Koenraad M.R. Audenaert

The quantum relative entropy between two quantum states rho and sigma, S(rho||sigma) = Tr rho(log rho - log sigma), is a non-commutative generalisation of the Kullback-Leibler distance between probability distributions. Because of its strong mathematical connections with von Neumann entropy, and its interpretation as an optimal asymptotic error rate in quantum hypothesis testing (in the context of Stein's lemma) relative entropy is widely used as a (non-symmetric) distance measure between states. One of its drawbacks, however, is that for non-faithful (rank-deficient) states the relative entropy can be infinite. We define the telescopic relative entropy (TRE) and prove sharp upper bounds on it in terms of the trace norm distance.