Two-Rail Photonic Qubit Utilizing the Quantum Holographic Imaging Idea

Abstract

We present the novel approach to physical implementation of qubits with the technology of photonic chips. Proposed multi-rail qubit model, called QBell, utilizes hyper-entanglement to work in Decoherence Free Subspace on physical layer. This makes this solution robust and can result in increasing fidelity of quantum circuit used in this model. We elaborate the two-rail case. We define the QBell and discuss its internal structure. We construct also one- and two-qubit gates to make the model comprehensive and ready to implement. Proposed model utilizes the early-stage ideas for optical quantum computation, but by using the polarization and position entanglement as the resource of computation allows to avoid the general problem of them, like heralded photon technique. The technology of photonic chips allows to brake other limitations that are pointed in the text. The presented model was inspired by quantum holographic imaging and uses the holographic technique for implementing the z-rotation operation. The final product will be the photonic quantum processor using multi-rail qubits. It will find the application in many domains (e.g., medical) on earth and in the space.

1. Introduction

Photonic qubits (quantum bits), which means qubits utilizing the quantum optical phenomena, was widely elaborated at the turn of XX and XXI centuries. The name photonic is a modern name, since in that time authors tell about optical qubits. However, this name belongs currently to ion-trapped qubits using standard energetic structure of atoms, e.g., Bruzewicz et al. [1]. The first one-qubit operation was described by Simon and Mukunda in [2] in the context of transformation between polarization and position logic of qubits. The early studies of this area assumed that the photonic qubits should utilize the linear optics phenomena, since the general name of such solutions is LOQC (Linear Optics Quantum Computing). The first protocol of quantum computing using optical circuit was elaborated by Knill, Laflamme, and Milburn and named after author’s initials KLM protocol, published in several articles [3, 4, 5]. Finally, Myers and Laflamme published a review and tutorial of KLM theory [6]. The fundamental problem with this scheme is that it uses as the basic building block so-called nonlinear sign NS gate. It is probabilistic gate, which means that it returns the correct result not always but with some probability depending on the specific implementation, e.g., Knill, Laflamme, and Milburn implementation success probability is only equal to $\frac{3-\sqrt{2}}{7}\sim 0.2265$! To inform which result is correct, one has to measure additional output. When on this output, there appears a photon, the result is correct. Currently such photons are called heralded [7]. Knill in [8] proved theoretically that the upper bounds of NS gate success probability are equal to $1/2$ and for conditionally NS gate with two modes equal to $3/4$. The problem with such “heralded” gates is that in case of high frequency of changes between correct and not correct results, the detection of the moment of correct result is very hard. Hence, the fidelity of such gates was quite low—at the level of 84%—Gasparoni et al. [9].

There was some improving KLM scheme protocols. The main improvement was include to linear systems nonlinear elements for generating the numbers of entangled photons. Yoran-Reznik [10] proposed a protocol based on “entanglement chains.” This is important effort for us, since it was the first time, when hyper-entanglement was used for computation. However, he still uses the probabilistic gates and heralded photons. There is also a class of protocols based on the cluster states, also known as multipartite entanglement [11]. It was, e.g., Nielsen [12] or Browne-Rudolph [13] protocol.

All those ideas were yet treated as “demonstratory” or even “simulation” of quantum computers. We think that we have no right to name it “simulation” since in those circuits there are real quantum effect and such a protocols are widely used in quantum security and quantum communication solutions. The problem that there was only demonstration of quantum computers lays in two facts. Firstly, they are not programmable—if you have new algorithm you have to rebuild the setup on the optical table, which is not practical in production application. Secondly, the number of devices needed to grow exponentially with the growth of qubit number. This is even harder limitation in practical solutions. Therefore, at about 2005, the number of works on linear optics application for quantum computing decreased rapidly, also by arising realistic superconducting and ion-trap solution in the area of quantum processors.

However, the idea of using quantum optics for computation is very tempting because it does not need extreme cooling like in mentioned superconducting solution (cooling below 1 K) or ion-trap (cooling to 10–35 K, depending on the components). The worst case for quantum dots-based optical current is temperature on the level of 213 K, which is much affordable. The most solution does not require cooling at all and potentially can work in any temperature. Therefore, when the tool for implementation appears, several solutions were developed. This tool is the technology of programmable photonic chips announced by Bogaerts et al. in [14]. There have been developed some quantum processors based on similar technology like Xanadu (Madsen et al. [15]) or QuiX (Taballione et al. [16], deGoede et al. [17]).

When we moving in the area of quantum holography, we have to pay attention on the nomenclature. That is because this notion is used for holographic data storage system [18] or even in psycholgy and social science [19]. In this chapter we will use the quantum holography notion as the tool for generating holographic images. There are two similar works of Defienne et al. [20] and Toepher et al. [21]. We will define works that inspired us in the elaboration of proposition of two-rail qubit, which will be described in more detail in Section 2. At this point it is enough to say that it uses the hyperentangled photons, like in Yoran-Roznik protocol of linear optics quantum computation.

Our contribution is the definition of the novel type of qubits that are consisting on more than one rail, to create more robust circuit with high fidelity. In this work we present the definition of two-rail qubit called two-rail Bell’s qubit, QBell. We discuss its internal structure, define and show how to construct one qubit gates for QBell needed for generation of universal gate. We also show how to physically implement controlled gate. This all constructions avail physical implementation of such qubits, especially on the photonic chips.

Our motivation in long-term perspective is to create multi-rail robust qubits with high fidelity, which allows to create robust photonic quantum circuit, where it will be possible to use one gate multiple times. The possibility of creating programmable quantum photonic chips is already available. Together with multiple usage of gates they will break, in long term, the mentioned limitation of photonic quantum computing. This work is one first step on this road.

2. Quantum holographic imaging

Two-rail optical qubit proposed by us is inspired by quantum holographic imaging, described by Defienne et al. in [20]. Like in classical case, it relies on extracting the differences in phase of light reflected from the object being imaged. Unlike the classical method in presented technique, the hyperentanglement is utilized in the channels of polarization and space.

Deffiene uses the output of SPDC (Spontaneous Parametric Down Conversion) type I process, which produces the states $\mid \mathit{VV}⟩+\mid \mathit{HH}⟩$¹. This formula describes the state, which is entangled in polarization channel, but there is no information about space entanglement. In quantum computing we pursue to narrow laser beam as possible, and we consider existing the photon on the “first” or “second” beam, therefore we can easily rewrite the state in four-slot bra or ket describing the state in Fock space. However, in the holographic imaging application, there are wider beams used, which illuminates the area of a given object and the group of pixels in the EMCCD camera playing the role of detector in the whole system. Therefore the position of the photon is given by its momentum $\mathbf{k}\mathbf{=}\left({\mathbf{k}}_{\mathbf{x}},{\mathbf{k}}_{\mathbf{y}}\right)$. The position entanglement means that if two photons have the same $k_y$ component, one of them has $k_x$ momentum component and the second one $-k_x$. we will denote this situation that one photon has $\mathbf{k}$ momentum while the second one $-\mathbf{k}$. Now, the hyperentangled state can be rewritten as: $\mid V⟩{}_{\mathbf{k}}\mid V⟩{}_{-\mathbf{k}}+\mid H⟩{}_{\mathbf{k}}\mid H⟩{}_{-\mathbf{k}}$. Since we mentioned about consideration of an area illuminated by a beam of hyperentangled photons, the final state equation is given by:

$$\mid A⟩=\sum _{\mathbf{k}}\left[|V⟩{}_{\mathbf{k}}|V⟩{}_{-\mathbf{k}}+|H⟩{}_{\mathbf{k}}|H⟩{}_{-\mathbf{k}}\right]$$E1

The procedure of obtaining the quantum holographic image is shown schematically on the Figure 1. The procedure starts with producing the hyperentangled state $\mid A⟩$, e.g., using the SPDC. Alice transmits its beam through the pretreated sample. On the scheme it is signed “Alice’s SLM” (Spatial Light Modulator), since in Defienne experiment this device was used for creating the images in phase domain. However, it can be any sample that we would to examine, e.g., medical preparation. Bob transmits its beam through another SLM. It plays the role of phase correction device. Before the examination, instead of sample there should be placed a device that has the exactly the same phase in all places. Because of imperfection of the instruments on the Bob’s and Alice’s cameras, the pictures can differ. Using Bob’s SLM we can mitigate these differences changing the phase shifts in all positions. We can consider this procedure as the setup calibration process. The pattern arised on the Bob’s SLM we call reference pattern. Having the sample and reference pattern, we can determine the function $\psi \left(\mathbf{k}\right)$, which is the difference between the phase shift of sample and reference SLM, which is shown on the right part of the scheme. The state after SLM is given by an equation:

Figure 1.

The general scheme of quantum holography imaging. On the picture, there are shown the states before and after going through SLM (Spatial Light Modulator). On the right side there is schematically shown the meaning of $\psi \left(\mathbf{k}\right)$ function.

$$\mid A^{\prime }⟩=\sum _{\mathbf{k}}\left[|V⟩{}_{\mathbf{k}}|V⟩{}_{-\mathbf{k}}+e^{\mathit{i\psi }\left(\mathbf{k}\right)}|H⟩{}_{\mathbf{k}}|H⟩{}_{-\mathbf{k}}\right]$$E2

At the end the beams are detected after passing polariser rotated by 45 degrees. The intensity correlations are measured between $\mathbf{k}$-th pixel on Bob’s camera and $-\mathbf{k}$ on Alice’s one. The procedure is repeated several times for different homogeneous phase shift on Bob’s SLM. In the Deffiene experiment, the iterations was made for phase shifts: $0,\pi /2,\pi$ and $3\pi /2$. Basing on those intensity correlation images, the original object can be reconstructed.

There are many possible application of such a technique, including medical (e.g, histopathology) preparations. Authors point the robustness of the system, which is very crucial for quantum computation as well.

In next sections we will show how to adopt this idea, especially the phase and polarization entanglement, in a quite similar way to the method described above.

3. Foundations of quantum computation

In the area of non-imaging optics, prominent place is taken by implementation of quantum computation. Quantum bit, called qubit, is the smallest piece of quantum information, just like bit is the realization of classical one. While bit can be in one of two states (0 or 1) during whole its life, qubit can be in the state that is superposition of 0 and 1. Two states, equivalent to 0 and 1 are denoted in bracket notation as $\mid 0⟩,\mid 1⟩$, as well as the qubit itself. Then the superposition for qubit $\mid q⟩$ is described by equation: $\mid q⟩=\alpha \mid 0⟩+\beta \mid 1⟩$, where $\alpha ,\beta \in \mathrm{ℂ}$ and ${\left|\beta \right|}^2+{\left|\beta \right|}^2=1$, called unit condition. This representation of qubit is, generally speaking, a vector in Hilbert space². At first glance, one can think it has four degrees of freedom (DoF), since four real numbers are needed to describe such system. However, if we include the unitary condition we can convince ourselves that: $\alpha =\mathit{cos}\left(\frac{\vartheta }{2}\right),\beta =e^{\mathit{i\varphi }}\mathit{sin}\left(\frac{\vartheta }{2}\right)$, thus it has two DoF: $\varphi ,\vartheta$. This formula is the general equation of qubit. It defines the representation of qubit on a Bloch Sphere in polar coordinates, considering unit length radius. On the north pole of the Bloch sphere, there is a representation of $\mid 0⟩$ and on the south pole of $\mid 1⟩$. Every other position is the superposition.

The bit state can be changed by a gate, which we identify as the logic (or Boolean) function. Similarly, the state of the qubit can be changed by a quantum gate, which is operator in Hilbert space. Because of unit condition, the gate has to be unitary, since unitary operators save the norm. There are plenty of different types of gate, e.g., rotation around $x,y$ and $z$ axis of Bloch sphere, X, Y, Z Pauli gates, phase shift gate, etc. However, for physical implementation ability to change the state by changing the coordinates on the Bloch Sphere is enough. There could be other set of operations depended on physical implementation, certainly. For example, IBM-Q uses rotation around the Z and X axis of Bloch sphere. Using such a rotation the general form of quantum gate can be constructed:

$$U\left(\theta ,\phi ,\lambda \right)=\left[\begin{array}{cc}\mathit{cos}\left(\frac{\theta }{2}\right)& -e^{\mathit{i\lambda }}\mathit{sin}\left(\frac{\theta }{2}\right)\\ e^{\mathit{i\phi }}\mathit{sin}\left(\frac{\theta }{2}\right)& e^{i\left(\phi +\lambda \right)}\mathit{cos}\left(\frac{\theta }{2}\right)\end{array}\right]$$E3

The multiqubit states can be constructed by tensor product of single qubits. In that case the gates are the tensor products of one qubit gates as well. The tensor products create the interference, which is the second quantum phenomenon used for computation. Nevertheless, there are states that cannot be expressed by tensor product of single qubit states. Such a states are called entangled state. Along the tensor product the system $\mid s⟩$ made of $S$ qubits has $2^S$ coordinates. They can be understood as something like $2^S$ parallel quantum cores for computation. The entanglement creates dependencies between such “cores” and interference allows to affect the chosen subset of them concurrently.

Measurement of a quantum system effects with a collapse of qubit to the so called eigen states, which are $\mid 0⟩$ and $\mid 1⟩$. It is not a determined process but each of eigenstate appearance is defined by probabilities equal to $⟨q|0⟩={\left|\alpha \right|}^2$ and $⟨q|1⟩={\left|\alpha \right|}^2$. The probability distribution function of eigen-state appearance is one of the possible results of quantum computation called quantum sampling. In multiqubit case there appears $2^S$ possible outputs, which would make such a procedure inapplicable in practical solutions. However the entanglement allows to limit the number of output states to the reasonable size.

Quantum computation can be divided into four parts: (1) initialization, which means to set up the qubit to state $\mid 0⟩$, (2) evolution of the qubit state with an evolution operator made of gates by composition and tensor product, (3) measurement, and (4) interpretation of the result.

To create one qubit:

1. We have to know what is the physical meaning of being in eigen state $\mid 0⟩$ or $\mid 1⟩$ for qubit.

2. We have to create such a setup that will cover all possible rotations on a Bloch Sphere. In other words, the setup has to implement the general quantum gate (Eq. (3)). It is obvious it implies that qubit can be superposed.

3. We have to be able to measure the qubit and to decide what state the qubit had collapsed to after measurement.

4. In case of multiqubit state we have to implement:

   1. the interference.

   2. the entanglement of qubits.

In this chapter we show how to build qubit and multiqubit systems using the linear and nonlinear optics. We will also discuss the possible practical implementation.

4. The concept of two-rail optical qubit

Two rail qubit employs the idea described in the Section 2. The process starts like in the holographic case—we create two streams of hyperentangled photons using nonlinear unit. In the next section we will discuss which materials can be used for efficient implementation of this process. The state after this process is (see [22]):

$$\frac{\mid \mathit{HV}⟩+e^{\mathit{i\delta }}\mid \mathit{VH}⟩}{\sqrt{2}}$$E4

In the equation above $e^{\mathit{i\delta }}$ is a local phase, which will be eliminated during qubit calibration process. Moreover this equation tell us, that there two possible states that system will collapse to, with equal ($0.5$) probability. Hence the state is in the superposition of those states. In both cases two photons generated by nonlinear element are on the separate rails and have different orthogonal polarizations (H and V). To establish qubit we will call the first one control rail TR and the second one computation rail CR. The photons on those two rails are entangled, since if on the TR there is horizontally polarized photon then on the CR is the one vertically polarized, and vice-versa.

Now, we will define the two rail qubit, using 4 points defined in the Section 3.

1. In our proposition we will use the polarization entanglement and temporal correlations of photon as a resource for creation the qubit. As was mentioned, we have only got two possible configuration of photons, initially: $\mid \mathit{HV}⟩$ and $\mid \mathit{VH}⟩$, which we consider as $\mid 0⟩$. Moreover this configuration is $\mid {\mathrm{\Psi }}^{+}⟩$ Bell state. However, it is easy to change polarization between Bell states, e.g., with half plates, the changes of phase are much more complicated. This operation produces the Bell states except of local phase. Therefore we will call them Bell-like states. Finally, we will consider ${\mathrm{\Psi }}^{\pm }$-like as $\mid 0⟩$ and ${\mathrm{\Phi }}^{\pm }$-like as $\mid 1⟩$.

2. Polarization rotator and phase modulator will be used for implementation of one qubit gates.

3. The measurement relies on splitting each rail of the qubit by polarization beam splitter and each of four rails will be measured by photo detectors. Then the time coincidence of photon detected will be determined. We will take into considerations only those detections that cover one of the Bell state. The rest of them are considered as noise.

4. The multiqubit systems will consists of $2n$ rails. Therefore the tensor product of them appears naturally on the output set of detectors.

   1. In the virtue of above the interference appears in the system spontaneously.

   2. The entanglement between qubits, will be established basing on the CNOT (Controlled NOT) gates designed for one-rail qubits in the polarization channel.

4.1 The representation of the states

For the formal description of two-rail qubit we use the Fock space (or second quantization) formalism owing to the nature of the light. Firstly we establish the symbols for creation and annihilation operators. There four pairs of such operators:

$$\begin{array}{l}{h}_{\mathit{tr}}^{†},h_{\mathit{tr}}{v}_{\mathit{tr}}^{†},v_{\mathit{tr}}{h}_{\mathit{cr}}^{†},h_{\mathit{cr}}{v}_{\mathit{cr}}^{†},v_{\mathit{cr}}\\ \text{horizontal},\mathrm{TR}\text{vertical},\mathrm{TR}\text{horizontal},\mathrm{CR}\text{vertical},\mathrm{CR},\end{array}$$E5

where $†$ means the creation operator and no upper index means annihilation one. The acting of operators from TR group on the state, creates the photon with H or V polarization on the control rail. Similarly, operators from CR group creates the photon on the computation rail. The description of state in Fock space is defined as follows:

$$\mid \mathrm{\Psi }⟩=\mid H_{\mathit{TR}}{V}_{\mathit{TR}},H_{\mathit{CR}}{V}_{\mathit{CR}}⟩,$$E6

where $X_Y$ means the number of photons with polarization $X$ (horizontal or vertical) on the rail $Y$ (TR or CR). For example, $h_{\mathit{tr}}^{†}{v}_{\mathit{cr}}^{†}\mid 0⟩=\mid 10,01⟩$, which represents the state $\mid \mathit{HV}⟩$ in Hilbert space.

4.2 The qubit

We can define the operator defining the output of SPDC process (Eq. 4), using the second quantization:

$$\widehat{S}=\frac{h_{\mathit{tr}}^{†}{v}_{\mathit{cr}}^{†}+v_{\mathit{tr}}^{†}{h}_{\mathit{cr}}^{†}}{\sqrt{2}}$$E7

Indeed, if this operator acts on the vacuum we obtain:

$$\begin{array}{l}\widehat{S}\mid 0⟩=\frac{1}{\sqrt{2}}\left(h_{\mathit{tr}}^{†}{v}_{\mathit{cr}}^{†}|0⟩+v_{\mathit{tr}}^{†}{h}_{\mathit{cr}}^{†}|0⟩\right)=\frac{1}{\sqrt{2}}\left(h_{\mathit{tr}}^{†}|00,01⟩+v_{\mathit{tr}}^{†}|00,10⟩\right)\\ =\frac{\mid 10,01⟩+\mid 01,10⟩}{\sqrt{2}}=\frac{\mid \mathit{HV}⟩+\mid \mathit{VH}⟩}{\sqrt{2}}.\end{array}$$

From the above equation we learn that the SPDC’s process output is the superposition of two eigen-states only: $\mid 10,01⟩$ and $\mid 01,10⟩$. It means that after measurement, the time-correlated two photons are detected on the different rails and has mutually orthogonal polarization. Moreover one of them has the same polarization as the input to the SPDC process. Now, we can formally define our proposition for an implementation of photonic qubit as follows.

Definition 4.1 The two-rail Bell’s qubit, QBell is the system of two hyperentangled photons, which is composed of, so-called Bell-like states:

$$\begin{array}{l}\mid {\mathrm{\Psi }}_l^{+}⟩=\frac{\mid 10,01⟩+e^{i{\lambda }_1}\mid 01,10⟩}{\sqrt{2}}\mid {\mathrm{\Phi }}_l^{+}⟩=\frac{\mid 10,10⟩+e^{i{\lambda }_2}\mid 01,01⟩}{\sqrt{2}}\\ \mid {\mathrm{\Psi }}_l^{-}⟩=\frac{\mid 10,01⟩-e^{i{\lambda }_3}\mid 01,10⟩}{\sqrt{2}}\mid {\mathrm{\Phi }}_l^{-}⟩=\frac{\mid 10,10⟩-e^{i{\lambda }_4}\mid 01,01⟩}{\sqrt{2}},\end{array}$$

where ${\lambda }_j\in \mathbb{R}$³ is local phase and can be any number from the range $\left[-2\pi ,2\pi \right]$. The Bell-like state $\mid {\mathrm{\Psi }}_l^{\pm }⟩$ is considered as the qubit eigen state $\mid 0⟩$, and $\mid {\mathrm{\Phi }}_l^{\pm }⟩$ as qubit eigen state $\mid 1⟩$. For convenience, we will denote the QBell states with Q in index behind the bra-ket: $\mid 0⟩{}_Q,\mid 1⟩{}_Q$.

The first part of Bell-like states’ component we call the leading, e.g. $\mid 10,01⟩$ for $\mid {\mathrm{\Psi }}_l^{\pm }⟩$. The second part we call complementary state. If we denote the leading state as $\mid A⟩$_, then the complementary state we denote $\mid \overline{A}⟩$.

Note that leading part always begins with horizontally polarized photon, while the complementary—with vertically polarized one.

Definition 4.2 Two QBells components $\mid q⟩{}_Q$ and $\mid p⟩{}_Q$ are considered as similar with phase exclusion $\mid q⟩{}_Q\sim \mid p⟩{}_Q$ (or just similar) if and only if:

$$\forall \delta \in \mathbb{R}\exists n,m\in \mathbb{R}s.t.:\mid q⟩{}_Q=\frac{\mid A⟩+e^{\mathit{in\delta }}\mid \overline{A}⟩}{\sqrt{2}}\wedge \mid p⟩{}_Q=\frac{\mid A⟩+e^{\mathit{im\delta }}\mid \overline{A}⟩}{\sqrt{2}}$$

The relation defined above, is the equivalence relation, certainly, which can be proved using elementary methods, hence we omit them. Therefore it divides the set of QBells’ components into equivalence classes. Finally each class can be considered as the QBell’s component itself. Therefore we can use it for deeper look on the coefficient inside the QBell (see Figure 2A). Namely the coefficient of the leading part is always real and equal to $1/\sqrt{2}$, and the coefficient of the complementary part can be placed on the complex plain on a circle with a center in the $\left(0,0\right)$ and passing through the coefficient of the leading point.

Figure 2.

The location on complex space of the leading and complementary coefficients and the local and global phase on the state.

Corollary 4.1 Acting on the Bell-like eigen-state with an operator changing the phase of leading state changes the global phase of this eigen-state.

Proof: Indeed, let us denote Bell-like state as $\mid X_l⟩=\frac{\mid A⟩+e^{\mathit{in\lambda }}\mid \overline{A}⟩}{\sqrt{2}}$ and consider that a state $\mid Y⟩={\widehat{O}}_{\alpha }\mid X_l⟩=\frac{e^{\mathit{i\alpha }}\mid A⟩+e^{\mathit{i\epsilon }}\mid \overline{A}⟩}{\sqrt{2}}$, where $\alpha ,\epsilon \ne 0$. In that case we can write:

$$\mid Y⟩=e^{\mathit{i\alpha }}\frac{\mid A⟩+e^{i\cdot 1\cdot \left(\epsilon -\alpha \right)}\mid \overline{A}⟩}{\sqrt{2}}=e^{\mathit{i\alpha }}\mid X_l⟩,n=1,\delta =\epsilon -\alpha$$

Looking on the Figure 2B, we see that the global phase rotates both leading and complementary coefficients around the circle of complementary coefficients (dotted line on picture B) on the complex space. As the effect the new state does not fulfill the QBell condition, that the leading coefficient has to be equal to $1/\sqrt{2}$. In algebraic sense it means that we factor out the global phase. In geometric representation we can imagine that it is a rotation (picture B on mentioned figure) of complex space by the phase in order to place the leading coefficient on $1/\sqrt{2}$ location up again. It means that the global phase of the given eigen-state becomes local phase of QBell.

Corollary 4.2 Acting on the Bell-like state with an operator changing the complementary phase does not change the QBell state.

Proof: Indeed, let us denote Bell-like state as $\mid X_l⟩=\frac{\mid A⟩+e^{\mathit{in\lambda }}\mid \overline{A}⟩}{\sqrt{2}}$ and consider that a state $\mid Y⟩={\widehat{O}}_{\alpha }\mid X_l⟩=\frac{\mid A⟩+e^{\mathit{i\alpha }}{e}^{\mathit{i\epsilon }}\mid \overline{A}⟩}{\sqrt{2}}$, where $\alpha ,\epsilon \ne 0$. In that case:

$$\mid Y⟩=\frac{\mid A⟩+e^{i\cdot 1\cdot \left(\epsilon +\alpha \right)}\mid \overline{A}⟩}{\sqrt{2}}=\mid X_l⟩,n=1,\delta =\epsilon +\alpha$$

In this case, shown on the Figure 2D we change the local phase of one of the QBell’s component, but not the QBell state itself. The original and changed component’s state are similar in the light of def. 4.2. Therefore, they represents the same component, so the QBell state stays untouched.

On the left hand, there is shown two types of Bell-like states. The potential existence of the photon on the rail is marked with a black dot with label describing which polarization state it concerns. The correlated states from control and computation rails are marked with a line. The same color of the lines means the same Bell-like state. Solid line is the leading state while the dashed one complementary state.

On the right hand, is the process of measurement. First the horizontal and vertical polarized photons are distributed on separate lines by polarized beam splitters. Nevertheless the entanglement is hold, which is shown by correlation lines. On the most right side the long rectangles symbolize the detectors. The circles inside them are the results of measurement- filled is the detection of photon while empty is the lack of detection. Solid fill means the detection of leading state, while dashed one - the complementary state. Each column through the detectors means the time correlation of measurement. The first two of them describes the situation that we interpret the result as $\mathrm{\Phi }$-like state, hence $\mid 0⟩{}_Q$, the last two - as $\mathrm{\Psi }$-like state, hence $\mid 1⟩{}_Q$.

Initialization. This qubit obtained in SPDC process is in the $\mid 0⟩{}_Q$ state already. Since the initialization state occurs automatically, so we do not need to make any operations to generate it, in opposite to other solutions.

4.3 The measurement

The measurement is performed by avalanche photo detectors preceded by division both rails using polarization beam splitter. In such a way we can check how many photons had received each polarization on each rail, during the wave function collapse caused by measurement. On theoretical level we consider two photons and their polarization-space configuration, like it is shown on the Figure 3. There are two photo-detectors for TR rail, marked with TRH for horizontal polarized photon and TRV for vertical polarized photon. The detectors for CR rail are marked similarly: CRH i CRV. Let us consider that our qubit is in state.

Figure 3.

Scheme of the two-rail Bell’s qubit, QBell and its measurement. There are shown three layers of abstraction. The most bottom one contains the physical optical current. The middle one—the state of the current in the Fock space. The upper one is the QBell logic, which is an interpretation of the current state in the perspective of QBell definition or measurement results.

$$\mid q⟩{}_Q=\alpha \mid 0⟩{}_Q+\beta \mid 1⟩{}_Q$$E8

It means that on physical, optic layer the system is in the state:

$$\frac{\alpha \mid 1010⟩+\alpha e^{i{\delta }_1}\mid 0101⟩}{\sqrt{2}}+\frac{\beta \mid 1001⟩+\beta e^{i{\delta }_2}\mid 0110⟩}{\sqrt{2}}$$

Thus, the probability that detectors TRH and CRH click in the same time is equal to $\frac{{\left|\alpha \right|}^2}{2}$ as well as TRV and CRV, which can be seen on mentioned Figure 3 where the clicks of detectors at the same time are arranged in columns. Hence we can say that the probability of event that the photons with the same polarization on separate rails was measured is equal to $\mid {\alpha }^2\mid$. It means that we measured $\mathrm{\Phi }$-like state. Hence our QBell is in $\mid 0⟩{}_Q$ state with probability $\mid {\alpha }^2\mid$, which is consistent with our assumption from Eq. (8). Such configuration of detectors’ clicks we will call alternating. Using the same line of reasoning, we conclude that probability of the event that photons on the TR and CR rails has different polarizations, is equal to ${\left|\beta \right|}^2$. Hence the probability of obtaining QBell in state $\mid 1⟩{}_Q$ is equal to ${\left|\beta \right|}^2$ as well, which is also consistent with the same assumption. This configuration of detector clicks we will call straight.

In the real application, in current state of the art, it is not possible to generate separate in space and time hyperentangled photon pairs efficiently in the process of SPDC. It’s because of very low efficiency of the process even in the sources considered as strong. Therefore we use extremely narrow laser pulses, nano- or even femtosecond long, for powering the SPDC sources, nevertheless photon count in such an impulse is on the order of ${10}^{12}$. Therefore, a detection of time correlated photo diodes clicks is one of the key point of the quantum computing systems.

4.4 One qubit quantum gates

For practical implementation of the complete set of one qubit quantum gates the universal $U\left(\theta ,\phi ,\lambda \right)$ has to be physically available. One of available forms of such a gate is as follows.

$$U\left(\theta ,\phi ,\lambda \right)=R_z\left(\phi +\frac{\pi }{2}\right)R_x\left(\theta \right)R_z\left(\lambda -\frac{\pi }{2}\right)$$E9

From equation above we see that we need to implement rotations around $z$ and $x$ axis of Bloch sphere.

There are two building blocks of the gates: polarization rotator and phase modulator.

4.5 Rotation around $x$ axis of Bloch sphere

In the terms of second quantization, polarization rotator $\widehat{R}\left(\alpha \right)$, where $\alpha$ is the rotation angle, has got the input creation operators $R_i^{h†},R_i^{v†}$ and output creation operators $R_o^{h†},R_o^{v†}$. The relation between them is defined as follows.

$$\begin{array}{c}{R}_i^{h†}\left(\alpha \right)=\mathit{cos}\left(\alpha \right)R_o^{h†}-\mathit{\text{isin}}\left(\alpha \right)R_o^{v†}\\ R_i^{v†}\left(\alpha \right)=\mathit{cos}\left(\alpha \right)R_o^{v†}-\mathit{\text{isin}}\left(\alpha \right)R_o^{h†}\end{array}$$E10

We apply the polarization rotator to the control rail. One photon on it can be in states: $\mid 10,\cdot \cdot ⟩$ in case of horizontal polarization and $\mid 01,\cdot \cdot ⟩$ in case of vertical polarization or in superposition of those states. The symbol $\cdot \cdot$ means that on the computation rail can be any allowed state ($01,10$) In fact after non-linear element, it is in homogeneous superposition. In case when no device is applied to a rail we use “no change” operator $N_i^{h†}=N_o^{h†},N_i^{v†}=N_o^{v†}$ that saves polarization. Hence the acting of polarization rotation on Bell states in second quantization terms are defined as follows.

$$\begin{array}{c}\widehat{R}\left(\alpha \right)\mid {\mathrm{\Psi }}_l^{+}⟩=\frac{1}{\sqrt{2}}\left[R_i^{h†}\left(\alpha \right)N_i^{v†}+R_i^{v†}{N}_i^{h†}\right]\\ \widehat{R}\left(\alpha \right)\mid {\mathrm{\Phi }}_l^{+}⟩=\frac{1}{\sqrt{2}}\left[R_i^{h†}\left(\alpha \right)N_i^{h†}+R_i^{v†}{N}_i^{v†}\right],\end{array}$$E11

As a result of action of above operator on a vacuum, we obtain the state as follows.

$$\begin{array}{c}\widehat{R}\left(\alpha \right)\mid {\mathrm{\Psi }}_l^{+}⟩=\frac{1}{\sqrt{2}}\left[\left(\mathit{cos}\left(\alpha \right)R_o^{h†}-\mathit{\text{isin}}\left(\alpha \right)R_o^{v†}\right)N_o^{v†}+\left(\mathit{cos}\left(\alpha \right)R_o^{v†}-\mathit{\text{isin}}\left(\alpha \right)R_o^{h†}\right)N_o^{h†}\right]\mid 0⟩\\ =\frac{1}{\sqrt{2}}\left[\mathit{cos}\left(\alpha \right)|10,01⟩-\mathit{\text{isin}}\left(\alpha \right)|01,01⟩+\mathit{cos}\left(\alpha \right)|01,10⟩-\mathit{\text{isin}}\left(\alpha \right)|10,10⟩\right]\\ =\frac{1}{\sqrt{2}}\left[\mathit{cos}\left(\alpha \right)\left(|10,01⟩+|01,10⟩\right)-\mathit{\text{isin}}\left(\alpha \right)\left(|10,10⟩+|01,01⟩\right)\right]\\ =\mathit{cos}\left(\alpha \right)\mid {\mathrm{\Psi }}_l^{+}⟩-\mathit{\text{isin}}\left(\alpha \right)\mid {\mathrm{\Phi }}_l^{+}⟩\end{array}$$

Similar reasoning for $\widehat{R}\left(\alpha \right)$ acting on remaining Bell states, will lead us to the result

$$\begin{array}{l}\mid 0⟩{}_Q=\mid {\mathrm{\Phi }}_l^{\pm }⟩\stackrel{\widehat{R}}{\to }\mathit{cos}\left(\alpha \right)\mid {\mathrm{\Phi }}_l^{\pm }⟩-\mathit{\text{isin}}\left(\alpha \right)\mid {\mathrm{\Psi }}_l^{\pm }⟩=\mathit{cos}\left(\alpha \right)\mid 0⟩{}_Q-\mathit{\text{isin}}\left(\alpha \right)\mid 1⟩{}_Q\\ \mid 1⟩{}_Q=\mid {\mathrm{\Phi }}_l^{\pm }⟩\stackrel{\widehat{R}}{\to }-\mathit{\text{isin}}\left(\alpha \right)\mid {\mathrm{\Psi }}_l^{\pm }⟩+\mathit{cos}\left(\alpha \right)\mid {\mathrm{\Phi }}_l^{\pm }⟩=-\mathit{\text{isin}}\left(\alpha \right)\mid 0⟩{}_Q+\mathit{cos}\left(\alpha \right)\mid 1⟩{}_Q\end{array}$$

The above formula can be rewritten in the context of Pauli matrices ${\sigma }_x,{\sigma }_y$ and ${\sigma }_z$, using famous Euler formula:

$$\widehat{R}\left(\alpha \right)=e^{-i{\sigma }_x\alpha }$$E12

In virtue of above, we say that polarization rotation makes the rotation around $x$ by angle $2\alpha$ on Bloch sphere where north pole is $\mid {\mathrm{\Psi }}^{+}⟩$ and south pole is $\mid {\mathrm{\Phi }}^{+}⟩$. In the light of the definition of QBell this Bloch sphere also represents this qubit. Hence polarization rotation is rotation around $x$ axis of QBell’s Bloch sphere by an angle $2\alpha$:

$$R_x\left(\alpha \right)=\widehat{R}\left(\frac{\alpha }{2}\right).$$E13

4.6 Rotation around $z$ axis of Bloch sphere

Phase modulator $\widehat{M}\left(\gamma \right)$ placed on the TR applies the phase shift operation, as follows.

$${\widehat{M}}_i^{†}=e^{-\mathit{i\gamma }}{\widehat{M}}_o^{†}\iff {\widehat{M}}_{i,\mathit{TR}}^{†}=e^{\mathit{i\gamma }}{\widehat{M}}_{o,\mathit{CR}}^{†}$$E14

The above equivalence is correct because $\gamma$ is a local phase considered as difference of phase between two rails, similar like in holography method and Mach-Zehnder interferometer. Global phase $\left(e^{\mathit{i\varphi }}\left(\alpha |0⟩+\beta |1⟩\right)\right)$ is ignored since it cannot be measured⁴: ${\left|e^{\mathit{i\varphi }}\alpha \right|}^2=e^{\mathit{i\varphi }}{e}^{-\mathit{i\varphi }}{\left|\alpha \right|}^2={\left|\alpha \right|}^2$. Thus, phase modulator represents the phase shift gate $P_{\gamma }$. It is obvious that phase modulator does not change the polarization, hence it is not possible to change the type of Bell state:

$$\begin{array}{l}\mid {\mathrm{\Psi }}^{\pm }⟩\stackrel{P_{\gamma }}{\to }{e}^{-\mathit{i\gamma }}\mid {\mathrm{\Psi }}^{\pm }⟩\\ \mid {\mathrm{\Phi }}^{\pm }⟩\stackrel{P_{\gamma }}{\to }{e}^{-\mathit{i\gamma }}\mid {\mathrm{\Phi }}^{\pm }⟩\end{array}$$E15

Therefore and from 4.1, the phase modulator itself change the global phase of QBell, not a local one, which is needed for $z$ rotation operator.

In order to obtain the rotation around $z$ axis on the Bloch Sphere we use the more general circuit called by us General Phase Shift circuit, GPSh and market with ${\mathcal{G}}_{\zeta ,\gamma }$, shown on Figure 4. We start the evolution with the polarization separation phase, which we divide the control rail into two sub-rails (red and yellow) with polarizing beam splitter (PBS). It works in such a way that horizontally polarized photons go straight (red trajectory on the mentioned figure) an vertically are reflected (yellow trajectory). After this operation, the system state, called separated state is roughly the same as before it. Nonetheless the horizontally and vertically TR polarization are now on the different rails. On the TR rail, there is indeed one photon in superposition of horizontal and vertical polarization, in general case. Thus after separation there is a superposition of the photons position on red and yellow sub-rail. Moreover the red sub-rail represents the horizontal eigen-state, while the yellow one - vertical eigen state. Indeed, we change the polarization basis into two sub-rail spatial basis. It allows us to apply any operations including phase shifts to each polarization eigen-state separately. Which we actually do in the computation phase with,red” adjustable phase shifter (with phase $\zeta$) applied to horizontally polarized sub-rail and the “yellow” one with shift $\gamma$. Then, the polarization reintegration phase changes the basis again to polarization one. Because of photon (on TR and CR) entanglement, phase shift affect the photon on CR crosswise against the entangled relation. It can be seen on the scheme since, in the resulting state, one red label is on the green relation and other one on the blue line. Yellow labels behave similarly. Furthermore, if we apply with GPSh to the input state $\mid q⟩{}_Q=\alpha \mid 0⟩{}_Q+\beta \mid 1⟩{}_Q$, which is rewritten in physical layer on the top of the scheme, we obtain:

Figure 4.

General Phase Shift Circuit (GPSh) diagram. The polarization separation and integration is made with two Polarization Beam splitters. The trajectory of horizontally polarized photon is market with red color, while vertically polarized with yellow one. In resulting state we see that phase shifts done for vertical ($\zeta$)/ horizontal ($\gamma$) polarized photons propagates on both $\mid \mathrm{\Psi }⟩,\mid \mathrm{\Phi }⟩$ Bell-like states. However $\zeta$ affects the leading part of $\mid \mathrm{\Phi }⟩$ representing $\mid 0⟩{}_Q$ on logic layer, while $\gamma$ its complementary part. In the case of $\mid \mathrm{\Psi }⟩$ phase shift propagation occurs in opposite way.

$$\begin{array}{l}\alpha \left[\frac{e^{\mathit{i\zeta }}\mid 10,10⟩+e^{i{\delta }_1}{e}^{\mathit{i\gamma }}\mid 01,01⟩}{\sqrt{2}}\right]+\beta \left[\frac{e^{\mathit{i\gamma }}\mid 10,01⟩+e^{i{\delta }_2}{e}^{\mathit{i\zeta }}\mid 01,10⟩}{\sqrt{2}}\right]\\ =\alpha e^{\zeta }\left[\frac{\mid 10,10⟩+e^{i{\delta }_1}{e}^{i\left(\gamma -\zeta \right)}\mid 01,01⟩}{\sqrt{2}}\right]+\beta e^{\mathit{i\gamma }}\left[\frac{\mid 10,01⟩+e^{i{\delta }_1}{e}^{i\left(\zeta -\gamma \right)}\mid 01,10⟩}{\sqrt{2}}\right]\end{array}$$

The resulting formula is the physical layer state, shown on the scheme as well. In the virtue of cor. 4.2 we can say that this is the state of the form:

$$\mid q^{\prime }⟩{}_Q={\mathcal{G}}_{\zeta ,\gamma }\mid q⟩{}_Q=\alpha e^{\mathit{i\zeta }}\mid 0⟩{}_Q+\alpha e^{\mathit{i\gamma }}\mid 1⟩{}_Q$$E16

Using the above circuit we can construct the rotation around $z$ axis of the Bloch sphere. In fact it is simply equal to:

$$R_z\left(\alpha \right)={\mathcal{G}}_{-\frac{\alpha }{2},\frac{\alpha }{2}}=e^{-i{\sigma }_z\alpha }$$E17

Now, we return to equations: 3, 9, especially to the second one. We see that there only appears $R_z$ and $R_x$ operators, which we have already elaborated above. The physical structure of the U Gate is presented on the Figure 5. The above consideration should be treated as the constructive proof, that proposed QBell qubit can be implemented physically. This is because w have shown how to construct the universal gate. It mean that all one-qubit gates can be constructed using it. However it does not mean that a particular gate cannot be implemented in other way, e.g. the phase shift gate $P_{\alpha }={\mathcal{G}}_{\mathrm{0,0.5}\alpha }$. In practical solution, there each gate implementation should be carefully optimized from the perspective of the number of devices needed, however this issue goes beyond the scope of this chapter.

Figure 5.

The scheme of the universal gate for QBell using the Eq. (9). The circuit consists of three phases. Two phase modulations sandwiches the polarization rotation phase. The execution of phases is in opposite direction to the equation, since operators are applied from right to left.

4.7 Two QBell systems

Not entangled states of QBell are obtained in very simple way. If we build two QBell qubits one next to other and we perform measurement (which means we detect time-correlations of clicks on eight detectors), we obtain four possibilities:

1st QBell detectors	2nd QBel detectors
Alternating	Alternating
Alternating	Straight
Straight	Alternating
Straight	Straight

If we add another one, the number of options will grow to eight. Thus, we see that it fulfills the pattern of tensor product. Therefore we can say, that we create the interference (which comes down to tensor product) of two QBell just by putting them one next to other and detect the time correlation of detectors’ click.

For the purpose of generating ${\mathit{\text{CNOT}}}_Q$ gate, we assume that we dispose Toffoli gate for one rail qubits. It is realistic assumption since such circuits are implemented using linear optics systems already, like Huang et al. in [23]. Toffoli circuit work in three qubits systems. It changes polarization of the photon of the third qubit if and only if both photons of the first and second qubit has, for the sake of argument, horizontal polarization⁵. The first two qubits are called controlling while the last one controlled. Of course in case of photons polarization superposition, this operation is applied to polarization eigen-states building it. On the Figure 6 there is a scheme of ${\mathit{\text{CNOT}}}_Q$ gate acting on the logic level on the two QBells’ system. Let us assume that the first QBell is controlling qubit, while the second one is controlled. We start with the tensor product of two QBells, which is shown on the left part of this scheme by two independent patterns of QBell states. Then there is the ${\mathit{\text{CNOT}}}_Q$ gate shown on the second part of the scheme. First we act with Toffoli gate where controlling rails are ${\mathit{CR}}_1$ and ${\mathit{TR}}_1$ and controlled ${\mathit{TR}}_2$. It results in changing the polarization of the photon on the ${\mathit{TR}}_2$ in case photons on ${\mathit{CR}}_1$ and ${\mathit{TR}}_1$ are vertically polarized. It means that the first QBell is in the $\mid 1⟩{}_Q$ state. In that case, the second QBell state will be changed from $\mid 0⟩{}_Q$ to $\mid 1⟩{}_Q$ and opposite. It occurs because the polarization of ${\mathit{TR}}_2$ is changed, while the polarization of ${\mathit{CR}}_2$ is not. If the first QBell has different polarizations, nothing happen. In that point one can think, that one Toffoli circuit fully implements the ${\mathit{\text{CNOT}}}_Q$ gate, but it is not true. Namely if first QBell is in complementary state, nothing will happen as well. At this point we have two options. Within the first one, shown on the scheme, we change the polarization of the first QBell with the polarization rotator with angle $\pi /2$, which acting was described in previous sections. Thus, if both photons was polarized vertically, now they are polarized horizontally. So we can use the same Toffoli gate once again. It changes the polarization of the second QBell in the same way as for its leading part. Finally we turn back the polarization to left the state of the first QBell untouched. The second option is to use the vertical-sensitive Toffoli gate, which changes the controlled state if the controlling qubits has the vertical polarization.

Figure 6.

${\mathit{\text{CNOT}}}_Q$ gate implementation on the logic level of two QBells system, using Toffoli one-rail qubit gates.

Now, we have the state, shown schematically in the last part of mentioned figure as the, resulting state”. We see there that for both “green states” (leading and complementary), the colors on the second QBell are changed, while in case of “blue states” the second QBell looks the same. It means that with operation described above, in the system arises a correlation, that is an entanglement between two QBells.

As we can see above, we use hyper-entanglement as the internal structure of proposed qubit. Thus, we utilize the notion of multipartite entanglement (see Szalay [11]). The system of $n$ qubits is the simple system of the first type consisting of biparities (Figure 7). The CNOT and in fact all controlled gates, generates the clusterization of second type (Szalay, ibidem), since Toffoli gate connect CR, TR from the first qubit and TR from the second one. So there is no connection for CR from the second qubit. Hence this is not full entanglement, thus the set $\left\{\mathit{CR}1,\mathit{TR}1,\mathit{CR}2,\mathit{TR}2\right\}$ is not a clique, so it is not coarser (see Szalay, ibidem) then sets $\left\{\mathit{CR}1,\mathit{TR}1\right\},\left\{\mathit{CR}2,\mathit{TR}2\right\}$. Generally we can say, that $n$-rail systems creates the $n$-partite structure of the first type and the controlled gate will cover it with the structure of the second type.

Figure 7.

The multipartite look at the QBell and possible four-rail solution. On the left side simple three QBels structure with no controlled gate applied. In the middle the second type of clusterization arising by CNOT gate application on the first two qubits. At the right side, there is a simple structure of four-rail qubits utilizing GHZ states for $\mid 0⟩{}_Q$ and alternating polarization arrangement of $\mid 1⟩{}_Q$.

5. Discussion and conclusions

In this chapter we have defined the qubit named QBell inspired by quantum holography imaging. We also defined all operation needed for quantum computation, articulated in Section 3. Namely, we have defined how to implement the rotation around $x$ and $z$ axis on the Bloch Sphere for QBell. We pointed how to build the universal, one qubit gate using those two rotations. Furthermore, we defined how to create the interference of two QBells and how to generate entanglement on the level of QBell. In the light of above, we are permitted to announce that we have constructed the two-rail Bell’s qubit, defined in def. 4.1 and we have constructively shown that it is physically implementable.

If we look at the internal structure of QBell once again, we see that all the time of QBell living, we shell ensure that there are two, time-correlated photons on two separate rails, one photon on one rail. But this is only subspace of Hilbert space representing any arrangement of two photons on two rails. Because of hyperentanglement, the system has the feature, that any external distortion of phase affects all eigenstates of QBell in the same way. It means that this is so called Decoherence Free Subspace DFS, described by Lidar et al. in [24]. This fact is also confirmed by Deffiene et al. (ibidem). Therefore QBell is robust in the perspective of dephasing type of decoherence.

Due to taking into consideration only time-correlated photons, there are also excluded the influence of noise induced by other reactions with environment, like scattering, damping, time de-correlation, devices imperfection, etc. In that case we lowering the efficiency but we exclude many incorrect results. Which also increase the robustness of our solution.

The next advantage we can see in a case of more than two rails. Let us consider $\mid 1⟩{}_Q$ as the GHZ-like (Greenberger-Horne-Zeilinger) state, which is an extension of Bell $\mathrm{\Phi }$-like state in case of two-rail QBell. $\mid 1⟩{}_Q$ as the superposition of the states consisting of alternating horizontally and vertically polarized photons, starting from horizontal polarization (leading part) and vertical polarization (complementary part), which is an extension of Bell $\mathrm{\Psi }$-like state. In that case any external influence touching the polarization has to concern a half of rails. The probability of such an event decreases with the growth of the rails. We can say that the computation subspace has always four dimension, while the dimension of the space where it is embedded grows. Indeed, it is an exponential growth. Therefor the mentioned probability also decreases exponentially. This advantage creates the robustness of proposed solution in the polarization channel as well.

This work opens the wide area of research on multi-rail qubits working in the DFS areas. The possibility of its implementation on photonic chips can lead to mature solutions in the area of photonic quantum processors braking the limitations of quantum optics implemented without this technology.

There are many applications of such technology, terrestrial and space. Quantum processors, with all needed apparatus can be fitted in a few components of nano satellites. It makes the idea of using quantum processors on earth orbit affordable, which is not possible in a case of superconducting and ion-trap implementation due to cooling modules size and weight.

Except of common hopes connected with quantum computation technology, like quantum advantage (or supremacy), photonic solution has some additional advantages. Namely they can be connected with other optical devices without in-between silicon devices. It can speed-up the computations in the area of quantum communication and security. It avail also straight connections with quantum sensors like quantum holographic cameras that was described in the beginning of this chapter.

Recapitulating, in this chapter we have presented the theoretical study on QBell, however it has a potential for great impact for the practical implementation. This, practical aspects of its physical implementation in photonic chips technology will be the subject of a work of our team in the future.

Acknowledgments

This publication was supported by the Department of Graphics, Computer Vision and Digital Systems, under research project for young scientists (Rau6, 2022) and statue research project (Rau6, 2022), Silesian University of Technology (Gliwice, Poland).