# Open neighborhood of an entangled state with non-decreasing Schmidt rank

Let $$\psi\in H_A \otimes H_B$$ be an entangled state, which means that it has Schmidt rank $$r \geq 2$$. Does there exist some $$\epsilon>0$$ for which all states $$\varphi$$ with $$\|\psi - \varphi\|< \epsilon$$ have Schmidt rank at least $$r$$? I think this must be studied somewhere but I'm unable to locate a reference or proof.

I assume $$H_A$$ and $$H_B$$ are finite dimensional.

TL;DR: Yes. The set of pure bipartite quantum states with Schmidt rank at least $$r$$ is open$$^1$$.
Let $$S_{r}$$ denote the set of all pure bipartite quantum states in $$H_A\otimes H_B$$ with Schmidt rank exactly $$r$$. Clearly, $$H_A\otimes H_B=\bigcup_{i=1}^d S_i$$ where $$d=\min(\dim H_A, \dim H_B)$$. For any $$k\in\mathbb{N}$$, define $$S_{\leq k}=\bigcup_{i=1}^k S_i$$ and $$S_{\geq k}=\bigcup_{i=k}^d S_i$$. I will show that all $$S_{\leq k}$$ are closed and all $$S_{\geq k}$$ are open.
Consider the map $$s:H_A\otimes H_B\to\mathbb{R}^d$$ which sends a pure bipartite quantum state to$$^2$$ the $$d$$-tuple of its Schmidt coefficients listed in decreasing order and padded$$^3$$ with zeros. Note that $$s$$ is continuous$$^4$$. Moreover, for any $$k\in\mathbb{N}$$ the set $$\mathbb{R}^{(k)}=\mathbb{R}^k\times\{0\}^{d-k}$$ is a closed subset of $$\mathbb{R}^d$$. But $$S_{\leq k}=s^{-1}\big[\mathbb{R}^{(k)}\big]$$. A continuous preimage of a closed set is itself closed, so $$S_{\leq k}$$ is closed. Therefore, its complement $$\big(S_{\leq k}\big)^c=H_A\otimes H_B\setminus S_{\leq k}=S_{\geq k+1}$$ is open. Consequently, if $$|\psi\rangle\in H_A\otimes H_B$$ has Schmidt rank $$r\geq 2$$, then there is an $$\epsilon>0$$ such that every $$|\phi\rangle$$ with $$\||\psi\rangle-|\phi\rangle\|<\epsilon$$ has Schmidt rank at least $$r$$.
$$^1$$ In the usual topology induced by any of the norms equivalent to the operator norm.
$$^2$$ The image of $$s$$ is actually a subset of the unit sphere in $$\mathbb{R}^d$$.
$$^3$$ Padding is needed since Schmidt coefficients are positive real numbers by definition.
$$^4$$ The squares of the Schmidt coefficients are the eigenvalues of the reduced density matrix. Partial trace, eigenvalues and square root are continuous.