I will present a review of our current understanding of the quantum states in a carbon nanotube quantum dot deduced from low temperature transport measurements in parallel and perpendicular magnetic fields. The observed energy spectrum is shown to be ordered in shells of two doublets consistent with a single-particle four-state model including spin-orbit interaction, valley mixing and an orbital g-factor . Furthermore, for certain shells, the two doublets are observed to be differently coupled to the leads, resulting in gate-dependent level renormalization. By comparison to the shell model this is shown to be a consequence of intra-shell valley mixing in the nanotube. Moreover, a parallel magnetic field is shown to reduce this mixing and thus suppress the effects of tunnel-renormalization .
Finally, I will give an idea of our on-going effort on nanotube Cooper pair splitters. We are particular interested in utilizing our understanding of the level structure and spin-orbit coupling presented above to fabricate devices, which are predicted to be ideal for testing the spin entanglement of split Cooper pairs .
 T. Sand Jespersen, K. Grove-Rasmussen, J. Paaske, K. Muraki, T. Fujisawa, J. Nygård, and K. Flensberg, Nat. Phys. 7, 348 (2011).
 K. Grove-Rasmussen, S. Grap, J. Paaske, K. Flensberg, S. Andergassen, V. Meden, H. I. Jørgensen, K. Muraki, and T. Fujisawa, Phys. Rev. Lett. 108, 176802 (2012).
 B. Braunecker, P. Burset, and A. Levy Yeyati, Phys Rev. Lett. 109, 166403 (2012)