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  • Letter
  • Open Access

Mixed state entanglement via the Cauchy-Schwarz inequality

Nishaant Jacobus1, Paul Brumer1,*, and Chern Chuang2,†

  • 1Chemical Physics Theory Group, Department of Chemistry, and Center for Quantum Information and Quantum Control, University of Toronto, Toronto, Ontario M5S 3H6, Canada
  • 2Department of Chemistry and Biochemistry, University of Nevada, Las Vegas, Las Vegas, Nevada 89154, USA

  • *Contact author: paul.brumer@utoronto.ca
  • †Contact author: chern.chuang@unlv.edu
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Phys. Rev. Research 8, L032014 – Published 23 July, 2026

DOI: https://doi.org/10.1103/3s44-6j39

Abstract

The entanglement properties of mixed states are of great importance in the study of open quantum systems and quantum information science, but common entanglement criteria such as the Peres-Horodecki condition can be difficult to relate to physical properties of the system. We introduce the Cauchy-Schwarz violation (CSV) condition for entanglement, which has a simple dependence on the populations and coherences of the density operator. A sufficient condition for entanglement, it provides a direct connection to the physical characteristics of the system such as its symmetries, yielding intuition that can be used to understand and engineer highly entangled states. We illustrate the often surprising insights gained from the CSV condition by applying it to the Jaynes-Cummings model, the quantum Rabi model, and an open-system quantum Rabi model.

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References (49)

  1. A. Einstein, Can quantum-mechanical description of physical reality be considered complete, Phys. Rev. 47, 777 (1935).
  2. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
  3. G. Benenti, G. Casati, D. Rossini, and G. Strini, Principles of Quantum Computation and Information (World Scientific, Singapore, 2018).
  4. E. Karimi and R. W. Boyd, Classical entanglement, Science 350, 1172 (2015).
  5. D. Paneru, E. Cohen, R. Fickler, R. W. Boyd, and E. Karimi, Entanglement: Quantum or classical, Rep. Prog. Phys. 83, 064001 (2020).
  6. N. Korolkova, L. Sánchez-Soto, and G. Leuchs, An operational distinction between quantum entanglement and classical non-separability, Philos. Trans. R. Soc. A 382, 20230342 (2024).
  7. M. Molina-Espíritu, R. O. Esquivel, S. López-Rosa, and J. S. Dehesa, Quantum entanglement and chemical reactivity, J. Chem. Theory Comput. 11, 5144 (2015).
  8. J. Li and S. Kais, Entanglement classifier in chemical reactions, Sci. Adv. 5, eaax5283 (2019).
  9. K. B. Whaley, M. Sarovar, and A. Ishizaki, Quantum entanglement phenomena in photosynthetic light harvesting complexes, Proc. Chem. 3, 152 (2011).
  10. F. Caruso, A. W. Chin, A. Datta, S. F. Huelga, and M. B. Plenio, Entanglement and entangling power of the dynamics in light-harvesting complexes, Phys. Rev. A 81, 062346 (2010).
  11. M. B. Plenio and S. S. Virmani, An introduction to entanglement theory, in Quantum Information and Coherence, edited by E. Andersson and P. Öhberg (Springer International Publishing, Cham, 2014), pp. 173–209.
  12. A. Elben, R. Kueng, H.-Y. R. Huang, R. van Bijnen, C. Kokail, M. Dalmonte, P. Calabrese, B. Kraus, J. Preskill, P. Zoller, and B. Vermersch, Mixed-state entanglement from local randomized measurements, Phys. Rev. Lett. 125, 200501 (2020).
  13. A. Peres, Separability criterion for density matrices, Phys. Rev. Lett. 77, 1413 (1996).
  14. M. Horodecki, P. Horodecki, and R. Horodecki, Separability of mixed states: Necessary and sufficient conditions, Phys. Lett. A 223, 1 (1996).
  15. F. Benabdallah, M. Y. Abd-Rabbou, M. Daoud, and S. Haddadi, Quantum information resources in spin-1 Heisenberg dimer systems, Quantum Inf. Proc. 24, 20 (2025).
  16. M. Brenes, B. Min, N. Anto-Sztrikacs, N. Bar-Gill, and D. Segal, Bath-induced interactions and transient dynamics in open quantum systems at strong coupling: Effective Hamiltonian approach, J. Chem. Phys. 160, 244106 (2024).
  17. G. Parez and W. Witczak-Krempa, Entanglement negativity between separated regions in quantum critical systems, Phys. Rev. Res. 6, 023125 (2024).
  18. C. D. Meyer, Matrix Analysis and Applied Linear Algebra (Society for Industrial and Applied Mathematics, PA, Philadelphia, 2000).
  19. A. Nitzan, Chemical Dynamics in Condensed Phases: Relaxation, Transfer, and Reactions in Condensed Molecular Systems (Oxford University Press, Oxford, 2014).
  20. G. A. Skorobagatko, Universal separability criterion for arbitrary density matrices from causal properties of separable and entangled quantum states, Sci. Rep. 11, 15866 (2021).
  21. C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Vol. 1: Basic Concepts, Tools, and Applications (John Wiley & Sons, Hoboken, NJ, 2019).
  22. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  23. F. G. S. L. Brandão, Quantifying entanglement with witness operators, Phys. Rev. A 72, 022310 (2005).
  24. G. Vidal and R. F. Werner, Computable measure of entanglement, Phys. Rev. A 65, 032314 (2002).
  25. M. Mirzaee and M. Batavani, Atom–field entanglement in the Jaynes–Cummings model without rotating wave approximation, Chin. Phys. B 24, 040306 (2015).
  26. S. Akella, K. Thapliyal, H. S. Mani, and A. Pathak, Dynamics of single-mode nonclassicalities and quantum correlations in the Jaynes-Cummings model, J. Opt. Soc. Am. B 39, 1829 (2022).
  27. K. Mandal, Entanglement dynamics in intensity-dependent double Jaynes-Cummings model for squeezed coherent thermal states, J. Phys. B: Atom. Mole. Opt. Phys. 58, 165401 (2025).
  28. E. Boukobza and D. J. Tannor, Entropy exchange and entanglement in the Jaynes-Cummings model, Phys. Rev. A 71, 063821 (2005).
  29. L.-B. Fan, Y.-H. Zhou, F. Zou, H. Guo, J.-F. Huang, and J.-Q. Liao, Quantum thermalization and vanishing thermal entanglement in the open Jaynes–Cummings model, Ann. Phys. 532, 2000134 (2020).
  30. Y.-T. Yang and H.-G. Luo, Characterizing superradiant phase of the quantum Rabi model, Chin. Phys. Lett. 40, 020502 (2023).
  31. M. Bina, S. M. Felis, and S. Olivares, Entanglement generation in the ultra-strongly coupled Rabi model, Int. J. Quantum Inf. 12, 1560016 (2014).
  32. Y.-Q. Shi, L. Cong, and H.-P. Eckle, Entanglement resonance in the asymmetric quantum Rabi model, Phys. Rev. A 105, 062450 (2022).
  33. Z.-J. Ying, M. Liu, H.-G. Luo, H.-Q. Lin, and J. Q. You, Ground-state phase diagram of the quantum Rabi model, Phys. Rev. A 92, 053823 (2015).
  34. W.-Y. Liu, L.-B. Fan, Y.-X. Zeng, J.-F. Huang, and J.-Q. Liao, Quantum thermalization and thermal entanglement in the open quantum Rabi model, Ann. Phys. 535, 2200502 (2023).
  35. D. Z. Rossatto, C. J. Villas-Bôas, M. Sanz, and E. Solano, Spectral classification of coupling regimes in the quantum Rabi model, Phys. Rev. A 96, 013849 (2017).
  36. C. Chuang, A. Kapulkin, A. Pattanayak, and P. Brumer, Parametric hypersensitivity in many-body bath-mediated transport: The quantum Rabi model, Phys. Rev. A 108, 022213 (2023).
  37. See Supplemental Material at http://link.aps.org/supplemental/10.1103/3s44-6j39 for further details. Section 1 presents the analytical results for the Jaynes-Cummings model that are used in the main text, Sec. 2 discusses the Redfield theory and computational methods used to determine steady states, and Sec. 3 provides a visualization of the density operators for the thermal quantum Rabi model, corroborating the discussion in the main text.
  38. M. Scala, R. Migliore, and A. Messina, Dissipation and entanglement dynamics for two interacting qubits coupled to independent reservoirs, J. Phys. A: Math. Theor. 41, 435304 (2008).
  39. M.-J. Wang and Y.-J. Xia, Steady-state entanglement and heat current of two coupled qubits in two baths without rotating wave approximation, Chin. Phys. B 28, 060303 (2019).
  40. K. Kawabata, R. Sohal, and S. Ryu, Lieb-Schultz-Mattis theorem in open quantum systems, Phys. Rev. Lett. 132, 070402 (2024).
  41. K. Kawabata, A. Kulkarni, J. Li, T. Numasawa, and S. Ryu, Symmetry of open quantum systems: Classification of dissipative quantum chaos, PRX Quantum 4, 030328 (2023).
  42. V. V. Albert and L. Jiang, Symmetries and conserved quantities in Lindblad master equations, Phys. Rev. A 89, 022118 (2014).
  43. B. Buča and T. Prosen, A note on symmetry reductions of the Lindblad equation: Transport in constrained open spin chains, New J. Phys. 14, 073007 (2012).
  44. L. Sá, P. Ribeiro, and T. Prosen, Symmetry classification of many-body Lindbladians: Tenfold way and beyond, Phys. Rev. X 13, 031019 (2023).
  45. Y.-N. Zhou, X. Li, H. Zhai, C. Li, and Y. Gu, Reviving the Lieb–Schultz–Mattis theorem in open quantum systems, Natl. Sci. Rev. 12, nwae287 (2025).
  46. W. T. Pollard and R. A. Friesner, Solution of the Redfield equation for the dissipative quantum dynamics of multilevel systems, J. Chem. Phys. 100, 5054 (1994).
  47. J.-Q. Liao, J.-F. Huang, and L.-M. Kuang, Quantum thermalization of two coupled two-level systems in eigenstate and bare-state representations, Phys. Rev. A 83, 052110 (2011).
  48. T. V. Tscherbul and P. Brumer, Quantum coherence effects in natural light-induced processes: Cis–trans photoisomerization of model retinal under incoherent excitation, Phys. Chem. Chem. Phys. 17, 30904 (2015).
  49. https://github.com/NAJacobus/CSV_Condition_Data.

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Phys. Rev. Research 8, L032014– Published 23 July, 2026
  • Received 3 August 2025
  • Revised 9 May 2026
  • Accepted 6 July 2026
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