Bounds on Dimension Reduction in the Nuclear Norm

$\newcommand{\schs}{\scriptstyle{\mathsf{S}}_1}$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.Comment: 16 page

Bounds on Dimension Reduction in the Nuclear Norm

For all n ≥ 1, we give an explicit construction of m × m matrices A_1,…,A_n with m = 2^([n/2]) such that for any d and d × d matrices A′_1,…,A′_n that satisfy ∥A_′i−A′_j∥S_1 ≤ ∥A_i−A_j∥S_1 ≤ (1+δ)∥A′_i−A′_j∥S_1 for all i,j∈{1,…,n} and small enough δ = O(n^(−c)), where c > 0 is a universal constant, it must be the case that d ≥ 2^([n/2]−1). This stands in contrast to the metric theory of commutative ℓ_p spaces, as it is known that for any p ≥ 1, any n points in ℓ_p embed exactly in ℓ^d_p 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