Let's assume we have a function $f: \mathbb{R}^n \longrightarrow \mathbb{R}$ that's the outcome of the composition of two functions $g: \mathbb{R}^n \longrightarrow \mathbb{C}^{2n}$ and $h: \mathbb{C}^{2n} \longrightarrow \mathbb{R}$ such that $f = h(g(x_1, x_2,...,x_n))$.
Is there a way to know whether the function $f$ has saddle points.
Context: the function $f$ is the objective function of an optimization problem. The function $g$ takes input from $\mathbb{R}^n$ to the Hilbert space $\mathbb{C}^{2n}$ and its output (a pure quantum state actually) is passed to $h$ which calculates the expectation and returns a single real number.
If $f$ is a convex function, every local minimum is a global minimum and one needn't worry about the optimizer getting stuck in local minimum. If not, then how would one show that it is not convex and may have saddle points?