# Mixed states

Mixing quantum states is a basic operation, by which several different preparations are combined by switching between different preparing procedures with a classical random generator. When the outcome *x* of the random generator occurs with probability *p**x*, and if *ρ**x* is the state prepared upon outcome *x*, then the overall state generated in this way is

*ρ* = ∑*x**p**x* *ρ**x* This expression is called a convex combination, or **mixture**, and since the *p**x* have to be nonnegative and add up to one, we can write convex combinations with only two terms as *ρ* = *p* *ρ*1 + (1 − *p*) *ρ*2. That the result of a mixture is again a state is expressed by saying that the state space of a quantum system is a convex set. Its extreme points, i.e., those states that cannot be represented as a mixture of different states with positive weights, are called pure, all other states are called **mixed states**.

The key features of quantum mechanics, as opposed to classical probability, can be discussed in terms of the convex structure of their state spaces, the prototypes being the Bloch sphere in the quantum case and a simplex in the classical case. The latter are characterized by the property that any two convex decompositions of the same state admit a common refinement, so that ultimately, every state has a unique finest convex decomposition into pure states. In contrast, points in the interior of the Bloch sphere have many different decompositions into surface points. Nevertheless there is a special decomposition, characterized by the property that the constituent pure states are mutually orthogonal. This is the spectral decomposition of the density operator. In the Bloch sphere this is the decomposition into antipodes. It is unique, except for the "unpolarized state" at the center of the sphere.

For many applications in quantum information it is useful to compare states with regards to their mixedness. We say that *ρ*2 is **more mixed than** *ρ*1, if we can write *ρ*2 as a mixture of unitary copies *U**x**ρ*1 *U**x* * of *ρ*1. The least mixed states are then the pure ones, and the unique most mixed state has density matrix proportional to the identity operator. States are equivalent with respect to this partial order if and only if they have the same eigenvalues. The comparison of mixedness is completely characterized by the majorization relation between the eigenvalue sequences.

Numerical measures of mixedness are functions on the state space, which are monotone with respect to this ordering. Among these the Renyi entropies, and especially the von Neumann entropy are the most relevant.