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

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...

 and the study of open quantum system

Open quantum system

In physics, an open quantum system is a quantum system which is found to be in interaction with an external quantum system, the environment...

s, the trace distance T is a metric

Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

 on the space of density matrices

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...

. It is just half of the trace norm of the difference of the matrices:


The purpose of the factor of two is to restrict the trace distance between two normalized density matrices to the range [0, 1] and to simplify formulas in which the trace distance appears.

Since density matrices are Hermitian,


where the are eigenvalues of the Hermitian, but not necessarily positive, matrix .