Bounds on Dimension Reduction in the Nuclear Norm

For all n >= 1, we give an explicit construction of m x m matrices A(1),...,A(n) with m = 2([n/2]) such that for any d and d x d matrices A(1)',..., A(n)' that satisfy parallel to A(i)' - A(j)'parallel to(S1) <= parallel to A(i) - A(j) parallel to(S1) <= (1 + delta) parallel to A(i)' - A(j)'parallel to S-1 for all i, j is an element of {1,..., n} and small enough delta = O(n(-c)), where c > 0 is a universal constant, it must be the case that d >= 2([n/2]-1). This stands in contrast to the metric theory of commutative l(p) spaces, as it is known that for any p >= 1, any n points in l(p) embed exactly in l(p)(d) for d = n(n - 1)/2. Our proof is based on matrices derived from a representation of the Clifford algebra generated by n anti-commuting Hermitian matrices that square to identity, and borrows ideas from the analysis of nonlocal games in quantum information theory.<br />