The physics states in Quantum mechanics is represented by vectors in Hilbert space, however in Heisenberg's picture, the equation of motion
$$ \frac{d}{dt}A_H(t) = \frac{i}{\hbar}[H,A_H(t)]+\frac{\partial}{\partial t}A_H(t) $$
only deals with operators' time evolution.
I am confused that what is the physical state in Heisenberg picture?
If you ask about what plays the role of the state of the system at some given time than the answer is: nothing. You talk only about the initial state and what you get in the measurements (expectation values or probabilities of outcomes for observables). The Heisenberg picture is very "Copenhagen" in its spirit and abstracts itself from what's happening with the system itself.
The physical states in the Heisenberg picture are frozen in time, and can be made to coincide with the Schrodinger-picture state at any given time $t_0$. In other words, $$|\psi(t_0)\rangle^S=|\psi\rangle^H$$ which doesn't evolve with time.
