## 6.1 Definitions

If you are an impatient mathematically minded person, who feels more comfortable when things are properly defined right from the beginning, here is your definition:^{85}

A **density operator**

It follows that any such ^{86}

An important example of a density operator is a rank one projector.^{87}
Any quantum state that can be described by the state vector **pure state**, can be also described by the density operator **mixed states**, can be always written as the convex sum of pure states:

A self-adjoint matrix

M is said to be**non-negative**, or**positive semi-definite**, if\langle v|M|v\rangle\geqslant 0 for any vector|v\rangle , or if all of its eigenvalues are non-negative, or if there exists a matrixA such thatM=A^\dagger A . (This is called a**Cholesky factorization**.)↩︎A subset of a vector space is said to be

**convex**if, for any two points in the subset, the straight line segment joining them is also entirely contained inside the subset.↩︎The

**rank**of a matrix is the number of its non-zero eigenvalues.↩︎