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

[MarioKrenn6240]

What is the conjecture

This is problem #32 on the Open Quantum Problems site, attributed there to Nicolas Gisin and based on his paper “Bell inequalities: many questions, a few answers”.

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)
• Open Quantum Problems master list (for numbering/metadata)
• N. Gisin, “Bell inequalities: many questions, a few answers”
  • 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

• 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
• This issue is up for grabs: I would like to see this conjecture added by somebody else

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