Energy–Information Bounds: Assumptions and Status

Energy–information results constrain different mathematical objects: cyclic work, a sampled stress tensor, a complete null integral, regional relative entropy, entropy shape derivatives, reset heat, or the energy balance of a measurement-and-feedback protocol. They cannot be ordered as versions of one universal bound. This page compares their hypotheses and strongest direct conclusions using primary literature checked through 26 August 2026.

Required background. The definitions used below come from passivity, quantum energy inequalities, ANEC, relative-entropy bounds, QNEC, quantum interest, species and regulators, Landauer erasure, and quantum energy teleportation.

Results with different mathematical objects

In the table, “theorem” always means a theorem in the stated domain. It does not mean that the assumptions hold in every QFT, background, state, regulator, or laboratory protocol. “Finite-model demonstration” means that the declared Hamiltonian and operations have been implemented or checked; it does not establish a continuum limit.

Energy–information results, their domains, and their strongest direct conclusions through 26 August 2026.

Result and primary source	Object constrained	Domain and assumptions	Strongest direct conclusion	Saturation or model check	What changes the conclusion	Status through 26 Aug 2026
Pusz–Woronowicz 1978	Work under cyclic operations; all finite tensor powers for complete passivity	A declared C*-dynamical system, state, generator, and allowed cyclic operations	Passive states yield no positive cyclic work; completely passive states have ground/KMS equilibrium structure under the theorem's hypotheses	Ground and KMS states provide the canonical completely passive cases; commuting operations give zero work	Change the operation class, endpoint Hamiltonian, charge constraints, or omit controller energy	Structural theorem; not a local stress-energy inequality
Ozawa 2002; Kuramochi–Tajima 2023	Measurement error versus conserved-quantity asymmetry	Additive conservation law, specified error, and Yanase-compatible pointer; continuous unbounded case needs the later theorem's domains	A projective measurement of an observable that does not commute with the system's conserved quantity cannot be exact under the stated conditions; approximate error is bounded by apparatus conserved-quantity fluctuations	Finite apparatus models test the variance–error tradeoff	Relax the Yanase condition, move the conserving controller across the boundary, or change how resource variance is converted to energy	Conditional no-go and resource tradeoff; no apparatus-independent localization-energy formula
Fewster–Eveson 1998	Timelike average of renormalized energy density	Specified free scalar field, spacetime dimension, inertial worldline, smooth sampler, reference prescription, and state domain	A state-independent lower bound for that sampled observable; in the four-dimensional massless case, the bound is proportional to minus the squared second derivative of the sampler	Explicit Hadamard states and numerical samplings check the inequality; optimality is a separate question	Use a point, top-hat, null sampler, boundary, curvature, interacting theory, or different reference without deriving a new bound	Rigorous field- and sampler-specific theorem
Faulkner–Leigh–Parrikar–Wang 2016	Integral of Tkk along a complete null generator	Relativistic QFT on Minkowski space with the modular, state, and convergence assumptions of the proof	The complete averaged null energy is nonnegative in the proved setting	Vacuum gives zero; positive-energy excitations provide simple checks	Replace the complete generator by a finite segment, introduce boundaries/defects without a new proof, or lose convergence	Broad continuum theorem with explicit hypotheses; not a finite-null-window inequality
Casini 2008	Regional relative entropy D(ρR‖σR) = Δ⟨Kσ⟩ − ΔS	Same local algebra/region and reference state; an admissible state pair with finite relative entropy; modular Hamiltonian Kσ identified	Positivity gives ΔS ≤ Δ⟨Kσ⟩, a regulator-safe state-comparison bound	First-order state perturbations obey the entanglement first law and saturate at linear order	Replace modular energy by unweighted total energy, compare different algebras, or subtract entropies with inconsistent regulators	Exact information-theoretic inequality; geometric simplifications are region- and theory-dependent
Balakrishnan–Faulkner–Khandker–Wang 2019	Local null energy versus a second null shape variation of entropy	Continuum QFT, a specified null deformation and transverse normalization, suitable states, and controlled renormalization	QNEC relates Tkk to the entropy second variation in the proved domain; it is not the pointwise classical NEC	Vacuum and special symmetric states can saturate; interacting examples test nonsaturation	Move the regulator with the cut, use a nonsmooth deformation, change the local algebra, or omit transverse-area normalization	General proof in the source's continuum framework; rigorous algebraic variants state their own state domains
Ford–Roman 1999	Compensation between negative- and positive-energy pulses	Massless scalar fields in specified two- and four-dimensional flat-spacetime pulse models, derived from quantum inequalities	In those models, delayed compensation must overpay the negative pulse, with restrictions on separation	Delta-pulse and controlled profile calculations establish the stated restricted result	Generalize to arbitrary interacting fields, curved backgrounds, or incomplete energy histories without a new theorem	Proved in the cited model classes; “quantum interest” is not one universal formula
Srednicki 1993	Vacuum entanglement entropy across a regulated boundary	A lattice-regulated free scalar field and a specified spatial bipartition	The calculation exhibits leading area scaling and ultraviolet sensitivity	Varying lattice spacing and region size reproduces the regulated scaling	Change species number, cutoff, factorization, gauge-center choice, or order of limits	Canonical model calculation; it does not establish an absolute regulator-independent entropy cap
Reeb–Wolf 2014	Memory entropy decrease and heat into a Gibbs reservoir	Identified memory/reservoir, initial product with a Gibbs bath, and joint unitary evolution; work needs additional Hamiltonian bookkeeping	Exact equality βQ = ΔS + final mutual information + reservoir relative entropy, hence the Landauer inequality	Increasing-reservoir quasistatic sequences approach the entropy-only value; finite baths have positive corrections for nontrivial erasure	Use a nonthermal or initially correlated bath, leave an environment uncounted, or confuse heat with controller work	Exact finite-system identity with controlled extensions; field reservoirs require localization and limit control
Hotta 2009; Rodríguez-Briones et al. 2023; Ikeda 2023; Ikeda 2026	Measurement injection, conditional local extraction, and classical feed-forward	Correlated state, declared local instrument and energy operator, causal message, conditional operation, and complete apparatus account	Positive remote local extraction occurs in controlled models while total energy remains nonnegative and the classical message remains causal	Finite spin models, an NMR bipartite implementation, an encoded superconducting-hardware implementation, and a 2026 lattice–continuum sector-matching calculation	Remove the message or relevant correlations, let Alice's excitation overlap Bob's operation and supply the output by ordinary absorption, omit EA, or mismatch lattice and continuum sectors	Valid finite protocols and model-specific continuum theory; the cited hardware papers concern finite bipartite or encoded Hamiltonian systems rather than direct spatially separated continuum-field extraction

Energy accounts must be compared before inequalities

An operational claim should begin with a closed accounting boundary. If a target system $S$, apparatus $A$, battery $B$, message carrier $C$, and reservoir $R$ are inside that boundary, the work supplied from outside satisfies

$$W_{\rm on} =\Delta E_S+\Delta E_A+\Delta E_B +\Delta E_C+\Delta E_R.$$

Each term is signed. A battery output means $\Delta E_B>0$; heat dumped into a bath means $\Delta E_R>0$ under the chosen convention. Passivity constrains work for a declared cyclic operation class. Landauer constrains a reservoir-energy change under Gibbs assumptions. A QEI constrains a sampled component of $\Delta E_S$, not the whole right-hand side. QET relates Alice’s injection and Bob’s conditional extraction inside a particular protocol. Combining them requires an explicit model that contains every object appearing in both statements.

A scientifically useful comparison records five items for every claim:

1. the operator or operational quantity on the left-hand side;
2. the state, region, sampler, and renormalization domain;
3. the allowed dynamics, instruments, and causal order;
4. the energy reference and accounting boundary;
5. one in-domain equality/model check and one outside-domain control.

Reproducible classification benchmark

Three short calculations show why the rows are not interchangeable.

For the four-dimensional massless-scalar QEI with normalized Gaussian square-root sampler, the bound is

$$\int dt\,g_\tau(t)^2 \langle{:}T_{00}{:}\rangle \ge-\frac{3}{64\pi^2\tau^4}.$$

It equals $-0.0047494305$ at $\tau=1$ and $-0.0002968394$ at $\tau=2$. These numbers constrain one timelike sampled energy density.

For a uniform bit reset with error $\epsilon=0.01$ in a Gibbs bath,

$$\beta Q_R\ge\ln2-h_2(0.01)=0.63714565.$$

This dimensionless number constrains bath heat multiplied by inverse temperature.

For the two-qubit QET model with $h=k=1$,

$$E_A=0.7071068, \qquad E_B=0.1147476, \qquad E_A-E_B=0.5923591.$$

These are Hamiltonian energies in a finite model. Comparing $0.637$ with $0.115$ numerically would be meaningless: the units, systems, operations, and constrained quantities differ. A valid combined calculation would first embed the QET record in a physical memory, choose a Gibbs bath, and add its reset heat to the same energy account.

How conclusions change when assumptions fail

Perturb the sampler or integration region. Halving the Gaussian QEI duration multiplies the lower-bound magnitude by $16$. Replacing a complete ANEC generator with a finite segment changes the theorem rather than merely its numerical input.

Perturb a saturation example. Relative entropy vanishes at the reference state, so $\Delta S=\Delta\langle K\rangle$ at first order. At second order, positive relative entropy generally opens a gap. Linear saturation is not evidence for exact saturation at finite perturbation.

Change the reservoir resource. A squeezed or population-inverted bath is not characterized by one Gibbs temperature. A temperature-only Landauer comparison can then mistake consumed nonequilibrium free energy for a violation.

Remove protocol inputs. In QET, optimize Bob’s operation again with no classical outcome and with the relevant cross-correlator removed. In localization, relax the Yanase condition or move the conserving controller across the accounting boundary. A conclusion that survives only because the control was not reoptimized is not decisive.

Move the regulator. Entropy differences, null shape derivatives, and lattice–continuum operator matching must use one frozen prescription. Changing the cutoff, species count, charge sector, or algebra while retaining an old bound changes the problem.

The chapter orientation, claim-validity table, and failure controls provide the shared entry points without duplicating their overview figures here.

Common pitfalls

Reading “proven” as “universal.” A proof transfers only with its field, state, geometric, operator-domain, and normalization assumptions.

Using one word for different energies. Total Hamiltonian energy, local renormalized energy density, heat, work, and battery energy are not interchangeable.

Treating a counterexample outside the domain as a refutation. It instead shows that an omitted hypothesis was essential. The converse error—using a restricted proof outside its domain—is equally serious.

Exercises

1. Classify four proposed uses

For each claim, name the relevant row and decide whether the cited result applies: (a) ANEC is used on a finite null segment; (b) a thermal bit is reset with known error; (c) Bob’s QET unitary is chosen before the message arrives; (d) a WAY bound is converted to joules without an apparatus Hamiltonian.

Solution

(a) The ANEC row does not apply because its integral is over a complete null generator; a finite segment needs another inequality. (b) The Landauer row can apply if the bath is genuinely Gibbs, initially uncorrelated, and the retained dynamics is unitary; the error fixes the entropy decrease. (c) This is not QET feed-forward: a pre-message operation cannot depend on Alice’s outcome through a causal channel. (d) The WAY result constrains error versus conserved-quantity fluctuation. Joules require a Hamiltonian that assigns energy to that fluctuation, plus a controller account.

2. Rescale the QEI benchmark

Starting from the value at $\tau=1$, find the lower bound at $\tau=1/2$ and $\tau=3$.

Solution

The magnitude scales as $\tau^{-4}$. At $\tau=1/2$ it is multiplied by $16$:

$$B_{1/2}=-0.07599089.$$

At $\tau=3$ it is divided by $81$:

$$B_3=-5.86349\times10^{-5}.$$

No ANEC, QNEC, or Landauer statement was used; this is a sampler rescaling within one QEI.

3. Combine QET with record reset correctly

The two-qubit benchmark has $E_A=0.7071068$ and $E_B=0.1147476$. Alice’s equiprobable one-bit record is reset in a Gibbs bath at $\beta=2$ with zero error. Ignore all other overhead. What heat is required ideally, and what is the delivery-stroke output before recovery of the residual system energy?

Solution

Ideal zero-error reset requires

$$Q_R\ge\frac{\ln2}{\beta}=0.3465736.$$

Counting Bob’s received energy as output and Alice’s injection plus reset heat as inputs gives the delivery-stroke balance

$$W_{\rm net}^{\rm out} \le0.1147476-0.7071068-0.3465736 =-0.9389328.$$

The system still contains $E_A-E_B=0.5923592$. Recovering all of it later would raise the ideal complete-cycle balance to $-0.3465736$, the reset cost. The calculation combines two rows only after adding a physical memory and bath to the QET protocol.

4. Explain first-order saturation

Let $\rho(\lambda)$ be a smooth state family with $\rho(0)=\sigma$. Use $D(\rho(\lambda)\|\sigma)=\Delta\langle K_\sigma\rangle-\Delta S$ to show the entanglement first law.

Solution

Relative entropy is nonnegative and has a minimum of zero at $\lambda=0$. Its first derivative therefore vanishes there:

$$0=\left.\frac{dD}{d\lambda}\right|_0 =\left.\frac{d}{d\lambda} \big(\Delta\langle K_\sigma\rangle-\Delta S\big) \right|_0.$$

Hence $\delta S=\delta\langle K_\sigma\rangle$. The second derivative of relative entropy is generally positive, so this linear equality does not imply finite-perturbation saturation.

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.
• Casini, Horacio. “Relative Entropy and the Bekenstein Bound.” Classical and Quantum Gravity 25 (2008): 205021. DOI.
• 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.
• Fewster, Christopher J., and Simon P. Eveson. “Bounds on Negative Energy Densities in Flat Spacetime.” Physical Review D 58 (1998): 084010. DOI.
• Ford, L. H., and Thomas A. Roman. “The Quantum Interest Conjecture.” Physical Review D 60 (1999): 104018. DOI.
• Hotta, Masahiro. “Quantum Energy Teleportation in Spin Chain Systems.” Journal of the Physical Society of Japan 78 (2009): 034001. DOI.
• Ikeda, Kazuki. “Quantum Energy Teleportation across a Lattice and the Continuum.” Physical Review D 114 (2026): 045016. DOI.
• Ikeda, Kazuki. “Demonstration of Quantum Energy Teleportation on Superconducting Quantum Hardware.” Physical Review Applied 20 (2023): 024051. DOI.
• Kuramochi, Yui, and Hiroyasu Tajima. “Wigner–Araki–Yanase Theorem for Continuous and Unbounded Conserved Observables.” Physical Review Letters 131 (2023): 210201. DOI.
• Ozawa, Masanao. “Conservation Laws, Uncertainty Relations, and Quantum Limits of Measurements.” Physical Review Letters 88 (2002): 050402. DOI.
• 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.
• Rodríguez-Briones, Nayeli A., Hemant Katiyar, Eduardo Martín-Martínez, and Raymond Laflamme. “Experimental Activation of Strong Local Passive States with Quantum Information.” Physical Review Letters 130 (2023): 110801. DOI.
• Srednicki, Mark. “Entropy and Area.” Physical Review Letters 71 (1993): 666–669. DOI.