After years of not having an intuitive interpretation of the unital condition on CP maps, I recently learned a beautiful one: unitality means the dynamics never decreases the state’s mixedness, in the sense of the majorization partial order.

Consider the Lindblad dynamics generated by a set of Lindblad operators , corresponding to the Lindbladian

(1)

and the resulting quantum dynamical semigroup . Let

(2)

be the Renyi entropies, with the von Neumann entropy. Finally, let denote the majorization partial order on density matrices: exactly when exactly when for all , where and are the respective eigenvalues in *decreasing* order. (In words: means is more mixed than .) Then the following conditions are equivalent:^{a }

- : “ is a unital map (for all )”
- for all , , and : “All Renyi entropies are non-decreasing”
- for all : “ is mixedness non-decreasing”
- for all and some unitaries and probabilities .

The non-trivial equivalences above are proved in Sec. 8.3 of Wolf, “Quantum Channels and Operations Guided Tour“.^{b }

Note that having all Hermitian Lindblad operators () implies, but is not implied by, the above conditions. Indeed, the condition of Lindblad operator Hermiticity (or, more generally, normality) is not preserved under the unitary gauge freedom (which leaves the Lindbladian invariant for unitary .) I was curious whether unital dynamics can always be expressed in terms of Hermitian operators, but based on some quick numerics it looks like this is not the case. I’d be interested to find a minimal and interpretable counterexample.

### Footnotes

(↵ returns to text)

- None of this depends on the dynamics being Lindbladian. If you drop the first condition and drop the “” subscript, so that is just some arbitrary (potentially non-divisible) CP map, the remaining conditions are all equivalent.↵
- See also “On the universal constraints for relaxation rates for quantum dynamical semigroup” by Chruscinski et al [2011.10159] for further interesting discussion.↵

Cool! I did not know this condition either. This seems to answer an old loophole in a famous paper on BH information loss by Banks, Peskin, and Susskind: https://inspirehep.net/literature/194105. After Eq. 9 they say (wrt a general Lindblad equation) “One might also insist that the entropy defined by \rho not decrease with time. We do not know what conditions are necessary to insure these properties, but we can state some simple sufficient conditions. “