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
\langle v|\rho|v\rangle\geqslant 0for all |v\rangle
- Trace one:
It follows that any such
An important example of a density operator is a rank-one projector:106
any quantum state that can be described by the state vector
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
Mis said to be non-negative, or positive semi-definite, if \langle v|M|v\rangle\geqslant 0for any vector |v\rangle, or if all of its eigenvalues are non-negative, or if there exists another matrix Asuch that M=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.↩︎