Lower Bound for Electron Spin Entanglement from Beam Splitter Current Correlations

We determine a lower bound for the entanglement of formation of pairs of electron spins injected into a mesoscopic conductor. The bound can be expressed in terms of experimentally accessible quantities, the zero-frequency current correlators (shot noise power or cross correlators) after transmission through an electronic beam splitter and can be used to gain information about the entanglement from experiment. Spin relaxation ($T_1$ processes) and decoherence ($T_2$) during the ballistic coherent transmission of carriers are taken into account within Bloch theory. A variable inhomogeneous magnetic field gives rise to a useful lower bound for the entanglement of arbitrary states. The decrease in entanglement due to thermally mixed states is studied. Both the entanglement of the output of a source (entangler) and $T_{1,2}$ can be determined from current correlators.

Figure 1

Inset: Proposed setup with two-electron scattering at a beam splitter (BS) with transmittivity $T$. Electrons are injected pairwise from the entangler (E) into the BS contacts 1,2. The mean current $I=⟨I_{\alpha }⟩$ and one of the correlators $S_{\alpha \beta }$ are measured at the contacts $\alpha ,\beta =3,4$. Plot: Entanglement of formation $E$ of the electron spins versus singlet fidelity $F$ and the reduced correlator $f=S_{33}/2eIT(1-T)$. The curve illustrates the exact relation for Werner states. For general states, the curve is a lower bound for $E$; allowed values for $E$ and $f$ (or $F$) are above the curve.

Figure 2

Homogeneous field $\delta h=0$ and $\tilde{P}=1$. (a) $f$ of the singlet state $|{\Psi }_{-}⟩$ after ballistic transmission through a BS as a function of $T_2/t_0$, where $t_0=L/v_F$. Different curves correspond to different values of the relaxation time $T_1/t_0$ (plotted only for $T_2\le 2T_1$). (b) $f$ for a triplet state $|{\Psi }_{+}⟩$. Since $f\le 1$, the lower bound on $E$ is zero, i.e., determination of triplet entanglement is impossible at $\delta h=0$. (c) Lower bound on $E$.

Figure 3

(a) Reduced current correlator $f$ versus field inhomogeneity $\delta h=h_1-h_2$ (units of $\hslash /t_{0}g{\mu }_B$, $t_0=L/v_F$, $g=\mathrm{\text{g factor}}$, ${\mu }_B=\mathrm{\text{Bohr magneton}}$) for injected singlet and $\tilde{P}=1$. Solid lines represent $T_1\gg t_0$ and $T_2=t_0,3t_0,10t_0,\infty$, gray dashed lines $T_1=3t_0$ and $T_2=t_0,3t_0$. For triplets, the plot is phase shifted by $\pi$, providing tight lower bounds at $\delta h=\pi$. Tight bounds for any input state are obtained by varying the direction of $\delta \mathbf{h}$. (b) Plot of $f$ for a thermally mixed initial state versus $T_2/t_0$ for $T_1\gg t_0$, $\tilde{P}=1$. Various curves correspond to $k_{B}T/J=0,0.25,0.33,0.5,1$, where $J$ and $T$ denote the exchange energy and temperature during state preparation. Inset: The maximal $f$ (at $T_1,T_2\gg t_0$) versus $k_{B}T/J$. Entanglement is absent ($f\le 1$, $E=0$) above ${\mathrm{T}}_c=0.91J/k_B$.