Electrical current noise of a beam splitter as a test of spin-entanglement

We investigate the spin entanglement in the superconductor-quantum dot system proposed by Recher, Sukhorukov and Loss, coupling it to an electronic beam-splitter. The superconductor-quantum dot entangler and the beam-splitter are treated within a unified framework and the entanglement is detected via current correlations. The state emitted by the entangler is found to be a linear superposition of non-local spin-singlets at different energies, a spin-entangled two-particle wavepacket. Colliding the two electrons in the beam-splitter, the singlet spin-state gives rise to a bunching behavior, detectable via the current correlators. The amount of bunching depends on the relative positions of the single particle levels in the quantum dots and the scattering amplitudes of the beam-splitter. The singlet spin entanglement, insensitive to orbital dephasing but suppressed by spin dephasing, is conveniently quantified via the Fano factors. It is found that the entanglement-dependent contribution to the Fano factor is of the same magnitude as the non-entangled, making an experimental detection feasible. A detailed comparison between the current correlations of the non-local spin-singlet state and other states, possibly emitted by the entangler, is performed. This provides conditions for an unambiguous identification of the non-local singlet spin entanglement.