Device-independent witnesses of entanglement depth from two-body correlators

TL;DR

This paper develops device-independent witnesses for entanglement depth in many-body quantum systems using Bell inequalities, enabling certification with collective measurements suitable for current large-scale experiments.

Contribution

It introduces new DIWEDs based on Bell inequalities that require only collective measurements, with methods to compute their bounds both variationally and via semi-definite programming.

Findings

Effective measurement of DIWEDs with collective measurements.

Numerical and analytical bounds for large systems.

Compatibility with current experimental setups.

Abstract

We consider the characterization of entanglement depth in a quantum many-body system from the device-independent perspective; i.e., certifying how many particles are genuinely entangled without relying on assumptions on the system itself nor on the measurements performed. We obtain device-independent witnesses of entanglement depth (DIWEDs) using the Bell inequalities introduced in [J. Tura et al., Science 344, 1256 (2014)] and compute their $k$-producibility bounds. To this end, we present two complementary methods: First, a variational one, yielding a possibly optimal $k$-producible state. Second, a certificate of optimality via a semi-definite program, based on a relaxation of the quantum marginal problem. Numerical results suggest a clear pattern on $k$-producible bounds for large system sizes, which we then tackle analytically in the thermodynamic limit. Contrary to existing DIWEDs, the ones we present here can be effectively measured by accessing only collective measurements and second moments thereof. These technical requirements are met in current experiments, which have already been performed in the context of detecting Bell correlations in quantum many-body systems of $5\cdot 10^2 \sim 5 \cdot 10^5$ atoms.