What is the conjecture
This is problem #33 in the later Open Quantum Problems collection, a community-maintained continuation of Reinhard F. Werner's original collection.
Status: this problem is listed as solved on the Solved Quantum Problems page (solved by Ji–Natarajan–Vidick–Wright–Yuen, 2020; see below).
Problem (Open Quantum Problem #33: “Bell inequalities and operator algebras”).
Quantum Bell-type inequalities are defined in terms of two (or more) subsystems of a quantum system. The subsystems may be treated either via local Hilbert spaces — tensor factors of a global Hilbert space — or via commuting local operator algebras on a single Hilbert space. The latter approach is less restrictive: it only requires that observables belonging to different subsystems commute.
Are these two approaches equivalent?
Concretely, fix finite question sets X,Y and answer sets A,B, and consider bipartite correlations p(a,b|x,y).
A correlation is in the tensor-product model if there exist Hilbert spaces H_A, H_B, a unit vector ψ ∈ H_A ⊗ H_B, and POVMs {E_x^a}_a ⊂ B(H_A), {F_y^b}_b ⊂ B(H_B) such that
p(a,b|x,y) = ⟨ψ, (E_x^a ⊗ F_y^b) ψ⟩.
A correlation is in the commuting-operator model if there exist a Hilbert space H, a unit vector ψ ∈ H, and POVMs {E_x^a}_a, {F_y^b}_b ⊂ B(H) with
[E_x^a, F_y^b] = 0 for all x,y,a,b, such that
p(a,b|x,y) = ⟨ψ, E_x^a F_y^b ψ⟩.
The original question asks whether every commuting-operator correlation is already realizable in the tensor-product model. A weaker / approximation version asks whether every commuting-operator correlation can at least be approximated arbitrarily well by finite-dimensional tensor-product correlations (closure taken in the usual finite-dimensional topology on the table of probabilities p(a,b|x,y)).
Solved statement to formalize (Ji–Natarajan–Vidick–Wright–Yuen, 2020):
The approximation question has a negative answer. There exist finite question/answer alphabets X,Y,A,B and a commuting-operator correlation p(a,b|x,y) that does not lie in the closure of the finite-dimensional tensor-product correlations for that scenario. In the standard notation for quantum correlation sets, for some finite Bell scenario one has
C_qa(X,Y,A,B) ⊊ C_qc(X,Y,A,B).
Equivalently, there exists a finite two-player nonlocal game (or Bell functional) whose optimal value in the commuting-operator model is strictly larger than the supremum achievable by finite-dimensional tensor-product strategies, even after taking limits of such strategies.
This settles the density question from the OQP page in the negative: the tensor-product model is not dense in the commuting-operator model.
Important structure / caveats:
- In the finite-dimensional bipartite setting, the tensor-product and commuting-operator pictures are equivalent.
- Slofstra (2016/2020) already showed exact nonequivalence: there are nonlocal games with perfect commuting-operator strategies but no perfect tensor-product strategies.
- Ji et al. resolve the stronger approximation / closure question, and via earlier work of Junge–Navascués–Palazuelos–Pérez-García–Scholz–Werner and Fritz this also gives a negative resolution of Connes’ embedding problem / Kirchberg’s QWEP conjecture.
Where to find the details / references
Primary sources:
Key solution reference (as cited on the OQP page):
- Z. Ji, A. Natarajan, T. Vidick, J. Wright, and H. Yuen, “MIP=RE”*, arXiv: 2001.04383; see also Communications of the ACM 64(11), 131–138 (2021)
Related / partial results mentioned on the OQP page (useful background for a formalization roadmap):
- B. S. Tsirelson, “Some results and problems on quantum Bell-type inequalities”, Hadronic Journal Supplement 8, 329–345 (1993)
- V. B. Scholz and R. F. Werner, “Tsirelson’s Problem”, arXiv: 0812.4305
- M. Junge, M. Navascués, C. Palazuelos, D. Pérez-García, V. B. Scholz, and R. F. Werner, “Connes’ embedding problem and Tsirelson’s problem”, J. Math. Phys. 52, 012102 (2011); arXiv: 1008.1142
- T. Fritz, “Tsirelson’s problem and Kirchberg’s conjecture”, Rev. Math. Phys. 24(5), 1250012 (2012); arXiv: 1008.1168
- N. Ozawa, “About the Connes embedding conjecture: algebraic approaches”, Jpn. J. Math. 8(1), 147–183 (2013)
- W. Slofstra, “Tsirelson’s problem and an embedding theorem for groups arising from non-local games”, J. Amer. Math. Soc. 33, 1–56 (2020); arXiv: 1606.03140
- M. Navascués, S. Pironio, and A. Acín, “Bounding the set of quantum correlations”, Phys. Rev. Lett. 98, 010401 (2007)
Prerequisites needed
- Bell scenarios / nonlocal games: conditional probability tables
p(a,b|x,y), Bell inequalities, game values
- Basic quantum measurement formalism: Hilbert spaces, tensor products, unit vectors / states, POVMs, Born rule
- Operator-algebraic formulation of bipartite systems: commuting subalgebras, C*-algebras or von Neumann algebras
- Basic convexity / topology of correlation sets: closure, density, separation by linear functionals
- (Background for the solution pathway) Tsirelson/Connes/Kirchberg equivalences and the NPA hierarchy / semidefinite relaxations
- ams-81 (Quantum theory)
- ams-46 (Functional analysis)
- ams-47 (Operator theory)
Choose either option
What is the conjecture
This is problem #33 in the later Open Quantum Problems collection, a community-maintained continuation of Reinhard F. Werner's original collection.
Status: this problem is listed as solved on the Solved Quantum Problems page (solved by Ji–Natarajan–Vidick–Wright–Yuen, 2020; see below).
Concretely, fix finite question sets
X,Yand answer setsA,B, and consider bipartite correlationsp(a,b|x,y).A correlation is in the tensor-product model if there exist Hilbert spaces
H_A,H_B, a unit vectorψ ∈ H_A ⊗ H_B, and POVMs{E_x^a}_a ⊂ B(H_A),{F_y^b}_b ⊂ B(H_B)such thatp(a,b|x,y) = ⟨ψ, (E_x^a ⊗ F_y^b) ψ⟩.A correlation is in the commuting-operator model if there exist a Hilbert space
H, a unit vectorψ ∈ H, and POVMs{E_x^a}_a, {F_y^b}_b ⊂ B(H)with[E_x^a, F_y^b] = 0for allx,y,a,b, such thatp(a,b|x,y) = ⟨ψ, E_x^a F_y^b ψ⟩.The original question asks whether every commuting-operator correlation is already realizable in the tensor-product model. A weaker / approximation version asks whether every commuting-operator correlation can at least be approximated arbitrarily well by finite-dimensional tensor-product correlations (closure taken in the usual finite-dimensional topology on the table of probabilities
p(a,b|x,y)).Solved statement to formalize (Ji–Natarajan–Vidick–Wright–Yuen, 2020):
The approximation question has a negative answer. There exist finite question/answer alphabets
X,Y,A,Band a commuting-operator correlationp(a,b|x,y)that does not lie in the closure of the finite-dimensional tensor-product correlations for that scenario. In the standard notation for quantum correlation sets, for some finite Bell scenario one hasC_qa(X,Y,A,B) ⊊ C_qc(X,Y,A,B).Equivalently, there exists a finite two-player nonlocal game (or Bell functional) whose optimal value in the commuting-operator model is strictly larger than the supremum achievable by finite-dimensional tensor-product strategies, even after taking limits of such strategies.
This settles the density question from the OQP page in the negative: the tensor-product model is not dense in the commuting-operator model.
Important structure / caveats:
Where to find the details / references
Primary sources:
answer(sorry)and asymptotic problems #33 appears on the later website)Key solution reference (as cited on the OQP page):
Related / partial results mentioned on the OQP page (useful background for a formalization roadmap):
Prerequisites needed
p(a,b|x,y), Bell inequalities, game valuesAMS categories
Choose either option