Given a two positive matrices $A,B$. For simplicity, let's assume that $Tr A=Tr B=1$. Assume that $\|A-B\|_1\leq\varepsilon$, for some small $\varepsilon>0$, where $\|\cdot\|_1$ is the $l_1$-norm, namely the sum of all its singular values.

My question is whether there is a generic transformation to move $A$ to $B$ for arbitrary such $A$ and $B$. The following are some simple observations. There are two simple transformations moving $A$ to its neighbours.

(1) Find an eigensystem (eigenvalues with corresponding eigenvectors) of $A$ and replace the eigenvalues $(\lambda_1(A),\cdots,\lambda_k(A))$ by a new sequence $(\theta_1,\cdots,\theta_k)\in[0,1]^k$ where $\sum_i\theta_i=1$ and $\sum_i|\lambda_i(A)-\theta_i|\leq\varepsilon$. Here we need the eigensystem of $A$ because the eigenvalues may not be unique.

(2) Choose a unitary matrix $U$ satisfying $\|U-I\|\leq\varepsilon$, where $\|\cdot\|$ is the spectral norm and replace $A$ by $UAU^*$.

It is easy to see neither single transformation (1) nor single transformation (2) is enough to cover all the neighbors.

Consider a naive example

$$ A=\left(\begin{array}{cc} \frac{1+\varepsilon}{2} & 0 \\ 0 & \frac{1-\varepsilon}{2} \end{array}\right) $$

And

$$B=(\frac{1}{2}+\varepsilon) \left(\begin{array}{c}\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}}\end{array}\right)\left(\begin{array}{cc}\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\end{array}\right)+(\frac{1}{2}-\varepsilon) \left(\begin{array}{c}\frac{1}{\sqrt{2}}\\ -\frac{1}{\sqrt{2}}\end{array}\right)\left(\begin{array}{cc}\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}}\end{array}\right).$$

$\|A-B\|_1\leq\varepsilon$ because $\|A-I/2\|_1\leq\varepsilon/2$ and $\|B-I/2\|_1\leq\varepsilon/2$, where $I$ is the $2\times 2$ identity matrix. Transformation (2) cannot bring $A$ to $B$ because the eigenvectors of $A$ and $B$ are far from each other. Also transformation (1) is not enough because the eigenvectors are different. But we can first apply transformmation (1) moving $A$ to $I/2$ and then apply transformation (2) moving $I/2$ to $B$ (There is freedom to choose the eigenvectors for $I/2$).

A more specific question. Can we move $A$ to all its $\varepsilon$-close (in $\|\cdot\|_1$-distance) neighbours by the composition of constant number of transformation (1) and (2)? Even more, can we do it with one transformation (1) and one transformation (2)?

I am not sure the question is research-level. Do not hesitate to close if it is not. Thank you.