Macroscopic quantum coherence and entanglement in mechanical systems

Benjamin Yadin; Matteo Fadel

doi:10.22331/q-2026-10-30-2257

Macroscopic quantum coherence and entanglement in mechanical systems

Benjamin Yadin¹ and Matteo Fadel^2,3

¹Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, Siegen 57068, Germany
²Institute of Quantum Science and Technology $\&$ Frontiers Science Center for Nano-Optoelectronics, School of Physics, Peking University, Beijing 100871, China
³Department of Physics, ETH Zürich, Zürich 8093, Switzerland

Published:	2026-10-30, volume 10, page 2257
Editor:	Borivoje Dakic
Eprint:	arXiv:2503.08324v4
Doi:	https://doi.org/10.22331/q-2026-10-30-2257
Citation:	Quantum 10, 2257 (2026).

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

The notion of a macroscopic quantum state must be pinned down in order to assess how well experiments probe the large-scale limits of quantum mechanics. However, the issue of quantifying so-called quantum macroscopicity is fraught with multiple approaches having varying interpretations and levels of computability and measurability. Here, we introduce two measures that capture independent macroscopic properties: i) extensive size, measuring the degree of coherence in position or momentum observables relative to atomic-scale units, and ii) entangled size, quantifying the number of entangled subsystems. These measures unify many desirable features of past proposals while having rigorously justified interpretations from quantum information theory in terms of coherence and multipartite entanglement. We demonstrate how to estimate them and obtain lower-bounds using experimental data from micro-mechanical oscillators and diffracting molecules. Notably, we find evidence for genuine multipartite entanglement of $10^5$ to $10^7$ atoms in recent mechanical superposition states.

Featured image: Extensive and entangled sizes of mechanical quantum experiments and proposals, from trapped ions and diffracting molecules to micro-oscillators and quantum-gravity tests.

Popular summary

Quantum mechanics should apply to arbitrarily large objects, yet we do not see quantum effects in our everyday lives. One good explanation is that the system loses quantum coherence by coupling to its environment, and the number of coupling channels grows with the system size. In addition, the precision required to detect quantum behaviour also becomes more demanding for large systems. These are only technical challenges, but it is also possible that new physics, such as effects from quantum gravity, sets a fundamental limit on how large a quantum superposition can be. Experiments can settle this question by probing ever-larger quantum regimes, and mechanical systems, from vibrating micro-resonators to diffracting molecules, have made remarkable progress on this in recent years. But how do we decide which of these very different experiments is “more macroscopic”? Neither “macroscopic” nor “quantum” has a unique meaning, and the many measures proposed so far are often hard to compare or apply only to specific systems. Here, we introduce two complementary measures, grounded in quantum information theory and directly estimable from experimental data. The extensive size quantifies the scale, in position or momentum relative to atomic units, over which quantum coherence extends. The entangled size counts how many atoms must be entangled to explain the observed behaviour. Placing experiments and proposals side by side on these two axes allows a transparent comparison of how close each comes to a true Schrödinger cat. It also reveals evidence for genuine entanglement of 10⁵ to 10⁷ atoms in recent mechanical superposition states.

► BibTeX data

@article{Yadin2026macroscopicquantum,
  doi = {10.22331/q-2026-10-30-2257},
  url = {https://doi.org/10.22331/q-2026-10-30-2257},
  title = {Macroscopic quantum coherence and entanglement in mechanical systems},
  author = {Yadin, Benjamin and Fadel, Matteo},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {10},
  pages = {2257},
  month = oct,
  year = {2026}
}

► References

[1] E. Schrödinger. ``Die gegenwärtige Situation in der Quantenmechanik''. Die Naturwissenschaften 23, 807–812 (1935).
https://doi.org/10.1007/BF01491891

[2] Wojciech Hubert Zurek. ``Decoherence, einselection, and the quantum origins of the classical''. Reviews of Modern Physics 75, 715–775 (2003).
https://doi.org/10.1103/RevModPhys.75.715

[3] Asher Peres and Nathan Rosen. ``Macroscopic bodies in quantum theory''. Physical Review 135, B1486 (1964).
https://doi.org/10.1103/PhysRev.135.B1486

[4] Johannes Kofler and Časlav Brukner. ``Classical world arising out of quantum physics under the restriction of coarse-grained measurements''. Physical Review Letters 99, 180403 (2007).
https://doi.org/10.1103/PhysRevLett.99.180403

[5] Sadegh Raeisi, Pavel Sekatski, and Christoph Simon. ``Coarse graining makes it hard to see micro-macro entanglement''. Physical Review Letters 107, 250401 (2011).
https://doi.org/10.1103/PhysRevLett.107.250401

[6] Pavel Sekatski, Nicolas Sangouard, and Nicolas Gisin. ``Size of quantum superpositions as measured with classical detectors''. Physical Review A 89, 012116 (2014).
https://doi.org/10.1103/PhysRevA.89.012116

[7] Michalis Skotiniotis, Wolfgang Dür, and Pavel Sekatski. ``Macroscopic superpositions require tremendous measurement devices''. Quantum 1, 34 (2017).
https://doi.org/10.22331/q-2017-11-21-34

[8] Florian Fröwis, Pavel Sekatski, Wolfgang Dür, Nicolas Gisin, and Nicolas Sangouard. ``Macroscopic quantum states: Measures, fragility, and implementations''. Reviews of Modern Physics 90, 025004 (2018).
https://doi.org/10.1103/RevModPhys.90.025004

[9] Akira Shimizu and Tomoyuki Morimae. ``Detection of macroscopic entanglement by correlation of local observables''. Physical Review Letters 95, 090401 (2005).
https://doi.org/10.1103/PhysRevLett.95.090401

[10] Chang Woo Lee and Hyunseok Jeong. ``Quantification of macroscopic quantum superpositions within phase space''. Physical Review Letters 106, 220401 (2011).
https://doi.org/10.1103/PhysRevLett.106.220401

[11] E. G. Cavalcanti and M. D. Reid. ``Criteria for generalized macroscopic and mesoscopic quantum coherence''. Physical Review A 77, 062108 (2008).
https://doi.org/10.1103/PhysRevA.77.062108

[12] Benjamin Yadin and Vlatko Vedral. ``General framework for quantum macroscopicity in terms of coherence''. Physical Review A 93, 022122 (2016).
https://doi.org/10.1103/PhysRevA.93.022122

[13] Wolfgang Dür, Christoph Simon, and J. Ignacio Cirac. ``Effective size of certain macroscopic quantum superpositions''. Physical Review Letters 89, 210402 (2002).
https://doi.org/10.1103/PhysRevLett.89.210402

[14] Benjamin Yadin and Vlatko Vedral. ``Quantum macroscopicity versus distillation of macroscopic superpositions''. Physical Review A 92, 022356 (2015).
https://doi.org/10.1103/PhysRevA.92.022356

[15] Jan I. Korsbakken, K. Birgitta Whaley, Jonathan Dubois, and J. Ignacio Cirac. ``Measurement-based measure of the size of macroscopic quantum superpositions''. Physical Review A 75, 042106 (2007).
https://doi.org/10.1103/PhysRevA.75.042106

[16] Florian Marquardt, Benjamin Abel, and Jan von Delft. ``Measuring the size of a quantum superposition of many-body states''. Physical Review A 78, 012109 (2008).
https://doi.org/10.1103/PhysRevA.78.012109

[17] Stefan Nimmrichter and Klaus Hornberger. ``Macroscopicity of mechanical quantum superposition states''. Physical Review Letters 110, 160403 (2013).
https://doi.org/10.1103/PhysRevLett.110.160403

[18] Björn Schrinski, Stefan Nimmrichter, Benjamin A. Stickler, and Klaus Hornberger. ``Macroscopicity of quantum mechanical superposition tests via hypothesis falsification''. Physical Review A 100, 032111 (2019).
https://doi.org/10.1103/PhysRevA.100.032111

[19] Björn Schrinski, Yu Yang, Uwe von Lüpke, Marius Bild, Yiwen Chu, Klaus Hornberger, Stefan Nimmrichter, and Matteo Fadel. ``Macroscopic quantum test with bulk acoustic wave resonators''. Physical Review Letters 130, 133604 (2023).
https://doi.org/10.1103/PhysRevLett.130.133604

[20] Anthony J. Leggett. ``Macroscopic quantum systems and the quantum theory of measurement''. Progress of Theoretical Physics Supplement 69, 80–100 (1980).
https://doi.org/10.1143/PTPS.69.80

[21] Florian Fröwis and Wolfgang Dür. ``Measures of macroscopicity for quantum spin systems''. New Journal of Physics 14, 093039 (2012).
https://doi.org/10.1088/1367-2630/14/9/093039

[22] E. Oudot, P. Sekatski, F. Fröwis, N. Gisin, and N. Sangouard. ``Two-mode squeezed states as Schrödinger cat-like states''. Journal of the Optical Society of America B 32, 2190–2197 (2015).
https://doi.org/10.1364/JOSAB.32.002190

[23] Anthony J. Leggett. ``Testing the limits of quantum mechanics: motivation, state of play, prospects''. Journal of Physics: Condensed Matter 14, R415–R451 (2002).
https://doi.org/10.1088/0953-8984/14/15/201

[24] Samuel L. Braunstein and Carlton M. Caves. ``Statistical distance and the geometry of quantum states''. Physical Review Letters 72, 3439–3443 (1994).
https://doi.org/10.1103/PhysRevLett.72.3439

[25] Akira Shimizu and Takayuki Miyadera. ``Stability of quantum states of finite macroscopic systems against classical noises, perturbations from environments, and local measurements''. Physical Review Letters 89, 270403 (2002).
https://doi.org/10.1103/PhysRevLett.89.270403

[26] F. Fröwis, N. Sangouard, and N. Gisin. ``Linking measures for macroscopic quantum states via photon–spin mapping''. Optics Communications 337, 2–11 (2015).
https://doi.org/10.1016/j.optcom.2014.07.017

[27] Iman Marvian. ``Operational interpretation of quantum Fisher information in quantum thermodynamics''. Physical Review Letters 129, 190502 (2022).
https://doi.org/10.1103/PhysRevLett.129.190502

[28] Benjamin Yadin. ``Resource theories of quantum coherence: foundations and applications''. DPhil thesis. University of Oxford. (2017).
https://doi.org/10.5287/ora-wx05gxk92

[29] Stefan Nimmrichter. ``Macroscopic matter wave interferometry''. Springer Theses. Springer International Publishing. Cham (2014).
https://doi.org/10.1007/978-3-319-07097-1

[30] 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, 073006 (2023).
https://doi.org/10.1088/1367-2630/ace1a0

[31] Otfried Gühne, Géza Tóth, and Hans J. Briegel. ``Multipartite entanglement in spin chains''. New Journal of Physics 7, 229 (2005).
https://doi.org/10.1088/1367-2630/7/1/229

[32] Anthony J. Leggett. ``Note on the ``size'' of Schroedinger cats'' (2016). arXiv:1603.03992.
arXiv:1603.03992

[33] L. Cartz. ``Thermal vibrations of atoms in cubic crystals II: The amplitude of atomic vibrations''. Proceedings of the Physical Society. Section B 68, 957–967 (1955).
https://doi.org/10.1088/0370-1301/68/11/321

[34] G. Björk and P. G. L. Mana. ``A size criterion for macroscopic superposition states''. Journal of Optics B: Quantum and Semiclassical Optics 6, 429–436 (2004).
https://doi.org/10.1088/1464-4266/6/11/001

[35] A. Steif, S. C. Tiersten, and S. C. Ying. ``Vibrational correlation functions for Si and Ge''. Physical Review B 35, 857–859 (1987).
https://doi.org/10.1103/PhysRevB.35.857

[36] Adriano Filipponi. ``Vibrational correlation function in ordered and disordered covalent solids''. Physical Review B 37, 7027–7037 (1988).
https://doi.org/10.1103/PhysRevB.37.7027

[37] Roman Tomaschitz. ``Effective real-space correlations of crystal lattice vibrations''. Journal of Physics and Chemistry of Solids 152, 109773 (2021).
https://doi.org/10.1016/j.jpcs.2020.109773

[38] Elliott W. Montroll. ``Theory of the vibration of simple cubic lattices with nearest neighbor interactions''. In Jerzy Neyman, editor, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Contributions to Astronomy and Physics. Volume 3, pages 209–246. Berkeley (1956). University of California Press.
https://doi.org/10.1525/9780520350694-014

[39] Massimiliano Rossi, Andrei Militaru, Nicola Carlon Zambon, Andreu Riera-Campeny, Oriol Romero-Isart, Martin Frimmer, and Lukas Novotny. ``Quantum delocalization of a levitated nanoparticle''. Physical Review Letters 135, 083601 (2025). arXiv:2408.01264.
https://doi.org/10.1103/2yzc-fsm3
arXiv:2408.01264

[40] J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W. Lehnert, and R. W. Simmonds. ``Sideband cooling of micromechanical motion to the quantum ground state''. Nature 475, 359–363 (2011).
https://doi.org/10.1038/nature10261

[41] E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg. ``Quantum-coherent coupling of a mechanical oscillator to an optical cavity mode''. Nature 482, 63–67 (2012).
https://doi.org/10.1038/nature10787

[42] M. Ringbauer, T. J. Weinhold, L. A. Howard, A. G. White, and M. R. Vanner. ``Generation of mechanical interference fringes by multi-photon counting''. New Journal of Physics 20, 053042 (2018).
https://doi.org/10.1088/1367-2630/aabb8d

[43] Mahdi Chegnizadeh, Marco Scigliuzzo, Amir Youssefi, Shingo Kono, Evgenii Guzovskii, and Tobias J. Kippenberg. ``Quantum collective motion of macroscopic mechanical oscillators''. Science 386, 1383–1388 (2024).
https://doi.org/10.1126/science.adr8187

[44] Igor Pikovski, Michael R. Vanner, Markus Aspelmeyer, M. S. Kim, and Časlav Brukner. ``Probing Planck-scale physics with quantum optics''. Nature Physics 8, 393–397 (2012).
https://doi.org/10.1038/nphys2262

[45] Germain Tobar, Sreenath K. Manikandan, Thomas Beitel, and Igor Pikovski. ``Detecting single gravitons with quantum sensing''. Nature Communications 15, 7229 (2024).
https://doi.org/10.1038/s41467-024-51420-8

[46] D. Kienzler, C. Flühmann, V. Negnevitsky, H.-Y. Lo, M. Marinelli, D. Nadlinger, and J. P. Home. ``Observation of quantum interference between separated mechanical oscillator wave packets''. Physical Review Letters 116, 140402 (2016).
https://doi.org/10.1103/PhysRevLett.116.140402

[47] Marius Bild, Matteo Fadel, Yu Yang, Uwe von Lüpke, Phillip Martin, Alessandro Bruno, and Yiwen Chu. ``Schrödinger cat states of a 16-microgram mechanical oscillator''. Science 380, 274–278 (2023).
https://doi.org/10.1126/science.adf7553

[48] Florian Fröwis. ``Lower bounds on the size of general Schrödinger-cat states from experimental data''. Journal of Physics A: Mathematical and Theoretical 50, 114003 (2017).
https://doi.org/10.1088/1751-8121/aa5a92

[49] Stefano Marti, Uwe von Lüpke, Om Joshi, Yu Yang, Marius Bild, Andraž Omahen, Yiwen Chu, and Matteo Fadel. ``Quantum squeezing in a nonlinear mechanical oscillator''. Nature Physics 20, 1448–1453 (2024).
https://doi.org/10.1038/s41567-024-02545-6

[50] Roger H. French, David J. Jones, and Stephen Loughin. ``Interband electronic structure of $\alpha$-alumina up to 2167 K''. Journal of the American Ceramic Society 77, 412–422 (1994).
https://doi.org/10.1111/j.1151-2916.1994.tb07009.x

[51] Klaus Hornberger, Stefan Gerlich, Philipp Haslinger, Stefan Nimmrichter, and Markus Arndt. ``Colloquium: Quantum interference of clusters and molecules''. Reviews of Modern Physics 84, 157–173 (2012).
https://doi.org/10.1103/RevModPhys.84.157

[52] Yaakov Y. Fein, Philipp Geyer, Patrick Zwick, Filip Kiałka, Sebastian Pedalino, Marcel Mayor, Stefan Gerlich, and Markus Arndt. ``Quantum superposition of molecules beyond 25 kDa''. Nature Physics 15, 1242–1245 (2019).
https://doi.org/10.1038/s41567-019-0663-9

[53] Sebastian Pedalino, Bruno E. Ramírez-Galindo, Richard Ferstl, Klaus Hornberger, Markus Arndt, and Stefan Gerlich. ``Probing quantum mechanics with nanoparticle matter-wave interferometry''. Nature 649, 866–870 (2026).
https://doi.org/10.1038/s41586-025-09917-9

[54] Florian Fröwis, Peter C. Strassmann, Alexey Tiranov, Corentin Gut, Jonathan Lavoie, Nicolas Brunner, Félix Bussières, Mikael Afzelius, and Nicolas Gisin. ``Experimental certification of millions of genuinely entangled atoms in a solid''. Nature Communications 8, 907 (2017).
https://doi.org/10.1038/s41467-017-00898-6

[55] Sougato Bose, Anupam Mazumdar, Gavin W. Morley, Hendrik Ulbricht, Marko Toroš, Mauro Paternostro, Andrew A. Geraci, Peter F. Barker, M. S. Kim, and Gerard Milburn. ``Spin entanglement witness for quantum gravity''. Physical Review Letters 119, 240401 (2017).
https://doi.org/10.1103/PhysRevLett.119.240401

[56] John Clarke and Frank K. Wilhelm. ``Superconducting quantum bits''. Nature 453, 1031–1042 (2008).
https://doi.org/10.1038/nature07128

[57] Jan I. Korsbakken, Frank K. Wilhelm, and K. Birgitta Whaley. ``Electronic structure of superposition states in flux qubits''. Physica Scripta T137, 014022 (2009).
https://doi.org/10.1088/0031-8949/2009/T137/014022

[58] Caspar H. van der Wal, A. C. J. ter Haar, F. K. Wilhelm, R. N. Schouten, C. J. P. M. Harmans, T. P. Orlando, Seth Lloyd, and J. E. Mooij. ``Quantum superposition of macroscopic persistent-current states''. Science 290, 773–777 (2000).
https://doi.org/10.1126/science.290.5492.773

[59] Jonathan R. Friedman, Vijay Patel, W. Chen, S. K. Tolpygo, and J. E. Lukens. ``Quantum superposition of distinct macroscopic states''. Nature 406, 43–46 (2000).
https://doi.org/10.1038/35017505

[60] Jan I. Korsbakken, Frank K. Wilhelm, and K. B. Whaley. ``The size of macroscopic superposition states in flux qubits''. EPL 89, 30003 (2010).
https://doi.org/10.1209/0295-5075/89/30003

[61] Harald Ibach and Hans Lüth. ``Solid-state physics: An introduction to principles of materials science''. Chapter 10. Springer. Berlin, Heidelberg (2009). 4 edition.
https://doi.org/10.1007/978-3-540-93804-0

[62] Florian Fröwis, Benjamin Yadin, and Nicolas Gisin. ``Insufficiency of avoided crossings for witnessing large-scale quantum coherence in flux qubits''. Physical Review A 97, 042103 (2018).
https://doi.org/10.1103/PhysRevA.97.042103

[63] Benjamin Morris, Benjamin Yadin, Matteo Fadel, Tilman Zibold, Philipp Treutlein, and Gerardo Adesso. ``Entanglement between identical particles is a useful and consistent resource''. Physical Review X 10, 041012 (2020).
https://doi.org/10.1103/PhysRevX.10.041012

[64] Davide Girolami. ``Observable measure of quantum coherence in finite dimensional systems''. Physical Review Letters 113, 170401 (2014).
https://doi.org/10.1103/PhysRevLett.113.170401

[65] Benjamin Yadin, Matteo Fadel, and Manuel Gessner. ``Metrological complementarity reveals the Einstein-Podolsky-Rosen paradox''. Nature Communications 12, 2410 (2021).
https://doi.org/10.1038/s41467-021-22353-3

[66] Ky Fan and Gordon Pall. ``Imbedding conditions for Hermitian and normal matrices''. Canadian Journal of Mathematics 9, 298–304 (1957).
https://doi.org/10.4153/CJM-1957-036-1

[67] Matteo G. A. Paris. ``Quantum estimation for quantum technology''. International Journal of Quantum Information 7, 125–137 (2009).
https://doi.org/10.1142/S0219749909004839

[68] Björn Granéli and Ulf Dahlborg. ``The vibrational motions in vitreous silica at high temperatures''. Journal of Non-Crystalline Solids 109, 295–310 (1989).
https://doi.org/10.1016/0022-3093(89)90043-4

[69] Benjamin A. Stickler and Klaus Hornberger. ``Molecular rotations in matter-wave interferometry''. Physical Review A 92, 023619 (2015).
https://doi.org/10.1103/PhysRevA.92.023619

[70] Yaakov Y. Fein. ``Long-baseline universal matter-wave interferometry''. PhD thesis. Universität Wien. Vienna (2020).
https://doi.org/10.25365/thesis.62746

Cited by

Could not fetch Crossref cited-by data during last attempt 2026-10-30 10:13:54: Could not fetch cited-by data for 10.22331/q-2026-10-30-2257 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-10-30 10:13:54: Cannot retrieve data from ADS due to rate limitations.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.