Minimum distortion for embedding finite metric spaces into Schatten S_p

Determine the minimum distortion D_{n,S_p}, as a function of n and p, required to embed any n-point metric space into the Schatten p-class S_p.

Background

The paper proves that all S_p-based notions of quantum expansion are equivalent, analogously to Matoušek’s equivalence between certain ℓ_p-based graph expansion notions used to derive lower bounds on embedding expanders into ℓ_p. Motivated by this analogy, the authors consider the problem of embedding finite metric spaces into the Schatten classes S_p and highlight a lack of existing literature on the distortion required for such embeddings.

They outline a route toward lower bounds similar to the ℓp case: define graph expansion notions based on S_p norms and show a relationship of the form \tilde h{S_p}(G)=Ωd(\tilde h{S_2}(G)/p). Since S_2 is isometric to ℓ2 and there exist expander families with \inf \tilde h{S_2}(G_n)>0, such a relation would imply \inf \tilde h_{S_p}(G_n)=Ω(1/p), yielding a lower bound D_{n,S_p}=Ω(log(n)/p). They also note that a matching upper bound would follow because ℓ_p embeds isometrically into S_p.