Finite-dimensional quantum relative entropy

Let $H$ be a finite-dimensional complex inner-product space and let $A,B:H\to H$ be non-negative operators. For such an operator $A$, set $\mathrm{supp}(A):=(\ker A{)}^{\perp }$, where $\ker A$ is the kernel of a linear map (Kernel and image of a linear map); this extends the density-operator support convention in Density operator. If $\mathrm{supp}(A)\subseteq \mathrm{supp}(B)$, define

$D(A\mathrm{\parallel }B):=\mathrm{tr}(A\ln A)-\mathrm{tr}(A\ln B)\mathrm{.}$

Here $\ln A$ is defined spectrally on the positive eigenspaces of $A$ with $0\ln 0:=0$, and $\ln B$ in the cross term is restricted to $\mathrm{supp}(B)$; the support condition ensures that $A$ vanishes on $\ker B$, so the omitted kernel contributes zero. If $\mathrm{supp}(A)⊈\mathrm{supp}(B)$, set $D(A\mathrm{\parallel }B):=+\mathrm{\infty }$. In particular, $D(0\mathrm{\parallel }B)=0$. For density operators this is the finite-dimensional quantum relative entropy. The logarithms are natural, so the value is in nats; the base-two value is $D_2(A\mathrm{\parallel }B):=D(A\mathrm{\parallel }B)\mathrm{/}\ln 2$.