  Let $G$ be a linear algebraic group acting linearly on a $G$-variety
$\mathcal{V}$, and let $k[\mathcal{V}]^G$ be the corresponding algebra of
invariant polynomial functions. A separating set $S \subseteq k[\mathcal{V}]^G$
is a set of polynomials with the property that for all $v,w \in \mathcal{V}$,
if there exists $f \in k[\mathcal{V}]^G$ separating $v$ and $w$, then there
exists $f \in S$ separating $v$ and $w$.
  In this article we consider the action of $G = \mathrm{GL}_2(\mathbb{C})$ on
the variety $\mathcal{M}_2^n$ of $n$-tuples of $2 \times 2$ matrices by
simultaneous conjugation. Minimal generating sets $S_n$ of
$\mathbb{C}[\mathcal{M}_2^n]^G$ are well-known, and $|S_n| = \frac16(n^3+11n)$.
In recent work, Kaygorodov, Lopatin and Popov showed that for all $n \geq 1$,
$S_n$ is a minimal separating set by inclusion, i.e. that no proper subset of
$S_n$ is a separating set. This does not necessarily mean that $S_n$ has
minimum cardinality among all separating sets for
$\mathbb{C}[\mathcal{M}_2^n]^G$. Our main result shows that any separating set
for $\mathbb{C}[\mathcal{M}_2^n]^G$ has cardinality $\geq 5n-5$. In particular,
there is no separating set of size $\dim(\mathbb{C}[\mathcal{M}_2^n]) = 4n-3$
for $n \geq 3$. Further, $S_3$ has indeed minimum cardinality as a separating
set, but for $n \geq 4$ there may exist a smaller separating set than $S_n$. We
show that for $n \geq 5$ there does, in fact, exist a smaller separating set
than $S_n$. We also prove similar results for the left-right action of
$\mathrm{SL}_2(\mathbb{C}) \times \mathrm{SL}_2(\mathbb{C})$ on
$\mathcal{M}_2^n$.
