Quantum entanglement for doubly-excited resonance states of the helium atom

It is well known that quantum entanglement is relevant to quantum information, quantum computation, quantum teleportation, and quantum cryptography. In our present work, quantum entanglement for doubly-excited resonance states are quantified by calculating the linear entropy ($S_{L})$ and von Neumann entropy ($S_{vN})$ for such states. The linear entropy is defined as $S_{L} =1-Tr\left( {\rho_{red}^{2} } \right)$ and the von Neumann entropy as $S_{vN} =-Tr\left( {\rho_{red} \log_{2} \rho_{red} } \right)$, where $\rho_{red} $ is the reduced density matrix, and \textit{Tr} denotes the trace of the matrix. In our previous works, we calculated the linear entropy for the bound states of the helium atom in free space [1]. Here, we employ the projection operator method [2] to calculate the energies and wave functions of doubly-excited resonance states in the helium atom. Using the projection operators $P$ and $Q$ with $P\left| \right\rangle =\left( {1-Q} \right)\left| \right\rangle $, we can evaluate the eigenvalues of $\left\langle {\mbox{\thinspace }} \right|QHQ\left| \right\rangle =\varepsilon_{res} \left\langle {\mbox{\thinspace }} \right|QQ\left| \right\rangle $, and such eigenvalues $\varepsilon_{res} $ approximate the resonance energies. Once the wave functions for the resonance states are obtained, we can use them to calculate the von Neumann and linear entropies of the doubly-excited resonance states. In the present work, we investigate the $^{\mathrm{1,3}}S^{\mathrm{e}}$, $^{\mathrm{1,3}}P^{\mathrm{o}}$, $^{\mathrm{1,3}}D^{\mathrm{e}}$, and $^{\mathrm{1,3}}F^{\mathrm{o}}$ resonance series in the helium atom lying below the He$^{\mathrm{+}}(N=$2) threshold. Our results indicate that different series will have different behaviors for their entropies. The detail of our findings will be presented at the meeting.