Who owns the diamond mines?

Your answer depends on the diamond mine you reference. Most diamond mines are owned by different companies.

For example, in Canada, from their Wikipedia page:

"The mine is owned by a joint venture between the Rio Tinto and Dominion Diamond Corporation, and is operated by Diavik Diamond Mines Inc., a subsidiary of Rio Tinto Group."

In Australia, the Argyle Mine is owned by the Rio Tinto Group.

In Russia, the Udachnaya pipe is owned by Alrosa.

In Africa, The (Botswana) Orapa diamond mine is owned by Debswana.