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 : `2 → `2 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-dimensional linear subspace of S1. C is not even a O(1)-Lipschitz quotient of any subset of any n-dimensional linear subspace of S1. Thus, S1 does not admit a dimension reduction result á 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. Such a statement was previously known with S1 replaced by the Banach space `1 of absolutely summable sequences via the work of Brinkman and Charikar (2003). 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 `1). 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 √ log dim(X). E-mail addresses: [email protected], [email protected], [email protected]. Date: October 25, 2017. 2010 Mathematics Subject Classification. 30L05, 46B85, 46B20, 46B80.