Purity of a quantum state

In quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, and especially quantum information theory

Quantum information

In quantum mechanics, quantum information is physical information that is held in the "state" of a quantum system. The most popular unit of quantum information is the qubit, a two-level quantum system...

, the purity of a state is a scalar defined as


where is the density matrix

Density matrix

In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...

 of the state. The purity can range between unity, corresponding to a completely pure state, and , corresponding to a completely mixed state. (Here, is the dimension

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

 of the density matrix.)

Purity is trivially related to the Linear entropy  of a state by

Projectivity of a measurement

For a quantum measurement, the projectivity is the purity of its pre-measurement state

Quantum tomography

Quantum tomography or quantum state tomography is the process of reconstructing the quantum state for a source of quantum systems by measurements on the systems coming from the source. The source may be any device or system which prepares quantum states either consistently into quantum pure states...

.
This pre-measurement state

Quantum tomography

Quantum tomography or quantum state tomography is the process of reconstructing the quantum state for a source of quantum systems by measurements on the systems coming from the source. The source may be any device or system which prepares quantum states either consistently into quantum pure states...

 is the main tool of the retrodictive approach of quantum physics, in which we make predictions about state preparations leading to a given measurement result. It allows us to determine in which kind of states the measured system was prepared for leading to such a result.