Q: In a partnership with a non-religious Jew or gentile, is chametz merchandise permissible after Pesach?

A: There are a number of cases of partnership (see Pischei Choshen, Shutfim 10:14[35]):

A Jew who sold his chametz whose partner with a gentile — the merchandise is certainly permissible, since all the chametz was owned by gentiles on Pesach.

A Jew who did not sell his chametz whose partner is a gentile — Shaagas Aryeh (#89-91) addresses this question at length. He concludes that there is a dispute about the application here of breirah (retroactive clarification), so regarding the Rabbinic prohibition of chametz after Pesach, we can be lenient and allow even the share of the Jew. Others disagree. Some allow only if the majority belongs to the gentile.

A Jew who sold his chametz whose Jewish partner did not sell it — Shaagas Aryeh permits the share of the Jew who sold the chametz (if they subsequently divide).

If a gentile partner bought chametz on Pesach it does not become prohibited, since the Jewish partner does not want him to purchase chametz on his behalf.