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 (T₁ processes) and decoherence (T₂) 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.