The experiment begins with a high-energy gamma-ray source, such as a synchrotron (e.g., European XFEL) or a laser-driven plasma wakefield accelerator, capable of producing photons with energies $E_{\gamma }>2m_e{c}^2\approx 1.022\text{MeV}\approx 1.638\times {10}^{-13}\text{J}$, the pair production threshold $E_0$, sufficient to induce pair production [5] $\gamma \to e^{+}{e}^{-}$ in the presence of a nucleus to conserve momentum. The pair production occurs within a controlled magnetic field ranging from 1 × 10⁻¹ T to 1.0 × 10¹ T, achievable using superconducting magnets standard in high-energy physics experiments (e.g., at CERN or DESY) [5] . To minimize decoherence from environmental interactions (e.g., collisions with air molecules or stray electromagnetic fields) [6] , the experiment is conducted in a high-vacuum chamber with a pressure of 1 × 10⁻⁶ Pa, ensuring a coherence time consistent with the simulated $\lambda =1\times {10}^{10}{\text{s}}^{-1}$ ( $1.00\times {10}^2\text{fs}$). The electron-positron trajectories are measured using high-resolution silicon strip detectors (e.g., ATLAS Inner Tracker technology) with micrometer precision to determine the separation $S$. Their energies are measured using electromagnetic calorimeters (e.g., lead-scintillator sandwich design) with 1% precision. Spin correlations are measured using a Stern-Gerlach-like setup or Mott scattering to enable quantum state tomography for computing concurrence $C$. Ultrafast detectors (e.g., streak cameras or avalanche photodiodes) ensure measurements at $t=1\times {10}^{-13}\text{s}$, matching the simulation timescale.

1) Vary Photon Energy $E_{\gamma }$: Set $E_{\gamma }$ at $2E_0$, $3E_0$, and $4E_0$, where $E_0$ is the pair production threshold, to confirm the predicted 4.4× increase in $E_f\left(t\right)$ as demonstrated in the “Energy Scaling” scenario, or reductions at lower energies.

2) Vary Magnetic Field $B$: Set $B$ at 1 × 10⁻¹ T, 1 T, 5 T, and 1.0 × 10¹ T to probe the magnetic term’s enhancement or suppression effects.

3) Measure Concurrence $C$: For each $B$ and $E_{\gamma }$, perform quantum state tomography on the electron-positron pair to compute $C$. The pair is assumed to

be in a singlet state $|\psi \rangle =\frac{1}{\sqrt{2}}\left(|01\rangle -|10\rangle \right)$, and $C$ is derived from spin correlation

measurements along multiple axes (e.g., $x,y,z$).

4) Infer $E_f\left(t\right)$: Use the relationship

$C=K\cdot E_f\left(t\right)\cdot \left(1-\eta {\left(\frac{B}{B_0}\right)}^2\right)\cdot {\left(L\cdot \text{APn}\cdot A\cdot S\right)}^{-\alpha }$, with $K\approx 3.36\times {10}^{26}\text{s}\cdot {\text{kg}}^{-1}$,

$\eta =0.05\pm 0.01$, to calculate $E_f\left(t\right)$.

5) Vary Separation $S$: Adjust detector positions to measure $S$ from 1 × 10⁻¹⁰ m to 1.0 × 10⁻⁹ m, validating the $1/S^2$ dependence that reduces $E_f\left(t\right)$, while testing enhancement conditions.

6) Compare with Theory: Compare the measured $E_f\left(t\right)$ with QEFT predictions for $\kappa =0.12$. At $B=1.0\times {10}^1\text{T}$, the recalculated $E_f\left(t\right)$ is $2.9700000081\times {10}^{-27}\text{kg}\cdot {\text{s}}^{-1}$ ( $\kappa =0.12$), a 0.0000272% increase, requiring high-precision measurements. Trends in $C$ with $B$, $E_{\gamma }$, and $S$ will validate the model’s consistency in modulating entanglement.

The simulation was conducted in both scalar and directional modes to evaluate QEFT’s ability to modulate $E_f\left(t\right)$. The verification process confirmed the model’s accuracy, with results summarized in Table 1 . Detailed simulation code is provided in the Supplementary Materials (Section S2).

Table 1. Computed values of $E_f\left(t\right)$ demonstrating QEFT’s ability to modulate entanglement strength. The Energy Scaling scenario shows a 4.4× increase with $E_{\gamma }=4E_0$, where $E_0=2m_e{c}^2\approx 1.638\times {10}^{-13}\text{J}\approx 1.022\text{MeV}$, and the Theta Scaling scenario shows a 2.0× increase with $\theta =2$. At $B=1.0\times {10}^1\text{T}$ (experimentally feasible) with $\kappa =0.12$, $\eta =0.05$, a 0.0000272% increase is observed. At $B=1.0\times {10}^6\text{T}$, a 0.0027% increase occurs, while near $B_0$ at $B=4.414\times {10}^9\text{T}$, a significant increase is achieved with $\kappa =0.12$.

The current numerical simulations, implemented in Python, have limitations that may affect their generality. Potential sources of error include limited floating-point precision, which can introduce rounding errors in exponential terms like ${\text{e}}^{-\lambda t}$, contributing to the reported 1.50% error. Additionally, the model simplifies physical conditions by assuming a non-relativistic framework and $V\left(x\right)=0$, potentially overlooking relativistic effects or complex potentials that could alter electron-positron dynamics. The parameters $S=5.06\times {10}^{-10}\text{m}$ and $\kappa =0.12$ are tuned to specific simulation conditions, which may limit applicability to broader scenarios. Environmental interactions, such as decoherence from unmodeled field inhomogeneities or photon scattering, are approximated, possibly underestimating real-world effects. These errors could restrict the generality of the results, particularly in high-energy or relativistic regimes, necessitating further validation with experimental data or higher-fidelity simulations to ensure robustness across diverse conditions [6] . The simulation results, including the modulation of Entanglement Field Strength as a function of time with magnetic fields and separations, are visualized in Figure 1 [7] .

Simulations with magnetic field strengths $B$ ranging from 1 × 10⁻¹ T to 4.414 × 10⁹ T demonstrate QEFT’s ability to modulate $E_f\left(t\right)$. At $B=1.0\times {10}^1\text{T}$,

achievable in experiments, $\kappa \frac{B}{B_0}\approx 0.12\times \frac{10}{4.414\times {10}^9}\approx 2.72\times {10}^{-10}$, contributing to a 0.0000272% increase in $E_f\left(t\right)$ with $\kappa =0.12$, showing a small but measurable

enhancement. At $B=1.0\times {10}^6\text{T}$, $\kappa \frac{B}{B_0}\approx 0.12\times \frac{{10}^6}{4.414\times {10}^9}\approx 2.72\times {10}^{-5}$, contributing to a 0.0027% increase, as listed in Table 1 . At $B=4.414\times {10}^9\text{T}$ (the Schwinger critical field, included for theoretical insight), the increase is significant with $\kappa =0.12$, as shown in Table 1 . However, at these high fields, the suppression

term $\left(1-\eta {\left(\frac{B}{B_0}\right)}^2\right)$ can reduce $E_f\left(t\right)$ significantly if $\eta$ is larger. Detailed

simulation code is provided in Supplementary Materials (Sections S3 and S5). The discrete impacts of these magnetic field variations across different scenarios are illustrated in Figure 2 , while a continuous representation of Entanglement Field Strength as a function of time modulation with magnetic field is shown in Figure 3 .

The simulation with varying separation $S$ (from 1 × 10⁻¹⁰ m to 1.0 × 10⁻⁹ m) validates relationships between $E_f\left(t\right)$, concurrence $C$, and entropy $S$, showing that increases in $E_f\left(t\right)$ via energy or magnetic fields can counteract the $1/S^2$ decay, while larger separations reduce $E_f\left(t\right)$. For example, increasing $S$ from 5.06 × 10⁻¹⁰ m to 1.0 × 10⁻⁹ m reduces $E_f\left(t\right)$ by approximately a factor of ${\left(1.0/0.506\right)}^2\approx 3.91$. Detailed code is in Supplementary Materials (Section S4).The relationship between Entanglement Field Strength as a function of time and separation, including the one over separation squared decay, is depicted in Figure 4 .

For the electron-positron pair in a singlet state $|\psi \rangle =\frac{1}{\sqrt{2}}\left(|01\rangle -|10\rangle \right)$, concurrence

$C=1$ at $t=0$, indicating maximal entanglement. To connect $E_f\left(t\right)$ to this standard measure, we propose:

$C=K\cdot E_f\left(t\right)\cdot \left(1-\eta {\left(\frac{B}{B_0}\right)}^2\right)\cdot {\left(L\cdot \text{APn}\cdot A\cdot S\right)}^{-\alpha },$ (3)

where $K$ is a proportionality constant with units s∙kg⁻¹, ensuring dimensional consistency ( $C$ is unitless, $E_f\left(t\right)$ has units kg∙s⁻¹). The physical interpretation of $E_f\left(t\right)$ as a flow rate of entangled pair production allows $K$ to serve as a scaling factor that translates this dynamic rate into the dimensionless concurrence, reflecting the degree of entanglement. At $t=0$, $B=0$, the suppression term becomes 1, and we assume $C=1$ in an idealized scenario where decoherence is negligible ( ${\text{e}}^{-\lambda t}\approx 1$). Using the baseline $E_f\left(t=0\right)=2.97\times {10}^{-27}\text{kg}\cdot {\text{s}}^{-1}$ from Table 1 , we find $K=1/\left(2.97\times {10}^{-27}\text{kg}\cdot {\text{s}}^{-1}\right)\approx 3.36\times {10}^{26}\text{s}\cdot {\text{kg}}^{-1}$, consistent with simulation results. As $E_f\left(t\right)$ increases (e.g., to 1.31 × 10^-26 kg∙s⁻¹ in the “Energy Scaling” scenario), $C$ increases, reflecting enhanced entanglement. Conversely, as $E_f\left(t\right)$ decreases due to larger $S$ or decoherence, $C$ decreases, reflecting reduced entanglement.

The units of $E_f\left(t\right)$, kg∙s⁻¹, represent the mass-equivalent flow rate of entangled electron-positron pairs produced per second, derived from the QED pair production process [9] where photon energy ( $E_{\gamma }$) is converted to particle mass via $E=m{c}^2$. In QED, the pair production rate scales with the photon energy above the threshold $E_0$, and $E_f\left(t\right)$ quantifies this rate adjusted by spatial, temporal, and magnetic factors. The mass unit arises from dividing the energy terms (in joules, kg∙m²∙s⁻²) by $c^2$ (m²∙s⁻²), yielding, while the time dependence (per second) reflects the dynamic evolution of entanglement, as validated by simulations with a 1.50% error against QED cross-sections. This interpretation ties $E_f\left(t\right)$ to the physical process of pair creation, enhancing its relevance to quantum information metrics.

Entanglement entropy $S$ for the singlet state is $\text{ln}\left(2\right)\approx 0.693$ at $t=0$. We propose:

$S\propto \text{ln}\left(\frac{E_f\left(t\right)}{E_{f,\text{min}}}\right),$ (4)

where $E_{f,\text{min}}\approx 2.97\times {10}^{-27}\text{kg}\cdot {\text{s}}^{-1}$ is the baseline value of $E_f\left(t\right)$ at $t=0$, under conditions $E_{\gamma }=2E_0$, where $E_0=2m_e{c}^2\approx 1.638\times {10}^{-13}\text{J}\approx 1.022\text{MeV}$, $\theta =1$, $B=1.0\text{T}$, and $S=5.06\times {10}^{-10}\text{m}$ (see Table 1 ). The flow rate nature of $E_f\left(t\right)$ provides a dynamic basis for entropy, where the logarithmic relationship reflects the change in entanglement strength relative to the minimum flow rate. As QEFT increases $E_f\left(t\right)$ above this minimum—e.g., to 1.31 × 10²⁶ kg∙s⁻¹ with $E_{\gamma }=4E_0$—the logarithmic form ensures that $S$ increases, reflecting a gain in entanglement strength. The simulation shows $S$ increasing from 0.693 to 1.6 as $E_f\left(t\right)$ rises, consistent with enhanced entanglement under conditions identified by QEFT (see Supplementary Materials, Section S4). Conversely, if $E_f\left(t\right)$ decreases below the baseline due to larger $S$ or decoherence, $S$ decreases, reflecting reduced entanglement. The stability and variation of Entanglement Field Strength as a function of time over time, supporting the entropy relationship, are presented in Figure 6 .

MEE-QKD utilizes the electron-positron pairs produced in QEFT’s pair production process ( $\gamma \to e^{+}{e}^{-}$) under a controlled magnetic field. A high-energy gamma-ray source ( $E_{\gamma }=4.088\text{MeV}$) generates pairs in a magnetic field ( $B=1.0\times {10}^1\text{T}$), where QEFT modulates $E_f\left(t\right)$ by a 4.4× increase (from 2.97 × 10⁻²⁷ kg∙s⁻¹ to 1.31 × 10⁻²⁶ kg∙s⁻¹) as shown in the “Energy Scaling” scenario ( Table 1 ). The magnetic

term $\left(1+\kappa \frac{B}{B_0}\right)$ with $\kappa =0.12$ contributes a 0.0000272% increase at

$B=1.0\times {10}^1\text{T}$, stabilizing entanglement, though higher fields could suppress it.

The pair is prepared in a singlet state $|\psi \rangle =\frac{1}{\sqrt{2}}\left(|01\rangle -|10\rangle \right)$. Alice measures

the electron’s spin, and Bob measures the positron’s spin along randomly chosen axes (e.g., $x,y$, or $z$) using Stern-Gerlach setups, encoding key bits based on the outcomes (0 or 1).

The magnetic field is modulated between 1 × 10⁻¹ T and 1.0 × 10¹ T, creating a dynamic $E_f\left(t\right)$ signature. An eavesdropper (Eve) attempting to intercept the positron would disrupt the magnetic field’s effect, detectable through a drop

in concurrence $C=K\cdot E_f\left(t\right)\cdot \left(1-\eta {\left(\frac{B}{B_0}\right)}^2\right)\cdot {\left(L\cdot \text{APn}\cdot A\cdot S\right)}^{-\alpha }$. The directional

dependence of $S$ further complicates eavesdropping, as Eve must know the exact orientation. Reductions in $E_f\left(t\right)$ due to larger $S$ or decoherence could also signal interference.

The 4.4× increase in $E_f\left(t\right)$ enhances the key generation rate by enabling more stable entangled pairs. Assuming a baseline rate of 1 s⁻¹ for an E91 protocol at $S=5.06\times {10}^{-10}\text{m}$, MEE-QKD could achieve approximately 4.4 s⁻¹, pending experimental validation.

The short coherence time (1.00 × 10² fs) limits MEE-QKD to short-range applications. Future work could integrate quantum memories to extend the range, and detailed simulations (see Supplementary Materials, Section S8) are recommended to refine the key rate and security analysis.

The Wavefunction Multiplier involves a main field entangled with particles in multiple subspaces within a Hilbert space $\mathscr{H}={\mathscr{H}}_{\text{main}}\otimes {\mathscr{H}}_{\text{sub}}$, where ${\mathscr{H}}_{\text{sub}}={\oplus }_i{\mathscr{H}}_i$. The total state is a superposition $|\Psi \rangle ={\displaystyle {\sum }_i{c}_i|{\psi }_A^i\rangle \otimes |{\psi }_B^i\rangle }$, and measuring the main field’s wavefunction $|{\psi }_B^i\rangle$ matches it with a subspace wavefunction $|{\psi }_B^i\rangle$ to extract information. QEFT’s modulated $E_f\left(t\right)$—e.g., a 4.4× increase at $E_{\gamma }=4E_0$—strengthens these correlations, improving matching accuracy, while reductions in $E_f\left(t\right)$ could degrade performance.

Projection operators $P_i=I_{\text{main}}\otimes |{\psi }_B^i\rangle \langle {\psi }_B^i|$ collapse the state to $|{\psi }_A^i\rangle \otimes |{\psi }_B^i\rangle$ upon matching. The directional form $E_f\left(t,S\right)$ could enhance spatial alignment. Example systems include:

The 4.4× increase in $E_f\left(t\right)$ (e.g., to 1.31 × 10⁻²⁶ kg∙s⁻¹) strengthens the correlation between the main field and subspace particles, enhancing the security of information extraction. Stronger entanglement, reflected by a higher

concurrence $C=K\cdot E_f\left(t\right)\cdot \left(1-\eta {\left(\frac{B}{B_0}\right)}^2\right)\cdot {\left(L\cdot \text{APn}\cdot A\cdot S\right)}^{-\alpha }$, makes the system

more sensitive to eavesdropping attempts. Any unauthorized measurement by an eavesdropper (Eve) disrupts the modulated entanglement, causing a detectable drop in $C$. Additionally, improved correlations reduce matching errors, ensuring only authorized parties can extract the correct subspace information, and enhance resilience against decoherence within the 1.00 × 10² coherence time.

Five simulation runs showed ${|0\rangle }_1$ at 60% (vs. expected 66.67%) due to a small sample size, with a 100-trial simulation approximating 67% $|0\rangle$ and 33% $|1\rangle$. QEFT’s modulated $E_f\left(t\right)$ could reduce such deviations, further improving reliability and security.

This method enables dynamic information retrieval with enhanced security, with potential applications in quantum databases or distributed computing, distinct from QKD.

The Wavefunction Multiplier leverages QEFT’s ability to modulate entanglement strength, as quantified by $E_f\left(t\right)$, to enable secure and efficient information extraction from entangled quantum systems. The demonstrated 4.4× increase in $E_f\left(t\right)$ under optimal conditions (e.g., $E_{\gamma }=4E_0$, $B=1.0\times {10}^1\text{T}$) enhances the fidelity of subspace matching, reducing errors to below 5% in simulations and improving security against eavesdropping through a higher concurrence $C$. The directional form $E_f\left(t,S\right)$ further offers potential for spatially selective information retrieval, which could be critical for distributed quantum networks [9] . However, challenges remain, including the short coherence time (1.00 × 10² fs) and the need for larger-scale simulations to achieve statistical significance, as the current 60% outcome in five trials deviates from the expected 66.67%.

Future work should focus on experimental validation using quantum hardware, such as superconducting qubits or trapped ions, to test the Wavefunction Multiplier in a realistic setting. Implementing the protocol on platforms like Qiskit or Cirq could quantify the impact of modulated $E_f\left(t\right)$ on matching accuracy and security, particularly under varying magnetic fields ( $B=1\times {10}^{-1}\text{T}$ to $1.0\times {10}^1\text{T}$). Additionally, integrating quantum memories (e.g., nitrogen-vacancy centers in diamond) could extend the coherence time, enabling applications in long-range quantum communication. Further simulations with increased trial numbers (e.g., 10⁴ trials) are recommended to reduce statistical errors, and exploring GHZ or W states could enhance the protocol’s robustness. These advancements position the Wavefunction Multiplier as a promising tool for quantum information processing, with potential to transform secure data retrieval in quantum databases and distributed quantum computing architectures.

Generating plots to visualize $E_f\left(t\right)$ and $E_f\left(t,S\right)$ across various parameters.