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arXiv:2404.12323 · New Journal of Physics

Entropy estimation in a spin-1 Bose-Einstein condensate

Yannick Deller, Martin Gärttner, Tobias Haas, Markus K. Oberthaler, Moritz Reh, Helmut Strobel

Published Apr 1, 2025• 3 citations
arXivHTMLPDF

Abstract

Abstract We investigate the information extractable from measurement distributions of two non-commuting spin observables in a multi-well spin-1 Bose–Einstein condensate. We provide a variety of analytic and numerical evidence that suitably chosen classical entropies and classical mutual informations thereof contain the typical feature of quantum entropies known in quantum field theories, that is, the area law, even in the non-Gaussian regime and for a non-zero temperature. Towards a feasible experimental implementation, we estimate entropic quantities from a finite number of samples without any additional assumptions on the underlying quantum state using k-nearest neighbor estimators.

DOI
10.1088/1367-2630/adc32a

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Abstract We investigate the information extractable from measurement distributions of two non-commuting spin observables in a multi-well spin-1 Bose–Einstein condensate. We provide a variety of analytic and numerical evidence that suitably chosen classical entropies and classical mutual informations thereof contain the typical feature of quantum entropies known in quantum field theories, that is, the area law, even in the non-Gaussian regime and for a non-zero temperature. Towards a feasible experimental implementation, we estimate entropic quantities from a finite number of samples without any additional assumptions on the underlying quantum state using k-nearest neighbor estimators.
The scaling of local quantum entropies is of utmost interest for characterizing quantum fields, many-body systems, and gravity. Despite their importance, theoretically and experimentally accessing quantum entropies is challenging as they are nonlinear functionals of the underlying quantum state. Here, we show that suitably chosen classical entropies capture many features of their quantum analogs for an experimentally relevant setting. We describe the post-quench dynamics of a multi-well spin-1 Bose-Einstein condensate from an initial product state via measurement distributions of spin observables and estimate the corresponding entropies using the asymptotically unbiased kkk-nearest neighbor method. We observe the dynamical build-up of quantum correlations signaled by an area law, as well as local thermalization revealed by a transition to a volume law, both in regimes characterized by non-Gaussian distributions. We emphasize that all relevant features can be observed at small sample numbers without reconstructing the underlying state or measurement distributions, rendering our method directly applicable to a large variety of models and experimental platforms.
The area law-like scaling of local quantum entropies is the central characteristic of the entanglement inherent in quantum fields, many-body systems, and spacetime. Whilst the area law is primarily associated with the entanglement structure of the underlying quantum state, we here show that it equally manifests in classical entropies over measurement distributions when vacuum contributions dictated by the uncertainty principle are subtracted. Using the examples of the Gaussian ground and thermal states, but also the non-Gaussian particle state of a relativistic scalar field, we present analytical and numerical area laws for the entropies of various distributions and unveil how quantities of widespread interest such as the central charge and the (local) temperature are encoded in classical observables. With our approach, quantum entropies are no longer necessary to probe quantum phenomena, thereby rendering area laws and other quantum features directly accessible to theoretical models of high complexity as well as state-of-the-art experiments.
In this paper we study the entanglement properties of free nonrelativistic Bose gases. At zero temperature, we calculate the bipartite block entanglement entropy of the system and find that it diverges logarithmically with the particle number in the subsystem. For finite temperatures, we study the mutual information between the two blocks. We first analytically study an infinite-range hopping model, then numerically study a set of long-range hopping models in one dimension that exhibit Bose-Einstein condensation. In both cases we find that a Bose-Einstein condensate, if present, makes a divergent contribution to the mutual information which is proportional to the logarithm of the number of particles in the condensate in the subsystem. The prefactor of the logarithmic divergent term is model dependent.
In this work we study an ideal bosonic quantum field system at finite temperature, and in a canonical and a grand canonical ensemble. For a simple spatial partition we derive the corresponding mutual information, a quantity that measures the total amount of information of one of the parts about the other. In order to find it, we first derive the von Neumann entropy that corresponds to the spatially separated subsystem (i.e. the geometric entropy) and then we subtract its extensive part which coincides with the thermal entropy of the subsystem. In the framework of the grand canonical description, we examine the influence of the underlying Bose-Einstein condensation on the behaviour of the mutual information, and we find that its derivative with respect to the temperature possesses a finite discontinuity at exactly the critical temperature.
Information shared between parties quantifies their correlation. The encoding of correlations across space and time characterises the structure, history, and interactions of systems. One of the most fundamental properties that emerges from studies of information is the area law, which states that information shared between spatial subregions typically scales with the area of their boundary rather than their volume. In non-interacting, quantum many-body systems, where Gaussian statistics apply, the scaling of information measures is well understood. Within interacting systems, the readout of information measures is impeded by the complexity of state reconstruction. As such, no measurements beyond small quantum systems (e.g., composed of few, localised particles) have been made. Here, we fill this gap by experimentally demonstrating the area law of mutual information in an ultra-cold atom simulator of quantum fields with tuneable interaction strength. Our results detail the scaling of mutual information with subsystem volume, boundary area, and separation between spatial regions at finite temperature. Moreover, we quantify the total effect of non-Gaussian correlations using an information-theoretic measure - relative entropy. Our presented approach is data-driven, model agnostic, and readily applicable to other platforms and observables, thus constituting a universal toolkit for probing information in high-dimensional quantum systems and its role in shaping quantum matter and spacetime.
We study the area-dependent entropy and two-site entanglement for two-state Bose-Einstein condensates in a two-dimensional optical lattice. We consider the case where the array of two-component condensates behaves like an ensemble of spin-half particles with the interaction to its nearest neighbors and next-nearest neighbors. We show how the Hamiltonian of their Bose-Einstein condensate lattice with nearest-neighbor and next-nearest-neighbor interactions can be mapped onto a harmonic lattice. We use this to determine the entropy and entanglement content of the lattice.