  $ \newcommand{\schs}{\scriptstyle{\mathsf{S}}_1} $For all $n \ge 1$, we give
an explicit construction of $m \times m$ matrices $A_1,\ldots,A_n$ with $m =
2^{\lfloor n/2 \rfloor}$ such that for any $d$ and $d \times d$ matrices
$A'_1,\ldots,A'_n$ that satisfy \[ \|A'_i-A'_j\|_{\schs} \,\leq\,
\|A_i-A_j\|_{\schs}\,\leq\, (1+\delta) \|A'_i-A'_j\|_{\schs} \] for all
$i,j\in\{1,\ldots,n\}$ and small enough $\delta = O(n^{-c})$, where $c> 0$ is a
universal constant, it must be the case that $d \ge 2^{\lfloor n/2\rfloor -1}$.
This stands in contrast to the metric theory of commutative $\ell_p$ spaces, as
it is known that for any $p\geq 1$, any $n$ points in $\ell_p$ embed exactly in
$\ell_p^d$ for $d=n(n-1)/2$.
  Our proof is based on matrices derived from a representation of the Clifford
algebra generated by $n$ anti-commuting Hermitian matrices that square to
identity, and borrows ideas from the analysis of nonlocal games in quantum
information theory.
