Dimensionless entropy, mutual and conditional information

All Hilbert spaces in this item are nonzero and finite-dimensional over $\mathbb{C}$. A density operator $\rho$ on $H$ is a Hermitian positive semidefinite endomorphism with $\mathrm{Tr}\rho =1$. Its dimensionless von Neumann entropy, in nats, is

$S(\rho ):=-\mathrm{Tr}(\rho \log \rho )=-{\sum }_{j:{\lambda }_j>0}{\lambda }_j\log {\lambda }_j,$

where ${\lambda }_j$ are the eigenvalues of $\rho$, the logarithm is natural, and $0\log 0:=0$. For any positive semidefinite endomorphisms $X,Y$ on one finite-dimensional $H$, let $\mathrm{supp}X:=(\ker X{)}^{\perp }$. Define

$D(X\mathrm{\parallel }Y):=\mathrm{Tr}(X\log X)-\mathrm{Tr}(X\log Y)$

when $\mathrm{supp}X\subseteq \mathrm{supp}Y$, taking each logarithm spectrally on its positive support and setting zero-eigenvalue contributions to zero; set $D(X\mathrm{\parallel }Y):=+\mathrm{\infty }$ otherwise. For density operators this is the finite-dimensional quantum relative entropy in nats.

For a density operator ${\rho }_{AB}$ on the ordered product $H_A\otimes H_B$, define ${\rho }_A:={\mathrm{Tr}}_B{\rho }_{AB}$ and ${\rho }_B:={\mathrm{Tr}}_A{\rho }_{AB}$, where in orthonormal product bases $({\mathrm{Tr}}_{B}R{)}_{i{i}^{\mathrm{\prime }}}={\sum }_j{R}_{(i,j),(i^{\mathrm{\prime }},j)}$ and the first factor is retained. Define

$I(A:B{)}_{\rho }:=S({\rho }_A)+S({\rho }_B)-S({\rho }_{AB})\mathrm{.}$

For a density operator ${\rho }_{ABC}$ on $H_A\otimes H_B\otimes H_C$, define

${\rho }_{AB}:={\mathrm{Tr}}_C{\rho }_{ABC},{\rho }_{BC}:={\mathrm{Tr}}_A{\rho }_{ABC},{\rho }_B:={\mathrm{Tr}}_A({\mathrm{Tr}}_C{\rho }_{ABC}),$

where each partial trace contracts the indicated tensor factor and retains the other factors in their displayed order.

$I(A:C\mid B{)}_{\rho }:=S({\rho }_{AB})+S({\rho }_{BC})-S({\rho }_B)-S({\rho }_{ABC}),$

using the partial traces in the displayed factor order. For any fixed positive conversion factor $k_B$ with units J/K, the corresponding physical entropy is $S_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(\rho ):=k_{B}S(\rho )$; no numerical value of $k_B$ is specified here.

Density operator

Finite-dimensional von Neumann entropy

Finite-dimensional quantum relative entropy

Partial trace over a finite tensor factor

Partial trace over the first tensor factor

Partial traces of a finite bipartite state are states