I am studying measures of closeness of two matrices such as trace distance and fidelity.
I read that the Frobenius norm of a matrix $A$ is a measure of the "size" of $A$, defined as the square root of the sum of the absolute squares of its elements.
Now I wonder whether the Frobenius norm defined for two matrices is a measure of closeness? I need reference for that.
The Frobenius norm is an instance of the Schatten $p$-norms: Starting from the singular value decomposition $A=\sum_js_j|v_j\rangle\langle w_j|$ of a matrix $A\in\mathbb C^{n\times n}$ (you can think of $A$ as $\rho-\sigma$ for states $\rho,\sigma$, if you wish) one defines its Schatten-$p$ norm as $$ \|A\|_p:=\Big(\sum_j s_j^p\Big)^{1/p}, $$ where $p\in[1,\infty]$. For $p=1$ this reproduces the trace norm $\|A\|_1={\rm tr}(\sqrt{A^\dagger A})$, and for $p=2$ this is equal to the Frobenius norm $\|A\|_2=\sqrt{{\rm tr}(A^\dagger A)}$ (i.e., "the square root of the sum of the absolute squares of its elements").
Now the reason this unifying framework is important is that there are inequalities relating different Schatten-$p$ norms, cf. page 32 in the book "The Theory of Quantum Information" by Watrous (alt link). For the trace norm and the Frobenius norm this reduces to $$ \|A\|_2\leq\|A\|_1\leq\sqrt{{\rm rank}(A)}\|A\|_2 $$ for all $A\in\mathbb C^{n\times n}$. In this case one says that the norms are equivalent, i.e., given two density matrices $\rho,\sigma$ (and, again, inserting $A=\rho-\sigma$) this shows that $\rho$ and $\sigma$ are close in trace norm if and only if they are close in Frobenius norm.
To conclude: yes, the Frobenius norm—and in fact every Schatten-$p$ norm—is a measure of closeness of two quantum states.