Trace distance

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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:

T(\rho,\sigma) := \frac{1}{2}||\rho - \sigma||_{1} = \frac{1}{2} \mathrm{Tr} \left[ \sqrt{(\rho-\sigma)^\dagger (\rho-\sigma)} \right] .

(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,

T(\rho,\sigma) = \frac{1}{2} \mathrm{Tr} \left[ \sqrt{(\rho-\sigma)^2} \right] = \frac{1}{2} \sum_i | \lambda_i | ,

where the \lambda_i are eigenvalues of the Hermitian, but not necessarily positive, matrix (\rho-\sigma).

Relationship to the fidelity[edit]

The fidelity of two quantum states F(\rho,\sigma) is related to the trace distance T(\rho,\sigma) by

1-F(\rho,\sigma) \le T(\rho,\sigma) \le\sqrt{1-F(\rho,\sigma)^2} \, .

Relationship to the total variation distance[edit]

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.