Information-Measure Domain and Comparison Atlas

No single information measure ranks every QFT state, channel, and protocol. The right quantity is selected by the physical task, available algebra, copy and energy resources, and continuum prescription. This page provides a domain test rather than a universal hierarchy.

Required background. Use Relative Entropy for QFT States and Infinite-Dimensional and Energy-Constrained Channel Distances. Helpful background. Hypothesis Testing and Asymptotic Distinguishability and Mutual Information and Regulator-Independent Correlations supply the main state-comparison and correlation tasks.

Select by operational question

Use algebraic relative entropy for asymmetric comparison to a reference on a fixed local algebra. Use $D_H^\varepsilon$ for a one-shot binary test with type-I tolerance. Use smooth min- or max-relative entropy when the task has a one-shot resource theorem formulated in that smoothing metric. Use fidelity for symmetric closeness, relative-entropy variance for second-order independent-copy asymptotics, conditional mutual information for tripartite Markov structure, and an energy-constrained diamond norm for channel discrimination.

The chapter diagram is a compact selection tree. Its central lesson is that branches share inequalities but answer different questions.

State comparison is not a single numerical ranking. Choose the branch from the decision task: asymmetric reference comparison, one-shot or many-copy discrimination, correlation, recovery, or channel discrimination. Schematic.

Four representative domains

For a vacuum versus coherent excitation on one interval, Araki relative entropy is intrinsic and modular energy often gives a calculable representation. For a detector deciding between two smeared Gaussian signals once, $D_H^\varepsilon$ or bounded-observable bias is direct. For correlations between separated regions, mutual information is finite under matched algebra and regulator choices. For an ideal versus truncated bosonic evolution, the energy-constrained diamond norm is the channel-level quantity.

Fidelity can support any of these analyses as a closeness bound, but does not replace their task definition. A recovery fidelity also depends on the chosen recovery channel. Smooth quantities require a declared purified-distance or trace-distance ball. Relative-entropy variance is useful only when the logarithmic moment exists and the copy model is justified.

Domain matrix

Information measures selected by physical task and resource domain.

Question	Appropriate quantity	Necessary declaration	Typical failure
Compare a local state with a reference	Araki relative entropy	algebra and support	treating it as symmetric
Optimize one binary decision	DHε	effect algebra and ε	hiding detector or smoothing changes
Bound state closeness	fidelity	square convention	comparing conventions
Quantify separated total correlation	mutual information	two algebras and separation	calling it distillable entanglement
Test approximate Markov structure	conditional mutual information and recovery fidelity	ordered tripartition and channel domain	inferring locality or feasibility
Distinguish field channels	‖Φ − Ψ‖⋄, E, H	H, E, ancilla, cutoff	promoting to an unconstrained norm

The failure map supplies a final adversarial check.

Before comparing numbers, verify that their algebras, supports, channel conventions, logarithms, energy sets, and regulators match. A measure evaluated outside its domain is not rescued by a precise numerical value. Schematic.

An adversarial check asks whether a different detector algebra, reference support, energy Hamiltonian, or cutoff would reverse the apparent ranking. If so, report the result as task-specific rather than intrinsic.

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.
• Ogawa, Tomohiro, and Hiroshi Nagaoka. “Strong Converse and Stein’s Lemma in Quantum Hypothesis Testing.” IEEE Transactions on Information Theory 46 (2000): 2428–2433. DOI.
• Shirokov, M. E. “Energy-Constrained Diamond Norms and Their Use in Quantum Information Theory.” Problems of Information Transmission 54 (2018): 20–33. DOI.