Measurement-dependence cost for Bell nonlocality: causal vs retrocausal models

TL;DR

This paper investigates the minimal measurement dependence cost needed to simulate Bell nonlocal correlations, comparing causal and retrocausal models, and finds retrocausal models can be more efficient, raising foundational questions about causality in quantum physics.

Contribution

It constructs the most efficient causal simulation of Bell nonlocality in the CHSH scenario and compares it with retrocausal models, highlighting differences in measurement dependence costs.

Findings

Maximal quantum violation requires ~0.080 bits of mutual information in causal models.

Retrocausal models can simulate maximal violation with only ~0.046 bits.

Retrocausality may be more efficient, challenging traditional causal assumptions.

Abstract

Device independent protocols based on Bell nonlocality, such as quantum key distribution and randomness generation, must ensure no adversary can have prior knowledge of the measurement outcomes. This requires a measurement independence assumption: that the choice of measurement is uncorrelated with any other underlying variables that influence the measurement outcomes. Conversely, relaxing measurement independence allows for a fully `causal' simulation of Bell nonlocality. We construct the most efficient such simulation, as measured by the mutual information between the underlying variables and the measurement settings, for the Clauser-Horne-Shimony-Holt (CHSH) scenario, and find that the maximal quantum violation requires a mutual information of just $\sim 0.080$ bits. Any physical device built to implement this simulation allows an adversary to have full knowledge of a cryptographic key or `random' numbers generated by a device independent protocol based on violation of the CHSH inequality. We also show that a previous model for the CHSH scenario, requiring only $\sim 0.046$ bits to simulate the maximal quantum violation, corresponds to the most efficient `retrocausal' simulation, in which future measurement settings necessarily influence earlier source variables. This may be viewed either as an unphysical limitation of the prior model, or as an argument for retrocausality on the grounds of its greater efficiency. Causal and retrocausal models are also discussed for maximally entangled two-qubit states, as well as superdeterministic, one-sided and zigzag causal models.