[Paper Review] Correlations and randomness generation based on energy constraints

Thomas van Himbeeck, Stefano Pironio|arXiv (Cornell University)|May 22, 2019

Molecular Communication and Nanonetworks28 references4 citations

TL;DR

This paper introduces a semi-device-independent quantum random number generator (QRNG) protocol based on energy constraints, where randomness is certified by observing quantum correlations under a bounded average energy. Using semidefinite programming, the authors derive a lower bound on the Shannon entropy of measurement outcomes, proving the protocol's soundness even with finite statistics and black-box measurement devices.

ABSTRACT

In a previous paper, we introduced a semi-device-independent scheme consisting of an untrusted source sending quantum states to an untrusted measuring device, with the sole assumption that the average energy of the states emitted by the source is bounded. Given this energy constraint, we showed that certain correlations between the source and the measuring device can only occur if the outcomes of the measurement are non-deterministic, i.e., these correlations certify the presence of randomness. In the present paper, we go further and show how to quantify the randomness as a function of the correlations and prove the soundness of a QRNG protocol exploiting this relation. For this purpose, we introduce (1) a semidefinite characterization of the set of quantum correlations, (2) an algorithm to lower-bound the Shannon entropy as a function of the correlations and (3) a proof of soundness using finite trials compatible with our energy assumption.

Motivation & Objective

• To develop a practical, semi-device-independent QRNG protocol that certifies randomness without requiring entanglement or loophole-free Bell tests.
• To quantify the randomness generated by quantum correlations under energy constraints, enabling rigorous entropy certification.
• To provide a soundness proof for a QRNG protocol under finite statistical trials while maintaining energy-based assumptions.
• To establish a semidefinite programming framework for characterizing quantum correlations and bounding entropy in this scenario.

Proposed method

• Introduces a semidefinite programming (SDP) characterization of the set of quantum correlations in a prepare-and-measure scenario with energy constraints.
• Develops an algorithm to compute converging lower bounds on the Shannon entropy of measurement outcomes using SDP optimization.
• Uses a dual formulation of the entropy-optimization problem to enable efficient numerical computation of entropy bounds.
• Applies a linearization technique to approximate the entropy function in the optimization, enabling tractable computation.
• Maps the quantum correlation set in the energy-constrained scenario to the quantum set in the Bell-CHSH scenario for theoretical consistency.
• Proves soundness of the QRNG protocol under finite trials by showing that non-classical correlations (outside deterministic models) imply positive entropy.

Experimental results

Research questions

• RQ1 Can quantum correlations under energy constraints be used to certify randomness in a semi-device-independent manner?
• RQ2 How can the amount of extractable randomness be quantitatively bounded from observed correlations in the energy-constrained scenario?
• RQ3 What is the relationship between the quantum correlation set in this energy-constrained prepare-and-measure scenario and the standard Bell-CHSH set?
• RQ4 Can a finite-sample soundness proof be established for a QRNG protocol relying only on energy bounds and not on full device modeling?
• RQ5 How can semidefinite programming be used to efficiently compute lower bounds on the entropy of measurement outcomes under energy constraints?

Key findings

• The authors derive a new semidefinite programming characterization of quantum correlations in the energy-constrained prepare-and-measure scenario, enabling precise entropy quantification.
• A converging series of lower bounds on the Shannon entropy of the measurement outcomes is computed via an SDP-based algorithm, with convergence guaranteed in the limit.
• The protocol is proven sound under finite statistics, ensuring that observed non-classical correlations imply positive randomness even with limited data.
• The minimum guessing probability $ G^ ext{max} $ is bounded via an optimization problem over classical models with energy constraints, and the resulting entropy $ -\log_2 G^ ext{max} $ is strictly positive for non-classical correlations.
• The set of classical (deterministic) correlations is fully characterized: for max-average energy constraints, classicality holds iff $ |E_1 + E_2| \leq 2(\omega_{\text{avg},1} + \omega_{\text{avg},2}) $, and for max-peak constraints, iff $ E_1 = E_2 $ in the non-trivial regime.
• The framework allows for quantum side information via the bound $ S(A|X,\Lambda) \geq -\log_2 G(A|X,\Lambda) $, extending the applicability to more general security models.