Impossibility of dimension reduction in the nuclear norm
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Let S_1 (the Schatten--von Neumann trace class) denote the Banach space of all compact linear operators T:ℓ_2→ℓ_2 whose nuclear norm T_S_1=∑_j=1^∞σ_j(T) is finite, where {σ_j(T)}_j=1^∞ are the singular values of T. We prove that for arbitrarily large n∈N there exists a subset C⊆S_1 with |C|=n that cannot be embedded with bi-Lipschitz distortion O(1) into any n^o(1)-dimensional linear subspace of S_1. C is not even a O(1)-Lipschitz quotient of any subset of any n^o(1)-dimensional linear subspace of S_1. Thus, S_1 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 S_1 replaced by the Banach space ℓ_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 S_1. The challenge is to demonstrate that C cannot be faithfully realized in an arbitrary low-dimensional subspace of S_1, while Brinkman and Charikar obtained such an assertion only for subspaces of S_1 that consist of diagonal operators (i.e., subspaces of ℓ_1). We establish this by proving that the Markov 2-convexity constant of any finite dimensional linear subspace X of S_1 is at most a universal constant multiple of √(dim(X)).
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