[Paper Review] The Relationship Between Discrete and Continuous Entropy in EPR-Steering Inequalities

James Schneeloch|arXiv (Cornell University)|Dec 9, 2013

Molecular Junctions and Nanostructures1 references3 citations

TL;DR

This paper establishes a rigorous mathematical connection between discrete and continuous entropy in the context of EPR-steering inequalities, demonstrating how discrete measurements can be used to derive and experimentally test continuous-variable EPR-steering inequalities. The key contribution is a strengthened steering inequality (20) that holds without assuming statistical independence across spatial degrees of freedom, enhancing the robustness of experimental violations of local hidden state models.

ABSTRACT

This document expands upon the relationship between discrete and continuous entropy given in (Phys. Rev. Lett. 110 130407), \Violating Continuous Variable Einstein-Podolsky-Rosen Steering with Discrete Measurements". We provide a detailed derivation for the inequality relating the continuous conditional entropy to its discrete approximation, and show how this connection works between discrete and continuous entropic quantities in general. In addition, we use this connection to show how to derive the continuous variable Einstein-Podolsky-Rosen steering inequality with discrete measurements as seen in (Phys. Rev. Lett. 110 130407), and make an additional comment which strengthens this result.

Motivation & Objective

• To formalize the relationship between discrete and continuous entropy in quantum information, particularly for EPR-steering scenarios.
• To provide a rigorous derivation of how discrete measurements approximate continuous conditional entropies in continuous-variable systems.
• To strengthen the EPR-steering inequality derived in prior work by removing the assumption of statistical independence across spatial degrees of freedom.
• To enable more robust experimental verification of continuous-variable EPR steering using discrete measurement outcomes.
• To clarify the connection between entropic uncertainty relations and EPR-steering in multi-dimensional systems.

Proposed method

• Derives the fundamental inequality $ h(y|x) \leq H(Y|X) + \log(\Delta y) $, linking continuous conditional entropy to its discrete approximation.
• Uses the decomposition of continuous entropy into conditional entropies over discrete measurement windows to relate $ h(x) $, $ H(X) $, and window size $ \Delta x $.
• Applies Jensen’s inequality and the concavity of entropy to bound the entropy of conditional distributions within discrete bins.
• Extends the entropy connection to joint and conditional entropies, including mutual information, for multiple variables.
• Uses the entropic uncertainty relation $ h(\vec{x}) + h(\vec{k}) \geq n\log(\pi e) $ as a foundation for the multi-dimensional steering inequality.
• Derives the final steering inequality $ H(\vec{X}_B|\vec{X}_A) + H(\vec{K}_B|\vec{K}_A) \geq \sum_{i=1}^n \log\left(\frac{\pi e}{\Delta x_{Bi}\Delta k_{Bi}}\right) $ without assuming independence across dimensions.

Experimental results

Research questions

• RQ1 How can discrete measurements be used to approximate continuous conditional entropies in EPR-steering experiments?
• RQ2 What is the precise mathematical relationship between discrete and continuous entropy in the context of continuous-variable quantum systems?
• RQ3 Can the EPR-steering inequality derived from continuous variables be rigorously derived using only discrete measurement outcomes?
• RQ4 Does the assumption of statistical independence across spatial degrees of freedom need to be imposed in multi-dimensional EPR-steering inequalities?
• RQ5 Can the strength of the EPR-steering inequality be enhanced by removing such assumptions?

Key findings

• The paper derives the inequality $ h(y|x) \leq H(Y|X) + \log(\Delta y) $, showing that continuous conditional entropy is bounded above by the discrete conditional entropy plus a term dependent on the measurement resolution.
• The fundamental entropy connection $ h(x) = \sum_{\ell} P(X_\ell) h_\ell(x) + H(X) $ is established, linking continuous and discrete entropies via binning.
• The derived steering inequality (20) is valid without assuming statistical independence across spatial degrees of freedom, strengthening its physical significance.
• The inequality $ h(\vec{x}^B|\vec{x}^A) + h(\vec{k}^B|\vec{k}^A) \geq n\log(\pi e) $ is shown to hold generally, based on the entropic uncertainty relation.
• The discrete steering inequality (20) is proven to be stronger than previously thought, as it does not rely on independence assumptions.
• The connection between discrete and continuous entropies extends naturally to mutual information, with $ h(x:y) \geq H(X:Y) $, and to multi-variable systems via $ h(\vec{x}|\vec{y}) \leq H(\vec{X}|\vec{Y}) + \log\left(\prod_{i=1}^n \Delta x_i\right) $.