Bounds on Dimension Reduction in the Nuclear Norm

Abstract

For all $n\ge 1$, we give an explicit construction of $m\times m$ matrices $A_1,\dots ,A_n$ with $m=2^{\lfloor n/2\rfloor }$ such that for any $d$ and $d\times d$ matrices $A_1^{\prime },\dots ,A_n^{\prime }$ that satisfy \[ \|A'_i-A'_j\|_{\schs} \,\leq\, \|A_i-A_j\|_{\schs}\,\leq\, (1+δ) \|A'_i-A'_j\|_{\schs} \] 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 $d\ge 2^{\lfloor n/2\rfloor -1}$. This stands in contrast to the metric theory of commutative ${\ell }_p$ spaces, as it is known that for any $p\ge 1$, 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.

16 pages