[Paper Review] Tests for quantum contextuality in terms of $q$-entropies

Alexey E. Rastegin|arXiv (Cornell University)|Oct 25, 2012

Statistical Mechanics and Entropy18 references5 citations

TL;DR

This paper introduces a novel entropic formulation of Bell’s theorem using conditional Tsallis $q$-entropies for $q \geq 1$, generalizing standard entropic inequalities based on Shannon entropy. It derives a family of $q$-entropic Bell-type inequalities under the noncontextuality hypothesis, demonstrating quantum violations in both CHSH and KCBS scenarios, and shows improved robustness to detection inefficiencies—especially for $q=2$, which minimizes required detection efficiency in realistic setups.

ABSTRACT

The information-theoretic approach to Bell's theorem is developed with use of the conditional $q$-entropies. The $q$-entropic measures fulfill many similar properties to the standard Shannon entropy. In general, both the locality and noncontextuality notions are usually treated with use of the so-called marginal scenarios. These hypotheses lead to the existence of a joint probability distribution, which marginalizes to all particular ones. Assuming the existence of such a joint probability distribution, we derive the family of inequalities of Bell's type in terms of conditional $q$-entropies for all $q\geq1$. Quantum violations of the new inequalities are exemplified within the Clauser-Horne-Shimony-Holt (CHSH) and Klyachko-Can-Binicioǧlu-Shumovsky (KCBS) scenarios. An extension to the case of $n$-cycle scenario is briefly mentioned. The new inequalities with conditional $q$-entropies allow to expand a class of probability distributions, for which the nonlocality or contextuality can be detected within entropic formulation. The $q$-entropic inequalities can also be useful in analyzing cases with detection inefficiencies. Using two models of such a kind, we consider some potential advantages of the $q$-entropic formulation.

Motivation & Objective

• To extend the entropic formulation of Bell’s theorem beyond Shannon entropy to generalized Tsallis $q$-entropies for $q \geq 1$.
• To derive new Bell-type inequalities in terms of conditional $q$-entropies under the noncontextuality hypothesis.
• To analyze the robustness of these inequalities under detection inefficiencies, particularly in the CHSH and KCBS scenarios.
• To identify optimal $q$ values that minimize required detection efficiency for experimental verification.
• To demonstrate that $q$-entropic inequalities expand the class of probability distributions for which contextuality can be detected.

Proposed method

• Derives properties of conditional $q$-entropy for $q \geq 1$, including a generalized chain rule and a nonnegativity lemma.
• Applies the noncontextuality hypothesis to assume a joint probability distribution that marginalizes to all observed distributions.
• Constructs a family of $q$-entropic Bell-type inequalities based on the chain rule and nonnegativity of conditional $q$-entropy.
• Tests quantum violations in the CHSH and KCBS scenarios using known quantum states and measurement settings.
• Models detection inefficiencies via two schemes: single- and two-detector models, evaluating required detection efficiency $\eta$.
• Evaluates the ratio $r_q(\eta)$ to quantify robustness, showing that $q=2$ minimizes required efficiency for $\eta > 0.99$.

Experimental results

Research questions

• RQ1 Can the noncontextuality hypothesis be reformulated using conditional Tsallis $q$-entropies for $q \geq 1$?
• RQ2 Do $q$-entropic inequalities detect contextuality in scenarios like CHSH and KCBS, and how do they compare to Shannon-based formulations?
• RQ3 How does the choice of $q$ affect the robustness of entropic inequalities under detection inefficiencies?
• RQ4 Is there an optimal $q$ value that minimizes the required detection efficiency for observing quantum violations?
• RQ5 Can $q$-entropic inequalities detect contextuality in a broader class of probability distributions than standard entropic formulations?

Key findings

• The paper derives a family of $q$-entropic Bell-type inequalities for $q \geq 1$ based on the chain rule and nonnegativity of conditional $q$-entropy, generalizing the Braunstein–Caves inequality.
• Quantum violations of the $q$-entropic inequalities are demonstrated in both the CHSH and KCBS scenarios, confirming their validity for detecting contextuality.
• For the KCBS scenario, the maximum violation $\max \mathcal{C}_q$ is nearly maximal at $q=2$, making it a preferred choice for experimental testing.
• The required detection efficiency $\eta$ for observing violations is significantly reduced when using $q$-entropic inequalities, especially for $q=2$, where $r_q(0.99) \approx 0.3641$.
• In the two-detector model of detection inefficiency, $q=2$ yields the smallest ratio $r_q(\eta)$ for $\eta = 0.99$, indicating superior robustness.
• The $q$-entropic formulation allows detection of contextuality in a broader class of probability distributions than standard entropic approaches, offering an alternative to shared randomness models.