Entropy subadditivity

In plain terms

Entropy measures missing information, meaning how much you don't know about the state of a system. For quantum systems the relevant quantity is the von Neumann entropy, and the basic sanity check on it is subadditivity, which says the entropy of a joint system is at most the sum of the entropies of its two parts, so correlations between the parts can only make the whole more predictable than its parts suggest, never less.

Every textbook states the inequality, and the proof leans on two steps that are easy to say and laborious to formalise. The first is Klein's inequality, which says relative entropy, a measure of how distinguishable one state is from another, is never negative, and it needs a careful argument in terms of the eigenvalues of the two states and a matrix of overlaps between their eigenvectors, recording how much each eigenvector of one state has in common with each eigenvector of the other, whose rows and columns each add up to one. The second is the splitting of the logarithm of a tensor product, the joint operator formed from independent operators on the two parts, into a sum of the logarithms of its factors, which requires the way functions are applied to operators to respect the product structure, and this is exactly the step informal proofs wave through.

Both steps are done, so the inequality stands on checked ground, including for pure entangled states, the case in which the whole has zero entropy while its parts don't, which is the quantum signature classical entropy forbids.

In CSD

The entropy tier serves the programme's thermodynamics track and its accounting of decoherence. Entropy production in measurement, the entropy ledger of the chaos diagnostics, and the touchpoint with Landauer's principle all cite the von Neumann entropy machinery this inequality anchors.

Deferred pieces are named rather than hidden. The strong version, subadditivity with three parties, is proved conditionally on the data processing inequality, which is stated as an explicit hypothesis, and the purification construction needed for the triangle inequality is recorded as the missing step.

Mathematically

The theorem vonNeumannEntropy_subadditive states that S(rho_AB) <= S(rho_A) + S(rho_B) for a bipartite density with positive-definite marginals, the joint state merely PSD.

The route runs through Klein's inequality via the doubly-stochastic overlap expansion, the Kronecker-log split log(A (x) B) = log A (x) I + I (x) log B via a diagonalisation-respecting functional calculus, and the reduced-trace identities collapsing the cross terms. Araki-Lieb's one side is proved under its stated hypotheses.

Related entries

• von Neumann entropyA quantum whole can be perfectly known while its parts are as uncertain as they could possibly be, and entropy is where that stops being a metaphor and becomes arithmetic.
• Klein's inequalityOne quantum state is never negatively distinguishable from another, as the relative entropy of a state with respect to a second is at least zero, and that single inequality drives the second law, subadditivity, concavity and the Holevo bound in the corpus.
• Landauer's principleForgetting costs heat at a rate set by temperature alone, and that's the point at which information stops being an abstraction and acquires a physical price.
• Partial traceMathlib has no partial trace for matrices, so the corpus built one, the staple operation of quantum information, with its working properties proved and staged for contribution upstream.
• Holevo boundA quantum system can be prepared in infinitely many distinct ways, yet no reader can extract more classical information from it than from a classical system of the same size, and the measurement channel of CSD is proved to pass exactly one bit per qubit.