Flow of entropy in the evolution of the B B system: Upper bound on CP violation from unidirectionality

We have previously studied [1,6] the time dependence of a B B mixture in terms of its density matrix ðtÞ 1⁄4 NðtÞð1þ ~ ðtÞ ~ Þ=2. The requirement that ðtÞ, the absolute value of the Stokes vector ~ ðtÞ, should evolve monotonically from its initial value ð0Þ 1⁄4 0 to its final value ð1Þ 1⁄4 1 was shown to lead to an upper bound on the CP violating overlap 1⁄4 hBLjBSi. In the present paper, we consider the entropy variable S 1⁄4 trð~ %log2ð~ %ÞÞ, where ~ % 1⁄4 %=N, as an alternative measure of mixing. We show that exactly the same upper bound emerges from the requirement that the flow of entropy is unidirectional (dS=dt < 0). We compare the entropic current dS=dt with and without CP violation and identify certain physical features that appear when the bound on is violated.