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Necessary and Sufficient Condition for Randomness Certification from Incompatibility

Yi Li, Yu Xiang, Jordi Tura, Qiongyi He

Abstract

Quantum randomness can be certified from probabilistic behaviors demonstrating Bell nonlocality or Einstein-Podolsky-Rosen steering, leveraging outcomes from uncharacterized devices. However, such nonlocal correlations are not always sufficient for this task, necessitating the identification of required minimum quantum resources. In this work, we provide the necessary and sufficient condition for nonzero certifiable randomness in terms of measurement incompatibility and develop approaches to detect them. Firstly, we show that the steering-based randomness can be certified if and only if the correlations arise from a measurement compatibility structure that is not isomorphic to a hypergraph containing a star subgraph. In such a structure, the central measurement is individually compatible with the measurements at branch sites, precluding certifiable randomness in the central measurement outcomes. Subsequently, we generalize this result to the Bell scenario, proving that the violation of any chain inequality involving $m$ inputs and $d$ outputs rules out such a compatibility structure, thereby validating all chain inequalities as credible witnesses for randomness certification. Our results point out the role of incompatibility structure in generating random numbers, offering a way to identify minimum quantum resources for the task.

Necessary and Sufficient Condition for Randomness Certification from Incompatibility

Abstract

Quantum randomness can be certified from probabilistic behaviors demonstrating Bell nonlocality or Einstein-Podolsky-Rosen steering, leveraging outcomes from uncharacterized devices. However, such nonlocal correlations are not always sufficient for this task, necessitating the identification of required minimum quantum resources. In this work, we provide the necessary and sufficient condition for nonzero certifiable randomness in terms of measurement incompatibility and develop approaches to detect them. Firstly, we show that the steering-based randomness can be certified if and only if the correlations arise from a measurement compatibility structure that is not isomorphic to a hypergraph containing a star subgraph. In such a structure, the central measurement is individually compatible with the measurements at branch sites, precluding certifiable randomness in the central measurement outcomes. Subsequently, we generalize this result to the Bell scenario, proving that the violation of any chain inequality involving inputs and outputs rules out such a compatibility structure, thereby validating all chain inequalities as credible witnesses for randomness certification. Our results point out the role of incompatibility structure in generating random numbers, offering a way to identify minimum quantum resources for the task.
Paper Structure (4 sections, 5 theorems, 71 equations, 3 figures)

This paper contains 4 sections, 5 theorems, 71 equations, 3 figures.

Table of Contents

  1. Proof of Proposition 1
  2. Proof of Result 2
  3. Proof of Result 4
  4. Steering weight and steering-based randomness

Key Result

Proposition 1

Given an assemblage $\{\sigma_{a|x}^{\rm obs}\}$, the guessing probability satisfies $P_g^{\rm S}(x^*) < 1$ iff it lies outside the set $\mathcal{R}^{\rm S}_{x^*}$ for any $x^*\in\{1,\cdots,m\}$, where $\mathcal{R}^{\rm S}_{x^*}$ is defined as:

Figures (3)

  • Figure 1: Randomness certification based on quantum nonlocality. (a) The Bell scenario, involves both Alice and Bob receiving $m$ inputs and producing $d$ outputs, where each of them implements untrusted local measurements. Then, their behavior is described by the joint probability distributions $\{ p^{\rm obs}(ab|xy)\}$. (b) The steering scenario, which is similar to (a) but Bob performs trusted measurements, such that he holds the information of assemblage $\{\sigma_{a|x}^{\rm obs}\}$. In these scenarios, randomness on the outcomes of $x^*$ and $y^*$ can be certified based on the observed statistics. By constructing an incompatibility structure of the $m$ inputs, we clarify the necessary and sufficient condition for this task. Further, we prove that any chain inequality is a credible witness for randomness certification.
  • Figure 2: The compatibility structures of measurements. (a) A star graph $K_{1,m-1}\left(x^*\right)$. This means the measurement $x^*$ is compatible with every other measurement. (b) A hypergraph indicating two subsets of compatible measurements: $\left\{1, 2, 4\right\}$ and $\left\{1,3\right\}$. (c) The set of measurements $\{1,2\}$, together with every other measurement, is compatible.
  • Figure D1: The guessing probability and its lower bounds certified from the state $\rho_{AB}^{p,\theta}$. The red surface represents the exact value of $\min_{x^*}P_g^{\rm S}(x^*)$. The green and blue surfaces represent the lower bounds of $P_g^{\rm S}(x = {\rm argmin}_{x^*} P_g^{\rm S}(x^*))$ given by Eq. \ref{['eq:C5']} and Eq. \ref{['eq:C3']} respectively. For the same $\{\sigma_{a|x}^{\rm obs}\}$, the lower bound given by Eq. \ref{['eq:C5']} is tighter than the lower bound given by Eq. \ref{['eq:C3']}. Further, the thresholds for nonzero certifiable randomness on the green surface are the same as that of the red surface, which is consistent with Result \ref{['result 1']}.

Theorems & Definitions (19)

  • Proposition 1
  • proof
  • proof
  • proof
  • proof
  • Definition 1
  • Lemma 1
  • proof
  • Definition 2
  • Lemma 2
  • ...and 9 more