Encryption, decryption, and control with fractional quantum bits, quantum chiral states and pyramidal quantum bits switching in graphene

In quantum mechanics, quantum entanglement is a counterintuitive concept, conceptualized initially by Einstein, Podolsky, and Rosen (EPR),¹ followed by further elaboration of the EPR paradox by Erwin Schrödinger.² Quantum entanglement is a complex behavior of space and time for subatomic particles in ways such that the quantum state of each particle cannot be described independently; instead, a quantum state must be described for the entire system. The spin or polarization of entangled subatomic particles in different directions has been reported to produce violations of Bell’s inequality,^3,4 whereas the correlation of linear polarizations has been reported with time-dependent analyzers.^5,6 Entangled particles can be used, in quantum encryption, to form a categorical secure key and, in quantum teleportation, to transfer quantum information.^4,6 Entanglement is considered as a resource that cannot carry information itself, but can help in this context by diminishing the classical communication complexity.^7,8

To the best of our knowledge, there is no evidence in the literature about encryption and decryption of data using f-qubits. With the discovery of GMR,⁹ it has been possible to enhance the memory of mesoscopic fields with a single electron by using magnetic excitations due to fractional quantization. The ket-state functions are considered to be an analog for realizable quantum encryption.^10,11 In a recent experimental observation, electron spinning is evidenced at vertices of honeycomb graphene.¹² We agree with the notions of the Kitaev model and the Majorana fermions.¹³ The fractional quantum Hall effect (FQHE) has revealed numerous distinctive quantum phenomenon and behavior of two-dimensional electrons.^13,14 Diverse microscopic theories of FQHE emerged,^15,16 preferably analogous to Ginzburg–Landau theory¹⁷ of superconductivity but there is no proper explanation about how quasiparticles have fractional charges distributed and quantized. Laughlin previously deliberated on incompressible quantum fields with fractionally charged excitations.¹⁸ 

When a photon collides with an electron, the morphology of the bound electron is changed due to inelastic collision. An electron quantum, which is bound, can change its morphology in the momentum space. The bound electron is first stretched, twisted, and then twigged. The hypothetical periphery or wall of electron quanta before stretching behaves like an adiabatic wall from where some energy of electrons is released and absorbed in the surrounding matrix. This suggests that bound or quantized matter or subatomic particles with their corresponding exchange fields exist in a more stable degenerate quantum state. In other words, their degenerate quantum states are woven in a string of curvatures of space and time. More likely, the degenerate quantum states with their exchange fields are warped in topological space and time; the curvature of time is responsible for exchange fields. The twigged electron subquanta will have orientations up and down but with zero rotational momentum or spinning. This paradox is supported with magnetic excitations due to characteristic orientations as a manifestation of gyroscopic behavior.^19–21 Silicene is a 2D allotrope of silicon with a hexagonal honeycomb lattice similar to that of graphene.²² We envisage that the delivery of energy in a 2D honeycomb lattice of graphene in the momentum space from the surrounding matrix and without any scattering will change the configuration of the quantized, stretched, twisted, and twigged electron into six Dirac bosons on a ring (equivalent to a hexagonal lattice) and three Dirac fermions. The delivery of energy in the momentum space is conjectured as a manifestation of quantum entanglement. The partial increase in mass of broken subquanta of an electron is also due to a “knot” in a quantum braided spinless garland liquid soup. This is why magnetic ordering and fractional excitations in α-RuCl₃ are observed.¹² Anyons become invariant due to inter- and intra-woven curvatures of topological space and time. There is contrasting observation of spin–orbit coupling.^23,24 With Dirac bosons or subquanta on a single electron especially for 0.4 ≤ n_f ≤ 0.9 in underdamped conditions, precessional or wobbling (Larmor frequencies) varying frequencies are produced but tied with twisting effect as manifestations of gyroscopic behavior, i.e., 0.02 ≤ g²/ℏc ≤ 0.08. Encryption is accomplished for each of these six subquanta states. A six word (each word is equivalent to a vector tuple on a ring) with fractional states for encryption on a single electron can work as a quantum byte, i.e., a six-word f-qubits. These fractional quantum states will have orientations “up” and “down” resulting from twisting and that the spinning is gradually arrested. These subquanta of an electron are fixed at knots but oscillating at different precessional frequencies and tied with each other. Therefore, the concept of a twisted electron wire or braid is coined. Each of such subquanta works as “anyons” due to fractional charge quantization.⁸ 

With gyroscopic behavior, the azimuth angle 0° < ∅ < 90° increases, thereby decreasing the rotational momentum or spinning (gradual process) about its own axis. Eigenfunctions, $Φ$ for magnetic excitations, change exponentially with azimuth angles, i.e., Φ = e^±im∅, where ∅ is the azimuth angle and “m” is the magnetic moment due to varying basal area of the cone. With increasing ∅, the basal area of the cone increases and hence the magnetic excitations due to magnetic moments at their characteristic precessional frequencies.

A new method based on f-qubits, using “ring and field” operators and “chiral operators” with their corresponding Dirac addresses for bosons and fermions followed by Dirac Hermitian Hamiltonian for encryption and decryption, is proposed. Memory enhancement and data processing rate are exploited in the honeycomb lattice of graphene based on magnetic excitations of Dirac bosons. Each boson on a ring will have a different magnitude of magnetic excitation due to precessing, whereas fermions wrapped at the edges on a ring will accomplish pyramidal switching. This is how a 2D graphene lattice will be used for encryption, decryption, and control. A quantized, stretched, twisted, and twigged electron in the momentum space without scattering with other electron(s) changes its configuration to produce a new quantum state—a garland with six precessing Dirac bosons and three spinning fermions wrapped at a ring but all connected with a braid (stretched electron quanta-wire). Mathematical relations are developed and exploited for nanotechnology based on encryption, decryption, and control with switching. This is a paradigm shift in terms of f-qubits and their applications. Our wave functions at 0.1 ≤ n_f ≤ 0.9 are different from Laughlin wave functions¹⁸ because we are using a vector matrix of n_fxg²/ℏc, where 0.02 ≤ g²/ℏc ≤ 0.08 for “quantum chiral states” following topological phases.

An eigenfunction (with warped curvatures of topological space and time) for stretched and twisted electron quanta is obtained by using Fourier transforms, Gaussian behavior of wave packets, complex contour integrations, theory of residues, and Parseval’s theorem. Electrons when quantized exist in the form of stretched, twisted, and twigged states, i.e., topological quantum states.^25,26 Two or more than two electrons in the quantized states are warped in space–time as quasiparticles in the form of quantum braids. The free electron has a size of 5 × 10⁻¹⁴ m. When quantized, its size is stretched in the range of 10–15 × 10⁻¹³ m. Previously, we had obtained eigenfunctions for a twisted and twigged quasiparticle (a quantized electron in the momentum space).²⁶ 

Recently, eigenfunctions for a single quantized electron with beaded quantized states of fractional masses and for the fractional charges under different topological quantum states were obtained.²⁵ The 2016 Nobel Prize in Physics was awarded for work on topological phase transitions and topological phases of matter. They studied phase transitions and quantum Hall conductance in two-dimensional periodic potential, quantum Hall effect without Landau level, fractional quantization of the Hall effect and its fractional statistics, and critical properties of topological phases with universal jump in the superfluid density of two-dimensional superfluids.^27–33 

Previously, stripped or edge states were observed in the quantum Hall effect.³⁴ Kane and Melle studied the quantum spin Hall effect in graphene.³⁵ This was followed by a composite fermion approach for the fractional quantum Hall effect.³⁶ Another group demonstrated Bose–Einstein condensation of excitons in bi-layer systems and the vanishing Hall resistance at high magnetic fields.^37,38 Fu and Kane studied the superconducting proximity effect and Majorana fermions at the surface of a topological insulator.³⁹ Kitaev observed unpaired Majorana fermions in quantum wires.⁴⁰ Weyl fermions were observed in semimetals with topological phase transitions.^41–44 All such studies have centered around topological states and phase transitions in Hall effects, superfluids, semimetals, superconductors, and in quantum wires with emphases on Majorana and Weyl fermions.^27–44 Different kinds of phase transitions, such as excitations, Bose–Einstein condensation, and proximity effects, were considered.

Our approach is different from the above-mentioned because we consider a non-interacting single quantized electron, which undergoes topological phase transitions, such as stretching, twisting, and twigging with quantum knots.⁴⁵ We build up on our prior studies of topological behavior of electrons in the quantum Hall effect, giant-magnetoresistance, quantum conductance, electrodynamic theory of quantum Hall effect, and quantum theory of mesoscopic fractional electric fields in a cavity of viscous media like dielectric materials.^46–48 

The total eigenfunction was recently obtained as follows:⁴⁵ 

for the quantum phase transitions, that is, for six Dirac bosons and three Dirac fermions. Quantization of knots defines quantization of events. We considered time as imaginary and can be defined in a plane for conformal mapping.

Recently, superconductivity in graphene with different physical conditions was discovered, which showed an overall highly quantized weak fermionic system due to pseudo-gaps.^49–51 The spinless and spinning bosons with pseudo-gaps of d-wave electrons are responsible for a highly quantized weak fermionic system for superconductivity.^52,53 Thus, graphene is a suitable material for safe quantum computation and the tasks mentioned above because a highly magnetized applied field will not affect graphene’s internal properties.

The question of how a wave packet of an electron changes with time is resolved with cumbersome mathematical calculations. By using theory of residues, complex integration, Parseval’s theorem, and fractional quantization, n_f, we arrive

where α is the twisting angle, i.e., 0.17 ≤ α ≤ 1.53 rad,

σ is the standard deviation, and

Equation (1) describes the eigenfunction for any electron, in the momentum space, which is quantized, stretched, twisted, and twigged. Writing Cauchy’s integral formula for the fractional states,

Using Eq. (3) for fractional quantized states, complex integration, and theory of residues, we observed that the Gaussian wave packet is changing with time. Thus,

With twisting of stretched electron quanta (skewed Gaussian behavior), the concept of “quantization of events” is suggested. With established results of “fractional charge quantization,” we get $enf=∑nf=0.10.9nfe$ (where 0.1 ≤ n_f ≤ 0.9 and e is the charge of an electron and its integrated oscillatory effect is defined by $Hnf=2nf$⁠, where $Hnf$ is the Hermite function for fractional quantization). The mass of an electron is also fractionally distributed due to the overwhelming effect of 0.02 ≤ g²/ℏc ≤ 0.08 (gyroscopic behavior) as compared to charge coupling $e2ℏc=1137$ (electromagnetic behavior) but coupled with a braid (electroweak fields). The quantized fractional masses due to gyroscopic behavior are $mnf=∑nf=0.10.9nfme$⁠. In a dispersive medium, the index of refraction, μ, is considered as the quantum mechanical analog of the classical momentum of the particle.

With this “analogy,” p_q ≡ μ. Hence, $μ=ℏkqua=ℏ2mnf(E−V)ℏ=2mnf(E−V)$ and $μ=ℏkfrac=ℏ2m(E)ℏ=2mE$⁠.

The probability density of the Gaussian wave packet changing with time is obtained as the absolute square of $ψr,t$⁠, i.e.,

When $ψr,t2$ has the form $e−r2σ2$⁠, we have

Using Eqs. (5) and (6), we have

Considering Eq. (7), when t = 0, we get

for Eq. (6).

Reconsidering Eq. (7), i.e., $2Δropσ=1+ℏ2tnf2σ4m2nf$⁠, when t = 0,

With “stretching electron quanta” and “quantization of events,” one can ascertain that

The “quantization of event” is also related to the twisting angle, α. A theoretical result is obtained for eigenfunction pertaining to how fractional charge on an electron in the momentum space is quantized.^25,26 With arbitrary values of 0.1 ≤ n_f ≤ 0.9 and 0.17 ≤ α ≤ 1.53 rad, the empirical result is

Considering

the unitary operator, $∂∂t$⁠, is responsible for twisting of a stretched electron quantum.

We now put σ = 0, consider ψ(α) ≡ α, and apply an energy operator on it, i.e., on Eq. (10). Using E_op from Eq. (10) on (8), we have E_op $ψstretchingα=Eopα$⁠,

where ν_precession is the precessional frequency of Dirac bosons. Equation (11) shows that the twisting angle is directly proportional to logarithmic behavior of quantization of events and inversely proportional to logarithmic behavior of broken masses of electron quanta (Dirac bosons coupled with a twisted braid). The energy profile obtained through the energy operator shows logarithmic behavior of “quantization of events.” The eigenfunction with $e−α2$ for a newly developed morphological state of electron quanta represents a resonant behavior at a twisting angle of 0.5 rad, as shown in Fig. 1, when $e−α2$ is considered along with the final result,

Figure 2 represents the plot of $ψnf$ vs 0.25 ≤ α ≤ 1.53 rad for 0.1 ≤ n_f ≤ 0.9 obtained from fractional Fourier transform. This plot shows logarithmic profiles for overdamped, 0.1 ≤ n_f ≤ 0.3, and underdamped, 0.4 ≤ n_f ≤ 0.9, conditions, respectively. Stable degeneracy is obtained at 0.4 ≤ n_f ≤ 0.9 for six vector tuples on a ring.

The electrodynamic behavior of the quantum Hall effect⁴⁸ tells us that charge or fractional charge becomes invariant, which is the cause of producing a “knot” with a vector tuple (sub-loop) and making subquanta stable only in underdamped conditions. For 0.1 ≤ n_f ≤ 0.3, we shall have overdamped varying oscillations of each of these three fractional quantum states, and therefore, “knots” are unstable. However, there is a transient behavior of n_f in between 0.4 and 0.5, 0.4 ≤ n_f ≤ 0.5, which, to our understanding, is associated with “inertial energy oscillations” as shown in Fig. 2.

Saddle point or zero-point oscillating energy or inertial oscillatory energy is confined with new morphology of electron quanta as manifested in Fig. 1. Theoretically, we consider $e−α2$ to converge to unity to maintain its identity with a unitary operator. The eigenfunction when plotted without $e−α2$ with respect to the fractional quantum number, n_f, reveals a change in curvature at n_f = 0.5, as shown in Fig. 2.

In Fig. 2, “$ψnf$” exhibits a transient behavior at n_f = 0.4–0.5; there appears a transient state, which shows that the potential energy of the fractional state is changing with time, i.e., $∂Vnf∂t$⁠. At n_f > 0.5, underdamped stable degeneracy (permanent), whereas at n_f < 0.4, overdamped unstable degeneracy (reversible) are produced. At n_f = 0.5, natural (inertial) or damped degeneracy (permanent) is obtained, i.e., $∂Vnf∂t0.5$ = constant or threshold degeneracy.

A similar scenario of damping is also obvious with eigenfunction when plotted without $e−α2$ with respect to “α,” which shows a Gaussian behavior. This reflects that the inertial oscillating energy of an electron quantum yields overdamped (n_f ≤ 0.3) and underdamped (n_f ≥ 0.4) fractional quantized states, as a result of which stretching, twisting, and twigging occur. The twisting angles change with the curvature of time, i.e., $∂∂t$⁠, and hence the quantization of events, i.e., ħ$tnf$⁠.

The inertial energy oscillations with six f-qubits will make the electron a stable source for information storage. Such morphological states of an electron, as we know, are produced either due to photons or quantum entanglement in a momentum space. The information encrypted as a function of varying frequencies for each of these subquanta exploits “f-qubits” with a word size of six fractional bits, whereas n with 0.4 will work as a parity quantum bit. Such a morphological state of an electron with six f-qubits will work effectively for encryption, decryption, and control.

As mentioned, spinning is gradually arrested at fractional quantum states due to “knots” with fractional charge quantization. This is how a form of braid is produced. We agree with the notion of Majorana fermions because fermions break the “charge symmetry” due to an appropriate spinning pair of electrons. Majorana fermions comprise electron–hole pairs with opposite spins to each other. Hole is a “vacant quantum state” of an electron. Weyl fermions behave like Majorana fermions but with the difference that they exist at Dirac points, hop from their Dirac points following the left–right and vice versa symmetry especially in topological insulators.^41–43 Dirac points exist at the bottom of the valleys within the gap of valence and conduction bands (topological insulators).^41–43 Both Majorana and Weyl fermions obey Dirac fermions and their statistics.^41–44 In some cases, we could have gapless semimetals or semiconductors, which are categorized as topological materials with proximity effects.^39,40 The conduction mechanism in such a case will follow a Schottky barrier (following alternating dissimilar charge behavior) in semimetals or in pseudo-gap superconductors. In our case, the twisting of a stretched electron quantum produces “quantization of events,” i.e., twigs or subquanta (vector tuple, which is a sub-loop) with fractional charge quantization due to “knots.” Each of these subquanta is tied with each other with electroweak interactions in between them. Such a braided configuration of quantized electrons in the momentum space is useful for quantum teleportation. Many electron systems can be compiled with such a morphological behavior.

Theoretical explanations can be exploited by using Dirac δ-operators with symmetrical and asymmetrical states, group theory, and rings and fields (hyperspace, which is six-dimensional due to six vector tuples on a ring). The paired oriented “up and down” subquanta with zero rotational momentum can be used for encryption based either on the precessional frequency or on the twisting angle at the lateral surface of a stretched electron quantum. Each of these paired oriented “up and down” subquanta is encrypted as a function of either precessional frequency or rotation angle at the lateral surface of the stretched electron quanta. Thus, a single electron will have consecutive three pairs of oriented “up and down” subquanta, which will emerge simultaneously with six quantum gates for varying switching times in AND, NAND, OR, NOR, etc., with combinations of logic, i.e., RTL, DTL, and TTL, of course, following DeMorgan’s theorem in the form of combinatorial, sequential, neurological, genetic, synergistic, and parallel logic through quantum grid computing. Expert systems require “f-qubits,” which can be dealt with bra–ket state functions. Fortunately, the electron morphology in the momentum space is providing fractional quantum bits. Recent experimental observations of breaking up electron quanta into its segments and occupying positions at the vertices of a honeycomb of graphene¹² provide a clue for encryption. However, the experimental interpretations based on predictive modeling are different from ours. Each of the Dirac quantum states—whether for bosons or fermions—obeys quantum chiral states for both “precessing” and “spinning” of quasiparticles (quantized fractional masses of electrons) spontaneously with f-qubits. With vector matrix of $nfxg2ℏc$⁠, we get 9 × 8 = 72 f-qubits following 9 fractional states, i.e., 0.1 ≤ n_f ≤ 0.9, and eight gyroscopic states, i.e., 0.02 ≤ $g2ℏc$ ≤ 0.08. Each of the gyroscopic states produces magnetic moment. There are nine (9) chiral states in the overdamped conditions (0.1 ≤ n_f ≤ 0.3), whereas sixty three (63) chiral states in the underdamped conditions (0.4 ≤ n_f ≤ 0.9). Sixty-three quantum chiral states can be used for quantum information at six vertices of graphene because they are knotted at 0.4 ≤ n_f ≤ 0.9. These knots are quantum states with the same energy and termed Dirac bosons because the sixty-three chiral states would have different characteristic precessional frequencies.

Let the main loop or ring be considered as a 2D honeycomb lattice of graphene with six vector tuples (subquanta of an electron) at its vertices following fractional quantum bits, i.e., 0.4 ≤ n_f ≤ 0.9, for encryption and decryption. Dirac-δ operators will be used for encryption and decryption following either the fractional Fourier transform or the Heaviside function. Encryption will be accomplished in the underdamped conditions with a logarithmic profile. The fractional qubits, i.e., 0.4 ≤ n_f ≤ 0.9, obey Dirac-δ symmetry unitary operators. The remaining fractional qubits, i.e., 0.1 ≤ n_f ≤ 0.3, obey Dirac-δ asymmetry operators in the overdamped conditions following a logarithmic profile where chaos or periodic unstable chaos are expected with attractors (Lyapunov exponents).

With six vector tuples (subquanta of an electron) for 0.4 ≤ n_f ≤ 0.9 on the honeycomb lattice, one would expect $P66$ = 720 possible ways for selection of encryption. Let the up-orientation of vector tuples be defined as “1” (switching ON state) and the down orientation of vector tuples as “0” (switching OFF state). One would have a keying option of three pairs, each with ON and OFF state following keying with f-qubits. Thus, one would expect $P36$ = 120 possible ways for selection of either ON-state or OFF-state. This resulted in $C36$ = 20 possible ways for encrypting and decrypting the data with their characteristic frequencies only in the underdamped conditions.

In the case of parallel stacks of graphene³⁵ with its six vertices on each, say ten, we can have sixty knots at these vertices as bosons with fractional masses of electron. Of course, with fractional charges precessing at their characteristic frequencies in ten folds following quantum topology and quantum information, the remaining forty-two unknotted fractional masses with their fractional charges will have fourteen (14) pyramidal switching quantum states as Dirac fermions. Remember that the Dirac bosons usually termed here “quantum knots” (same energies), holding precessing fractional masses and charges of diverse categories, have negligible spin–orbit coupling due to gyroscopic behavior. Dirac fermions with fourteen pyramidal switching quantum states have strong spin–orbit coupling in the overdamped states. Fourteen (14) chiral states, each with three Dirac fermions, are composite fermions and obey quantum topology. The composite fermions^34,36 are produced in the overdamped states at 0.1 ≤ n_f ≤ 0.3. These composite fermions work alternatively at three edges of the hexagon.^39,40 The scheme for quantum encryption and decryption and control for “quantum chirality” will remain the same as for rings, vector tuples, and fields. Following are the steps needed for encryption and decryption:

The quantized, stretched, twisted, and twigged electron (subquanta due to quantization of events at 0.1 ≤ n_f ≤ 0.9) in the momentum space gains energy due to harmonic and time-dependent adiabatic perturbations. As a result, Dirac bosons and Dirac fermions are produced. A new configuration of a quantum state for 0.1 ≤ n_f ≤ 0.9 from a single electron, in the momentum space, is theoretically suggested, which follows with the Kitaev model⁴⁰ and Majorana fermions.^39,40 Eigenfunctions with warped topological curvatures of space and time for stretched, twisted, and twigged electron quanta are obtained following mathematical calculations. With stretching of electron quanta, a twisting angle is produced, which is a manifestation of quantization of events, i.e., $ℏtnf$⁠. Encryption and decryption methods are proposed by utilizing the magnetic excitations from six precessing Dirac bosons. The remaining spinning Dirac fermions at 0.1 ≤ n_f ≤ 0.3, which are not vector tupled but wrapped at the edges of a ring, are used for pyramidal switching. A new method based on f-qubits followed by Dirac Hermitian Hamiltonian provides addresses to fractional quantum states, i.e., for 0.1 ≤ n_f ≤ 0.9. The quantized, stretched, twisted, and twigged electron changes its configuration due to harmonic and time-dependent adiabatic perturbations and without scattering with other electron(s) in the momentum space, in the form of six Dirac bosons held at knots in a ring and three Dirac fermions wrapped at the edges of the ring. This “ring and field” theory for a hexagonal honeycomb lattice of graphene may also be applicable to honeycomb lattices with different atoms, for instance, the buckled honeycomb lattice of silicene atoms and their corresponding electrons changing their configurations. The consecutive wrapping and unwrapping of a quasi-Dirac fermion (spontaneous change of its quantum state into chiral quantum states and fermionic quantum states at only resonant frequencies or inertial oscillations) at the edges of the honeycomb-like structure³⁴ accomplishes switching action, that is, triangular pyramidal switching, which is also a quantum topological action.

With Dirac bosons or subquanta on a single electron especially for 0.4 ≤ n_f ≤ 0.9 in underdamped conditions, precessional or wobbling (Larmor frequencies) varying frequencies are produced but tied with twisting effect as a manifestation of gyroscopic behavior, i.e., 0.02 ≤ $g2ℏc$ ≤ 0.08. Encryptions are accomplished for each of these six subquanta states. A six word (each word is equivalent to vector tuples on a ring) with fractional states for encryption on a single electron can work as a quantum byte—a six-word f-qubits. The scheme for encryption, decryption, and control will remain the same for 72 quantum chiral states but with parallel stacks of graphene or with buckled honeycomb lattices of silicene.

There are no experimental data available with us. We do not have access to experimental data elsewhere. The mechanisms of encryption, decryption, and control by using quantum physics were established theoretically. We deciphered qubits and fractional qubits for eigenfunctions using both combinational and sequential logics. We also suggested appropriate electronic circuits for triangular pyramidal switching for spinning fermions at the alternate edges of graphene and six spinning Dirac bosons at the vertices of graphene. Switching for both qubits and f-qubits can be triggered with electronic jittering circuits, phase locking, and possibly with bistable multivibrator using coincidence and anticoincidence electronic circuits to achieve coherent states of energy field fluids following “ring and field” theory and quantum physics.

To our knowledge, this is a unique theoretically determined approach that would help in developing quantum computers at room temperature.

We are thankful to the libraries of Federal Urdu University and the University of Balochistan.

The authors have no conflicts to disclose.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.