (if the relative phase of two consecutive pulses is ‘0’, Bob will flip it to $`\pi '$. On the other hand, if it is $`\pi '$, he will flip it to ‘0’). When the modulation pulse is low, the Bob phase modulator will not do anything (or apply the zero phase). This forms the input file of Bob’s phase modulator. It is not generated as in the case of Alice.

The total phase of the optical pulses would be modified in this process. Note that, all the detections on the Alice node corresponding to the high modulation pulse should be flipped and the detections corresponding to a low modulation pulse should be unchanged. Since only Alice has the initial phase information, she can determine whether the detection is flipped or not. For any other party, the detections would be random. Alice will determine the modulation frequency of Bob from the detection timestamps and the frequency of detection flips. The list of the parameters of the Bob phase modulator is mentioned in Table 10.

Table 10 List of the parameters in phase modulator module

1.2 2. Delay line interferometer (DLI)

DLI, also known as Mach–Zehnder demodulator, encompasses the interference of adjacent pulses to demodulate the stored information in the phase of each pulse. A typical DLI comprises two 50:50 beam splitters (BS) each with two inputs and two outputs (BS1 assumes a vacuum state as the second input). The path-splitting operations of BS are parametrised by reflectivity (r) and transmissivity (t) which follows $|r|^2 + |t|^2 = 1.$

$$\begin{aligned}{} & {} \begin{bmatrix} \hat{a}_0^{\dagger } \\ \hat{a}_1^{\dagger } \end{bmatrix} \rightarrow \begin{bmatrix} t &{} \iota r \\ \iota r &{} t \end{bmatrix} \begin{bmatrix} \hat{a}_2^{\dagger } \\ \hat{a}_3^{\dagger } \end{bmatrix}. \end{aligned}$$

(C1)

$$\begin{aligned}{} & {} \begin{bmatrix} \hat{a}_0 \\ \hat{a}_1 \end{bmatrix} \rightarrow \begin{bmatrix} t &{} - \iota r \\ - \iota r &{} t \end{bmatrix} \begin{bmatrix} \hat{a}_2 \\ \hat{a}_3 \end{bmatrix}. \end{aligned}$$

(C2)

Eqs. (C1) and (C2) provide the correlation between the corresponding state creation and annihilating operators. Employing the above equations with the displacement operator, the input and output states are given by:

$$\begin{aligned} \mathinner {|{in}\rangle }= & {} \mathinner {|{\alpha }\rangle }_1 \mathinner {|{\beta }\rangle }_0. \end{aligned}$$

(C3)

$$\begin{aligned} \mathinner {|{out}\rangle }= & {} e^{\iota \phi } \mathinner {|{t \beta + \iota r \alpha }\rangle }_2 \mathinner {|{t \alpha + \iota r \beta }\rangle }_3. \end{aligned}$$

(C4)

Eq. (C4) represents the interference of the adjacent pulses in idealistic cases. However, the optical signal is comprised of multiple sidebands resulting from external modulation techniques. Thus, a multi-modal interference occurs at the second beam splitter, and the resultant power is given by:

$$\begin{aligned} P(t) = \Big |\sum _i E_x^{(i)}(t) + \sum _j E_y^{(j)}(t) \Big |^2. \end{aligned}$$

(C5)

where $E_x^{(i)}(t)$ and $E_y^{(i)}(t)$ are the $i^{th}$ mode of electric field in x and y polarisation field, respectively. The central frequency is represented by $E_x^{(0)}(t)$ and $E_y^{(0)}(t)$. This imperfect interference at the DLI causes a reduction in the output visibility defined by:

$$\begin{aligned} V = \frac{I_{\text {max}}-I_{\text {min}}}{I_{\text {max}}+I_{\text {min}}}. \end{aligned}$$

(C6)

The list of the parameters is mentioned in Table 11.

Table 11 List of the parameters in delay line interferometer (DLI) module

1.3 3. Single-photon detector

In QSDC, Bob’s set-up comprises of two single-photon detectors (SPD) placed at the two outputs of DLI to determine the QBER of the forward channel by detecting fractions of the incoming pulses.

Similarly, on Alice’s side, single-photon detectors are used to detect Bob’s encoded message and the check bits to determine the QBER of the backward quantum channel.

For a prepare and measure QSDC, measuring the quantum state is the most decisive step in the whole protocol. Since the information is encoded in the pulse train with a mean photon less than unity, we need single-photon detectors for measuring the qubit sent by Alice and Bob. Mostly, a Single-Photon Avalanche Detector (SPAD) is used for this application due to its relative simplicity and affordability as compared to superior Superconducting Nano-wire Single-Photon Detectors (SNSPD). In this work, we try to model and simulate the SPAD incorporating its complex behaviour depending on the parameters such as deadtime, detection efficiency, dark count rate, afterpulse probability, temperature, and time jitter.

In this simulation, the SPAD works at the clock frequency of 1 GHz, thus giving a detection event depending upon the energy of each input optical pulse. The detector block integrates the incoming power signal to obtain the energy contained in each pulse and is factored with the energy of a single photon. The mean photon number (μ) obtained is used to represent the coherent state of the pulse. SPAD detection inefficiency can be considered as the transmission losses, thus reducing the MPN to $\eta \mu$. The Poisson distribution of the photon is used to calculate the probability of multi-photons in the pulse.

$$\begin{aligned} \mathinner {|{\alpha }\rangle } = e^{-\eta \mu /2} \sum _n \frac{(\eta \mu )^{n/2}}{\sqrt{n!}} \mathinner {|{n}\rangle }. \end{aligned}$$

(C7)

A detection event in SPAD is governed by other phenomena such as background photons, dark counts and after pulsing. To model the after-pulsing effect as a function of a number of intervals pulses n, we can use an exponential distribution function given by:

$$\begin{aligned} P_{ap}(n) = p_0 e^{-an}. \end{aligned}$$

(C8)

The two parameters $p_0$ and a are set to 0.0317 and 0.00115, respectively. Given the dark count probability ($P_d$) from the detector specification sheet the probability of registering a detection event, $P_{\text {click}}$ can be calculated as:

$$\begin{aligned} P_{\text {click}} = P_p + P_{ap} + P_d - P_p P_{ap} - P_{ap} P_d - P_d P_p + P_p P_{ap} P_d. \end{aligned}$$

(C9)

To replicate the probabilistic nature of photon detection, a uniform random number is generated between 0 and 1. If the random number is less than $p_c$, a click is registered else no detection event. Once a photon click is registered, the detector goes ‘off’ for some time interval, known as detector dead time ($\tau$) and the $P_{\text {click}}$ is set to zero for the interval. After each iteration, the after-pulse probability is updated. The click timestamps from both detectors are sent for post-processing after adding time jitter noise. The list of the parameters is mentioned in Table 12.

Table 12 List of the parameters in single-photon detector (SPD) module