  We introduce symmetric states and quantum symmetric states on universal
unital free product C*-algebras an arbitrary unital C*-algebra A with itself
infinitely many times, as a generalization of the notions of exchangeable and
quantum exchangeable random variables. We prove existence of conditional
expectations onto tail algebras in various settings and we define a natural
C*-subalgebra of the tail algebra, called the tail C*-algebra. Extending and
building on the proof of the noncommutative de Finetti theorem of Koestler and
Speicher, we prove a de Finetti type theorem that characterizes quantum
symmetric states in terms of amalgamated free products over the tail
C*-algebra, and we provide a convenient description of the set of all quantum
symmetric states on the free product C*-algebra in terms of C*-algebras
generated by homomorphic images of A and the tail C*-algebra. This description
allows a characterization of the extreme quantum symmetric states. Similar
results are proved for the subset of tracial quantum symmetric states, though
in terms of von Neumann algebras and normal conditional expectations. The
central quantum symmetric states are those for which the tail algebra is in the
center of the von Neumann algebra, and we show that the central quantum
symmetric states form a Choquet simplex whose extreme points are the free
product states, while the tracial central quantum symmetric states form a
Choquet simplex whose extreme points are the free product traces.
