Number-phase uncertainty relations and bipartite entanglement detection in spin ensembles

Giuseppe Vitagliano1,2, Matteo Fadel3,4, Iagoba Apellaniz2,5,6, Matthias Kleinmann7,2, Bernd Lücke8, Carsten Klempt8,9, and Géza Tóth2,5,10,11,12

1Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, AT-1090 Vienna, Austria
2Theoretical Physics, University of the Basque Country UPV/EHU, ES-48080 Bilbao, Spain
3Department of Physics, ETH Zürich, CH-8093 Zürich, Switzerland
4Department of Physics, University of Basel, CH-4056 Basel, Switzerland
5EHU Quantum Center, University of the Basque Country UPV/EHU, Barrio Sarriena s/n, ES-48940 Leioa, Biscay, Spain
6Mechanical and Industrial Manufacturing Department, Mondragon Unibertsitatea, ES-20500 Mondragón, Spain
7Naturwissenschaftlich-Technische Fakultät, Universität Siegen, DE-57068 Siegen, Germany
8Institut für Quantenoptik, Leibniz Universität Hannover, DE-30167 Hannover, Germany
9Deutsches Zentrum für Luft- und Raumfahrt e.V. (DLR), Institut für Satellitengeodäsie und Inertialsensorik, DLR-SI, Callinstraße 36, DE-30167 Hannover, Germany
10Donostia International Physics Center (DIPC), ES-20080 San Sebastián, Spain
11IKERBASQUE, Basque Foundation for Science, ES-48013 Bilbao, Spain
12Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, HU-1525 Budapest, Hungary

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We present a method to detect bipartite entanglement based on number-phase-like uncertainty relations in split spin ensembles. First, we derive an uncertainty relation that plays the role of a number-phase uncertainty for spin systems. It is important that the relation is given with well-defined and easily measurable quantities, and that it does not need assuming infinite dimensional systems. Based on this uncertainty relation, we show how to detect bipartite entanglement in an unpolarized Dicke state of many spin-1/2 particles. The particles are split into two subensembles, then collective angular momentum measurements are carried out locally on the two parts. First, we present a bipartite Einstein-Podolsky-Rosen (EPR) steering criterion. Then, we present an entanglement condition that can detect bipartite entanglement in such systems. We demonstrate the utility of the criteria by applying them to a recent experiment given in K. Lange et al. [Science 360, 416 (2018)] realizing a Dicke state in a Bose-Einstein condensate of cold atoms, in which the two subensembles were spatially separated from each other. Our methods also work well if split spin-squeezed states are considered. We show in a comprehensive way how to handle experimental imperfections, such as the nonzero particle number variance including the partition noise, and the fact that, while ideally BECs occupy a single spatial mode, in practice the population of other spatial modes cannot be fully suppressed.

Due to the rapid technological development, experiments keep aiming at the observation of quantum effects in larger and larger systems. In particular, particles called “bosons” can be cooled close to absolute zero to form a Bose-Einstein condensate, namely an unusual state of matter which behaves as a single “giant” quantum particle. The sophisticated level of control achieved in such systems allowed for preparing and detecting strong quantum correlations (like entanglement) among its constituents. These correlations are of crucial importance for quantum technologies, as they enable tasks inaccessible with classical resources, such as secure communication, quantum state teleportation, and quantum metrology. For this reason, investigating correlations in Bose-Einstein condensates is extremely interesting for both fundamental research and technical applications.

In our work, we consider the scenarios in which the particles in the condensate are spatially split into two groups, "a" and "b." We then investigate the correlations between these two parts for an experimentally relevant situation, by utilizing versions of the celebrated Heisenberg uncertainty relations. The latter, loosely speaking, say that one cannot measure simultaneously and with arbitrary precision two complementary quantities, such as angular momentum components along different directions.

Our first result is an uncertainty relation for spin systems. This is derived from a number-phase uncertainty relation, but adapted in such a way that it involves practical measurements in finite-dimensional systems. We consider measurements of the collective spin, which can be seen as a vector in three-dimensional space, where the phase is defined from its projection on the xy-plane. The relation we present, essentially, says that when the phase is well determined, then the z spin component cannot be well determined as well.

Based on the uncertainty relation just mentioned, we present a comprehensive theory for detecting bipartite entanglement between two ensembles of spins. In particular, our methods are especially suited for the so-called unpolarized Dicke states, namely highly entangled states similar to Greenberger-Horne-Zeilinger (GHZ) states that are routinely prepared experimentally. Moreover, the methods we present are robust to common forms of noise, such as particle number fluctuations, and other experimental imperfections.

Our work paves the way for the application of Bose-Einstein condensates in quantum information tasks, such as entanglement distillation, quantum teleportation, and more stringent Bell inequality tests in cold gases.

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[1] Shuheng Liu, Matteo Fadel, Qiongyi He, Marcus Huber, and Giuseppe Vitagliano, "Bounding entanglement dimensionality from the covariance matrix", Quantum 8, 1236 (2024).

[2] Manish Chaudhary, Ebubechukwu O. Ilo-Okeke, Valentin Ivannikov, and Tim Byrnes, "Macroscopic maximally-entangled-state preparation between two atomic ensembles", Physical Review A 108 3, 032420 (2023).

[3] Jiajie Guo, Fengxiao Sun, Qiongyi He, and Matteo Fadel, "Assisted metrology and preparation of macroscopic superpositions with split spin-squeezed states", Physical Review A 108 5, 053327 (2023).

[4] Irénée Frérot, Matteo Fadel, and Maciej Lewenstein, "Probing quantum correlations in many-body systems: a review of scalable methods", Reports on Progress in Physics 86 11, 114001 (2023).

[5] Matteo Fadel, Benjamin Yadin, Yuping Mao, Tim Byrnes, and Manuel Gessner, "Multiparameter quantum metrology and mode entanglement with spatially split nonclassical spin ensembles", New Journal of Physics 25 7, 073006 (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-06-17 10:57:20) and SAO/NASA ADS (last updated successfully 2024-06-17 10:57:21). The list may be incomplete as not all publishers provide suitable and complete citation data.