Let S1β (the Schatten--von Neumann trace class) denote the Banach
space of all compact linear operators T:β2βββ2β whose nuclear norm
β₯Tβ₯S1ββ=βj=1ββΟjβ(T) is finite, where
{Οjβ(T)}j=1ββ are the singular values of T. We prove that
for arbitrarily large nβN there exists a subset
CβS1β with β£Cβ£=n that cannot be
embedded with bi-Lipschitz distortion O(1) into any no(1)-dimensional
linear subspace of S1β. C is not even a O(1)-Lipschitz
quotient of any subset of any no(1)-dimensional linear subspace of
S1β. Thus, S1β 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. Such a statement was previously known with S1β
replaced by the Banach space β1β 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 S1β. The challenge is
to demonstrate that C cannot be faithfully realized in an arbitrary
low-dimensional subspace of S1β, while Brinkman and Charikar
obtained such an assertion only for subspaces of S1β that consist of
diagonal operators (i.e., subspaces of β1β). We establish this by proving
that the Markov 2-convexity constant of any finite dimensional linear subspace
X of S1β is at most a universal constant multiple of logdim(X)β