Impossibility of dimension reduction in the nuclear norm

Understand

Let $\mathsf{S}_1$ (the Schatten--von Neumann trace class) denote the Banach space of all compact linear operators $T:\ell_2\to \ell_2$ whose nuclear norm $\|T\|_{\mathsf{S}_1}=\sum_{j=1}^\infty\sigma_j(T)$ is finite, where $\{\sigma_j(T)\}_{j=1}^\infty$ are the singular values of $T$.

• We prove that for arbitrarily large $n\in \mathbb{N}$ there exists a subset $\mathcal{C}\subseteq \mathsf{S}_1$ with $|\mathcal{C}|=n$ that cannot be embedded with bi-Lipschitz distortion $O(1)$ into any $n^{o(1)}$-dimensional linear subspace of $\mathsf{S}_1$.
• $\mathcal{C}$ is not even a $O(1)$-Lipschitz quotient of any subset of any $n^{o(1)}$-dimensional linear subspace of $\mathsf{S}_1$.
• Thus, $\mathsf{S}_1$ does not admit a dimension reduction result \'a la Johnson and Lindenstrauss (1984), which complements the work of Harrow, Montanaro and Short (2011) on the limitations of quantum dimension reduction under the assumption that the embedding into low dimensions is a quantum channel.