Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Deterministic randomness extraction for semi-device-independent quantum random number generation

Pablo Tikas Pueyo, Tomás Fernández Martos, Gabriel Senno

TL;DR

This work develops deterministic randomness extraction for quantum entropy sources in a semi-device-independent prepare-and-measure setting with a state-overlap bound $\delta$. By deriving operator inequalities from the dual of Eve's guessing-probability SDP, the authors show that the DI deterministic extractors from Foreman_2025 remain valid in the semi-DI framework, enabling seedless randomness with finite rounds. They construct a spot-checking randomness-generation protocol and prove security bounds that relate trace-distance security to dual-SDP operators, complemented by numerical simulations on a novel behavior family $\mathbf{p}_\delta$ demonstrating positive key rates for as few as $7\times 10^{3}$ rounds and reasonable noise tolerance. The results offer a practical path to semi-DI randomness generation, while highlighting efficiency gaps relative to seeded extractors and outlining avenues for extension to other semi-DI settings.

Abstract

It is a well-known fact in classical information theory that no deterministic procedure can extract close-to-ideal randomness from an arbitrary entropy source. On the other hand, if additional knowledge about the source is available -- e.g., that it is a sequence of independent Bernoulli trials -- then deterministic extractors do exist. For quantum entropy sources, where in addition to classical random variables we consider quantum side information, the use of extra knowledge about their structure was pioneered in a recent publication [C. Foreman and L. Masanes, Quantum 9, 1654 (2025)]. In that work, the authors provide deterministic extractors for device-independent randomness generation with memoryless devices achieving a sufficiently high CHSH score. In this work, we extend their construction to the prepare-and-measure scenario. Specifically, we prove that the considered functions are also extractors for memoryless devices in a semi-device-independent setting under an overlap assumption on the prepared quantum states. We then simulate the resulting randomness generation protocol on a novel and experimentally relevant family of behaviors, observing positive key rates already for $7\times 10^3$ rounds.

Deterministic randomness extraction for semi-device-independent quantum random number generation

TL;DR

This work develops deterministic randomness extraction for quantum entropy sources in a semi-device-independent prepare-and-measure setting with a state-overlap bound . By deriving operator inequalities from the dual of Eve's guessing-probability SDP, the authors show that the DI deterministic extractors from Foreman_2025 remain valid in the semi-DI framework, enabling seedless randomness with finite rounds. They construct a spot-checking randomness-generation protocol and prove security bounds that relate trace-distance security to dual-SDP operators, complemented by numerical simulations on a novel behavior family demonstrating positive key rates for as few as rounds and reasonable noise tolerance. The results offer a practical path to semi-DI randomness generation, while highlighting efficiency gaps relative to seeded extractors and outlining avenues for extension to other semi-DI settings.

Abstract

It is a well-known fact in classical information theory that no deterministic procedure can extract close-to-ideal randomness from an arbitrary entropy source. On the other hand, if additional knowledge about the source is available -- e.g., that it is a sequence of independent Bernoulli trials -- then deterministic extractors do exist. For quantum entropy sources, where in addition to classical random variables we consider quantum side information, the use of extra knowledge about their structure was pioneered in a recent publication [C. Foreman and L. Masanes, Quantum 9, 1654 (2025)]. In that work, the authors provide deterministic extractors for device-independent randomness generation with memoryless devices achieving a sufficiently high CHSH score. In this work, we extend their construction to the prepare-and-measure scenario. Specifically, we prove that the considered functions are also extractors for memoryless devices in a semi-device-independent setting under an overlap assumption on the prepared quantum states. We then simulate the resulting randomness generation protocol on a novel and experimentally relevant family of behaviors, observing positive key rates already for rounds.

Paper Structure

This paper contains 16 sections, 5 theorems, 83 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Setting and assumptions
  3. Operator inequalities from dual solutions to the guessing probability
  4. Single- and multi-bit deterministic extractors
  5. Single-bit extraction.
  6. Multi-bit extraction.
  7. Private randomness generation protocol
  8. XOR extraction.
  9. Multi-bit extraction.
  10. Numerical simulations
  11. Conclusions
  12. Dual of the SDP for Eve's guessing probability in Eq. \ref{['eq:classical-pguess-povm']}
  13. Proof of Thm. \ref{['thm: ancilla operator inequality']}
  14. Proof of Thm. \ref{['thm: single-bit extraction security proof']}
  15. Proof of Thm. \ref{['thm: multi-bit extraction security proof']}
  16. ...and 1 more sections

Key Result

Theorem 1

For every $\delta\in[0,1]$, if $\langle \{\nu_{a,x}\}_{a,x=0}^1,\{H_\lambda\}_{\lambda=0}^1\rangle$ is a solution to Eq. eq:dual for some behavior $\mathbf{p}$, then, for any $\{\rho_{S}^x\}_x$ such that $F(\rho_S^0,\rho_S^1)\geq\delta$ and for any $\{\Pi_{SM}^a\}_{a=0}^1$, the following inequality with $G_M \coloneqq 2\sum_{a,x} \nu_{a,x} \Tr_{S}[(\rho_{S}^x \otimes \mathbbm{1}_{M}) \Pi_{SM}^a]

Figures (4)

  • Figure 1: Setup scheme. Schematic representation of the semi-device-independent model considered for the entropy source based on the overlap framework Brask_2017, featuring two trusted preparations with a known lower bound on their fidelity and an uncharacterized and untrusted measurement.
  • Figure 2: Single-round $P_{{\rm guess}}$. We plot $P_{{\rm guess}}(A|E,X=0,\mathbf{p}_\delta,\delta)$ for the behaviors $\mathbf{p}_\delta$ in Eq. \ref{['eq:behaviors']}.
  • Figure 3: Finite-size rates vs. $\delta$. For different values of the overlap $\delta$, we numerically lower bound the finite-size rates in Eq. \ref{['eq:finite-size-rate']} for an honest IID device with single-round behavior $\mathbf{p}_\delta$ in Eq. \ref{['eq:behaviors']}. Lower bounds to the asymptotic rates given by Eq. \ref{['eq:asym-rate-mul']} are also displayed with dashed lines.
  • Figure 4: Rate's noise tolerance. For $\delta=1/2$, we plot lower bounds to the finite-size rates achievable by a honest but noisy device with input-output statistics given by Eq. \ref{['eq:noisy-behavior']} and as a function of the uncoloured noise rate $\gamma$.

Theorems & Definitions (7)

  • Theorem 1
  • Definition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 1
  • proof