Quantum information sharing between topologically distinct platforms

Can topological quantum entanglement between anyons in one topological medium “stray” into a different, topologically distinct medium? In other words, can quantum information encoded nonlocally in the combined state of non-Abelian anyons be shared between two distinct topological media? For one-dimensional topological superconductors with Majorana bound states at the end of system, the quantum information store in those Majorana bound states can be transfered by directly coupling nearby Majorana bound states. However, coupling of two one-dimensional Majorana states will produce a gap, indicating that distinct topological regions of one-dimensional wires unite into a single topological region through the information transfer process. In this paper, we consider a setup with two two-dimensional $p$-wave superconductors of opposite chirality adjacent to each other. Even two comoving chiral modes at the domain wall between them cannot be gapped through interactions; we demonstrate that information encoded in the fermionic parity of two Majorana zero modes, originally within the same superconducting domain, can be shared between the domains or moved entirely from one domain to another provided that vortices can tunnel between them in a controlled fashion.

Figure 1

(a) Idealized setup involving moving a vortex across the domain wall between $p_x+i{p}_y$ and $p_x-i{p}_y$ superconductors. Red arrows indicate chiral Majorana edge states at the boundaries of the superconducting domains. ${\gamma }_1$ denotes a Majorana zero mode associated with the vortex being moved, while ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2$, and ${\mathrm{\Gamma }}_3$ represent the stationary modes which, along with ${\gamma }_1$, encode a qubit. (b) The schematic setup of the model we used to “mimic” the move of a vortex across the domain wall. Majorana modes ${\mathrm{\Gamma }}_1,...,{\mathrm{\Gamma }}_3$ are decoupled from the edge states, while ${\gamma }_1$ and ${\gamma }_2$ are coupled to chiral Majorana edge states with time-dependent coupling constants ${\alpha }_1\left(t\right)$ and ${\alpha }_2\left(t\right)$ at $x=x_1$ and $x=x_2$ along the edge, respectively. Here, we assume $x_2>x_1$. (c) A schematic setup considered in Ref. [9]: two Majorana modes ${\gamma }_1$ and ${\gamma }_2$ coupled to a single chiral Majorana edge with time-dependent coupling constants ${\alpha }_1\left(t\right)$ and ${\alpha }_2\left(t\right)$, respectively.

Figure 2

Contour plot shows the weight $W(t_0,t,\mathrm{\Delta }x)$ as a function of dimensionless parameters, ${\mathrm{\Lambda }}^2/\beta {\hslash }^2{v}_m$ and $\beta (\mathrm{\Delta }x/v_m-\mathrm{\Delta }t)$. The weight function is given in Eq. (3.5) and is evaluated using the time-dependent coupling strengths given in Eq. (4.1). The inset shows the weight function $W$ along the line cut $\beta (\mathrm{\Delta }x/v_m-\mathrm{\Delta }t)=0$ as indicated by the dashed horizontal line.