### Summary

Physicists often define a Lindbladian superoperator as one whose action on an operator can be written as

(1)

for some operator with positive anti-Hermitian part, , and some set of operators . But how does one efficiently *check* if a given superoperator is Lindbladian? In this post I give an “elementary” proof of a less well-known characterization of Lindbladians:

*A superoperator generates completely positive dynamics , and hence is Lindbladian, if and only if , i.e.,*

for all . Here “” denotes a partial transpose, is the “superprojector” that removes an operator’s trace, is the identity superoperator, and is the dimension of the space upon which the operators act.

Thus, we can efficiently check if an arbitrary superoperator is Lindbladian by diagonalizing and seeing if all the eigenvalues are positive.

### A quick note on terminology

The terms **superoperator**, **completely positive** (CP), **trace preserving** (TP), and **Lindbladian** are defined below in Appendix A in case you aren’t already familiar with them.

Confusingly, the standard practice is to say a superoperator is “positive” when it is **positivity preserving**: . This condition is logically independent from the property of a superoperator being “positive” in the traditional sense of being a positive operator, i.e., for all operators (matrices) , where

is the Hilbert-Schmidt inner product on the space of matrices. We will refer frequently to this latter condition, so for clarity we call it **op-positivity**, and denote it with the traditional notation .

### Intro

It is reasonably well known by physicists that Lindbladian superoperators, Eq. (1), generate CP time evolution of density matrices, i.e., is completely positive when and satisfies Eq.… [continue reading]