  Recent experiments on Cooper pair splitters using superconductor-quantum dot
hybrids have embarked on creating entanglement in the solid-state, by
engineering the sub-gap processes in the superconducting region. Using the
thermoelectric Cooper pair splitter setup [Nat. Comm., 12, 21, (2021)] as a
prototype, we develop a detailed analysis of the observed transport signal to
bring out vital insights into the regimes of operation and establish the
non-local nature of the correlations arising from the crossed Andreev
processes. As a striking consequence, contact induced level broadening of the
quantum dot's discrete energy spectrum and its hybridization with the
superconducting segment, results in a parity reversal of the thermoelectric
current along with shifted resonances of the crossed Andreev processes. We
conclusively establish the presence of non-local correlations by making a clear
nexus with quantum discord. Our detailed analysis thereby provides insights
into the gate voltage control of the entanglement generation in
superconducting-hybrid Cooper pair splitters.
