When is the variance of one observable less than or equal to that of another with respect to all quantum states?

In quantum mechanics, the well-known Loewner order expresses that one observable’s expectation value is less than or equal than that of another with respect to all quantum states. In this paper we propose and study a similar order relation in terms of the variance, and we prove two theorems. Our first result states that one observable’s variance is less than or equal than that of another with respect to all quantum states if and only if the former is a $1$-Lipschitz function of the latter. The other main result we prove characterises the order automorphisms with respect to this proposed order relation. It turns out that in some sense these automorphisms have a more rigid form than in the case of the Loewner order.