Necessary and sufficient positivity conditions for the Banks–Susskind–Peskin master equation

Determine the necessary conditions on the Hermitian coefficient matrix in the Banks–Susskind–Peskin time-local master equation that ensure preservation of positivity and the associated physical properties of the density operator.

Background

Banks, Susskind, and Peskin considered a general time-local master equation for a finite-dimensional density operator that preserves Hermiticity and trace. Their analysis sought conditions ensuring that the density matrix remains positive under the evolution, but they established only simple sufficient conditions, such as positivity of the coefficient matrix.

The unresolved issue is to characterize necessary conditions for positivity preservation, rather than merely sufficient ones. This problem is distinct from complete positivity and arises in the historical discussion of phenomenological pure-to-mixed dynamics and its compatibility with physical requirements.