Coarse graining is a common imperfection of realistic quantum measurement, obstructing the direct observation of quantum features. Under highly coarse-grained measurement, we experimentally certify the continuous-variable nonclassicality of both Gaussian and non-Gaussian states. Remarkably, we find that this coarse-grained measurement outperforms the conventional fine-grained measurement for nonclassicality certification: it detects nonclassicality beyond the reach of the variance criterion, and furthermore, it exhibits stronger statistical significance than the high-order moments method. Our work demonstrates the usefulness of coarse-grained measurement by providing a reliable and efficient way of nonclassicality certification for quantum technologies.
Uncertainty relations involving incompatible observables are one of the cornerstones of quantum mechanics. Aside from their fundamental significance, they play an important role in practical applications, such as detection of quantum correlations and security requirements in quantum cryptography. In continuous variable systems, the spectra of the relevant observables form a continuum and this necessitates the coarse graining of measurements. However, these coarse-grained observables do not necessarily obey the same uncertainty relations as the original ones, a fact that can lead to false results when considering applications. That is, one cannot naively replace the original observables in the uncertainty relation for the coarse-grained observables and expect consistent results. As such, several uncertainty relations that are specifically designed for coarse-grained observables have been developed. In recognition of the 90th anniversary of the seminal Heisenberg uncertainty relation, celebrated last year, and all the subsequent work since then, here we give a review of the state of the art of coarse-grained uncertainty relations in continuous variable quantum systems, as well as their applications to fundamental quantum physics and quantum information tasks. Our review is meant to be balanced in its content, since both theoretical considerations and experimental perspectives are put on an equal footing.
We ask which is the best strategy to reveal uncertainty relations between complementary observables of a continuous variable system for coarse-grained measurements. This leads to the derivation of new uncertainty relations for coarse-grained measurements that are always valid, even for detectors with low precision. These relations should be particularly relevant in experimental demonstrations of squeezing in quantum optics, quantum state reconstruction, and the development of trustworthy entanglement criteria.
If the phase space-based Glauber-Sudarshan distribution,
Pϱ, has negative values the quantum state,
ϱ, it describes is nonclassical. Due to
P's singular behavior, this simple criterion is impractical to use. Recent work has presented a general, sensitive, and noise-tolerant certification functional for the detection of nonclassical behavior of quantum states
ϱ. There, it was shown that when this functional takes on negative values somewhere in phase space,
ξP(x,p)<0, this is sufficient to certify the nonclassicality of a state. Here we give examples where this certification fails. We investigate states which are known to be nonclassical but the certification function is nonnegative,
ξP(x,p)≥0, everywhere in phase space. We generalize
ξ, giving it an appealing form,
S, which allows for slight improvements in certification, but
S also fails for mixed very weakly nonclassical states. More important than a slight improvement in sensitivity of
S over
ξ is that we showed how very sensitive
ξ and
S are, and more important still is our simple derivation of
S allowing us to generalize
ξ and
S to multiple modes.