# Photon Enhanced Interaction and Entanglement in Semiconductor Position-Based Qubits
## 3 Position-Based Semiconductor Qubits in the Frame of Semiconductor Photon Communication
### Maximally Entangled States and Entanglement Entropy  
To investigate the entanglement and the dynamics of the system of two qubits, we will use the Von Neumann entanglement entropy \(S_{\mathrm{N}}\). The density operator is given by the expression:  
\[\hat{\rho}_{AB}=\left|\psi\right\rangle\left\langle\psi\right| \tag{21}\]  
Then, the Von Neumann entanglement entropy \(S_{\mathrm{N}}\) is defined as follows:  
\[S_{\mathrm{N}}=-\mathrm{tr}(\hat{\rho}_{A}\ln\hat{\rho}_{A})=-\mathrm{tr}(\hat {\rho}_{B}\ln\hat{\rho}_{B}) \tag{22}\]  
where, operators \(\hat{\rho}_{A}\) and \(\hat{\rho}_{B}\) are the reduced density operators, which can be found via the partial trace as  
\[ \begin{split}\hat{\rho}_{A}=\left\langle 0_{g}^{B}\right|\hat{ \rho}_{AB}\left|0_{g}^{B}\right\rangle+\left\langle 0_{e}^{B}\right|\hat{ \rho}_{AB}\left|0_{e}^{B}\right\rangle+\left\langle 1_{g}^{B}\right|\hat{ \rho}_{AB}\left|1_{g}^{B}\right\rangle+\left\langle 1_{e}^{B}\right|\hat{ \rho}_{AB}\left|1_{e}^{B}\right\rangle\\ \hat{\rho}_{B}=\left\langle 0_{g}^{A}\right|\hat{\rho}_{AB} \left|0_{g}^{A}\right\rangle+\left\langle 0_{e}^{A}\right|\hat{\rho}_{AB} \left|0_{e}^{A}\right\rangle+\left\langle 1_{g}^{A}\right|\hat{\rho}_{AB}\left|1_{g}^{A} \right\rangle+\left\langle 1_{e}^{A}\right|\hat{\rho}_{AB}\left|1_{e}^{A}\right\rangle\end{split}  \tag{23}\]  
Let us now assume the maximally entangled Bell state [31]:  
\[\left|\Phi_{AB}\right\rangle=\frac{1}{\sqrt{2}}\left(\left|0^{A}0^{B}\right\rangle +\left|1^{A}1^{B}\right\rangle\right) \tag{24}\]  
Figure 5: (**a**) Simulated quantum path as a result of transitions from the localized ground state \(\left|g\right\rangle\), to the localized excited state \(\left|e\right\rangle\). The transitions take place in a probabilistic manner, following the time evolution determined from the solution of (15). (**b**) Number of emitted/absorbed photons as a result of transitions between the ground \(\left|g\right\rangle\) and the excited \(\left|e\right\rangle\) states for the simulated quantum path. It is evident that transitions from the ground state \(\left|g\right\rangle\) to the excited state \(\left|e\right\rangle\) correspond to an absorption of a photon, while the transitions from the excited state \(\left|e\right\rangle\) to the ground state \(\left|g\right\rangle\) correspond to an emission of a photon. which is defined generally between any two systems \(A\) and \(B\). Writing this state in the basis of our system with system-A (qubit #1) and system-B (qubit #2), we get:  
\[\left|\Phi_{AB}\right\rangle=\frac{1}{\sqrt{2}}\Big{(}\Big{|}0_{g} ^{A}0_{g}^{B}\Big{\rangle}+\Big{|}0_{g}^{A}0_{e}^{B}\Big{\rangle}+\Big{|}0_{e }^{A}0_{g}^{B}\Big{\rangle}+\Big{|}0_{e}^{A}0_{e}^{B}\Big{\rangle}+\] \[\qquad\qquad+\Big{|}1_{g}^{A}1_{g}^{B}\Big{\rangle}+\Big{|}1_{g} ^{A}1_{e}^{B}\Big{\rangle}+\Big  
However, we also plot in Figure 6b the evolution of the probability with time of the localized state \(\left|0^{A}\right\rangle\) of system-A, \(P_{\left|0^{A}\right\rangle}(t)=P_{\left|0^{A}_{S}\right\rangle}(t)+P_{\left|0^{ A}_{S}\right\rangle}(t)\), and the evolution of the probability with time of the localized state \(\left|0^{\text{B}}\right\rangle\) of system-B, \(P_{\left|0^{B}\right\rangle}(t)=P_{\left|0^{B}_{S}\right\rangle}(t)+P_{\left|0^ {B}_{S}\right\rangle}(t)\). In this case, the anti-correlation is not trivial, and the result will depend on initial conditions. This is expected since we assumed that there is no electrostatic interaction between the two particles, i.e., the Coulomb force is negligible. Considering  
Figure 6: **(a)** Evolution of probability as a function of time of the ground state of system-A, \(P_{\left|g^{A}\right\rangle}=P_{\left|0^{A}_{S}\right\rangle}(t)+P_{\left|1^{ A}_{S}\right\rangle}(t)\), together with that of the ground state of system-B, \(P_{\left|g^{B}\right\rangle}=P_{\left|0^{B}_{S}\right\rangle}(t)+P_{\left|1^{ B}_{S}\right\rangle}(t)\). Remarkably, the energy states of the two qubits are anti-correlated.