Impossibility of Dimension Reduction in the Nuclear Norm

Let S₁ (the Schatten–von Neumann trace class) denote the Banach space of all compact linear operators T: ℓ₂→ ℓ₂ whose nuclear norm ‖T‖S1=∑j=1∞σj(T) is finite, where {σj(T)}j=1∞ are the singular values of T. We prove that for arbitrarily large n∈ N there exists a subset C⊆ S₁ with | C| = n that cannot be embedded with bi-Lipschitz distortion O(1) into any n^{o ( 1 )}-dimensional linear subspace of S₁. C is not even a O(1)-Lipschitz quotient of any subset of any n^{o ( 1 )}-dimensional linear subspace of S₁. Thus, S₁ does not admit a dimension reduction result á la Johnson and Lindenstrauss (Conference in modern analysis and probability, American Mathematical Society, Providence, 1984), which complements the work of Harrow et al. (Automata, languages and programming (ICALP 2011), Springer, Heidelberg, 2011) on the limitations of quantum dimension reduction under the assumption that the embedding into low dimensions is a quantum channel. Such a statement was previously known with S₁ replaced by the Banach space ℓ₁ of absolutely summable sequences via the work of Brinkman and Charikar (J ACM 52(5):766–788, 2005). In fact, the above set C can be taken to be the same set as the one that Brinkman and Charikar considered, viewed as a collection of diagonal matrices in S₁. The challenge is to demonstrate that C cannot be faithfully realized in an arbitrary low-dimensional subspace of S₁, while Brinkman and Charikar obtained such an assertion only for subspaces of S₁ that consist of diagonal operators (i.e., subspaces of ℓ₁). We establish this by proving that the Markov 2-convexity constant of any finite dimensional linear subspace X of S₁ is at most a universal constant multiple of logdim(X).