Relative Entropy, Distinguishability, and Recovery

Relative entropy is the continuum-safe center of state comparison in QFT. Around it sit operational tasks—binary testing, overlap, correlation, channel discrimination, and recovery—whose meanings depend on the observable algebra, reference support, copy model, energy bound, and regulator. This chapter develops those branches without importing finite-dimensional conclusions beyond their domains.

Helpful background. Review restricted local states, regulated subregion entropy, operator algebras and positive functionals, Markov semigroups, and trace ideals as needed.

From algebraic comparison to information tasks

For normal states $\omega$ and $\varphi$ on a local von Neumann algebra $\mathfrak M$, Araki relative entropy $S(\omega\Vert\varphi)$ needs no trace or tensor factor. It is positive, monotone under restriction, and compatible with type-I density-matrix formulas when a regulator or split realization supplies them. These features make it the right reference point for continuum distinguishability.

Operational interpretations require more. A Stein exponent assumes independent copies and a test algebra. Fidelity requires a square convention. Mutual information requires two algebras and a product reference. Conditional mutual information requires an ordered tripartition. Recovery requires a channel domain and error metric. Infinite-dimensional channel discrimination requires an input Hamiltonian and energy set.

The diagram below is the chapter dictionary. Follow a branch only after its label has become part of the problem statement.

Relative entropy is the common algebraic comparison. Copy resources select hypothesis testing; support and convention select overlap measures; chosen subalgebras select correlation measures; and channel, recovery, and energy data select operational reconstruction. Schematic.

What the chapter establishes

The opening sequence defines Relative Entropy for QFT States, explains Araki Relative Entropy and Regulated Limits, and proves the meaning of Positivity, Monotonicity, and Data Processing. The next three pages separate Hypothesis Testing and Asymptotic Distinguishability, Fidelity, Chernoff Bounds, and State Overlap, and Operational Distinguishability and Continuity Bounds.

Correlation structure begins with Mutual Information and Regulator-Independent Correlations and Strong Subadditivity and Entropic Inequalities, then sharpens through Conditional Mutual Information and Quantum Markov Structure. The recovery sequence treats Recovery Maps and Approximate Markovianity and Petz, Rotated, and Universal Recovery Maps. The final pair makes resources explicit through Infinite-Dimensional and Energy-Constrained Channel Distances and the Information-Measure Domain and Comparison Atlas.

Complete operator-algebraic proofs belong to the theorem-first mathematical treatment; here the focus is QFT meaning, regulated comparison, examples, and operational consequences. The chapter excludes unconstrained infinite-dimensional continuity claims, hidden postselection, and abstract Shannon theory detached from field algebras and physical resources.

Information tasks, measures, and domain checks

Physical question	Primary quantity	Data that must be fixed	What does not follow
Compare a local state with a reference	Araki relative entropy	von Neumann algebra, normal states, support	A density matrix or symmetric distance
Optimize a binary decision	DHε or bounded-observable bias	test algebra, error tolerance, copies, energy	An unlimited-copy or detector-independent rate
Quantify separated correlations	mutual or conditional mutual information	region algebras, ordering, common regulator	Distillable entanglement or causal recovery
Reconstruct lost information	recovery fidelity and Petz-type map	reference, channel, adjoint, support, domain	Uniqueness, locality, or finite energy
Compare field channels	energy-constrained diamond norm	Hamiltonian, energy budget, ancilla, cutoff	An unconstrained norm or capacity statement

A reliable route

Readers new to type-III algebras should take the pages in order through data processing. For an operational route, continue through hypothesis testing, fidelity, and channel bounds. For a correlation-and-recovery route, move from mutual information through strong subadditivity, conditional mutual information, and Petz recovery. In either route, finish with the domain atlas before comparing numerical values from different measures.

A recurring example is a vacuum or Gaussian field state restricted to intervals, with lattice or mode regulators used only as controlled approximants. Every computation declares which combination is expected to remain finite, how supports are handled, and which ultraviolet or high-energy tail is bounded.

Validity before inference

The second figure summarizes the domain checks. Its four upper boxes are theorem inputs; each dashed lower box names a way to leave the theorem’s domain.

State comparison is valid only on a fixed algebra and support. Data processing additionally needs a physical positive map, while testing and channel claims need fixed errors, copies, Hamiltonians, energy budgets, ancillas, and regulators. Schematic.

For example, small conditional mutual information guarantees an abstract recovery map in a stated fidelity metric. It does not imply that the map is generated within a causal diamond or has bounded energy cost. Likewise, a finite regulated relative entropy approaches the continuum quantity only under compatible algebra embeddings and normal state convergence. These are not technical footnotes: they decide which conclusion is true.

Check your preparation

You are ready to use the chapter if you can:

1. Explain why Araki relative entropy survives for a type-III local algebra while local von Neumann entropy need not.
2. State the support and channel hypotheses behind data processing.
3. Distinguish a one-shot hypothesis test from the independent-copy Stein limit.
4. Name the ordered algebras and recovery metric in an approximate Markov claim.
5. Define the Hamiltonian and ancillary resources in an energy-constrained channel distance.

Further reading

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11 (1976): 809–833. DOI.
• Fawzi, Omar, and Renato Renner. “Quantum Conditional Mutual Information and Approximate Markov Chains.” Communications in Mathematical Physics 340 (2015): 575–611. DOI.
• Petz, Dénes. “Sufficient Subalgebras and the Relative Entropy of States of a von Neumann Algebra.” Communications in Mathematical Physics 105 (1986): 123–131. DOI.