Trace norm

The trace norm $\backslash |\backslash rho\; \backslash |\_1$ of a matrix $\backslash rho$ is the sum of the singular values of $\backslash rho$. The singular values are the roots of the eigenvalues of $\backslash rho\; \backslash rho^\backslash dagger$. $\backslash |\backslash rho\; \backslash |\_1\; =\; \backslash text\{Tr\}\; \backslash sqrt\{\backslash rho\; \backslash rho^\backslash dagger\}\; .$ The trace norm is the a special case p=1 of the class of [[Schatten norm|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 $\backslash rho$ 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. [[Category:Mathematical Structure]] [[Category:Linear Algebra]]