A concept related to the entropy is studied. Let A and B be two density matrices, with eigenvalues a1, a2,… and b1, b2,…, arranged in decreasing order and repeated according to multiplicity. Then A is said to be “more mixed”, or “more chaotic”, than B, if a1⩽b1, a1+a2⩽b1+b2,…,a1+…+am⩽b1+…+bm,…; It turns out that if A is more mixed than B, then the entropy of A is larger than the entropy of B. However, more generally, let v be an arbitrary concave function, ⩾0, and vanishing at 0. Then, if A is more mixed than B, tr v(A)⩾tr v(B). It is shown that also the converse is true. Furthermore, a variety of other characterizations of the relation “A is more mixed than B” is obtained, and several applications to quantum statistical mechanics are given.
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