Energy–Information Bounds: Assumptions and Status

Energy–information claims can be compared only after their mathematical inputs and evidence classes are separated. The matrix below records theorem domain, saturation mechanism, model evidence, and characteristic counterexample. It reflects literature available through 10 August 2026 and should be read together with the detailed pages.

Required background. Review passivity, QEIs, ANEC, relative-entropy bounds, QNEC, quantum interest, species and regulators, Landauer erasure, and QET before using this comparison.

Status matrix

Statement	Mathematical status in stated domain	Main hypotheses	Equality or saturation	Stability question	Known route to failure outside domain
Passivity	Structural theorem for declared cyclic operations	Dynamics, energy domain, passive state	Ground/KMS examples; operation-dependent equality	Restrict or enlarge local operation algebra	Omit controller work or allow a forbidden collective operation
Complete passivity $\Leftrightarrow$ KMS/ground structure	Structural theorem under $C^*$-dynamical assumptions	Every finite tensor power, common dynamics	Equilibrium state on every copy number	Catalysts, charges, infinite-energy limits	Test only product unitaries or change composition rules
Free-field timelike QEI	Rigorous field- and sampler-specific lower bounds	Renormalized stress tensor, smooth sampler, state class	Optimized states or limiting sequences in special cases	Mass, curvature, boundary, interaction	Take a pointwise limit without scaling the bound
ANEC	Proven in broad but hypothesis-dependent relativistic QFT settings	Complete affine null line, convergence, unitarity/causality or modular assumptions	State- and theory-dependent	Defects, boundaries, noncomplete generators	Use a finite null segment
Regional relative-entropy bound	Exact positivity identity	Same region, state pair, known modular Hamiltonian	First-order perturbations saturate the first law	Reference, region, species, algebra	Replace weighted modular energy by unweighted total energy
QNEC	General continuum proofs under stated smoothness and renormalization assumptions	Null shape variation, renormalized $T_{kk}$, compatible normalization	Vacuum and special states can saturate	Defects, nonsmooth cuts, gauge-algebra choices	Retain regulator motion in $S''$
Quantum interest	Derived in specified models from QEIs	Smooth pulse profiles and full energy history	Optimized profiles or limiting configurations	Interaction, boundary, sampling class	Use delta pulses or omit late positive tails
Absolute entropy bound at fixed cutoff	Regulator-dependent statement, not universal QFT theorem	Species, cutoff, factorization convention	Scheme dependent	Large species number and limit order	Change $N$ or $\epsilon$ at fixed long-distance state
Landauer cost	Exact relative-entropy balance with ideal and finite-reservoir refinements	Memory, bath, reset error, full cycle	Quasistatic infinite-bath limit	Finite time, reservoir depletion, charges	Consume bath athermality or omit record reset
Quantum energy teleportation	Valid finite-model protocol; continuum and experimental scope model specific	Correlated state, local instrument, causal message, conditional operation	Model-dependent optimization	Noise, separation, local passivity, continuum limit	Remove message/correlation or omit injected energy

No row licenses a claim outside its hypothesis column. A counterexample to a heuristic extension does not refute the restricted theorem.

How the statements connect

The map groups results by principal input. Connections between branches require additional arguments; no vertical or horizontal position represents theorem strength. The diagram is schematic.

Passivity and Landauer reasoning use equilibrium structure and operation classes; the exact complete-passivity characterization is Pusz and Woronowicz 1978, Theorems 1.1 and 1.4, pp. 275–281. QEIs and ANEC use averaged stress energy, with the sampler and state domains reviewed in Fewster 2012, §§ 2–4, pp. 6–20. Regional bounds and QNEC use modular or entropic data; a general QNEC proof and its assumptions are given by Balakrishnan et al. 2019, §§ 2–5. QET is an operational protocol constrained by, but not equivalent to, any one of these inequalities.

Falsification sequence

For any proposed new bound:

1. state the spacetime, field, state and operator domain;
2. write the precise sampling, region, or shape derivative;
3. freeze the regulator, species and charge sector;
4. declare the allowed instrument and causal feed-forward;
5. account separately for injected, extracted, switching and reset energy;
6. reproduce a saturating case and a known outside-domain counterexample;
7. perturb the saturation example and test numerical stability.

Domain, averaging, renormalization, and operational accounting are independent gates. A claim that survives only because one gate was not varied is not stable evidence for a new bound. The map is schematic.

Interpretation of evidence

“Proven” means a theorem under stated hypotheses, not experimentally realized. “Model-checked” means evaluated in a controlled Hamiltonian or QFT example. “Hardware executed” means a finite encoded protocol ran on a device; it does not by itself validate a continuum limit. “Conjectural” should identify both the unproved statement and known restricted results. A result can be rigorous, saturable, and still too narrow for a proposed application.

References

• Balakrishnan, Souvik, Thomas Faulkner, Zuhair U. Khandker, and Huajia Wang. “A General Proof of the Quantum Null Energy Condition.” Journal of High Energy Physics 2019, no. 9 (2019): 020. DOI.
• Fewster, Christopher J. “Lectures on Quantum Energy Inequalities.” In Quantum Field Theory and Gravity, edited by Felix Finster et al., 2012. arXiv.
• Pusz, Władysław, and Stanisław L. Woronowicz. “Passive States and KMS States for General Quantum Systems.” Communications in Mathematical Physics 58 (1978): 273–290. DOI.