In quantum mechanics, and especially quantum information and the study of open quantum systems, the trace distance T is a metric on the space of density matrices and gives a measure of the distinguishability between two states. It is the quantum generalization of the Kolmogorov distance for classical probability distributions.
Contents
Definition
The trace distance is just half of the trace norm of the difference of the matrices:
(The trace norm is the Schatten norm for p=1.) 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
Physical interpretation
It can be shown that the trace distance satisfies the equation
where the maximization can be carried either over all projectors
For example, suppose Alice prepares a system in either the state
of correctly identifying in which state Alice prepared the system.
Properties
The trace distance has the following properties
For qubits, the trace distance is equal to half the Euclidean distance in the Bloch representation.
Fidelity
The fidelity of two quantum states
The upper bound inequality becomes an equality when
Total variation distance
The trace distance is a generalization of the total variation distance, and for two commuting density matrices, has the same value as the total variation distance of the two corresponding probability distributions.