[Paper Review] Tight analytic bound on the trade-off between device-independent randomness and nonlocality

Lewis Wooltorton, Peter Brown+1 more|arXiv (Cornell University)|Apr 29, 2022

Quantum Mechanics and Applications44 references31 citations

TL;DR

This paper establishes tight analytic bounds on the maximum device-independent (DI) randomness that can be certified from quantum correlations with a given CHSH value. It introduces two families of Bell inequalities that self-test quantum strategies achieving maximal randomness: up to 2 bits for CHSH values in (2, 3√3/2], and a smooth, monotonically decreasing randomness for higher values up to 2√2. The key contribution is a complete characterization of the optimal trade-off between nonlocality and certifiable randomness, resolving an open question on whether 2 bits of randomness can be achieved without approaching the local set.

ABSTRACT

Two parties sharing entangled quantum systems can generate correlations that cannot be produced using only shared classical resources. These nonlocal correlations are a fundamental feature of quantum theory but also have practical applications. For instance, they can be used for device-independent (DI) random number generation, whose security is certified independently of the operations performed inside the devices. The amount of certifiable randomness that can be generated from some given non-local correlations is a key quantity of interest. Here we derive tight analytic bounds on the maximum certifiable randomness as a function of the nonlocality as expressed using the Clauser-Horne-Shimony-Holt (CHSH) value. We show that for every CHSH value greater than the local value ($2$) and up to $3\sqrt{3}/2\approx2.598$ there exist quantum correlations with that CHSH value that certify a maximal two bits of global randomness. Beyond this CHSH value the maximum certifiable randomness drops. We give a second family of Bell inequalities for CHSH values above $3\sqrt{3}/2$, and show that they certify the maximum possible randomness for the given CHSH value. Our work hence provides an achievable upper bound on the amount of randomness that can be certified for any CHSH value. We illustrate the robustness of our results, and how they could be used to improve randomness generation rates in practice, using a Werner state noise model.

Motivation & Objective

• To resolve the open question of whether 2 bits of device-independent (DI) randomness can be achieved in the 2-input, 2-output scenario without requiring the CHSH violation to approach the local boundary.
• To derive an achievable upper bound on the maximum DI randomness that can be certified for any given CHSH value, addressing the non-trivial trade-off between nonlocality and randomness.
• To construct two families of Bell inequalities that self-test quantum strategies achieving the maximum possible randomness for their respective CHSH value ranges.
• To demonstrate the robustness of the constructions under noise, particularly in the Werner state model, and to show practical advantages over standard CHSH-based protocols.

Proposed method

• Introduces a first family of Bell expressions that self-test two-qubit strategies achieving exactly 2 bits of global randomness for all CHSH values in the interval (2, 3√3/2].
• Proposes a second family of Bell inequalities for CHSH values in [3√3/2, 2√2], which certify a smooth, monotonically decreasing amount of randomness as a function of the CHSH value.
• Uses semidefinite programming (SDP) duality to derive upper bounds on the conditional von Neumann entropy H(AB|X=0,Y=0,E), which quantifies the DI randomness.
• Employs self-testing techniques to show that the derived bounds are tight and achievable via explicit quantum strategies parameterized by an angle γ ∈ [0, π/12].
• Derives an explicit parametric expression for the randomness R(s) as a function of the CHSH value s, using trigonometric identities and inverse functions.
• Analyzes robustness under a Werner state noise model, comparing the new constructions to tilted CHSH inequalities and showing both are robust at practical noise levels.

Experimental results

Research questions

• RQ1 Can 2 bits of device-independent randomness be certified in the 2-input, 2-output scenario without the CHSH violation tending toward the local boundary (i.e., without requiring the strategy to be close to classical correlations)?
• RQ2 What is the maximum amount of DI randomness that can be certified for any given CHSH value, and how does this maximum vary with nonlocality?
• RQ3 Are there Bell inequalities other than the extremal CHSH inequality that can certify more randomness, and if so, what are their self-testing properties?
• RQ4 How does the robustness of the new randomness certification protocols compare to existing ones, such as those based on tilted CHSH inequalities, under realistic noise models?

Key findings

• For all CHSH values in the interval (2, 3√3/2], there exist quantum strategies that certify exactly 2 bits of global device-independent randomness.
• The maximum achievable randomness drops smoothly and monotonically for CHSH values in [3√3/2, 2√2], with the exact functional form given by R(s) = 1 + H_bin(1/2 + s/2 - 3√2/2 * cos(1/3 arccos(-s/(2√2)))), where H_bin is the binary entropy.
• The upper bound on randomness is tight and achievable: the constructed Bell inequalities self-test the quantum states and measurements that achieve the maximum randomness for each CHSH value.
• The construction achieves maximal randomness without requiring extra measurements, full distribution constraints, or strategies approaching the local set, resolving a key open question from prior work.
• The new protocols are robust under the Werner state noise model, and at any given noise level, there exists an optimal CHSH-based statistic that outperforms the standard CHSH inequality for practical randomness generation.
• The paper proves that the derived bound is the true maximum by showing that the upper bound from SDP duality coincides with a lower bound derived from an explicit family of self-testing strategies.