The Golden-Thompson trace inequality is complemented

AbstractWe prove a class of trace inequalities which complements the Golden-Thompson inequality. For example, Tr(epA#epB)2/p⩽ Tr eA+B holds for all p > 0 when A and B are Hermitian matrices and # denotes the geometric mean. We also prove related trace inequalities involving the logarithmic function; namely p−1Tr X log Yp/2XpYp/2⩽ Tr X(log X+log Y) ⩽ p−1Tr X log Xp/2YpXp/2 for all p > 0 when X and Y are nonnegative matrices. These inequalities supply lower and upper bounds on the relative entropy