The purity of a normalized quantum state satisfies
is the dimension of the Hilbert space upon which the state is defined.
is a projection, which defines a pure state, then the upper bound is saturated:
The lower bound is obtained by the completely mixed state, represented by the matrix
The purity of a quantum state is conserved under unitary transformations acting on the density matrix in the form
Specifically, it is conserved under the time evolution operator
[1][2] A pure quantum state can be represented as a single vector
are d orthonormal vectors that constitute a basis of the Hilbert space.
[3] On the Bloch sphere, pure states are represented by a point on the surface of the sphere, whereas mixed states are represented by an interior point.
Thus, the purity of a state can be visualized as the degree to which the point is close to the surface of the sphere.
For example, the completely mixed state of a single qubit
A graphical intuition of purity may be gained by looking at the relation between the density matrix and the Bloch sphere,
is the vector representing the quantum state (on or inside the sphere), and
Since Pauli matrices are traceless, it still holds that tr(ρ) = 1.
which agrees with the fact that only states on the surface of the sphere itself are pure (i.e.
Purity is trivially related to the linear entropy
The linear entropy then is obtained by expanding ln ρ = ln (1−(1−ρ)), around a pure state, ρ2 = ρ; that is, expanding in terms of the non-negative matrix 1−ρ in the formal Mercator series for the logarithm,
Both the linear and the von Neumann entropy measure the degree of mixing of a state, although the linear entropy is easier to calculate, as it does not require diagonalization of the density matrix.
Some authors[4] define linear entropy with a different normalization
The degree in which it is entangled is related to the purity of the states of its subsystems,
Using this and Peres–Horodecki criterion (for 2-qubits), a state is entangled if its partial transpose has at least one negative eigenvalue.
In the context of localization, a quantity closely related to the purity, the so-called inverse participation ratio (IPR) turns out to be useful.
, the square of the momentum space wave function
, or some phase space density like the Husimi distribution, respectively.
[6] The smallest value of the IPR corresponds to a fully delocalized state,
In one dimension IPR is directly proportional to the inverse of the localization length, i.e., the size of the region over which a state is localized.
Localized and delocalized (extended) states in the framework of condensed matter physics then correspond to insulating and metallic states, respectively, if one imagines an electron on a lattice not being able to move in the crystal (localized wave function, IPR is close to one) or being able to move (extended state, IPR is close to zero).
The IPR basically takes the full information about a quantum system (the wave function; for a
values, the components of the wave function) and compresses it into one single number that then only contains some information about the localization properties of the state.
Even though these two examples of a perfectly localized and a perfectly delocalized state were only shown for the real space wave function and correspondingly for the real space IPR, one could obviously extend the idea to momentum space and even phase space; the IPR then gives some information about the localization in the space at consideration, e.g. a plane wave would be strongly delocalized in real space, but its Fourier transform then is strongly localized, so here the real space IPR would be close to zero and the momentum space IPR would be close to one.