Universal $N$-Partite $d$-Level Pure-State Entanglement Witness Based on Realistic Measurement Settings

Entanglement witnesses are operators that are crucial for confirming the generation of specific quantum systems, such as multipartite and high-dimensional states. For this reason, many witnesses have been theoretically derived which commonly focus on establishing tight bounds and exhibit mathematical compactness as well as symmetry properties similar to that of the quantum state. However, for increasingly complex quantum systems, established witnesses have lacked experimental achievability, as it has become progressively more challenging to design the corresponding experiments. Here, we present a universal approach to derive entanglement witnesses that are capable of detecting the presence of any targeted complex pure quantum system and that can be customized towards experimental restrictions or accessible measurement settings. Using this technique, we derive experimentally optimized witnesses that are able to detect multipartite $d$-level cluster states, and that require only two measurement settings. We present explicit examples for customizing the witness operators given different realistic experimental restrictions, including witnesses for high-dimensional entanglement that use only two-dimensional projection measurements. Our work enables us to confirm the presence of probed quantum states using methods that are compatible with practical experimental realizations in different quantum platforms.

Figure 1

Complexity of the measurement settings vs noise tolerance of the witness. In any experiment, the measurement settings introduce an experimental noise floor (gray area), which increases with the measurement complexity. The optimal theoretical witness (orange square) has the highest noise tolerance, but it demands intricate and often unfeasible measurements. The optimal experimental witness (green circle) requires reduced measurement complexity, at the cost of having a lower noise tolerance. The goal is to maximize $\sigma$ (the shaded green or orange area), which represents the measurement confidence, given by the distance between the noise floor and the witness noise tolerance. Such maximization provides the highest confidence to measure a negative expectation value of the witness.