Proposal for Production and Detection of Entangled Electron-Hole Pairs in a Degenerate Electron Gas

We demonstrate theoretically that the shot noise produced by a tunnel barrier in a two-channel conductor violates a Bell inequality. The nonlocality is shown to originate from entangled electron-hole pairs created by tunneling events—without requiring electron-electron interactions. The degree of entanglement (concurrence) equals $2(T_1{T}_2{)}^{1/2}(T_1+T_2{)}^{-1}$, with $T_1,T_2\ll 1$ the transmission eigenvalues. A pair of edge channels in the quantum Hall effect is proposed as an experimental realization.

Figure 1

Schematic description of the method to produce and detect entangled edge channels in the quantum Hall effect. The thick black lines indicate the boundaries of a two-dimensional electron gas. A strong perpendicular magnetic field $B$ ensures that the transport near the Fermi level $E_F$ takes place in two edge channels, extended along a pair of equipotentials (thin solid and dashed lines, with arrows that give the direction of propagation). A split gate electrode (shaded rectangles at the center) divides the conductor into two halves, coupled by tunneling through a narrow opening (dashed arrow, scattering matrix $S$). If a voltage $V$ is applied between the two halves, then there is a narrow energy range $eV$ above $E_F$ in which the edge channels are predominantly filled in the left half (solid lines) and predominantly empty in the right half (dashed lines). Tunneling events introduce filled states in the right half (black dots) and empty states in the left half (open circles). The entanglement of these particle-hole excitations is detected by the violation of a Bell inequality. This requires two gate electrodes to locally mix the edge channels (scattering matrices $U_L,U_R$) and two pairs of contacts $1,2$ to separately measure the current in each transmitted and reflected edge channel.