## 6.1 Definitions

If you are an impatient, more mathematically minded person, who feels most comfortable when things are properly defined right from the beginning, here is your definition:^{105}

A **density operator**

**Hermitian:**\rho^\dagger=\rho **Non-negative:**\langle v|\rho|v\rangle\geqslant 0 for all|v\rangle **Trace one:**\operatorname{tr}\rho=1 .

It follows that any such **convex sum** of the two:
**convex set**: 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.

An important example of a density operator is a rank-one projector:^{106}
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:

Convex spaces show up in many areas of mathematics: combinatorists and discrete geometers are often interested in **convex polytopes**, and the special case of **simplices** is even more fundamental, turning in up in algebraic topology, higher algebraic geometry, and, more generally, higher category theory.
Closer to what we are studying, the notion of **entropy** in (classical) information theory is somehow inherently convex (see e.g. arXiv:1106.1791).

The specific type of convex polytope that we are interested in turns out to be a **convex hull**, which are also found all throughout mathematics.

Now that we have settled the mathematical essentials, we will turn to physical applications.

Recall that a Hermitian 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 another matrixA such thatM=A^\dagger A . (This is called a**Cholesky factorization**.)↩︎Recall that the rank of a matrix is equal to the number of its non-zero eigenvalues, or (equivalently) the dimension of its image.↩︎