The trace norm ∥ρ∥1 of a matrix ρ is the sum of the singular values of ρ. The singular values are the roots of the eigenvalues of ρρ † .
$\|\rho \|_1 = \text{Tr} \sqrt{\rho \rho^\dagger} .$
The trace norm is the a special case p=1 of the class of Schatten p-norms.
Special cases
For a Hermitian matrix, like a density matrix, the absolute value of the eigenvalues are exactly the singular values, so the trace norm is the sum of the absolute value of the eigenvalues of the density matrix.
Applications
The trace norm is used e.g. in the definition of the logarithmic negativity, which is a measure of the entanglement possed by a state with density matrix ρ or fidelity between quantum states. Note that in this case one is taking the trace norm of a partially transposed density matrix, which may have negative eigenvalues.