The quantum relative entropy of density matrices 
and 
on the same finite-dimensional Hilbert space is
![D(rhoOsigma)=Tr[rho(lnrho-lnsigma)],](/images/equations/QuantumRelativeEntropy/NumberedEquation1.svg) |
provided the column space of 
is contained in the column space
of 
.
Otherwise its value is infinite. The matrix trace
is interpreted on the column space of 
, with 
. This convention handles zero eigenvalues
without applying a logarithm to a singular matrix.
Quantum relative entropy is nonnegative and vanishes iff 
. If the two density
matrices commute, matrix diagonalization
in a common orthonormal basis reduces it to
classical relative entropy with natural
logarithms. It is neither symmetric nor a metric (Watrous
2018).
For a joint density matrix 
, its quantum mutual
information is 
, where 
and 
are the reduced density
matrices. This quantity is the correction for final correlations in Landauer's
principle.
