Finite-temperature Bell test for quasiparticle entanglement in the Fermi sea

We demonstrate that the Bell test cannot be realized at finite temperatures in the vast majority of electronic setups proposed previously for quantum entanglement generation. This fundamental difficulty is shown to originate in a finite probability of quasiparticle emission from Fermi-sea detectors. In order to overcome the feedback problem, we suggest a detection strategy, which takes advantage of a resonant coupling to the quasiparticle drains. Unlike other proposals, the designed Bell test provides a possibility to determine the critical temperature for entanglement production in the solid state.

Figure 1

A generic beam splitter for entanglement production in the solid state. The voltage bias applied between the sources S1 and S2 generates an entangled outgoing state at the detectors D1 and D2 provided the temperature in the sources $T$ is smaller than a critical temperature $T_c$.

Figure 2

The spin cross correlator $\mathcal{C}$ obtained from the density matrix of the final scattering state [solid lines; cf. Eq. (20)], and its generalization ${\mathcal{C}}^M$, evaluated numerically from Eqs. (9, 10, 11, 12, 13, 14, 15) for different values of the detection time $eV{t}_{\det }∕h=0.01$ (●), 0.1 (▴), 1 (◼), 5 (◆) [dashed lines; see Eqs. (21, 22)].

Figure 3

The case of resonant detector coupling. The short-dashed line shows ${\mathcal{C}}^M$ from Eq. (26), while the long-dashed lines are numerical results for the Breit–Wigner resonances [Eq. (24)] with finite width $\Gamma =0.01eV$, detector voltage $V_{\mathrm{D}}=-V$, and different detection times $\Gamma t_{\det }∕h=0.01\left(●\right),0.1(▴)$. The measurement is useless for $t_{\det }\gtrsim 0.1h∕\Gamma$. The solid line shows the correlator $\mathcal{C}$ from Eq. (20).