Formalize Open Quantum Problem #12: Bell Inequalities for long range vacuum correlations

### What is the conjecture

This is problem [#12](https://oqp.iqoqi.oeaw.ac.at/bell-inequalities-for-long-range-vacuum-correlations) in [Reinhard F. Werner’s collection](https://arxiv.org/abs/quant-ph/0504166), later [collected by a community of quantum researchers](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems).

> **Problem (Open Quantum Problem #12: “Bell Inequalities for long range vacuum correlations”).**
> It is well known [Summers–Werner] that vacuum fluctuations maximally violate the CHSH‑Bell inequalities for suitable spacelike separated observables, and that this violation goes to zero as the two localization regions are moved apart.
> Decide whether some (necessarily small) violation of the inequalities is possible for regions arbitrarily far apart. For definiteness, consider a massive scalar free relativistic Bose field.

A standard formalization framework for this question is **algebraic quantum field theory (AQFT)**:

- To each (bounded) spacetime region `O` one associates a local *C*‑ or von Neumann algebra `𝒜(O)` of observables.
- If `O_A` and `O_B` are **spacelike separated**, then the corresponding algebras commute: `𝒜(O_A)` and `𝒜(O_B)` form a bipartite system in the sense of commuting subalgebras.
- The **vacuum state** `ω₀` of the (massive) free scalar field is a state on the global algebra, whose restriction to `𝒜(O_A) ∨ 𝒜(O_B)` encodes the vacuum correlations between the two regions.

For `A, A' ∈ 𝒜(O_A)` and `B, B' ∈ 𝒜(O_B)` that are **self-adjoint contractions** (e.g. `||A||,||A'||,||B||,||B'|| ≤ 1`), define the usual **CHSH expression**
`S(A,A',B,B') := ω₀( A (B + B') + A' (B - B') )`.

- In any local hidden variable (LHV) model one has the CHSH bound `|S| ≤ 2`.
- Quantum mechanically, `|S|` can reach `2√2` (Tsirelson bound).

It is convenient to package this as the **maximal CHSH/Bell correlation parameter** for the state `ω₀` between regions `O_A, O_B`:
`β(ω₀; O_A, O_B) := (1/2) sup_{A,A',B,B'} |S(A,A',B,B')|`,
where the supremum ranges over self-adjoint contractions localized in `O_A` / `O_B`.

Then:
- `β ≤ 1` means **no CHSH violation** (CHSH inequalities hold),
- `β > 1` means a **CHSH violation**.

**Open question (one natural precise form):**
Fix a bounded region `O` and translate it by a spacelike displacement of length `L` to get `O_L`. For the vacuum `ω₀` of the **massive free scalar field**, decide whether
- for **arbitrarily large** `L` one still has `β(ω₀; O, O_L) > 1` (possibly with `β - 1` extremely small),
or equivalently,
- whether there exists some `L₀` beyond which `β(ω₀; O, O_L) = 1` (no CHSH violation at sufficiently large separation).

The Open Quantum Problems page emphasizes that although the CHSH violation can be maximal at suitable spacelike separations, **the strength of violation tends to zero** as the two localization regions are moved far apart; the unresolved part is whether it becomes exactly zero (no violation) for sufficiently large separations, or remains positive (some violation) at every finite separation.

### Where to find the details / references

Primary sources:
- [Open Quantum Problems site (Problem #12)](https://oqp.iqoqi.oeaw.ac.at/bell-inequalities-for-long-range-vacuum-correlations)
- [Open Quantum Problems master list (for numbering/metadata)](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems)
- O. Krüger & R. F. Werner, “Some Open Problems in Quantum Information Theory” (Problem 12):
- https://arxiv.org/abs/quant-ph/0504166
- https://arxiv.org/pdf/quant-ph/0504166 (see “Problem 12”)

Key references cited on the OQP page:
- S. J. Summers and R. F. Werner, “The vacuum violates Bell’s inequalities” (Phys. Lett. A 110, 257–259 (1985)).
- H. Halvorson and R. Clifton, “Generic Bell correlation between arbitrary local algebras in quantum field theory” (J. Math. Phys. 41, 1711–1717 (2000)), arXiv: math-ph/9909013.
- R. Verch and R. F. Werner, “Distillability and positivity of partial transposes in general quantum field systems” (Rev. Math. Phys. 17, 545–576 (2005)), arXiv: quant-ph/0403089.

(Background recorded on the OQP page:)
- The vacuum is **not separable at any distance** (entanglement persists across arbitrary spacelike separations).
- An AQFT analogue of the **PPT** condition can fail for arbitrary regions at any distance.
- Nevertheless, the “long-range CHSH violation” question above remains open in the stated form.

### Prerequisites needed

- Bell inequalities (especially **CHSH**): LHV models, Tsirelson bound, and the operator-algebraic CHSH formulation with commuting subalgebras
- Basic operator algebra / functional analysis: *C*‑algebras or von Neumann algebras, states as positive normalized linear functionals, operator norm, self-adjoint contractions
- AQFT basics (Haag–Kastler viewpoint): local algebras `𝒜(O)`, locality/commutativity for spacelike separation, vacuum state
- Free (massive) scalar field basics (at a high level): Klein–Gordon field, smeared observables/Weyl operators, vacuum correlations and clustering
- (Optional, for the background remarks) entanglement notions beyond finite dimensions: separability in commuting-algebra bipartite systems, PPT analogues, distillability in AQFT

### [AMS categories](https://github.com/google-deepmind/formal-conjectures/labels?q=ams-)

- ams-81 (Quantum theory)
- ams-46 (Functional analysis)
- ams-47 (Operator theory)

### Choose either option

- [ ] I plan on adding this conjecture to the repository
- [X] This issue is up for grabs: I would like to see this conjecture added by somebody else
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