A wavefunction of a quantum system is the system's state written in a particular form - no more nor any less than that. Here the word state has an exactly analogous meaning to the state of a classical system insofar that the state at any time uniquely defines the state at any other time and contrariwise. The state's evolution with time in either a quantum or classical system is utterly deterministic. Any linear system - classical or quantum(see footnote 1) - has a state whose time evolution is defined by a first order equation of the form dtψ=AψdtψAψ - in the case of a quantum system the operator AA is skew-self-adjoint - i.e. of the form iHiH where HH is self-adjoint - so that the state vector's L2L2 (Pythogorean) length is preserved(see footnote 2).

The difference between a classical and quantum system is that for the classical system, knowledge of the state uniquely defines the system's behavior and what measurements we shall take from the system, whereas for a quantum system the state defines probability distributions for measurements. Moreover, straight after a measurement, the state of a quantum system is uniquely determined by the actual measurement one gets.

By dint of the experimentally observed behavior that a quantum system is in an eigenstate of the measurement operator (observable) straight after the measurement, it is convenient to write the state vector with the eigenvectors of the measurement operator in question as basis - in such a basis the measurement operator is diagonal (a "multiplication operator") with the measurement values (eigenvalues) along its diagonal. 

Some people reserve the word "wavefunction" strictly for a one-particle state and when the state is written with the eigenvectors of the position operator as basis. In that case the wavefunction's square modulus can be loosely interpreted as a probability to find the "particle" in question at the position named by the eigenvalue. This is what people mean, for example, when they say that the photon has no wavefunction, because it is very difficult to define a position operator for a relativistic particle, and there is no nonrelativistic description of the photon. However, many other people (myself included) simply take the wavefunction as a synonym for "quantum state". You need to be aware of both usages in reading.

Hamilton formulation: classical state: (p,q), point in phase space. product state: ((p,q),(r,s)). Lagrange: same with (q, dq/dt). For fields: (phi,pi)(x), canonical field variables. Lagrange: (phi, dphi/dt)(x). phi(x): real field amplitude. Now QFT: state: complex wave functional PSI[phi(x)]

have a nice day

Can provide some reference information ？

This is a good reference:

sham, Chris J (1995). Lectures on Quantum Theory: Mathematical and Structural Foundations. Imperial College Press. ISBN 978-1-86094-001-9.

@Wang Minjie: nothing is better than: Gerald Rosen, Formulations of Classical and Quantum Dynamical Theory; Acad. Press 1969. The best: only 90(!) pages (without appendices)

As Information Scientist I am not yet  so fix in Mathematics and Physics as all above.

For me you use mixed terms in your question, Wang:

a) classical states: I add the words "of a system" or "of a product";

you see that a product  is or can be a part of a system;

the differentiation classical state to product state is per se not important.

b) it's well known that's not possible to mix Quantum Mechanics with Classical Physics.

So the term state is different in both areas of Physics too.

I wouldn't try to find unification between these two areas - even not in any mathematical theory.