What is the conjecture
This is problem #26 in Reinhard F. Werner’s collection, later collected by a community of quantum researchers.
Problem (Open Quantum Problem #26: “Bell inequalities holding for all quantum states”).
In an (N,M,K) Bell scenario, compare the local polytope C, the quantum correlation set Q, and the no-signalling polytope P.
Problem 26.A asks whether every genuinely quantum boundary point of Q can already be realized using pure states, local K-dimensional Hilbert spaces, and complete K-outcome projective measurements.
Problem 26.B asks whether every nontrivial facet Bell inequality for C is violated by some quantum correlation, or equivalently whether C and Q can share any facet that is not already inherited from P.
A standard formalization is in terms of Bell scenarios and the nested sets
C_{N,M,K} ⊆ Q_{N,M,K} ⊆ P_{N,M,K}:
- Fix integers
N,M,K ≥ 2.
- A behavior is a family of conditional probabilities
p(a_1,...,a_N | x_1,...,x_N),
where x_i ∈ {1,...,M} is the measurement choice of party i and a_i ∈ {1,...,K} is its outcome, satisfying
p(a|x) ≥ 0 and ∑_a p(a|x) = 1 for every x.
p is no-signalling iff every marginal distribution for a subset of parties depends only on the measurement choices of those parties.
The set of all such behaviors is the no-signalling polytope P_{N,M,K}.p is local / classical iff it admits a local hidden-variable decomposition
p(a|x) = ∑_λ q(λ) ∏_{i=1}^N p_i(a_i | x_i, λ).
Equivalently, it lies in the convex hull of deterministic local strategies. This is the local polytope C_{N,M,K}.p is quantum iff there exist local Hilbert spaces H_1,...,H_N, a state ρ on H_1 ⊗ ··· ⊗ H_N, and local POVMs {E_i^{x_i}(a_i)} such that
p(a|x) = Tr[ρ (E_1^{x_1}(a_1) ⊗ ··· ⊗ E_N^{x_N}(a_N))].
The set of all such behaviors is the quantum set Q_{N,M,K}.
A Bell inequality is a linear functional
B(p) = ∑_{a,x} β_{a,x} p(a|x) ≤ L.
It is tight / proper if it defines a facet of C_{N,M,K} and is not merely a trivial inequality coming from positivity, normalization, or no-signalling (equivalently, the corresponding face of C is not already a face of P).
The title phrase “holding for all quantum states” can then be formalized as:
for every quantum behavior q ∈ Q_{N,M,K}, one has B(q) ≤ L.
With this notation, Problem 26.B becomes:
- Main question / open geometric formulation: for every proper Bell inequality
B(p) ≤ L for C_{N,M,K}, does there exist a quantum behavior q ∈ Q_{N,M,K} with B(q) > L?
- Equivalently, do
C_{N,M,K} and Q_{N,M,K} share any nontrivial facets/supporting hyperplanes beyond those inherited from P_{N,M,K}?
- Equivalently again, does there exist a proper Bell inequality with
sup_{q ∈ Q_{N,M,K}} B(q) = L,
i.e. a nontrivial Bell inequality that is satisfied by all quantum behaviors?
For Problem 26.A, one natural formalization is to define Q_{N,M,K}^{min} as the set of behaviors realizable by pure states on local K-dimensional Hilbert spaces using complete K-outcome projective measurements, and ask whether every genuinely quantum boundary behavior already lies in Q_{N,M,K}^{min}.
The current status is that the two sub-problems behave differently:
- Problem 26.A has a negative answer in general.
- Problem 26.B has a negative answer for some multipartite scenarios (
N ≥ 3), but remains open in its natural bipartite form.
- So the central remaining open version is:
determine whether every proper bipartite Bell inequality is violated by some bipartite quantum correlation.
Where to find the details / references
Primary sources:
Key related references:
- Ll. Masanes, “Extremal quantum correlations for
N parties with two dichotomic observables per site” (arXiv: quant-ph/0512100)
(proves Problem 26.A in the special case (N,M,K) = (N,2,2))
- T. Vértesi and K. F. Pál, “Generalized Clauser-Horne-Shimony-Holt inequalities maximally violated by higher dimensional systems” (Phys. Rev. A 77, 042106 (2008); arXiv: 0712.4225)
(negative answer to Problem 26.A in general)
- M. L. Almeida, J.-D. Bancal, N. Brunner, A. Acín, N. Gisin, and S. Pironio, “Guess your neighbour’s input: a multipartite non-local game with no quantum advantage” (Phys. Rev. Lett. 104, 230404 (2010); arXiv: 1003.3844)
(negative answer to Problem 26.B for more than two parties)
- N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality” (Rev. Mod. Phys. 86, 419 (2014); arXiv: 1303.2849)
(standard review/background, including the bipartite facet question)
- R. Ramanathan, M. T. Quintino, A. B. Sainz, G. Murta, and R. Augusiak, “On the tightness of correlation inequalities with no quantum violation” (Phys. Rev. A 95, 012139 (2017); arXiv: 1607.05714)
(shows important bipartite no-quantum-advantage correlation inequalities are not facets)
- W. Slofstra, “The set of quantum correlations is not closed” (Forum Math. Pi 7, e1 (2019); arXiv: 1703.08618)
(shows finite input/output Bell scenarios can require limiting / infinite-dimensional quantum strategies)
- L. Escolà-Farràs, J. Calsamiglia, and A. Winter, “All tight correlation Bell inequalities have quantum violations” (Phys. Rev. Research 2, 012044 (2020); arXiv: 1908.06669)
(settles the two-party XOR / correlation-facet case positively)
- R. Ramanathan, “All two-party facet Bell inequalities are violated by Almost Quantum correlations” (Phys. Rev. Research 3, 033100 (2021); arXiv: 2004.07673)
(shows the stronger “almost-quantum” relaxation violates every two-party facet Bell inequality)
- T. P. Le, C. Meroni, B. Sturmfels, R. F. Werner, and T. Ziegler, “Quantum Correlations in the Minimal Scenario” (Quantum 7, 947 (2023); arXiv: 2111.06270)
(modern detailed study of the smallest bipartite scenario (2,2,2))
Prerequisites needed
- Finite probability distributions and conditional probability tables
- Bell scenarios: local hidden-variable models, no-signalling, and quantum behaviors
- Basic convex geometry: convex hulls, polytopes, faces/facets, supporting hyperplanes
- Basic linear algebra / quantum mechanics: Hilbert spaces, tensor products, density operators, POVMs, projective measurements
- (Optional, for partial results and computational approaches) semidefinite programming, Tsirelson bounds, XOR games / nonlocal games, and dimension witnesses
- ams-81 (Quantum theory)
- ams-52 (Convex and discrete geometry)
- ams-68 (Computer science)
- ams-94 (Information and communication theory)
Choose either option
What is the conjecture
This is problem #26 in Reinhard F. Werner’s collection, later collected by a community of quantum researchers.
A standard formalization is in terms of Bell scenarios and the nested sets
C_{N,M,K} ⊆ Q_{N,M,K} ⊆ P_{N,M,K}:N,M,K ≥ 2.p(a_1,...,a_N | x_1,...,x_N),where
x_i ∈ {1,...,M}is the measurement choice of partyianda_i ∈ {1,...,K}is its outcome, satisfyingp(a|x) ≥ 0and∑_a p(a|x) = 1for everyx.pis no-signalling iff every marginal distribution for a subset of parties depends only on the measurement choices of those parties.The set of all such behaviors is the no-signalling polytope
P_{N,M,K}.pis local / classical iff it admits a local hidden-variable decompositionp(a|x) = ∑_λ q(λ) ∏_{i=1}^N p_i(a_i | x_i, λ).Equivalently, it lies in the convex hull of deterministic local strategies. This is the local polytope
C_{N,M,K}.pis quantum iff there exist local Hilbert spacesH_1,...,H_N, a stateρonH_1 ⊗ ··· ⊗ H_N, and local POVMs{E_i^{x_i}(a_i)}such thatp(a|x) = Tr[ρ (E_1^{x_1}(a_1) ⊗ ··· ⊗ E_N^{x_N}(a_N))].The set of all such behaviors is the quantum set
Q_{N,M,K}.A Bell inequality is a linear functional
B(p) = ∑_{a,x} β_{a,x} p(a|x) ≤ L.It is tight / proper if it defines a facet of
C_{N,M,K}and is not merely a trivial inequality coming from positivity, normalization, or no-signalling (equivalently, the corresponding face ofCis not already a face ofP).The title phrase “holding for all quantum states” can then be formalized as:
for every quantum behavior
q ∈ Q_{N,M,K}, one hasB(q) ≤ L.With this notation, Problem 26.B becomes:
B(p) ≤ LforC_{N,M,K}, does there exist a quantum behaviorq ∈ Q_{N,M,K}withB(q) > L?C_{N,M,K}andQ_{N,M,K}share any nontrivial facets/supporting hyperplanes beyond those inherited fromP_{N,M,K}?sup_{q ∈ Q_{N,M,K}} B(q) = L,i.e. a nontrivial Bell inequality that is satisfied by all quantum behaviors?
For Problem 26.A, one natural formalization is to define
Q_{N,M,K}^{min}as the set of behaviors realizable by pure states on localK-dimensional Hilbert spaces using completeK-outcome projective measurements, and ask whether every genuinely quantum boundary behavior already lies inQ_{N,M,K}^{min}.The current status is that the two sub-problems behave differently:
N ≥ 3), but remains open in its natural bipartite form.determine whether every proper bipartite Bell inequality is violated by some bipartite quantum correlation.
Where to find the details / references
Primary sources:
https://arxiv.org/abs/quant-ph/0504166
https://arxiv.org/pdf/quant-ph/0504166
Key related references:
Nparties with two dichotomic observables per site” (arXiv: quant-ph/0512100)(proves Problem 26.A in the special case
(N,M,K) = (N,2,2))(negative answer to Problem 26.A in general)
(negative answer to Problem 26.B for more than two parties)
(standard review/background, including the bipartite facet question)
(shows important bipartite no-quantum-advantage correlation inequalities are not facets)
(shows finite input/output Bell scenarios can require limiting / infinite-dimensional quantum strategies)
(settles the two-party XOR / correlation-facet case positively)
(shows the stronger “almost-quantum” relaxation violates every two-party facet Bell inequality)
(modern detailed study of the smallest bipartite scenario
(2,2,2))Prerequisites needed
AMS categories
Choose either option