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arxiv: 2605.18640 · v1 · pith:4EAO54CTnew · submitted 2026-05-18 · ✦ hep-th · quant-ph

Modular Lower Bounds on Reeh-Schlieder State Preparation

Javier Blanco-Romero , Florina Almenares Mendoza This is my paper

Pith reviewed 2026-05-20 09:12 UTC · model grok-4.3

classification ✦ hep-th quant-ph
keywords Reeh-Schlieder theoremmodular Hamiltonianstate preparationTomita-Takesaki estimatepostselection overheadlocal operatorstype III algebrasquantum field theory
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The pith

States with deeply negative modular energy require large local operators and postselection overhead for Reeh-Schlieder approximation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper isolates a standard Tomita-Takesaki estimate to turn the qualitative Reeh-Schlieder theorem into a quantitative lower bound on the cost of preparing target states from the vacuum with operators localized in small spacetime regions. Targets carrying deeply negative modular energy with respect to the vacuum modular Hamiltonian demand local operators of large norm. Rescaling those operators to physical contractions yields a lower bound on the overhead from postselection. The bound is model-independent yet becomes explicit in geometries with known modular Hamiltonians, such as wedges via the Bisognano-Wichmann theorem or bounded regions in conformal field theories via the Casini-Huerta-Myers formula. Local unitaries are restricted to states of nonnegative modular energy, so negative sectors necessarily involve nonunitary or postselected outcomes that complement vacuum embezzlement arguments in type III local algebras.

Core claim

This note isolates a standard Tomita-Takesaki estimate as a model-independent preparation bound. Targets with deeply negative modular energy require large local operators. After rescaling such an operator to a physical contraction, the same estimate becomes a lower bound on postselection overhead. In geometries where the modular Hamiltonian is known, the bound becomes explicit. Bisognano-Wichmann turns it into a boost energy statement for wedges, and the Casini-Huerta-Myers formula gives a stress-tensor version for bounded regions of conformal field theories. Local unitaries can only reach states of nonnegative modular energy. Negative modular sectors require nonunitary or postselected, post

What carries the argument

The modular energy with respect to the vacuum modular Hamiltonian, which directly controls the minimal norm of a local operator that can approximate a given target vector in the Reeh-Schlieder construction.

If this is right

  • Local unitaries can only prepare states of nonnegative modular energy.
  • Negative modular sectors require nonunitary or postselected outcomes.
  • The bound becomes an explicit boost-energy statement for wedge geometries.
  • The bound becomes a stress-tensor integral statement for bounded regions in conformal field theories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same modular-energy cost may constrain the resources needed to prepare certain entangled field states in quantum-information protocols that use local operations.
  • The bound offers a complementary quantitative handle to vacuum-embezzlement constructions, suggesting that total preparation overhead cannot be made arbitrarily small for sufficiently negative targets.

Load-bearing premise

The standard Tomita-Takesaki estimate can be directly isolated and reinterpreted as a model-independent lower bound on preparation cost without additional model-dependent corrections or assumptions about the algebra representation.

What would settle it

An explicit calculation in a free scalar field theory on a wedge, computing the minimal norm of a local operator approximating a target state whose modular energy is known to be negative and checking whether that norm matches the predicted lower bound.

read the original abstract

The Reeh-Schlieder theorem says that every target vector can be approximated from the vacuum by an operator localized in an arbitrarily small spacetime region, but it gives no quantitative cost for doing so. This note isolates a standard Tomita-Takesaki estimate as a model-independent preparation bound. Targets with deeply negative modular energy require large local operators. After rescaling such an operator to a physical contraction, the same estimate becomes a lower bound on postselection overhead. In geometries where the modular Hamiltonian is known, the bound becomes explicit. Bisognano-Wichmann turns it into a boost energy statement for wedges, and the Casini-Huerta-Myers formula gives a stress-tensor version for bounded regions of conformal field theories. Local unitaries can only reach states of nonnegative modular energy. Negative modular sectors require nonunitary or postselected outcomes, giving a preparation cost bound that complements vacuum embezzlement in type III local algebras.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper isolates a standard Tomita-Takesaki estimate ||Δ^{1/2} (A Ω)|| = ||A^* Ω|| ≤ ||A|| for A in the local algebra and reinterprets it as a lower bound on the operator norm ||A|| for approximating target states ψ with large ||Δ^{1/2} ψ|| (deeply negative modular energy). After rescaling to a contraction, this yields a lower bound on postselection overhead. In geometries with known modular Hamiltonians the bound is made explicit via the Bisognano-Wichmann theorem for wedges and the Casini-Huerta-Myers formula for CFT regions. The note also observes that local unitaries cannot reach negative-modular-energy sectors.

Significance. If the derivation holds, the result supplies the first quantitative, model-independent lower bound on the preparation cost of arbitrary states under the Reeh-Schlieder theorem, complementing the qualitative density statement and the vacuum-embezzlement phenomenon in type-III algebras. Explicit realizations in wedges and CFTs make the bound falsifiable and potentially useful for discussions of postselection and non-unitary operations in algebraic QFT.

major comments (1)
  1. [The derivation following the Tomita-Takesaki estimate (near the statement of the main bound)] The central step isolates the Tomita-Takesaki identity and claims that Hilbert-space approximation AΩ ≈ ψ directly implies ||A|| ≳ ||Δ^{1/2} ψ||. Because Δ^{1/2} is unbounded, norm convergence of AΩ to ψ does not control the difference ||Δ^{1/2}(AΩ − ψ)||; additional domain or approximation-sequence control is required to justify the lower bound without representation-dependent corrections. This point is load-bearing for the model-independent claim and needs explicit treatment.
minor comments (2)
  1. [Introduction and notation paragraph] Notation for the modular operator Δ and the modular Hamiltonian K = −log Δ should be introduced once with a brief reminder of the domain issues that arise when Δ^{1/2} acts on non-core vectors.
  2. [Paragraph on postselection overhead] The rescaling argument that converts the operator-norm bound into a postselection-overhead bound would benefit from an explicit inequality relating the contraction factor to the success probability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for identifying a technical subtlety in the central derivation. We address the comment below and have revised the manuscript to strengthen the rigor of the argument while preserving its model-independent character.

read point-by-point responses

  1. Referee: [The derivation following the Tomita-Takesaki estimate (near the statement of the main bound)] The central step isolates the Tomita-Takesaki identity and claims that Hilbert-space approximation AΩ ≈ ψ directly implies ||A|| ≳ ||Δ^{1/2} ψ||. Because Δ^{1/2} is unbounded, norm convergence of AΩ to ψ does not control the difference ||Δ^{1/2}(AΩ − ψ)||; additional domain or approximation-sequence control is required to justify the lower bound without representation-dependent corrections. This point is load-bearing for the model-independent claim and needs explicit treatment.

    Authors: We agree that the original presentation would benefit from greater explicitness on this point. In the revised manuscript we have added a dedicated paragraph immediately following the statement of the main bound. We now specify that the lower bound is understood in the sense of the infimum of ||A|| over bounded operators A satisfying ||AΩ − ψ|| < ε, with ε taken to zero after the estimate is applied. For any target ψ in the domain of Δ^{1/2} we select the approximating sequence so that A_n Ω → ψ and, by the closedness of the Tomita-Takesaki operator, Δ^{1/2} A_n Ω → Δ^{1/2} ψ whenever the latter exists. The identity ||Δ^{1/2} A_n Ω|| = ||A_n^* Ω|| ≤ ||A_n|| then passes to the limit, yielding liminf ||A_n|| ≥ ||Δ^{1/2} ψ|| without invoking representation-specific properties. This control is available within the Reeh-Schlieder framework because the relevant dense subspaces admit such approximating sequences. The revised text makes the domain and sequential requirements fully explicit while leaving the model-independent claim intact. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation isolates standard external Tomita-Takesaki estimate without self-referential reduction

full rationale

The paper's central step reinterprets the known Tomita-Takesaki identity ||Δ^{1/2}(AΩ)|| = ||A^* Ω|| ≤ ||A|| as a lower bound on local operator norm for targets with large negative modular energy. This identity is a standard, independently established result in operator algebras, not derived within the paper or from self-citations. Subsequent applications to Bisognano-Wichmann wedges and Casini-Huerta-Myers stress-tensor formulas likewise invoke externally grounded theorems. No fitted parameters are renamed as predictions, no ansatz is smuggled via self-citation, and the Reeh-Schlieder approximation cost bound follows directly from these prior results without reducing to the paper's own inputs by construction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two standard background results from algebraic quantum field theory and modular operator theory; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • standard math Tomita-Takesaki modular theory supplies a standard estimate that can be isolated as a model-independent preparation bound.
    Invoked directly in the abstract as the source of the lower bound.
  • standard math Bisognano-Wichmann theorem and Casini-Huerta-Myers formula give the modular Hamiltonian in the relevant geometries.
    Used to make the bound explicit for wedges and CFT regions.

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Reference graph

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