[Paper Review] An Entropic Einstein-Podolsky-Rosen Criterion

S. P. Walborn, A. Salles+3 more|ArXiv.org|Jul 24, 2009

Spectral Theory in Mathematical Physics2 references3 citations

TL;DR

This paper proposes an entropic Einstein-Podolsky-Rosen (EPR) criterion based on conditional Shannon entropy to detect quantum nonlocality in continuous-variable systems. It demonstrates experimentally that this criterion detects EPR nonlocality in non-Gaussian quantum states where the traditional variance-based EPR criterion fails, showing a violation of the entropic inequality by over 3 standard deviations while the variance criterion remains unviolated.

ABSTRACT

We propose an EPR inequality based on an entropic uncertainty relation for complementary continuous variable observables. This inequality is more sensitive than the previously established EPR inequality based on inferred variances, and opens up the possibility of EPR tests of quantum nonlocality in a wider variety of quantum states. We experimentally test the inequality using spatially entangled photons. For a particular quantum state, our experimental results show a violation of the entropic EPR inequality, while the variance EPR inequality is not violated.

Motivation & Objective

• To develop a more sensitive EPR criterion for detecting quantum nonlocality in continuous-variable systems, especially for non-Gaussian states.
• To address the limitation of the variance-based EPR criterion, which fails to detect nonlocality in certain non-Gaussian entangled states.
• To establish a connection between entropic uncertainty relations and EPR-type nonlocality, extending the scope of quantum nonlocality tests.
• To experimentally validate the new criterion using spatially entangled photons, demonstrating its superiority in detecting nonlocality.
• To provide a theoretical and experimental foundation for using entropy-based inequalities in quantum information protocols such as continuous-variable quantum key distribution.

Proposed method

• Formulates an EPR criterion based on the differential Shannon entropy of conditional probability distributions for complementary observables X and P.
• Derives the entropic EPR inequality: $ h(X_A|X_B) + h(P_A|P_B) \\_geq \ln \pi e $, where $ h(R_A|R_B) = -\int dr_B \mathcal{P}(r_B) h(R_A|r_B) $.
• Applies the entropic uncertainty relation $ h(X) + h(P) \geq \ln \pi e $ to conditional distributions under the assumption of local realism.
• Uses experimental data from spatially entangled photon pairs to compute discrete entropies from coincidence count distributions.
• Converts discrete entropies to differential entropies via $ h(Z) \approx H(Z) + \ln(z_{\text{step}}) $, accounting for spatial discretization.
• Calculates conditional differential entropies using $ h(R_i|R_j) = h(R_i,R_j) - h(R_j) $, and tests the entropic EPR inequality against the theoretical bound.

Experimental results

Research questions

• RQ1 Can an EPR criterion based on entropy detect nonlocality in non-Gaussian quantum states where the variance-based criterion fails?
• RQ2 How does the entropic EPR criterion compare in sensitivity to the traditional variance-based EPR criterion for continuous variables?
• RQ3 Is the entropic EPR inequality experimentally testable with current photonic setups using spatial degrees of freedom?
• RQ4 Does violation of the entropic EPR criterion guarantee a non-zero secret key rate in continuous-variable quantum key distribution?
• RQ5 Can entropic uncertainty relations be systematically used to derive new EPR-type nonlocality criteria beyond the variance-based approach?

Key findings

• The entropic EPR inequality was experimentally violated with $ h(X_A|X_B) + h(P_A|P_B) = 1.94 \pm 0.04 $, below the theoretical bound $ \ln \pi e \approx 2.145 $, indicating a clear violation by more than 3 standard deviations.
• The variance-based EPR criterion did not detect nonlocality for the same quantum state, demonstrating that the entropic criterion is more sensitive to non-Gaussian entanglement.
• Theoretical prediction of the sum of conditional entropies was $ 1.91 $, showing strong agreement with the experimental result.
• The entropic EPR criterion successfully revealed EPR nonlocality in a specific non-Gaussian state that remained undetected by the variance-based criterion.
• The violation of the entropic EPR inequality implies that local realism is inconsistent with quantum mechanics for this state, confirming the presence of nonlocality.
• The result establishes that the entropic EPR criterion provides a broader framework for detecting nonlocality in continuous-variable systems, particularly in non-Gaussian states.