 However, one should be careful not to conclude that the divergence of the quantum relative entropy formula_13 implies that the states formula_11 and formula_15 are orthogonal or even very different by other measures.Specifically, formula_13 can diverge when formula_11 and formula_15 differ by a "vanishingly small amount" as measured by some norm.For example, let formula_15 have the diagonal representation

formula_20

with formula_21 for formula_22 and formula_23 for formula_24 where formula_25 is an orthonormal set.