## 8.1 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.
Recall that a Hermitian matrix **non-negative**, or **positive semi-definite**, if ^{154} there exists another matrix

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 density operator ^{155} are all real, non-negative, and sum to **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:^{156}
any quantum state that can be described by the state vector **pure states**.
Pure states are the extremal points in the convex set of density operators: they cannot be expressed as a non-trivial convex sum of other elements in the set.
In contrast, all other states, called **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. Baez, Fritz, and Leinster’s “A Characterization of Entropy in Terms of Information Loss”, arXiv:1106.1791.

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

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