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 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 if154 there exists another matrix A such that M=A^\dagger A.

A density operator \rho on a Hilbert space \mathcal{H} is a non-negative Hermitian operator with trace equal to one:

  • 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 \rho can always be diagonalised, and that the eigenvalues155 are all real, non-negative, and sum to 1. Moreover, given two density operators \rho_1 and \rho_2, we can always construct another density operator as a convex sum of the two: \rho = p_1\rho_1 + p_2\rho_2 where p_1,p_2\geqslant 0 are such that p_1+p_2=1. You should check that the resulting \rho has all the defining properties of a density matrix, i.e. that it is Hermitian, non-negative, and that its trace is 1. This means that density operators form a 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 |\psi\rangle can be also described by the density operator \rho=|\psi\rangle\langle\psi|; such states are called 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: \sum_i p_i |\psi_i\rangle\langle\psi_i| for some p_i\geqslant 0 with \sum_i p_i=1.

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.

  1. (This is called a Cholesky factorization.)↩︎

  2. Note that these properties are exactly saying that we can interpret the eigenvalues as probabilities.↩︎

  3. Recall that the rank of a matrix is equal to the number of its non-zero eigenvalues, or (equivalently) the dimension of its image.↩︎