### 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:

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]