Operational and Algebraic Formulations

Gravity does not automatically provide a tensor product between “inside” and “outside.” Constraints and gravitational dressing tie local-looking excitations to asymptotic data. The information problem can instead be stated with observable algebras and operational channels. Its entropy contradictions survive when the named algebras behave as independent subsystems to controlled accuracy; they can fail when an assumed interior factor is actually encoded in, or shares a center with, the exterior algebra.

Required background. Quantum Information supplies states on algebras, channels, distinguishability, recovery, and entropy inequalities used below.

Helpful background. Semiclassical Gravity supplies the weak-backreaction regime; Quantum Fields in Curved Spacetime supplies local field algebras and horizon states; Causality and Global Structure supplies the regions and domains of dependence to which observables are assigned.

From factors to observable algebras

In an ordinary finite-dimensional bipartite system,

$$\mathcal H=\mathcal H_R\otimes\mathcal H_B, \qquad \mathcal A_R=\mathcal B(\mathcal H_R)\otimes I_B, \qquad \mathcal A_B=I_R\otimes\mathcal B(\mathcal H_B).$$

The two algebras commute, each is the other’s commutant, and a global state $\rho$ restricts to $\mathcal A_R$ through the reduced density operator $\rho_R=\operatorname{Tr}_B\rho$. This special type-I setting licenses familiar subsystem entropies.

An algebraic formulation begins instead with a state, a physical algebra $\mathcal A$, and subalgebras $\mathcal A_R,\mathcal A_B\subseteq\mathcal A$. It asks which expectation values $\omega(O)$ are assigned to each algebra and whether the subalgebras commute, have a nontrivial common center, or redundantly represent the same logical information. In continuum QFT, local algebras are generally not finite matrix algebras, so a cutoff-dependent density-matrix entropy is not intrinsic. Relative entropy, mutual information for separated regions, or a declared regulator may be better defined.

Gauge constraints already warn against naive factorization; gravity strengthens the warning because a gauge-invariant excitation requires gravitational dressing extending to a relational or asymptotic anchor. Donnelly and Giddings show explicitly that such dressings obstruct exact locality and subsystem factorization Donnelly and Giddings 2016, §§ II–IV. The obstruction is parametrically small in suitable weak-gravity regimes, but “small” must be compared with the accuracy of the entropy contradiction.

The operational question

An operational formulation specifies a preparation channel $\mathcal P$, physical evolution $\mathcal N$, an access channel $\mathcal A$ to the degrees of freedom an observer controls, and a decoder $\mathcal R$. The allowed observable algebra constrains $\mathcal A$; an algebra itself is not a channel that can be composed with $\mathcal N$. For a message $D$ and equally large reference $Q$, let $\Phi_{DQ}$ denote their normalized maximally entangled density operator. One recovery diagnostic is

$$\left\lVert \bigl[(\mathcal R\circ\mathcal A\circ\mathcal N\circ\mathcal P)\otimes\mathrm{id}_Q\bigr](\Phi_{DQ}) -\Phi_{DQ} \right\rVert_1\leq\varepsilon.$$

The decoder output is identified with $D$. This is a reference-entanglement test for the chosen input; a uniform guarantee over every input needs a channel norm or the dimension-dependent conversion to one. Access must preserve coherent quantum records if quantum information is to be recovered: a channel that only measures and stores classical outcomes cannot preserve arbitrary diary entanglement. The allowed operations must state collection time, detector energy, gravitational dressing, computational resources, and backreaction. Formal existence of a recovery channel is weaker than an observer completing it before causal contact is lost.

Conversely, operational inaccessibility does not prove that the complete state is mixed. Correlations can exist in an algebra even if no feasible observer measures them. The microscopic, algebraic, and operational formulations therefore ask, respectively: is the complete evolution invertible; which observables contain the correlations; and can a bounded observer extract them?

When the paradox survives

The AMPS entropy argument can be written for early radiation $E$, late exterior mode $B$, and interior partner $A$ whenever they are represented by independent finite-dimensional factors, or by commuting algebras with an applicable entropy inequality and one common state. Under those hypotheses, changing vocabulary from “subsystems” to “algebras” changes nothing: late purification and near-vacuum $AB$ entanglement still compete Almheiri et al. 2013, § 2.

The conclusion can change if $A$ is not independent of $E$, if the relevant algebras overlap, if a center carries shared classical data, or if the interior operator exists only on a restricted code subspace. But each move rejects a premise and incurs obligations. The proposed algebra must reproduce low-energy near-horizon correlators, preserve linearity on its domain, respect approximate commutators to the claimed order, and avoid assigning incompatible answers on overlapping state spaces. Algebraic language diagnoses the factorization assumption; it does not automatically supply a consistent interior.

Recent competing work gives massless-gravity examples with no auxiliary nongravitating reservoir, including radiation at null infinity, and argues that their Page curves are accessible through sufficiently controlled experiments on many copies Antonini et al. 2025, §§ 2–3. Separately, it argues that compactly supported gauge-invariant matter observables can be constructed to all orders in perturbation theory around generic backgrounds without spacetime isometries Antonini et al. 2025, § 4. These results challenge the inference that gravitational dressing alone universally forbids a useful approximate radiation algebra. They depend on the stated background symmetry, semiclassical algebra, perturbative expansion, and many-copy access; they neither establish an exact nonperturbative factorization nor erase the Donnelly–Giddings obstruction in symmetric backgrounds.

Harlow’s entropy review and firewall discussion make explicit that strong subadditivity applies only after the systems and common state have been identified Harlow 2016, Appendix C and § 7.1. This is the precise point at which a gravitational correction must enter.

A constrained two-qubit model

Let the physical subspace satisfy $Z_1Z_2=+1$, so

$$\mathcal H_{\mathrm{phys}}=\operatorname{span}\{\lvert00\rangle,\lvert11\rangle\}.$$

Within this subspace, $Z_1$ and $Z_2$ represent the same logical observable. The operator $X_1$ alone leaves the physical subspace, while the gauge-preserving logical flip is $X_1X_2$. For $\lvert\psi\rangle=\alpha\lvert00\rangle+\beta\lvert11\rangle$, the naive one-qubit reduced entropy is the binary entropy of $|\alpha|^2$. Yet the algebra generated by $Z_1$ contains only the classical sector label and has no independent conjugate operator localized on qubit 1.

This finite model is an adverse check on tensor-factor reasoning: two labels need not denote two independent physical systems. It is not a model of gravity, and it does not show that black-hole interior partners are encoded in early radiation. It shows exactly what a nonfactorization proposal must demonstrate in the full theory.

Results and open questions as of 2026-08-19

Whether algebraic reformulation resolves a black-hole contradiction remains unresolved and depends on the operator algebra, factorization accuracy, code subspace, and measurement regime. The credible positions range from approximate semiclassical factorization to nonlocal encoding; neither is a universal verdict.

Evidence or formulation	Assumptions	Controlled result	Limitation
Finite-dimensional subsystem entropy Harlow 2016, Appendix C	One normalized density operator on declared finite tensor factors	Standard entropy inequalities hold for the named factors	Exact tensor factorization is an input, not a consequence of gravity
Gravitational dressing Donnelly and Giddings 2016, §§ II–IV	Perturbative gravity, a chosen dressing, gauge, and asymptotic anchor	Gauge-invariant observables acquire long-range dressing and nonzero commutators	The perturbative obstruction does not by itself define the exact evaporation Hilbert space
Massless-gravity island examples Antonini et al. 2025, §§ 2–3	The paper’s AdS, null-infinity, or gravitating-radiation algebra and sufficiently controlled many-copy measurements	Argues that island Page curves can be operationally defined without an auxiliary nongravitating reservoir	Algebra choice, copy complexity, semiclassical control, and nonperturbative corrections remain qualifications
Perturbatively compact observables Antonini et al. 2025, § 4	A generic background without spacetime isometries and an order-by-order expansion in $G$	Gives an existence argument for compactly supported gauge-invariant matter observables to all perturbative orders	Symmetric backgrounds are obstructed, and the argument is not an exact nonperturbative factorization theorem
AMPS algebraic core Almheiri et al. 2013, § 2	Independent $E$, $B$, and $A$ systems, one common state, and controlled EFT errors	The late-purification and smooth-pair entropy requirements are inconsistent	If independence or the common-state assignment fails, a replacement algebra must be constructed
State-independent code comparison Harlow 2017, § 5	A fixed finite-dimensional encoding, logical algebra, erased region, and exact correctability condition	One logical algebra can have multiple consistent physical representatives on a code subspace	The theorem does not supply state-varying interior assignments or control overlaps between different codes

The conservative position uses approximate factors wherever gravitational commutators are below the error budget. The competing encoding position treats some interior labels as nonindependent exterior information. An operational position may further deny that one observer can compare all representations. These positions change different assumptions and should not be conflated.

The conclusions could change after an algebraic theorem controls gravitational factorization errors at the entropy scale, or after a microscopic model constructs interior and radiation algebras across complete evaporation, or after a recovery protocol survives dressing, resource, causal, and backreaction constraints.

Exercise

In the constrained two-qubit model, may one apply entanglement monogamy by treating qubit 1, qubit 2, and a reference as three independent physical subsystems?

Solution

Not without changing the model. On the physical subspace, the two qubit labels share the same logical $Z$ information, and a complete logical $X$ operation acts on both. They therefore do not supply two independent full matrix algebras. A three-factor monogamy theorem assumes the factorization it would be used to test. One may instead state an algebraic inequality for the actual subalgebras, including their center, or enlarge the Hilbert space and impose the constraint explicitly.

References

• Almheiri, Ahmed, Donald Marolf, Joseph Polchinski, and James Sully. 2013. “Black Holes: Complementarity or Firewalls?” Journal of High Energy Physics 2013 (02): 062. Open text.
• Antonini, Stefano, Chang-Han Chen, Henry Maxfield, and Geoff Penington. 2025. “An Apologia for Islands.” arXiv preprint 2506.04311. Open text.
• Donnelly, William, and Steven B. Giddings. 2016. “Observables, Gravitational Dressing, and Obstructions to Locality and Subsystems.” Physical Review D 94: 104038. Open text.
• Harlow, Daniel. 2016. “Jerusalem Lectures on Black Holes and Quantum Information.” Reviews of Modern Physics 88: 015002. Open text.
• Harlow, Daniel. 2017. “The Ryu–Takayanagi Formula from Quantum Error Correction.” Communications in Mathematical Physics 354: 865–912. Open text.