Formalize Open Quantum Problem #32: Bell inequalities: many questions, a few answers

### What is the conjecture

This is problem [#32](https://oqp.iqoqi.oeaw.ac.at/bell-inequalities-many-questions-a-few-answers) on the [Open Quantum Problems site](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems), attributed there to Nicolas Gisin and based on his paper [“Bell inequalities: many questions, a few answers”](https://arxiv.org/abs/quant-ph/0702021).

> **Problem (Open Quantum Problem #32: “Bell inequalities: many questions, a few answers”).**

Unlike several earlier OQP entries, this is not a single conjecture but a **bundle of Bell-nonlocality problems**. A faithful formalization is to encode the shared Bell-scenario framework and then state the individual subquestions. In substance, the page asks:

* **A. Hidden nonlocality:** does every entangled state become Bell-nonlocal after suitable local filtering? Is there an example where one needs a sequence of local filters rather than just one?
* **B. Superactivation:** can a Bell-local state `ρ` become Bell-nonlocal after taking finitely many tensor copies `ρ^{⊗ n}`?
* **C. Universal multipartite detection:** for each `n`, find genuinely `n`-party Bell inequalities violated by every `n`-party pure entangled state.
* **D. POVM advantage:** find a Bell inequality whose optimal quantum violation on some state genuinely requires POVMs and cannot already be attained with projective measurements.
* **E. Real vs complex quantum theory:** determine whether Bell data can distinguish real-Hilbert-space quantum mechanics from complex-Hilbert-space quantum mechanics.
* **F. Bound entanglement and Bell violation:** decide whether bound-entangled states can violate Bell inequalities.
* **G. Detecting nonlocality:** given a multipartite state `ρ`, provide an effective procedure to decide or certify whether `ρ` is Bell-nonlocal.
* **H. Experiment-friendly inequalities:** find Bell inequalities that are practical for current experiments while closing the standard loopholes.
* **I. Homodyne/optics inequalities:** find Bell inequalities adapted to simple optical states and homodyne detection.
* **J. One-bit simulation:** find a Bell inequality satisfied by every correlation simulable with one classical bit of communication, but violated by some partially entangled two-qubit state.
* **K. Two-PR-box simulation:** characterize inequalities satisfied by all correlations obtainable using two PR boxes.
* **L. Finite non-signalling simulation:** find a finitely described non-signalling box that can simulate partially entangled states.
* **M. Nonlocality vs secrecy:** ask whether every nonlocal correlation yields distillable secret key.

A natural common formalization framework is the language of Bell scenarios (typically with finite inputs/outputs, or finite-output binnings when continuous-variable measurements are involved):

* There are `N` parties. Party `i` receives an input `x_i` from a finite set `X_i` and returns an output `a_i` in a finite set `A_i`.
* A **behavior** is a conditional probability table `p(a_1,\dots,a_N | x_1,\dots,x_N)`.
* `p` is **local** iff there exists a shared hidden variable `λ` with distribution `μ` and local response functions such that
`p(a_1,\dots,a_N | x_1,\dots,x_N) = ∫ dμ(λ) ∏_{i=1}^N p_i(a_i | x_i, λ)`.
* A **Bell inequality** is a linear inequality `L(p) ≤ β` satisfied by all local behaviors.
* A **quantum behavior** is one of the form
`p(a_1,\dots,a_N | x_1,\dots,x_N) = Tr[ρ (M_{a_1|x_1}^{(1)} ⊗ ··· ⊗ M_{a_N|x_N}^{(N)})]`,
where `ρ` is a multipartite state and the `M_{a_i|x_i}^{(i)}` are local POVM elements.
* A **non-signalling** behavior is one whose marginals on any subset of parties do not depend on the measurement choices of the complementary parties.
* A **PR-box** is the standard bipartite non-signalling box with binary inputs/outputs `x,y,a,b ∈ {0,1}` and
`p(a,b|x,y) = 1/2` when `a ⊕ b = x y`, and `0` otherwise.
* **Local filtering** means applying local trace-nonincreasing operations and conditioning on success before the Bell test.

Within this shared framework, Problem #32 asks for existence / classification / algorithmic results about:
* activation of nonlocality under filtering or tensor powers (A,B),
* multipartite Bell inequalities detecting all pure entangled states of a fixed number of parties (C),
* separations between projective measurements and general POVMs (D),
* separations between real and complex quantum realizations (E),
* Bell violations by bound-entangled states (F),
* algorithms for nonlocality detection (G),
* experimentally friendly Bell inequalities (H,I),
* simulation power of bounded communication or finitely many PR boxes (J,K,L),
* and the relation between nonlocality and secret-key distillation (M).

Because Problem #32 is a package rather than a single theorem, the most faithful formalization target is a **meta-issue** encoding the common Bell-scenario definitions together with the list of subproblems A–M.

The OQP page also records substantial partial progress:
* Problem A: partial negative progress—there are entangled two-qubit Werner states that remain local even after local filtering.
* Problems B, D, E, F, H, I: marked there as solved.
* Problem G: marked there as essentially solved algorithmically, up to `δ`-approximation.
* Problem J: marked there as partially advanced; one bit suffices for sufficiently weakly entangled two-qubit pure states `\sqrt{p}|00⟩ + \sqrt{1-p}|11⟩` with `p ≥ 0.835`, and a trit suffices for arbitrary entangled two-qubit pure states under local projective measurements.
* Thus the most obviously unresolved parts on the current OQP page are C, K, L, M, together with the unresolved aspects of A and J.

### Where to find the details / references

Primary sources:

* [Open Quantum Problems site (Problem #32)](https://oqp.iqoqi.oeaw.ac.at/bell-inequalities-many-questions-a-few-answers)
* [Open Quantum Problems master list (for numbering/metadata)](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems)
* N. Gisin, [“Bell inequalities: many questions, a few answers”](https://arxiv.org/abs/quant-ph/0702021)
* [https://arxiv.org/pdf/quant-ph/0702021](https://arxiv.org/pdf/quant-ph/0702021)

Key related references (as listed on the OQP page):

* S. Popescu, “Bell’s inequalities and density matrices. Revealing hidden nonlocality.” (Phys. Rev. Lett. 74, 2619–2622 (1995); arXiv: quant-ph/9502005)
(hidden nonlocality; relevant for A)
* F. Hirsch, M. Túlio Quintino, J. Bowles, T. Vértesi, and N. Brunner, “Entanglement without hidden nonlocality” (New J. Phys. 18, 113019 (2016); arXiv: 1606.02215)
(partial negative result for A)
* C. Palazuelos, “Superactivation of quantum nonlocality” (Phys. Rev. Lett. 109, 190401 (2012); arXiv: 1205.3118)
(solves B)
* S. Khot and N. K. Vishnoi, “The Unique Games Conjecture, Integrality Gap for Cut Problems and Embeddability of Negative Type Metrics into `ℓ₁`” (Proc. 46th IEEE FOCS, 53–62 (2005))
(the game used in Palazuelos’ solution of B)
* T. Vértesi and E. Bene, “A two-qubit Bell inequality for which POVM measurements are relevant” (Phys. Rev. A 82, 062115 (2010); arXiv: 1007.2578)
(solves D)
* K. F. Pál and T. Vértesi, “Efficiency of higher-dimensional Hilbert spaces for the violation of Bell inequalities” (Phys. Rev. A 77, 042105 (2008); arXiv: 0712.4320)
(bipartite part of E)
* M. McKague, M. Mosca, and N. Gisin, “Simulating quantum systems using real Hilbert spaces” (Phys. Rev. Lett. 102, 020505 (2009); arXiv: 0810.1923)
(multipartite extension for E)
* T. Vértesi and N. Brunner, “Quantum nonlocality does not imply entanglement distillability” (Phys. Rev. Lett. 108, 030403 (2012); arXiv: 1106.4850)
(tripartite part of F)
* T. Vértesi and N. Brunner, “Bell nonlocality from bipartite bound entanglement” (Nat. Commun. 5, 5297 (2014); arXiv: 1405.4502)
(bipartite part of F)
* F. Hirsch, M. T. Quintino, T. Vértesi, M. F. Pusey, and N. Brunner, “Algorithmic construction of local hidden variable models for entangled quantum states” (Phys. Rev. Lett. 117, 190402 (2016); arXiv: 1512.00262)
(algorithmic progress for G)
* J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories” (Phys. Rev. Lett. 23, 880–884 (1969))
(CHSH; relevant for H)
* B. Hensen et al., “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres” (Nature 526, 682–686 (2015))
(first loophole-free Bell test; relevant for H)
* D. Cavalcanti, N. Brunner, P. Skrzypczyk, A. Salles, and V. Scarani, “Large violation of Bell inequalities using both particle and wave measurements” (Phys. Rev. A 84, 022105 (2011); arXiv: 1012.1916)
(solves I)
* M. J. Renner and M. T. Quintino, “The minimal communication cost for simulating entangled qubits” (Quantum 7, 1149 (2023); arXiv: 2207.12457)
(recent progress on J)

(There are also newer papers on one-bit simulation, nonlocality activation, and algorithmic local-model detection, but the OQP page above plus Gisin’s paper are the core references for writing the issue.)

### Prerequisites needed

* Quantum information / foundations: Bell scenarios, Bell inequalities, local hidden-variable models, quantum states, entanglement, and local measurements (projective and POVM)
* Finite probability distributions and conditional distributions; non-signalling constraints
* Convex geometry / polyhedral combinatorics of local, quantum, and non-signalling correlation sets
* Multipartite entanglement / nonlocality, local filtering, tensor powers, and bound entanglement
* (Optional, for several subproblems) communication complexity, PR boxes / nonlocal boxes, and device-independent cryptography

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

* ams-81 (Quantum theory)
* ams-94 (Information and communication theory)
* ams-52 (Convex and discrete geometry)
* ams-68 (Computer science)

### 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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