Minimal Entanglement Witness from Electrical Current Correlations

Despite great efforts, an unambiguous demonstration of entanglement of mobile electrons in solid state conductors is still lacking. Investigating theoretically a generic entangler-detector setup, we here show that a witness of entanglement between two flying electron qubits can be constructed from only two current cross correlation measurements, for any nonzero detector efficiencies and noncollinear polarization vectors. We find that all entangled pure states, but not all mixed ones, can be detected with only two measurements, except the maximally entangled states, which require three. Moreover, detector settings for optimal entanglement witnessing are presented.

Figure 1

Schematic of the generic entangler-detector setup, consisting of an entangler and two detector systems $A$ and $B$. The entangler generates split pairs of entangled electrons, in either spin or orbital degrees of freedom, at a rate $\mathrm{\Gamma }$. Each detector system consists of two detector terminals ($\pm$) and a beam splitter with efficiency $0\le {\zeta }_A,{\zeta }_B\le 1$ and unit-magnitude polarization vector $\mathbf{a}$, $\mathbf{b}$. An entanglement witness $W$ can be constructed from only two current cross correlation measurements, with respective polarization vectors ${\mathbf{a}}_1$, ${\mathbf{a}}_2$ at $A$ (${\mathbf{b}}_1$, ${\mathbf{b}}_2$ at $B$). The local coordinate systems are defined such that ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ (${\mathbf{b}}_1$ and ${\mathbf{b}}_2$) lie in the $xz$ plane, symmetrically about the $z$ axis, with relative local angles ${\theta }_A$ (${\theta }_B$). The correlations can be measured between currents at two terminals only (here $A+$ and $B+$).

Figure 2

Left: The maximal detection margins ${\tilde{\mathrm{\Delta }}}_{\rho }^{\pm }$ of $W^{(2)}$ for pure states $\rho$ with concurrence $0\le C\le 1$. Right: Symmetric detector settings ${\theta }_A={\theta }_B=\theta$ (top) and ${\zeta }_A={\zeta }_B=\zeta$ (bottom) giving ${\tilde{\mathrm{\Delta }}}_{\rho }^{\pm }$ of $W^{(2)}$ in left panel. In all plots, solid blue (dashed green) line corresponds to ${\tilde{\mathrm{\Delta }}}_{\rho }^{+}$ (${\tilde{\mathrm{\Delta }}}_{\rho }^{-}$).

Figure 3

(a)–(d) The detection margin ${\mathrm{\Delta }}_{\rho }^{W+}$ for symmetric setups, ${\theta }_A={\theta }_B=\theta$ and ${\zeta }_A={\zeta }_B=\zeta$, with mixed states, $\rho =(1-\lambda )|\psi ⟩⟨\psi |+\lambda \mathbb{1}/4$, $0\le \lambda \le 1$, where $|\psi ⟩$ has concurrence $0\le C\le 1$. (e) ${\tilde{\mathrm{\Delta }}}_{\rho }^{+}$ for $\rho$ with different $C$ and $\lambda$. The cases in (a)–(d) are marked with black dots. The color scale in (e) holds also for panels (a)–(d).