**2. Implementation of classical QKD schemes with frequency encoding**

Let us consider implementation of various modulation schemes, relying on the chronology of QKD systems with frequency encoding. The protocols BB84 [12] and B92 [11] are two main protocols used for their construction. During the BB84 protocol realization, Alice prepares and sends to Bob a lot of random qubits, chosen from the four main states:

$$\begin{cases} \left| \psi\_0 \right\rangle = \left| 0 \right\rangle \\ \left| \psi\_1 \right\rangle = \left| 1 \right\rangle \end{cases}$$

$$\begin{cases} \left| \psi\_+ \right\rangle = \frac{1}{\sqrt{2}} [\left| 0 \right\rangle + \left| 1 \right\rangle] \\ \left| \psi\_- \right\rangle = \frac{1}{\sqrt{2}} [\left| 0 \right\rangle - \left| 1 \right\rangle] \end{cases} \tag{1}$$

The first two states in (1) form one basis of two-dimensional quantum system, and the other two form second basis.

It is necessary to fulfill the terms ⟨ψ<sup>0</sup> │ψ<sup>1</sup> ⟩ = 0 and ⟨ψ<sup>+</sup> │ψ<sup>−</sup> ⟩ = 0, corresponding to the scalar production of their components. At the same time, the mentioned states of different bases are not orthogonal and maximum overlap [12]. Therefore, there is no measurement procedure, at which Eve can determine the state prepared by Alice and sent to Bob at 100% probability [21]. B92 protocol [11] is the modernization of BB84 protocol and is used to encode one of the two presented in (1) bases.

#### **2.1. PM-PM schemes**

We present in this chapter the results of the universal QKD system design, based on a tandem electro-optic AMPM-PMAM scheme built on microwave photonics principles applied to photon carrier modulation. It allows us to implement all of the above-mentioned classical symmetrical schemes PM-PM, AM-AM and meshed AM-PM (PM-AM) and also to review the requirements for building a promising tandem AM and phase commutation (PC) scheme with the possibility of implementing a nonclassical asymmetric structure with passive filter-

The chapter in the main is based on the results of analytical review of [1–19], materials of Morozov et al. [20] and additional and new results of theoretical and experimental researches in QKD theme and miscellaneous applications. Next chapter sections are organized as follows. The second section shows the principles of design of QKD systems with frequency encoding based on the classical approaches; key nodes involved for the implementation of PM-PM, AM-AM and meshed AM-PM (PM-AM) schemes are described; the descriptions of protocol bases and some experimental results are given; the advantages and disadvantages of classical schemes are evaluated, and the ways of its development are discussed. The third section discusses the design of promising universal tandem AMPM-PMAM scheme and its microwave photonic (MWP) basis; version of QKD system with tandem amplitude modulation and phase commutation of photons is proposed; the capabilities of re-modulation and possibilities of re-commutation procedures, or their absent and using only passive filtering structure realizations. In conclusion, the received results are analyzed and the key develop-

ment challenges for QKD systems with frequency encoding are highlighted.

**2. Implementation of classical QKD schemes with frequency** 

and sends to Bob a lot of random qubits, chosen from the four main states:

⎨ ⎪ ⎩ {

<sup>|</sup>*ψ*+⟩ <sup>=</sup> \_\_1 √ \_\_ 2

<sup>|</sup>*ψ*−⟩ <sup>=</sup> \_\_1 √ \_\_ 2

│ψ<sup>1</sup>

Let us consider implementation of various modulation schemes, relying on the chronology of QKD systems with frequency encoding. The protocols BB84 [12] and B92 [11] are two main protocols used for their construction. During the BB84 protocol realization, Alice prepares

> |*ψ*0⟩ = |0⟩ |*ψ*1⟩ = |1⟩ <sup>⎧</sup> <sup>⎪</sup>

The first two states in (1) form one basis of two-dimensional quantum system, and the other

⟩ = 0 and ⟨ψ<sup>+</sup>

production of their components. At the same time, the mentioned states of different bases are not orthogonal and maximum overlap [12]. Therefore, there is no measurement procedure, at which Eve can determine the state prepared by Alice and sent to Bob at 100% probability [21].

[|0⟩ + |1⟩]

[|0⟩ − |1⟩]

│ψ<sup>−</sup>

. (1)

⟩ = 0, corresponding to the scalar

ing (FBG/AWG) on Bob's side and suppressed carrier.

116 Advanced Technologies of Quantum Key Distribution

**encoding**

two form second basis.

It is necessary to fulfill the terms ⟨ψ<sup>0</sup>

One of the first PM-PM scheme variants is based on the B92 protocol [22]. Its OptiSystem model is presented in **Figure 1**.

Alice modulates photon |ω0 ⟩ in left PM by RF signal from sine generator with frequency Ω and phase Φ = ΦA, getting:

$$\|A\rangle = \sum\_{n \neq \omega}^{n \neq \omega} f\_n \exp^{|n \oplus \theta\_\perp|} \|\omega\_n\rangle. \tag{2}$$

where, for simplicity of display, the argument of the Bessel's function *J n* is not specified. On the receiving end, Bob modulates the input radiation synchronized with Alice RF signal from its sine generator (right) phase Φ=ΦB. At Bob's PM output, one will receive:

$$\left|B\right> = \sum\_{n,k} I\_n I\_{k-n} \exp^{\left|n\Phi\_s\right>} \exp^{\left(\left|k-n\right>\Phi\_s\right>} \left|\omega\_k\right> \tag{3}$$

It should be noted that modulation effect is to transfer energy from carrier on the sidebands (subcarriers). Its effectiveness depends on modulation and corresponding phases ΦA and ΦB. Transfer efficiency P(ω0 → ω0 ± Ω) is proportional to the function cos<sup>2</sup> (∆Φ/2), where ∆Φ = ΦB−ΦA, and is at maximum when ∆Φ = 0, which indicates the same basis chosen by Alice and Bob (**Figure 2**).

Further exchange of information between Alice and Bob allows them to set a secure connection with the implementation of the B92 protocol. The definition of a key with probability equal to one for Eve is impossible.

Determination of phase's compliance level in the scheme is actually implemented by the amplitude of the subcarriers. That is also the evidence of these scheme drawbacks, taking into account the small power of optical subcarriers and the presence of noise in the communication channel and single photon detector (SPD).

**Figure 1.** Modeling of PM-PM scheme for QKD system with frequency coding.

The |±;2⟩ states are determined without applying the re-modulation, by the use of filter sets based on FBG or AWG and logic conditions. Scheme decision shows the lowest QBER value. Only sideband SPD also works during the transfer of |±;1⟩ states, because at specified condi-

Universal Microwave Photonics Approach to Frequency-Coded Quantum Key Distribution

AM-AM schemes use for elimination of PM-PM ones shortcomings. One of them was imple-

The first AM-AM scheme is based on BB84 protocol [23]. Its OptiSystem model is presented in

It should be noted that the modulator on the of Alice's side is modulated according to the law cos(Ωt + ΦA), and on the Bob's side according to sin(Ωt + ΦB). Transfer efficiency P(ω0 → ω0 ± Ω)

bands, respectively, at ΦA = π/2 and ΦB = 3π/2. Determination of phase's compliance level in

AM-PM or PM-AM scheme implementation intuitively appears to be based on the principles set out, respectively, for AM-AM and PM-PM schemes. One of its OptiSystem model is pre-

Determination of phase's compliance level in the scheme is also implemented by the ampli-

It should be noted that we have some conflicting information about the possibility [13] and impossibility [23] of meshed AM-PM (PM-AM) scheme realization as for protocol BB84, so

(∆Φ/2) and sin<sup>2</sup>

**Figure 3**, and constructive and destructive interferences are shown in **Figure 4**.

the scheme is also implemented by the amplitude of the lateral components.

is equal to 0. The error

119

http://dx.doi.org/10.5772/intechopen.71974

(∆Φ/2) for the upper and lower side

tions of modulation and re-modulation the component at frequency ω0

level in transmission |±;1⟩ states is 4.7%.

in this case is proportional to function cos2

**2.3. Meshed AM-PM (PM-AM) schemes**

tude of the lateral components (**Figure 6**).

**Figure 3.** Modeling of AM-AM scheme for QKD system with frequency coding.

**2.2. AM-AM schemes**

sented in **Figure 5**.

mented only on the acousto-optic modulators [16].

**Figure 2.** Constructive ∆Φ = 0 (a) and destructive ∆Φ = π/2 (b), ∆Φ = 3π/2 (c), ∆Φ = π (d) interferences on the output of Bob's PM, when Alice's ΦA = 0.

A second version of the PM-PM scheme [14] was proposed for elimination of given drawbacks. It is based on nonlinear interaction of the RF signal and the photon in the electro-optic modulator and implements a more advanced BB84 protocol. Notch filter on ω0 frequency is set prior to sideband SPD, which reflects the carrier at the corresponding receiver, transmitting all the remaining subcarriers on ω0 ± Ω and ω0 ± 2 Ω frequencies.

For BB84 protocol realization, two bases are set as following:

$$\begin{cases} \text{from, two bases are set as following:}\\ \begin{cases} \left| +; \ 1 \right> = \frac{1}{\sqrt{2}} \left| \ 1 \right>\_{\omega\_{\circ}} + \frac{1}{2} \left| \ 1 \right>\_{\omega\_{\circ} \circ \Omega} - \frac{1}{2} \left| \ 1 \right>\_{\omega\_{\circ} \circ \Omega} \\ \left| -; \ 1 \right> = \frac{1}{\sqrt{2}} \left| \ 1 \right>\_{\omega\_{\circ}} - \frac{1}{2} \left| \ 1 \right>\_{\omega\_{\circ} \circ \Omega} + \frac{1}{2} \left| \ 1 \right>\_{\omega\_{\circ} \circ \Omega} \\ \begin{array}{c} \left| +; \ 2 \right> = \left| \ 1 \right>\_{\omega\_{\circ}} \\ \left| -; \ 2 \right> = \frac{1}{\sqrt{2}} \left| \ 1 \right>\_{\omega\_{\circ} \circ \Omega} - \frac{1}{\sqrt{2}} \left| \ 1 \right>\_{\omega\_{\circ} \circ \Omega} \end{array} \end{cases} \tag{4}$$

The |±;2⟩ states are determined without applying the re-modulation, by the use of filter sets based on FBG or AWG and logic conditions. Scheme decision shows the lowest QBER value. Only sideband SPD also works during the transfer of |±;1⟩ states, because at specified conditions of modulation and re-modulation the component at frequency ω0 is equal to 0. The error level in transmission |±;1⟩ states is 4.7%.

AM-AM schemes use for elimination of PM-PM ones shortcomings. One of them was implemented only on the acousto-optic modulators [16].
