State-Dependent Interior Operators

State-dependent interior proposals assign a mirror operator relative to a reference equilibrium state and a restricted algebra of simple excitations, rather than defining one operator on the entire microscopic Hilbert space. The construction is intended to reproduce smooth-horizon correlators on its chosen domain while denying an independent $EBA$ factorization; whether it does so consistently beyond the prescribed correlators is disputed. The central question is whether assignments on overlapping state spaces preserve linearity, commutators, correlators, and the Born rule.

Required background. Quantum Information supplies code subspaces, linear operators, mixtures, Born probabilities, and channel consistency used in the tests.

Helpful background. Semiclassical Gravity supplies the equilibrium black-hole regime; Quantum Fields in Curved Spacetime supplies the interior mirror-mode correlators; Black-Hole Information supplies the AMPS and complementarity constraints motivating the construction.

Mirror operators on a code subspace

Choose an equilibrium reference state $\lvert\Psi\rangle$ and a “small” exterior collection $\mathcal A$ of operators with bounded degree and energy. Often called the small algebra, this cutoff collection is not closed under arbitrarily many products; products are used only while they remain within the prescribed cutoff Papadodimas and Raju 2014, § 6.4. The associated state or code subspace is schematically

$$\mathcal C_\Psi =\operatorname{span}\{A_\alpha\lvert\Psi\rangle:A_\alpha\in\mathcal A\}.$$

For an exterior frequency mode $\mathcal O_\omega$, a mirror operator $\widetilde{\mathcal O}^{(\Psi)}_\omega$ is defined through its action on this domain, with relations of the form

$$\widetilde{\mathcal O}^{(\Psi)}_\omega A_\alpha\lvert\Psi\rangle =A_\alpha e^{-\beta\omega/2} \mathcal O_\omega^\dagger\lvert\Psi\rangle,$$

and prescribed commutators with simple exterior operators inside correlation functions. The thermal factor reproduces the entanglement relation of a smooth two-sided state. The superscript is essential: choosing a different reference state can change the operator assignment.

For this prescription to define a linear operator, every relation $\sum_\alpha c_\alpha A_\alpha\lvert\Psi\rangle=0$ among domain vectors must imply the same relation among the proposed output vectors. Otherwise one domain vector has two different images. A sufficiently small set with independent excitation vectors avoids this elementary obstruction; extending the domain demands a new check. Papadodimas and Raju construct mirrors on their restricted domain and argue for local bulk effective field theory in the prescribed correlators Papadodimas and Raju 2014, §§ 2–5. The proposal is a family of linear actions on restricted domains, together with state-dependent identification of which member represents the interior.

What the construction aims to change

The AMPS contradiction treats the interior partner $A$ as an independent state-independent factor alongside early radiation and the late mode. A mirror construction instead represents the interior through the same microscopic degrees of freedom, with the representation controlled by $\lvert\Psi\rangle$ and $\mathcal A$. The ordinary three-factor entropy theorem is then not applied to an independent $A$.

On one fixed $\mathcal C_\Psi$, the proposal must reproduce the near-horizon two-point function, the commutator between spacelike-separated simple operators to the stated order, the energy commutator of the mirror mode, and higher-point correlators generated within the small algebra. These are nontrivial achievements if all errors are controlled.

The limitation is domain size. Enlarging $\mathcal A$ too far can make the requested commutant trivial or overdetermine the mirror. Restricting it too far can exclude the measurements, superpositions, or evaporation processes whose consistency is being claimed. The energy and degree cutoff must therefore be physical and observer-independent within the application.

Linearity, Born rule, and overlap tests

Fixed-domain linearity. For $\lvert\chi_1\rangle,\lvert\chi_2\rangle\in\mathcal C_\Psi$, the mirror must act linearly on every allowed superposition. A rule that first identifies the unknown state and then chooses an action is not a quantum measurement channel.

Overlap consistency. If $\mathcal C_\Psi$ and $\mathcal C_\Phi$ overlap, the two assignments must give the same physical correlators on their intersection or state a controlled transition rule. Otherwise one vector has two interior interpretations. Harlow analyzes the construction, its $1/N$ treatment, and this measurement problem in Harlow 2014, §§ 2–5.

Born-rule consistency. Outcome probabilities must depend on the density operator and apparatus, not on an unrecorded ensemble decomposition. Marolf and Polchinski argue that state dependence strong enough to make generic states smooth produces large Born-rule tensions under their assumptions Marolf and Polchinski 2016, § 2 and Appendix B. This is an adverse result, not a theorem that every restricted code-subspace construction fails.

Algebra consistency. Mirror commutators, adjoints, products, and time evolution must agree across all correlators claimed to be semiclassical. Matching one two-point function can hide failures in a commutator or four-point function.

An overlapping-subspace counterexample

Let a three-dimensional Hilbert space have basis $\lvert0\rangle,\lvert1\rangle,\lvert2\rangle$. Define

$$\mathcal C_0=\operatorname{span}\{\lvert0\rangle,\lvert1\rangle\}, \qquad \mathcal C_+=\operatorname{span}\{\lvert+\rangle,\lvert2\rangle\},$$

where $\lvert+\rangle=(\lvert0\rangle+\lvert1\rangle)/\sqrt2$. Suppose the first domain represents an interior observable by $Z$ on $\mathcal C_0$, while the second assigns eigenvalue $-1$ to $\lvert+\rangle$. The shared vector $\lvert+\rangle$ then has expectation zero under the first assignment and $-1$ under the second.

The contradiction disappears only if the two symbols denote different measurements, the domains are physically disjoint, or an additional reference register selects a controlled linear operator. If none holds, the state-dependent operator domain gives incompatible Born probabilities. This toy model does not reproduce the PR construction; it is an independent overlap diagnostic every such construction must pass.

A second check compares decompositions of $I/2$. Assigning different number operators to the $Z$-basis and $X$-basis decompositions of the same density matrix is nonlinear unless the preparation label is physically recorded. Familiar background dependence avoids this because a relational field operator can still be assembled into one linear quantum operator on the enlarged system.

Results and open questions as of 2026-08-19

It remains unresolved whether state-dependent interior operators can describe generic old-hole infall while preserving all ordinary quantum predictions. Competing positions differ over code-subspace size, physical interpretation, measurement theory, and whether nonperturbative overlap conflicts occur.

Evidence or position	Assumptions	Controlled result	Limitation
Mirror-operator construction Papadodimas and Raju 2014, §§ 2–5	A reference equilibrium state, a sufficiently small algebra, large-$N$ control, and correlators evaluated on the associated domain	Constructs linear mirror actions on that domain and reproduces the prescribed smooth-interior correlators	State choice, algebra cutoff, and extension across generic states remain essential qualifications
Detailed proposal analysis Harlow 2014, §§ 2–5	The PR construction together with ordinary unitary measurement theory, including an apparatus	Clarifies the $1/N$ prescription and identifies ambiguous state interpretations and measurement conflicts	The critique is tied to those domains and does not exclude every approximate fixed-code construction
Generic Born-rule adverse test Marolf and Polchinski 2016, § 2 and Appendix B	Typical states and state dependence strong enough to assign smooth horizons to the generic state class considered	Derives parametrically large Born-rule violations under those assumptions	It is not a theorem that every restricted state-dependent assignment fails
State-independent code comparison Harlow 2017, § 5	One fixed finite-dimensional encoding, logical algebra, erased region, and exact correctability condition	Multiple physical representatives of one logical algebra are linear and mutually consistent on the fixed code	This theorem does not furnish PR’s state-varying assignments or compatibility maps between different code domains

The permissive position treats state dependence as a necessary feature of interior reconstruction and emphasizes its successful correlators. The conservative code position accepts reference-dependent reconstruction only on controlled domains. The adverse position argues that extension to generic states conflicts with ordinary measurement theory. The evidence does not establish one universal verdict.

The conclusions could change after a construction gives explicit compatible maps on overlapping code subspaces, or after a theorem bounds Born-rule deviations for all allowed superpositions and apparatus states, or after a microscopic model reproduces commutators and higher correlators through evaporation with controlled nonperturbative errors.

Scope

The page tests operator consistency rather than assuming a firewall or smooth interior. Generic code theory belongs to Quantum Information, local correlators to Quantum Fields in Curved Spacetime, and explicit AdS/CFT reconstruction to the holographic treatment. The decisive issue is not the phrase “state dependence” but the operator domain, linear measurement rule, and overlap agreement.

Exercise

Two mirror assignments agree on every basis vector of an overlap subspace but are each declared antilinear on complex coefficients. Do they define a valid quantum observable there?

Solution

No. Agreement on basis vectors is insufficient if the extension to $\alpha\lvert\psi\rangle+\beta\lvert\phi\rangle$ conjugates the coefficients. Quantum observables are represented by linear operators, while probabilities arise from the Born rule. An antilinear symmetry can be meaningful only with its own physical role and cannot serve as an ordinary Hermitian measurement operator. The superposition test therefore rejects the assignment despite basis agreement.

References

• Harlow, Daniel. 2014. “Aspects of the Papadodimas–Raju Proposal for the Black Hole Interior.” Journal of High Energy Physics 2014 (11): 055. Open text.
• Harlow, Daniel. 2017. “The Ryu–Takayanagi Formula from Quantum Error Correction.” Communications in Mathematical Physics 354: 865–912. Open text.
• Marolf, Donald, and Joseph Polchinski. 2016. “Violations of the Born Rule in Cool State-Dependent Horizons.” Journal of High Energy Physics 2016 (01): 008. Open text.
• Papadodimas, Kyriakos, and Suvrat Raju. 2014. “State-Dependent Bulk-Boundary Maps and Black Hole Complementarity.” Physical Review D 89: 086010. Open text.