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Charge and spin current correlations are analyzed in a source ofspin-entangled electrons built from a superconductor and two quantum dots inparallel. In addition to the ideal (crossed Andreev) channel, parasiticchannels (direct Andreev and cotunneling) and spin flip processes are fullydescribed in a density matrix framework. The way they reduce both theefficiency and the fidelity of the entangler is quantitatively described byanalyzing the zero-frequency noise correlations of charge current as well asspin current in the two output branches. Spin current noise is characterized bya spin Fano factor, equal to 0 (total current noise) and -1 (crossedcorrelations) for an ideal entangler. The violation of the Bell inequalities,as a test of non-locality (entanglement) of split pairs, is formulated in termsof the correlations of electron charge and spin numbers counted in a specifictime window $\tau$. The efficiency of the test is analyzed, comparing $\tau$ tothe various time scales in the entangler operation.Comment: 8 pages, 5 figures, references added, to appear in Phys. Rev.

Spin-current noise and Bell inequalities in a realistic superconductor-quantum dot entangler