Mathematically what is the difference between pure separable state and entangled state ?
Can anyone explain with equations?
Entangled States are the ones who doesn't have a classical analog. They are members of a tensor product of two Hilbert Spaces, say:
$$H_{A}\otimes H_{B}$$
So an entangled state of a composite system is the one state which can't be written as a simple product state. For instance:
Consider the entangled pure state:
$$\mid \psi^{+}\rangle = \frac{1}{\sqrt{2}}\Big(\mid 01\rangle_{AB} + \mid 10\rangle_{AB} \Big)$$
This is an example which you can't write as a product state.