Energy, Entropy, and Localization

Energy enters quantum field theory in several mathematically different ways. It can be the expectation of a Hamiltonian before and after a cyclic drive, a stress tensor averaged along a worldline or null generator, the modular energy of a spatial region, the heat deposited in a reservoir, or one term in the energy account of a measurement-and-feedback protocol. These energy-related quantities can differ in dimension, normalization, and operational meaning, and they do not obey one universal inequality. This chapter’s organizing rule is therefore simple: write down the operation and the energy functional before naming the bound.

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.

Enter this chapter

Start with passivity, work, and information: it fixes the Hamiltonian, the admissible cyclic operations, and the sign convention for extracted work. Then study complete passivity and KMS structure, where the question changes from one copy to every finite tensor power. A state can be passive for one copy yet become a work resource when several copies are driven collectively.

The stress-energy sequence begins with timelike quantum energy inequalities, continues to the averaged null energy condition, and then turns to information-theoretic statements through relative-entropy Bekenstein-type bounds and the quantum null energy condition. These pages remain separate because a sampled timelike average, a complete null integral, a regional modular Hamiltonian, and a local null shape derivative are inequivalent objects.

The final sequence asks what negative energy and information processing can do operationally. Quantum interest describes compensation constraints in specified free-field pulse models. Entropy bounds, species, and regulators explains why raw entanglement entropy is not a regulator-independent energy resource. Localization and measurement cost and Landauer erasure make the apparatus and reservoir explicit. Quantum energy teleportation then combines a correlated ground state, a local measurement, a causal classical message, and a conditional remote operation. The dated bounds status comparison closes the chapter.

The first figure is a dictionary. Read each row horizontally: the object on the left determines the mathematical statement in the middle and the conclusion on the right. Moving vertically changes the question rather than strengthening the same bound.

Work extraction, sampled-energy bounds, regional entropy inequalities, and measurement-and-feedback protocols begin from different data. Each arrow licenses only the conclusion on its own row; none of the rows is an omnibus energy condition. The diagram is schematic and not to scale.

Work and energy averages answer different questions

For a state $\rho$, Hamiltonian $H$, and admissible cyclic unitary $U$, define extracted work by

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

The cycle returns the externally controlled Hamiltonian to $H$; it need not return the state to $\rho$. A passive state satisfies $W_{\mathrm{ext}}\le 0$ for every unitary in the declared operation class. This statement already depends on the Hamiltonian, its energy domain, and which operations are admitted. Complete passivity asks for the same property of $\rho^{\otimes n}$ for every finite $n$ under collective cyclic operations. Under the standard $C^*$-dynamical hypotheses, completely passive states are ground or KMS states; the precise theorem is given by Pusz and Woronowicz 1978, Theorems 1.1 and 1.4, pp. 275–281.

A quantum energy inequality instead constrains a smeared stress-tensor expectation. A typical timelike worldline statement has the form

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

where $g$ is a smooth sampling function, $\mathcal T$ specifies the field theory and renormalization prescription, and $\gamma$ is the observer’s trajectory. It bounds a weighted energy density, not extractable work. For the free scalar field, Fewster and Eveson 1998 prove bounds for broad classes of smooth sampling functions; Fewster 2012, §§ 2–4 explains how the field, mass, dimension, trajectory, boundaries, and reference state enter.

The sampling scale is physical. If a normalized profile is narrowed according to

$$g_\tau(t)=\tau^{-1/2}g(t/\tau),$$

then $\int |g_\tau''(t)|^2dt=\tau^{-4}\int|g''(s)|^2ds$. Thus the four-dimensional massless-scalar bound weakens as $\tau^{-4}$ at short duration. A QEI does not imply pointwise positivity; it quantifies how the magnitude and duration of negative energy trade against one another within its theorem domain.

Null averages, modular energy, and entropy variation

Let $x^\mu(\lambda)$ be a complete affinely parametrized null generator with tangent $k^\mu=dx^\mu/d\lambda$. ANEC concerns

$$\mathcal E_k =\int_{-\infty}^{\infty}d\lambda\, \langle T_{\mu\nu}k^\mu k^\nu\rangle.$$

In broad classes of unitary Lorentz-invariant QFTs on Minkowski spacetime, $\mathcal E_k\ge0$ follows from modular-theoretic or causality arguments. Faulkner et al. 2016 obtain ANEC from deformed half-space modular Hamiltonians and monotonicity of relative entropy; Hartman, Kundu, and Tajdini 2017 derive it from microcausality and lightcone operator expansions under their stated interacting-QFT assumptions. A finite null segment is not a complete generator, and changing the affine normalization changes the numerical integral.

Relative entropy answers a regional comparison question. For states $\rho$ and $\sigma$ restricted to the same observable algebra,

$$S(\rho\Vert\sigma) =\Delta\langle K_\sigma\rangle-\Delta S \ge0,$$

so that $\Delta S\le\Delta\langle K_\sigma\rangle$ whenever both differences are defined. This becomes an energy-looking bound only when the reference modular Hamiltonian $K_\sigma$ has a local stress-tensor representation. For the vacuum half-space $x^1>0$ at $t=0$,

$$K_{\mathrm{vac}} =2\pi\int_{x^1>0}d^{d-1}x\,x^1T_{00}(0,\mathbf x) +\text{constant}.$$

The additive constant cancels in expectation-value differences. Positivity then yields a weighted regional energy–entropy inequality, not a universal statement $S\le 2\pi ER$ for every definition of $S$, $E$, and $R$. This regulator-safe interpretation and its resolution of the naive species problem are developed in Casini 2008, §§ 2–3.

QNEC is local in a different sense. Schematically, for a null deformation parameter $\lambda(y)$ of an entangling cut,

$$2\pi\langle T_{kk}(y)\rangle \ge \frac{1}{\sqrt h} \frac{\delta^2 S}{\delta\lambda(y)^2},$$

with the normalization of $k$, the entropy prescription, surface conditions, and diagonal shape variation fixed. Early proofs covered free or superrenormalizable bosonic theories on stationary null surfaces Bousso et al. 2016; later work established a much broader interacting-QFT result under explicit UV and modular-flow assumptions Balakrishnan et al. 2019. QNEC is not obtained by differentiating ANEC, and ANEC is not obtained by integrating QNEC without controlling boundary and off-diagonal terms.

Operational protocols require a complete energy account

An information-processing protocol introduces physical systems not present in a stress-tensor inequality: a memory, probe, controller, clock, reservoir, message register, or conditional actuator. Their energies cannot be silently assigned to the field.

For a memory $M$ initially uncorrelated with a thermal reservoir $R$ at inverse temperature $\beta$, an exact unitary evolution gives the Landauer identity

$$\beta Q_R =\Delta S_M +I(M':R') +D(\rho_R'\Vert\rho_R),$$

where $\Delta S_M=S(\rho_M)-S(\rho_M')$ is the memory’s entropy decrease and $Q_R$ is the reservoir’s energy gain. The two final terms are nonnegative, recovering $\beta Q_R\ge\Delta S_M$. Equality requires an ideal limit in which final correlations and reservoir disturbance vanish. Finite-reservoir corrections and the exact identity are derived in Reeb and Wolf 2014.

Quantum energy teleportation has another energy account. A local measurement by Alice injects energy $E_A$ into a correlated ground or stationary state. Its classical outcome travels causally to Bob, who applies an outcome-dependent local unitary and extracts $E_B$ from his neighborhood. The protocol must report at least

$$E_A, \qquad E_B, \qquad E_{\mathrm{control}}, \qquad E_{\mathrm{reset}}, \qquad E_{\mathrm{final}}.$$

The local field energy near Bob can decrease, but the full system does not acquire energy from nothing. Without Alice’s message, Bob’s outcome-averaged operation cannot use her result; without the initial correlations, the conditional advantage disappears. Hotta’s introductory review gives the finite-system protocol and its energy accounting. A qubit or spin-chain implementation is evidence for that regulated model, not by itself a continuum-QFT experiment.

Claim–validity comparison

The table compares the minimum information needed to state each result. It is a dictionary, not a ranking: a rigorous theorem can be narrow, while a laboratory implementation can realize a model without testing the corresponding continuum claim.

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

Failure controls that change the conclusion

The second figure is a diagnostic sequence. A proposed statement must survive five independent questions: Which quantity is being bounded? On which domain and geometry? With what normalization and renormalization? Which physical operations occur? Does the complete causal and energetic account support the advertised conclusion?

Independent checks prevent five common errors: inferring pointwise positivity from an averaged theorem, comparing differently normalized null quantities, treating raw cutoff-dependent entropy as finite, calling a local energy decrease extracted work, or calling correlation-assisted conditional extraction a superluminal signal. The map is schematic.

These controls can be applied without redoing an entire proof:

• Change the sampler or support. If a claimed QEI number does not scale when the sampling duration changes, its dimensions or normalization are wrong. If ANEC fails only after truncating the null line, the truncated observable is not ANEC.
• Change the reference or regulator. A regional relative-entropy inequality must survive a common regulator change. Raw entanglement entropy and raw energy separately need not.
• Remove the operational resource. A QET advantage must disappear when the classical message or the relevant correlations are removed. A Landauer claim must change when the reservoir is finite, nonthermal, or initially correlated.
• Close the energy account. A decrease of the field energy in one region is not automatically useful work. Include probe preparation, switching, control, readout, message storage, and reset before reporting a net gain.

Exercises

Sampling-scale check

Let $g_\tau(t)=\tau^{-1/2}g(t/\tau)$. Show that $g_\tau$ has the same $L^2$ norm as $g$ and that

$$\int dt\,|g_\tau''(t)|^2 =\tau^{-4}\int ds\,|g''(s)|^2.$$

Explain why a four-dimensional massless-scalar QEI with this derivative norm does not imply a pointwise lower bound.

Solution

With $s=t/\tau$ and $dt=\tau ds$,

$$\int dt\,|g_\tau(t)|^2 =\int \tau ds\,\tau^{-1}|g(s)|^2 =\int ds\,|g(s)|^2.$$

Two derivatives give $g_\tau''(t)=\tau^{-5/2}g''(t/\tau)$. Therefore

$$\int dt\,|g_\tau''(t)|^2 =\tau\tau^{-5}\int ds\,|g''(s)|^2 =\tau^{-4}\int ds\,|g''(s)|^2.$$

As $\tau\to0$, the permitted negative lower bound diverges like $-\tau^{-4}$. The theorem constrains finite-duration averages but supplies no finite pointwise lower bound.

Affine-normalization check

Rescale a future-directed affine parameter by $\lambda'=a\lambda$ with $a>0$. Determine how

$$\mathcal E_k=\int d\lambda\,T_{\mu\nu}k^\mu k^\nu$$

transforms. Why is the sign of ANEC invariant even though its numerical value is not?

Solution

The new tangent is $k'^\mu=dx^\mu/d\lambda'=k^\mu/a$, while $d\lambda'=a\,d\lambda$. Hence

$$\mathcal E_{k'} =\int a\,d\lambda\, T_{\mu\nu}\frac{k^\mu}{a}\frac{k^\nu}{a} =\frac{1}{a}\mathcal E_k.$$

Because $a>0$, the sign is unchanged. A numerical comparison between two ANEC integrals is meaningful only after their affine normalizations are matched.

Finite-reservoir Landauer account

An unbiased classical bit is erased while the reservoir inverse temperature is $\beta$. Suppose the final memory–reservoir mutual information is $0.03$ nats and the reservoir relative entropy is $0.02$ nats. What is the minimum value of $\beta Q_R$ allowed by the exact identity?

Solution

Perfectly erasing an unbiased bit decreases the memory entropy by $\Delta S_M=\ln2$. Therefore

$$\beta Q_R =\ln2+0.03+0.02 =0.743147\ldots.$$

The ideal Landauer value $\ln2$ is not reached because the final correlations and reservoir disturbance contribute an additional $0.05$ nats.

Scope boundary

This chapter treats nongravitational or fixed-background QFT. It does not derive curved-spacetime QEIs, the quantum focusing conjecture, black-hole generalized entropy, or holographic proofs; those belong to later volumes. “Bekenstein-type” means a regional inequality obtained from relative-entropy positivity with a specified modular Hamiltonian. “Quantum energy teleportation” means correlation-assisted local extraction with causal classical feed-forward. Neither phrase licenses a universal gravitational bound or superluminal transport.

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. Open preprint.
• Bousso, Raphael, Zachary Fisher, Jason Koeller, Stefan Leichenauer, and Aron C. Wall. “Proof of the Quantum Null Energy Condition.” Physical Review D 93 (2016): 024017. DOI. Open preprint.
• Casini, Horacio. “Relative Entropy and the Bekenstein Bound.” Classical and Quantum Gravity 25 (2008): 205021. DOI. Open preprint.
• Faulkner, Thomas, Robert G. Leigh, Onkar Parrikar, and Huajia Wang. “Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition.” Journal of High Energy Physics 2016, no. 9 (2016): 038. DOI. Open preprint.
• Fewster, Christopher J. “Lectures on Quantum Energy Inequalities.” Lecture notes, arXiv:1208.5399, 2012. Open preprint.
• Fewster, Christopher J., and Simon P. Eveson. “Bounds on Negative Energy Densities in Flat Spacetime.” Physical Review D 58 (1998): 084010. DOI. Open preprint.
• Hartman, Thomas, Sandipan Kundu, and Amirhossein Tajdini. “Averaged Null Energy Condition from Causality.” Journal of High Energy Physics 2017, no. 7 (2017): 066. DOI. Open preprint.
• Hotta, Masahiro. “Quantum Energy Teleportation: An Introductory Review.” arXiv:1101.3954, 2011. Open preprint.
• Pusz, Władysław, and Stanisław L. Woronowicz. “Passive States and KMS States for General Quantum Systems.” Communications in Mathematical Physics 58 (1978): 273–290. DOI.
• Reeb, David, and Michael M. Wolf. “An Improved Landauer Principle with Finite-Size Corrections.” New Journal of Physics 16 (2014): 103011. DOI. Open preprint.