Monotonicity of quantum relative entropy

Quantum relative entropy is the operator-algebraic analogue of Kullback–Leibler divergence for density operators .

For density operators $\rho,\sigma$ on the same finite-dimensional Hilbert space, define

Statement (data processing inequality)

For every completely positive trace-preserving (CPTP) map (quantum channel) $\Phi$,

Key hypotheses

• $\rho,\sigma$ are density operators; if $\mathrm{supp}(\rho)\nsubseteq \mathrm{supp}(\sigma)$ then $D(\rho\|\sigma)=+\infty$ by convention.
• $\Phi$ is linear, completely positive, and trace-preserving (CPTP).

Key conclusions

• Information cannot increase under coarse-graining: processing, measuring, adding noise, or discarding subsystems cannot increase $D$.

• Partial trace case: for bipartite states $\rho_{AB},\sigma_{AB}$,

  $$D(\rho_{AB}\|\sigma_{AB}) \ge D(\rho_A\|\sigma_A),$$

  where $\rho_A=\mathrm{Tr}_B\rho_{AB}$ and similarly for $\sigma_A$.

• Nonnegativity: choosing $\Phi$ to be a constant channel gives $0 = D(\Phi(\rho)\|\Phi(\sigma)) \le D(\rho\|\sigma)$, hence $D(\rho\|\sigma)\ge 0$.

• The inequality implies standard entropy consequences such as concavity of the von Neumann entropy .

Proof idea / significance

A common proof strategy is:

1. Stinespring dilation: represent $\Phi(\cdot)=\mathrm{Tr}_E\!\big(U(\cdot\otimes\tau_E)U^\dagger\big)$ for an ancilla state $\tau_E$ and a unitary $U$.
2. Use unitary invariance of $D$ and monotonicity under partial trace to reduce to the subsystem-discarding case.

In statistical mechanics and thermodynamics of open quantum systems, data processing formalizes irreversibility: when one loses access to degrees of freedom, the relative entropy to a reference state (e.g. equilibrium) cannot increase, yielding many “second-law type” bounds.