Source-linked AI summary
The contextual fraction as a measure of contextuality
Samson Abramsky, Rui Soares Barbosa, Shane Mansfield
TL;DR
The paper connects optimal linear-program solutions for contextual models to Bell inequalities. It also examines computational explorations, monotonicity, and task-related consequences of contextuality.
Problem
The paper studies how contextual models relate to Bell inequalities and task performance.
Method
It uses linear programming and its strong-duality relation to construct Bell inequalities, alongside analyses of qubit models and contextuality-preserving operations.
Results
An optimal linear-program solution yields a Bell inequality whose normalized violation equals the contextual fraction of the model.
Takeaways & Limitations
The contextual fraction can provide a quantitative link between contextuality, Bell-inequality violation, and success or failure probabilities in tasks.
Abstract
from arXiv · showhide
We consider the contextual fraction as a quantitative measure of contextuality of empirical models, i.e. tables of probabilities of measurement outcomes in an experimental scenario. It provides a general way to compare the degree of contextuality across measurement scenarios; it bears a precise relationship to violations of Bell inequalities; its value, and a witnessing inequality, can be computed using linear programming; it is monotone with respect to the "free" operations of a resource theory for contextuality; and it measures quantifiable advantages in informatic tasks, such as games and a form of measurement based quantum computing.
The Contextual Fraction as a Measure of Contextuality—Supplemental Material
The supplemental material is attributed to Samson Abramsky, Rui Soares Barbosa, and Shane Mansfield, with affiliations at Oxford and Edinburgh.
- Samson Abramsky, Rui Soares Barbosa, and Shane Mansfield are listed as the authors.
- Abramsky and Barbosa are affiliated with the University of Oxford's Department of Computer Science.
- Mansfield is affiliated with the University of Edinburgh's School of Informatics.
- The paper lists PACS numbers 03.65.Ud and 03.67.Mn.
A. Contextual fraction and violations of Bell inequalities (Proof of Theorem 1)
The proof connects contextual fraction to optimally violated Bell inequalities through linear programming and characterizes how the associated decomposition behaves. It also records technical caveats concerning trivial witnesses and non-unique decompositions.
- A. Contextual fraction and violations of Bell inequalities (Proof of Theorem 1): Theorem 1 states that contextual fraction equals the maximum normalized Bell-inequality violation attained by an empirical model.Every Bell-inequality violation is bounded above by CF(e), and an inequality attaining that bound exists when CF(e) > 0.
- A. Contextual fraction and violations of Bell inequalities (Proof of Theorem 1): For the optimal witness, the non-contextual component is tight while the strongly contextual component reaches the algebraic maximum.The corresponding values are a*·veNC = 0 and a*·veSC = 1 when the relevant fractions are nonzero.
- A. Contextual fraction and violations of Bell inequalities (Proof of Theorem 1): The proof bounds any Bell-inequality violation by decomposing e into non-contextual and strongly contextual components.Linearity combines the non-contextual contribution, bounded by R, with the strongly contextual contribution, bounded by the algebraic maximum.
- A. Contextual fraction and violations of Bell inequalities (Proof of Theorem 1): The linear program's optimal solution supplies inequality coefficients whose normalized violation is CF(e).Strong duality relates the primal optimum NCF(e) to the dual witness, yielding an inequality with bound 0 and algebraic bound 1.
- A. Contextual fraction and violations of Bell inequalities (Proof of Theorem 1): When CF(e) = 0, the constructed Bell inequality may be trivial rather than separating non-contextual from no-signalling models.This occurs for models in the relative interior of the non-contextual polytope, motivating the theorem's extra assumption CF(e) > 0.
- A. Contextual fraction and violations of Bell inequalities (Proof of Theorem 1): The maximal decomposition into non-contextual and strongly contextual parts need not be unique.Non-uniqueness can arise from parallel faces of the no-signalling and non-contextual polytopes; it does not arise in the (2, 2, 2) Bell scenario.
B. Computational explorations
The Mathematica package computes contextual fractions and related Bell inequalities for general empirical models, and its explorations compare Bell- and GHZ-state measurement families. Bell-state models can attain Tsirelson violation without strong contextuality, whereas suitable GHZ measurements attain strong contextuality.
- Computational tools: The package computes contextual fractions and Bell inequalities for arbitrary quantum states, Hilbert spaces, and compatible observables.It also accepts empirical models and measurement scenarios through general computational constructions.
- Bell-state explorations: The Bell-state maxima achieve the Tsirelson violation of CHSH, but the corresponding empirical models are not strongly contextual.Table III lists the models corresponding to the maxima in the Bell-state plot.
- Bell-state explorations: For Bell-state measurements, the contextual fraction varies with the chosen equatorial angles: (0, π/2) is local, while shifting both values by π/8 reaches maximum CHSH violation.The paper explains this difference through the relative phase introduced by equivalent rotations of the state.
- GHZ-state explorations: For three- and four-partite GHZ states, selected equatorial measurements reach CF(e) = 1, indicating strong contextuality.For three parties, the usual Pauli Y and X measurements give the GHZ–Mermin model; other maxima are equivalent up to relabelling.
- GHZ-state explorations: Equatorial measurements at the proposition’s specified angles reproduce the strongly contextual GHZ–Mermin n-partite model up to relabelling.The proof uses Z-axis rotations and accounts for the resulting relative phase through measurement and outcome relabellings.
C. Monotonicity (Proof of Theorem 2)
The proof establishes contextual-fraction monotonicity by constructing feasible primal or dual linear-program solutions under the allowed operations. It also proves the combining-operation identities needed for the resource-theoretic result.
- Theorem 2: Theorem 2 states that contextual fraction is invariant under relabellings and non-increasing under measurement translation and outcome coarse-graining.The theorem also gives additional properties for mixing, product, and choice operations.
- Translation and coarse-graining: Measurement translation and outcome coarse-graining preserve feasible LP weights, yielding the monotonicity of contextual fraction under both operations.The constructions pull back or aggregate subprobability distributions while retaining their weights.
- Mixing: Mixing models combines optimal primal solutions into a feasible solution for the mixture, establishing the corresponding inequality through LP optimality.The vector representation of a mixture is the same convex combination of the component vectors.
- Product: For product models, tensor-product primal and dual solutions establish the required LP relation for the combined empirical model.The product scenario’s global assignments and incidence matrix correspond to tensor products of the component structures.
- Choice: For the choice operation, the non-contextual fraction satisfies NCF(e1 & e2) = min{NCF(e1), NCF(e2)}.The proof constructs a joint subprobability distribution with the minimum component weight, using a coupling lemma for equal-weight subdistributions.
D. Contextual fraction and l2-MBQC (Proof of Theorem 3)
The contextual fraction bounds the average failure probability of Z2-linear measurement-based quantum computation by the non-contextual fraction of its resource and the task’s distance from linear functions. The proof decomposes the resource into non-contextual and strongly contextual parts, showing that deterministic non-contextual resources induce only linear computations.
- Model: An l2-MBQC uses parity-based classical control and an empirical model on an (n, 2, 2) Bell scenario as its resource.Its classical processing is described by matrices, including a strictly lower triangular flow matrix that orders adaptive measurements.
- Performance measures: The computation maps each input bit string to a probability distribution over output bit strings, with worst-case and average success probabilities satisfying pS ≤ ¯pS.The objective is a Boolean function f: 2^m → 2^l.
- Theorem 3: Average failure satisfies ¯pF ≥ NCF(e)˜ν(f), linking computation hardness, success probability, and the resource’s non-contextual fraction.Here ˜ν(f) is the average distance of the objective function from the closest linear function.
- Proof: The proof writes e = NCF(e)eNC + CF(e)eSC and bounds total failure below by failures arising from the non-contextual component.The strongly contextual component is treated as if it always succeeds, so only the non-contextual component contributes to the lower bound.
- Proof: For deterministic non-contextual resources, the induced computation is deterministic and implements a linear function of the input.This establishes the lower bound because the task’s deviation from linearity is at least ˜ν(f).
E. Contextual fraction and games
The paper formulates constraint-solving games as empirical models and relates their success to contextual fraction. For k-consistent constraint systems, the resulting theorem connects task hardness, failure probability, and contextuality through the corresponding Bell-inequality violation.
- Constraint games: A constraint system consists of variables V, a domain D, and formulae Γ, with each input formula requiring an assignment to its appearing variables.A probabilistic strategy supplies a distribution over assignments for each formula and must satisfy compatibility across formulae.
- Constraint games: If n formulae are k-consistent, at most k can be jointly satisfied, and (n−k)/n measures the task’s normalised hardness.The game seeks valid assignments satisfying the input formula as often as possible, assuming equally probable inputs.
- Connection to contextuality: These games are alternative formulations of measurement scenarios, with variables as measurements, domain values as outcomes, and formula-variable sets as contexts.In Bell scenarios, the formulation specializes to multi-player nonlocal games under a no-communication interpretation.
- Theorem 4: For a k-consistent constraint system, Theorem 4 relates the strategy’s average failure probability to the contextual fraction of its empirical model.The theorem applies to valid strategies with average success pS and failure pF = 1 − pS.
- Proof: The contextual-fraction bound follows by translating the constraint system into a Bell inequality and identifying its normalized violation with the strategy’s success.Substitution of pS = 1 − pF and CF(p) = 1 − NCF(p) yields the stated relation.