Energy, Entropy, and Localization

Energy bounds in QFT constrain different quantities: extractable work, sampled stress, modular energy, or the cost of a specified protocol. Choosing the energy functional and operation is essential before applying an information inequality.

Enter this chapter

Begin with Passivity, Work, and Information in QFT to define cyclic work extraction and passivity for a state.

Helpful background. Relative entropy in QFT supplies the algebraic comparison of states. Modular Hamiltonians and domains and the first law and quadratic response distinguish exact first-order identities from finite inequalities. The stress tensor and charge algebra, thermal KMS condition, and CFT stress tensor provide the operator input used below.

Topics

1. Passivity, Work, and Information in QFT — Determines whether a cyclic drive can extract work.
2. Complete Passivity, KMS Structure, and Resource Conversion — Tests passivity under multiple copies and relates it to equilibrium.
3. Quantum Energy Inequalities — Bounds suitably sampled negative energy on a worldline.
4. Averaged Null Energy Condition — Studies complete null averages with their geometric assumptions.
5. Relative Entropy and Bekenstein-Type Bounds — Derives a regional entropy bound from relative entropy.
6. Quantum Null Energy Condition — Relates null stress to a specified entropy shape derivative.
7. Quantum Interest and Negative-Energy Compensation — Constrains negative pulses through sampling inequalities.
8. Entropy Bounds, Species, and Regulator Dependence — Tracks species and cutoff dependence in entropy bounds.
9. Energy Cost of Localization and Measurement — Defines the resources of localized preparation and measurement.
10. Landauer Cost, Information Engines, and Field Reservoirs — Accounts for erasure using a physical reservoir and process.
11. Quantum Energy Teleportation — Tracks local measurement, communication, and feedback in an energy protocol.
12. Energy–Information Bounds: Assumptions and Status — Compares the domains of the different energy–information bounds.

How the ideas fit together

Passivity constrains work extracted by a cyclic operation; complete passivity asks that the constraint survive arbitrary tensor powers and is closely tied to thermal equilibrium Pusz and Woronowicz 1978, Theorems 1.1 and 1.4, pp. 275–281. A worldline energy inequality instead constrains a smeared stress expectation.

Relative-entropy bounds involve a reference modular Hamiltonian, which need not equal the physical energy of an apparatus. These distinctions matter in measurement and feedback protocols: negative local energy or a modular inequality cannot replace accounting for preparation work, communication, and reservoir changes.

The comparison below separates the assumptions and tests needed for these different conclusions.

Claim–validity comparison

Energy–information claims and the hypotheses that control their validity

Claim or protocol	Spacetime and state domain	Smearing or renormalization	Instrument, message, and feed-forward	Charge or sector	Energy quantity and account	Causal or geometric support	Governing condition	Evidence status	Decisive countercheck
Passivity	Stationary dynamics; state and cyclic unitaries in the energy domain	Same endpoint Hamiltonian and a consistent energy reference	No measurement required; operation class declared	Fixed representation or stated sector	Initial minus final system energy	Global or declared local operation algebra	Wext ≤ 0 for every allowed cycle	Structural theorem for the specified operation class	Add switching, control, or apparatus work
Complete passivity	Every finite tensor power with a specified composition rule	Same dynamics on each copy; finite-energy domain	Collective cyclic operation allowed	Product representation and conserved charges fixed	Total work across all copies	Composite-system operation algebra	Ground or KMS structure under standard hypotheses	Pusz–Woronowicz characterization	Activate a passive non-KMS state on several copies
Timelike QEI	Specified field and state class along a timelike curve or worldvolume	Smooth sampler and renormalized stress tensor	None	Theory dependent	Sampled energy density, not work	Sampling support and observer trajectory	Field-specific lower bound	Rigorous for important free-field settings	Narrow the sampler or change mass, boundary, or trajectory
ANEC	Complete affinely parametrized null generator; theorem-dependent state class	Renormalized null energy and convergent integral	None	Theorem dependent	Complete null integral, not device energy	Entire null line with fixed affine normalization	Integrated null energy is nonnegative	Broad flat-space QFT proofs under stated hypotheses	Truncate the line or rescale its affine parameter
Relative-entropy bound	State pair on one region with a known reference modular Hamiltonian	Common algebraic definition or regulator subtraction	None	Same observable algebra and sector	Modular-energy difference	Chosen region and reference state	Entropy change does not exceed modular-energy change	Exact consequence of relative-entropy positivity	Change the region, algebra, or reference state
QNEC	Suitable continuum state and null-deformed entangling cut	Renormalized stress tensor and entropy shape variation	None	Usually fixed sector	Local null energy density, not total work	Local cut and normalized null congruence	2π times null energy bounds the diagonal second variation	Free-field and broad interacting-QFT proofs with explicit assumptions	Retain cutoff artifacts or mix affine normalizations
Quantum interest	Specified free field and negative/positive pulse family	Sampling theorem and pulse idealization fixed	None	Usually neutral free field	Pulse area and separation, not extractable work	Timelike history of one observer	Positive repayment grows with allowed separation	Proved in important model classes, not universal across QFT	Smooth the pulses or change field, dimension, or boundary
Localization or measurement cost	Specified probe–field model and finite spacetime support	Switching and spatial profile control ultraviolet response	Probe preparation, coupling, readout, and reset declared	Coupling must respect charges and gauge constraints	Field, probe, switching, clock, and controller energy	Interaction support and causal factorization	Protocol-specific cost or error bound	No single universal localization-energy theorem	Change switching time while holding only spatial width fixed
Landauer erasure	Specified memory, reservoir, temperature, and final error	Finite entropy and energy accounting	Erasure channel and retained feedback record declared	Additional conserved quantities enter through generalized potentials	Reservoir heat plus work and control costs	Physical memory–reservoir coupling	βQ equals entropy decrease plus correlation and disturbance terms	Exact finite-reservoir identities and refinements	Use an initially correlated or nonthermal reservoir
Quantum energy teleportation	Correlated ground or stationary state in a specified model	Local Hamiltonian density and energy zero fixed	Local measurement, classical message, conditional remote operation	Model-dependent charges and control algebra	Measurement injection, remote extraction, control, and reset reported separately	Feed-forward remains inside the future light cone	Correlation-assisted conditional local extraction	Exact models and quantum-hardware demonstrations; continuum claims remain model specific	Remove the message, correlations, or injected-energy account

Where to go next

Continue to Entanglement Dynamics and Information Transport to specify the initial state, quench, subsystem, and regulator.

References

• Pusz, Władysław, and Stanisław L. Woronowicz. “Passive States and KMS States for General Quantum Systems.” Communications in Mathematical Physics 58 (1978): 273–290. DOI.