[Paper Review] Clauser-Horne inequality for qutrits

Dagomir Kaszlikowski, L. C. Kwek+3 more|ArXiv.org|Jun 3, 2001

Graph theory and applications3 citations

TL;DR

This paper introduces a new Clauser-Horne (CH)-type Bell inequality tailored for two entangled qutrits, derived from measurement probabilities rather than correlation functions. It demonstrates that a maximally entangled qutrit state violates this inequality when subjected to noise, with a threshold noise admixture of $ F_{\text{thr}} = \frac{11 - 6\sqrt{3}}{2} \approx 0.308 $, confirming strong nonlocality consistent with prior numerical and analytical results.

ABSTRACT

In this brief report we show the new Bell-Clauser-Horne inequality for two entangled three dimensional quantum systems (so called qutrits). This inequality is violated by a maximally entangled state of two qutrits observed via symmetric three input and three output port beamsplitter only if the amount of noise in the system equals ${11-6\sqrt3 \over 2}\approx 0.308$. This is in perfect agreement with the previous numerical calculations presented in Kaszlikowski {\it et. al.} Phys. Rev. Lett. {\bf 85}, 4418 (2000).

Motivation & Objective

• To derive a Bell inequality based on probabilities for two entangled qutrits, addressing the lack of a CH-type inequality for higher-dimensional systems.
• To provide an analytical framework that captures the nonlocality of qutrit entanglement more fundamentally than correlation-based inequalities.
• To verify that the proposed inequality is both necessary and sufficient for local realism, based on consistency with prior numerical and analytical results.
• To establish a quantitative threshold for noise admixture beyond which quantum correlations cannot be described by local hidden variables.

Proposed method

• Formulates a local realistic model using a joint probability distribution $ P_{\text{LR}}(a_1,a_2;b_1,b_2) $ that reproduces marginal probabilities.
• Derives a CH-type inequality based on measured probabilities, expressed as a linear combination of joint and single probabilities, with the constraint that the sum must be ≤ 0 for local realism.
• Uses a maximally entangled qutrit state $ |\psi\rangle = \frac{1}{\sqrt{3}}(|11\rangle + |22\rangle + |33\rangle) $ as the quantum state under test.
• Applies symmetric three-input, three-output beamsplitters (tritters) with tunable phase shifts to implement trichotomic measurements on both sides.
• Introduces a noise model via a mixed state $ \rho_F = (1-F)|\psi\rangle\langle\psi| + F \rho_{\text{noise}} $, where $ \rho_{\text{noise}} $ is maximally mixed.
• Computes quantum probabilities for specific phase configurations and evaluates the inequality’s violation as a function of noise parameter $ F $.

Experimental results

Research questions

• RQ1 Can a CH-type Bell inequality be formulated for two entangled qutrits based on measurement probabilities rather than correlation functions?
• RQ2 What is the maximum noise admixture $ F_{\text{thr}} $ for which a maximally entangled qutrit state still violates local realism?
• RQ3 Does the proposed inequality serve as a necessary and sufficient condition for local realism in the context of qutrit systems?
• RQ4 How does the nonlocality strength of qutrits compare to that of qubits under the same noise model?
• RQ5 Is the threshold noise value $ F_{\text{thr}} = \frac{11 - 6\sqrt{3}}{2} $ consistent with prior numerical and analytical findings?

Key findings

• The proposed CH-type inequality is violated by a maximally entangled qutrit state when the noise admixture $ F $ is below $ \frac{11 - 6\sqrt{3}}{2} \approx 0.308 $, indicating a breakdown of local realism.
• The threshold noise value $ F_{\text{thr}} \approx 0.308 $ matches the numerical result from Kaszlikowski et al. (2000), confirming consistency with prior work.
• The inequality is found to be consistent with the necessary and sufficient conditions for local realism established in earlier analytical proofs, suggesting it may also be necessary and sufficient.
• The quantum probabilities for the chosen phase settings are explicitly computed, showing specific values such as $ P_{QM}^{11}(1;2) = \frac{4 + 2\sqrt{3}}{27} $ and $ P_{QM}^{11}(2;2) = \frac{4 - 2\sqrt{3}}{27} $.
• Single probabilities are uniformly $ \frac{1}{3} $, indicating symmetric marginal distributions across all settings.
• The violation is robust under the specified measurement settings, with the noise threshold being the highest among all tested configurations, confirming optimal nonlocality.