We characterize metrologically useful multipartite entanglement by representing partitions with Young diagrams. We derive entanglement witnesses that are sensitive to the shape of Young diagrams and show that Dyson’s rank acts as a resource for quantum metrology. Common quantifiers, such as the entanglement depth and $k$-separability are contained in this approach as the diagram’s width and height. Our methods are experimentally accessible in a wide range of atomic systems, as we illustrate by analyzing published data on the quantum Fisher information and spin-squeezing coefficients.

Frequentist and Bayesian phase estimation strategies lead to conceptually different results on the state of knowledge about the true value of an unknown parameter. We compare the two frameworks and their sensitivity bounds to the estimation of an interferometric phase shift limited by quantum noise, considering both the cases of a fixed and a fluctuating parameter. We point out that frequentist precision bounds, such as the Cramér–Rao bound, for instance, do not apply to Bayesian strategies and vice versa. In particular, we show that the Bayesian variance can overcome the frequentist Cramér–Rao bound, which appears to be a paradoxical result if the conceptual difference between the two approaches are overlooked. Similarly, bounds for fluctuating parameters make no statement about the estimation of a fixed parameter.

A new class of superfluids and superconductors with spatially periodic modulation of the superfluid density is arising^1–12. It might be related to the supersolid phase of matter, in which the spontaneous breaking of gauge and translational symmetries leads to a spatially modulated macroscopic wavefunction^13–16. This relation was recognized only in some cases^1,2,5–9 and there is the need for a universal property quantifying the differences between supersolids and ordinary matter, such as the superfluid fraction, which measures the reduction in superfluid stiffness resulting from the spatial modulation^16–18. The superfluid fraction was introduced long ago¹⁶, but it has not yet been assessed experimentally. Here we demonstrate an innovative method to measure the superfluid fraction based on the Josephson effect, a ubiquitous phenomenon associated with the presence of a physical barrier between two superfluids or superconductors¹⁹, which might also be expected for supersolids²⁰, owing to the spatial modulation. We demonstrate that individual cells of a supersolid can sustain Josephson oscillations and we show that, from the current–phase dynamics, we can derive directly the superfluid fraction. Our study of a cold-atom dipolar supersolid⁷ reveals a relatively large sub-unity superfluid fraction that makes realistic the study of previously unknown phenomena such as partially quantized vortices and supercurrents^16–18. Our results open a new direction of research that may unify the description of all supersolid-like systems.

We provide general bounds of phase estimation sensitivity in linear two-mode interferometers. We consider probe states with a fluctuating total number of particles. With incoherent mixtures of states with different total number of particles, particle entanglement is necessary but not sufficient to overcome the shot noise limit. The highest possible phase estimation sensitivity, the Heisenberg limit, is established under general unbiased properties of the estimator. If coherences can be created, manipulated, and detected, the Heisenberg limit can only be set in the central limit, with a sufficiently large repetition of interferometric measurements.

Bayesian estimation is a powerful theoretical paradigm for the operation of the approach to parameter estimation. However, the Bayesian method for statistical inference generally suffers from demanding calibration requirements that have so far restricted its use to systems that can be explicitly modeled. In this theoretical study, we formulate parameter estimation as a classification task and use artificial neural networks to efficiently perform Bayesian estimation. We show that the network’s posterior distribution is centered at the true (unknown) value of the parameter within an uncertainty given by the inverse Fisher information, representing the ultimate sensitivity limit for the given apparatus. When only a limited number of calibration measurements are available, our machine-learning-based procedure outperforms standard calibration methods. Our machine-learning-based procedure is model independent, and is thus well suited to “black-box sensors”, which lack simple explicit fitting models. Thus, our work paves the way for Bayesian quantum sensors that can take advantage of complex nonclassical quantum states and/or adaptive protocols. These capabilities can significantly enhance the sensitivity of future devices.

We explore the possibility to overcome the standard quantum limit (SQL) in a free-fall atom interferometer using a Bose-Einstein condensate (BEC) in either of the two relevant cases of Bragg or Raman scattering light pulses. The generation of entanglement in the BEC is dramatically enhanced by amplifying the atom-atom interactions via the rapid action of an external trap, focusing the matter waves to significantly increase the atomic densities during a preparation stage—a technique we refer to as delta-kick squeezing (DKS). The action of a second DKS operation at the end of the interferometry sequence allows one to implement a nonlinear readout scheme, making the sub-SQL sensitivity highly robust against imperfect atom counting detection. We predict more than 30 dB of sensitivity gain beyond the SQL for the variance, assuming realistic parameters and ${10}^6\mathrm{atoms}$.