Bounds on Dimension Reduction in the Nuclear Norm

$$For all $n\ge 1$, we give an explicit construction of $m\times m$ matrices $A_1,\dots ,A_n$ with $m = 2^{\lfloor n/2 \rfloor}$suchthatforany$d$and$d \times d$matrices$A’_1,\ldots,A’_n$ that satisfy $$\mathrm{\parallel }{A}_i^{\mathrm{\prime }}-A_j^{\mathrm{\prime }}{\mathrm{\parallel }}_{\text{schs}}\le \mathrm{\parallel }{A}_i-A_j{\mathrm{\parallel }}_{\text{schs}}\le (1+\delta )\mathrm{\parallel }{A}_i^{\mathrm{\prime }}-A_j^{\mathrm{\prime }}{\mathrm{\parallel }}_{\text{schs}}$$ for all $i,j\in \{1,\dots ,n\}$ and small enough $\delta =O(n^{-c})$, where $c>0$ is a universal constant, it must be the case that $d\ge 2^{\lfloor n\mathrm{/}2\rfloor -1}$. This stands in contrast to the metric theory of commutative ${\mathrm{\ell }}_p$ spaces, as it is known that for any $p\ge 1$, any $n$ points in ${\mathrm{\ell }}_p$ embed exactly in ${\mathrm{\ell }}_p^d$ for $d=n(n-1)\mathrm{/}2$. 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.