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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. It 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 are eigenvalues of the Hermitian, but not necessarily positive, matrix .
Relationship to the fidelity
The fidelity of two quantum states is related to the trace distance by
Relationship to the 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.
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