Entropy Identities

Formula

S(\rho)=-\operatorname{Tr}(\rho\log\rho) =-\sum_i\lambda_i\log\lambda_i,

S(A|B)=S(AB)-S(B),

I(A:B)=S(A)+S(B)-S(AB) =D(\rho_{AB}\|\rho_A\otimes\rho_B),

D(\rho\|\sigma) =\operatorname{Tr}\rho(\log\rho-\log\sigma).

Principal inequalities are

|S(A)-S(B)|\le S(AB)\le S(A)+S(B),

S(ABC)+S(B)\le S(AB)+S(BC),

and, for a channel $\mathcal N$ ,

D(\mathcal N(\rho)\|\mathcal N(\sigma)) \le D(\rho\|\sigma).

For a pure bipartite state,

S(AB)=0, \qquad S(A)=S(B), \qquad I(A:B)=2S(A).

Assumptions and Conventions

State whether logarithms use base $2$ (bits) or base $e$ (nats).
Set $0\log0=0$ by continuity.
$D(\rho\|\sigma)$ is finite only when $\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma$ .
Subsystem states are obtained from the same joint state by partial trace.

Symbols

Symbol	Meaning
$S(A)$	entropy of reduced state $\rho_A$
$S(A	B)$
$I(A:B)$	quantum mutual information
$D(\rho\\|\sigma)$	quantum relative entropy
$\lambda_i$	eigenvalues of $\rho$

Validity and Warnings

Quantum conditional entropy can be negative.
Mutual information measures total correlation, not entanglement alone.
Relative entropy is neither symmetric nor a metric.
A channel need not increase subsystem entropy; reset and cooling are counterexamples. Data processing concerns distinguishability.
Finite-dimensional continuity bounds require a dimension. Infinite- dimensional entropy may be infinite without energy constraints.
Fine-grained entropy is invariant under closed unitary evolution, while subsystem or coarse-grained entropy can change.

Canonical Treatment

Definitions, proofs and interpretations, Rényi entropies, Holevo information, continuity and thermodynamic cautions, worked examples, exercises, and references are at Quantum Entropy.