Energy, Entropy, and Localization

Energy–information statements in QFT do not form a single hierarchy. Passivity restricts work extraction by cyclic operations; quantum energy inequalities constrain sampled stress energy; ANEC concerns a complete null generator; relative entropy and QNEC connect energy to entanglement under particular geometric and renormalization hypotheses. Localization and measurement introduce apparatus costs that none of those statements automatically includes.

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

Enter this chapter

Begin with passivity, work, and information, then sharpen one-copy passivity to complete passivity and KMS structure. The energy-bound sequence runs from timelike quantum energy inequalities to ANEC, relative-entropy Bekenstein bounds, and QNEC. These are deliberately separate routes because their averages, state domains, and geometric assumptions are inequivalent.

Next study the compensation of negative energy through quantum interest, and then test purported entropy bounds against species and regulator dependence. The operational sequence covers localization and measurement cost, Landauer erasure and information engines, and quantum energy teleportation. Finish with the dated bounds status map, which compares theorem strength without merging unlike claims.

Passivity, averaged-energy inequalities, entropy bounds, QNEC, and operational energy protocols answer different questions. Arrows indicate the principal input to each conclusion, not logical equivalence. The diagram is schematic and not to scale.

The variables that must remain visible

For a state $\rho$ and reference $\sigma$ on a region, relative entropy takes the form

$$S(\rho\Vert\sigma)=\Delta\langle K_\sigma\rangle-\Delta S\ge 0$$

when both terms are defined with a common regulator or algebraic subtraction. This becomes a Bekenstein-type inequality only when $K_\sigma$ has a useful energy representation. By contrast, a QEI has the schematic form

$$\int d\tau\, g(\tau)^2\, \langle T_{\mu\nu}u^\mu u^\nu\rangle_\psi \ge -\,\mathcal Q[g;m,\xi,\gamma],$$

where the field, sampling function, worldline, state class, and renormalization prescription determine $\mathcal Q$. ANEC instead integrates $T_{kk}$ along a complete affinely parametrized null generator. QNEC compares the local null energy with a second null shape derivative of renormalized entropy. Replacing one average or derivative by another changes the theorem.

For the regional relative-entropy derivation and its exact hypotheses, see Casini 2008, §§ 2–3. The state, sampling, and field-theory dependence of quantum energy inequalities is developed in Fewster 2012, §§ 2–4, pp. 6–20.

Work is equally conditional. For a cyclic unitary $U$ generated by an allowed perturbation of a Hamiltonian $H$, extracted work is

$$W_{\mathrm{ext}}=\operatorname{tr}(\rho H) -\operatorname{tr}(U\rho U^\dagger H).$$

A passive state has $W_{\mathrm{ext}}\le0$ for the declared operation class. Under the standard $C^*$-dynamical hypotheses, complete passivity characterizes ground and KMS states; Pusz and Woronowicz 1978, Theorems 1.1 and 1.4, pp. 275–281 give the precise statement. A local detector protocol must additionally count switching, control, memory reset, and any energy injected in preparing the measurement. The algebraic theorem does not include those apparatus costs.

Claim–validity comparison

Energy–information claims and the hypotheses that control their validity

Claim or protocol	Spacetime and state domain	Smearing or renormalization	Instrument and feed-forward	Charge sector	Injected and extracted energy	Causal support	Governing condition	Status	Decisive countercheck
Passivity	Stationary system; normal state in the energy domain	Fixed Hamiltonian zero and admissible cyclic perturbation	No measurement required	Fixed representation or stated sector	Difference of initial and final system energy	Global or declared local operation algebra	Extracted work Wext ≤ 0	Theorem for its operation class	Add switching or control work
Complete passivity	Every finite tensor power of the state	Same dynamics on each copy	Collective cyclic operation allowed	Composition rule fixed	Total work over all copies	Product-system operation	KMS or ground-state structure under standard hypotheses	Structural theorem	Test a passive non-KMS state on multiple copies
Timelike QEI	Hadamard-type states along a timelike curve in a specified field theory	Smooth sampling and renormalized stress tensor	None	Theory dependent	Bounds sampled energy density, not work	Sampling support	Field-specific lower bound	Rigorous in important free-field settings	Narrow the sampling function or change boundary conditions
ANEC	Complete affinely parametrized null generator; theorem-dependent state class	Renormalized null energy and convergence of the integral	None	Theorem dependent	Null integral, not device energy	Complete null line	Integrated null energy is nonnegative	Proven under several flat-space QFT hypotheses	Truncate the generator or change affine normalization
Relative-entropy bound	State pair restricted to a region with known modular Hamiltonian	Common algebraic or regulated subtraction	None	Same observable algebra	Modular-energy difference	Region dependent	Entropy change does not exceed modular-energy change	Exact consequence of positivity	Change the reference state or region
QNEC	Null-deformed entangling surface and suitable continuum state	Renormalized stress tensor and null entropy variation	None	Usually fixed sector	Local null energy, not total work	Local cut on a null congruence	2π times null energy bounds the second entropy variation	General proofs with stated assumptions	Reparametrize the null generator or retain a cutoff artifact
Landauer erasure	Specified memory, reservoir, temperature, and final error	Finite entropy and energy accounting	Erasure map; feedback record declared	Conserved quantities included through generalized potentials	Work or heat cost of reset	Physical reservoir coupling	βQ ≥ erased entropy in the ideal thermal limit	Exact refinements for finite reservoirs	Use a nonthermal finite bath
Quantum energy teleportation	Correlated ground or stationary state in a regulated model	Local energy density and Hamiltonian zero fixed	Local measurement, classical message, conditional remote unitary	Model dependent	Report measurement injection and remote extraction separately	Feed-forward inside the future light cone	Correlation-assisted local extraction	Model and hardware demonstrations; continuum claims remain model specific	Remove the message, correlation, or injected-energy account

The table is a semantic comparison, not a ranking. In particular, theorem status does not imply broader applicability, and experimental execution of a finite qubit protocol does not establish a continuum-QFT statement.

The most common failures occur before any numerical estimate: a pointwise claim is inferred from an averaged theorem, distinct bounds are conflated, the regulator or species limit changes, or apparatus and feed-forward costs are omitted. The map is schematic.

Scope boundary

This chapter stays in nongravitational or fixed-background QFT except where a result’s history motivates a name. It does not derive black-hole entropy, the quantum focusing conjecture, holographic entropy formulas, or curved-spacetime energy inequalities. “Bekenstein-type” labels a regional relative-entropy inequality here, not a universal gravitational bound. “Quantum energy teleportation” labels correlation-assisted local extraction with classical feed-forward, not superluminal energy transport.

References

• Casini, Horacio. “Relative Entropy and the Bekenstein Bound.” Classical and Quantum Gravity 25 (2008): 205021. 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.