The infinite-valued logic \(L_\infty\) (Lukasiewicz logic) was introduced as a generalization of classical logic. \textit{C. C. Chang} [Trans. Am. Math. Soc. 88, 467-490 (1958; Zbl 0084.00704)] introduced MV algebras in order to provide an algebraic proof of its completeness theorem. Just as MV algebras stand for nonidempotent extensions of Boolean algebras, quantum MV algebras, proposed in this paper, stand for nonidempotent extensions of orthomodular lattices. While every MV algebra is representable as a subdirect product of totally ordered MV algebras, the author shows that there exist quantum MV algebras that can not be represented as subdirect products of quasilinear quantum MV algebras. The paper ends with some open problems.