After years of not having an intuitive interpretation of what 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“.
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.… [continue reading]