The quantum relative entropy of density matrices 
and 
on the same finite-dimensional Hilbert space is
![D(rho‖sigma)=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.
