Entanglement between quantum dots transmitted via Majorana wire: Insights from the fermionic negativity, concurrence and quantum mutual information

Abstract

We study quantum entanglement in a system comprising two quantum dots interconnected through the short topological superconducting nanowire, which hosts overlapping boundary Majorana modes. Inspecting the fermionic negativity, we analyze the variation of entanglement against the position of the energy levels of quantum dots and their hybridization with the topological superconducting nanowire. In the absence of electron correlations, the optimal entanglement occurs when the energy levels coincide with the zero-energy Majorana modes, whereas upon increasing the hybridizations, the entanglement is gradually suppressed. Such monotonous behavior is no longer valid when the quantum dot levels are detuned from the zero-energy. Under these circumstances, the quantum dots become maximally entangled for a certain optimal hybridization. Moreover, we study the thermal concurrence to explore the entanglement properties at finite temperatures. We also compute the quantum mutual information and propose recipes for robust finite-temperature entanglement transmission via Majorana modes.

Figures (6)

• Figure 1: The schematic of the investigated setup with symmetric coupling $\lambda$ between the quantum dots (QDs) and topological superconducting nanowire, with Majorana modes located at its ends.
• Figure 2: Left and middle: The optimal couplings $\lambda_{i{\rm (opt)}}$ for the first and second quantum dot, respectively, as functions of QDs energies $\varepsilon_{1}$ and $\varepsilon_{2}$, corresponding to the maximum negativity. Right: The maximal negativity $N_{\rm max}$ as a function of quantum dot energies $\varepsilon_{1}$ and $\varepsilon_{2}$ determined at the optimal couplings $\lambda_{i{\rm (opt)}}$. Parameters are in units of $\varepsilon_M=1$.
• Figure 3: The optimal entanglement determined by the position of the logarithmic negativity maximum taken at fixed values of $\lambda^{}_1=\lambda^{}_2$ within the plane spanned by the quantum dots' energies $\{\varepsilon^{}_1,\varepsilon^{}_2\}$. All quantities are in units of the Majorana overlap $\omega=\varepsilon_M/2$.
• Figure 4: Left: The optimal logarithmic negativity calculated for the equal coupling strengths $\lambda_1=\lambda_2$ as a function of quantum dot energies. Right: The corresponding optimal coupling $\lambda_{\rm (opt)}$ ($\lambda_1=\lambda_2$) as function of quantum dot energies $\varepsilon^{}_1$ and $\varepsilon^{}_2$.
• Figure 5: (a)-(c) The negativity as a function of the coupling strength $\lambda_{1}=\lambda_{2}$ in the limit of small QDs energies $\varepsilon_{1}$ and $\varepsilon_2$, as indicated in the legends. In (a) and (b) $\varepsilon_{1}=\varepsilon_2$, whereas $\varepsilon_2=0.005$ in (c), while $\varepsilon_1$ is tuned. The blue line in (d) shows the negativity decay with increasing energies of the dots ($\varepsilon_{1}=\varepsilon_2$), while the red line describes the corresponding optimal value of the coupling strength. All energies are scaled in energy units of the Majorana overlap energy $\omega$.