Note on sampling without replacing from a finite collection of matrices

This technical note supplies an affirmative answer to a question raised in a recent pre-print in the context of a “matrix recovery” problem. Assume one samples m Hermitian matrices X1, . . . , Xm with replacement from a finite collection. The deviation of the sum X1+ · · ·+Xm from its expected value in terms of the operator norm can be estimated by an “operator Chernoff-bound” due to Ahlswede and Winter. The question arose whether the bounds obtained this way continue to hold if the matrices are sampled without replacement. We remark that a positive answer is implied by a classical argument by Hoeffding. Some consequences for the matrix recovery problem are sketched.