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  • Open Access

Achieving the Quantum Fisher Information Bound in Pseudo-Hermitian Sensors

Ievgen I. Arkhipov1,*, Franco Nori2,3, and Şahin K. Özdemir4

  • 1Joint Laboratory of Optics of Palacký University and Institute of Physics of CAS, Faculty of Science, Palacký University, 17. listopadu 12, 771 46 Olomouc, Czech Republic
  • 2Quantum Information Physics Theory Research Team, Quantum Computing Center, RIKEN, Wakoshi, Saitama 351-0198, Japan
  • 3Physics Department, The University of Michigan, Ann Arbor, Michigan 48109-1040, USA
  • 4Department of Electrical and Computer Engineering, Saint Louis University, St. Louis, Missouri 63103, USA

  • *Contact author: ievgen.arkhipov@upol.cz
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Phys. Rev. Lett. 136, 080802 – Published 24 February, 2026

DOI: https://doi.org/10.1103/qk5r-h851

Abstract

Non-Hermitian systems have attracted considerable interest over the last few decades due to their unique spectral and dynamical properties not encountered in Hermitian counterparts. An intensely debated question is whether non-Hermitian systems, described by pseudo-Hermitian Hamiltonians with real spectra, can offer enhanced sensitivity for parameter estimation when they are operated at or close to exceptional points. However, much of the current analysis and conclusions are based on mathematical formalism developed for Hermitian quantum systems, which is questionable when applied to pseudo-Hermitian Hamiltonians, whose Hilbert space is intrinsically nonflat. Here, we develop a covariant formulation of quantum Fisher information (QFI) defined on the deformed Hilbert space of pseudo-Hermitian Hamiltonians. This covariant framework ensures the preservation of the state norm and enables a consistent treatment of parameter sensitivity. We further show that the covariant QFI of pseudo-Hermitian systems is dual to the ordinary QFI of corresponding Hermitian systems. Importantly, this correspondence naturally imposes an upper bound on the covariant QFI and allows one to identify optimal projections which saturate the corresponding classical Fisher information to this ultimate limit. The developed framework also enables to set the criteria under which pseudo-Hermitian sensors can exhibit an advantage over their Hermitian counterparts of the same dimensionality.

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References (65)

  1. T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics (Springer, Berlin, 1995).
  2. J. Wiersig, Enhancing the sensitivity of frequency and energy splitting detection by using exceptional points: Application to microcavity sensors for single-particle detection, Phys. Rev. Lett. 112, 203901 (2014).
  3. J. Wiersig, Sensors operating at exceptional points: General theory, Phys. Rev. A 93, 033809 (2016).
  4. Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Parity-time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019).
  5. Z.-P. Liu, J. Zhang, Ş. K. Özdemir, B. Peng, H. Jing, X.-Y. Lü, C.-W. Li, L. Yang, F. Nori, and Y.-X. Liu, Metrology with PT-symmetric cavities: Enhanced sensitivity near the PT-phase transition, Phys. Rev. Lett. 117, 110802 (2016).
  6. W. Chen, Ş. K. Özdemir, G. Zhao, J. Wiersig, and L. Yang, Exceptional points enhance sensing in an optical microcavity, Nature (London) 548, 192 (2017).
  7. H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia, R. El-Ganainy, D. N. Christodoulides, and M. Khajavikhan, Enhanced sensitivity at higher-order exceptional points, Nature (London) 548, 187 (2017).
  8. H.-K. Lau and A. A. Clerk, Fundamental limits and non-reciprocal approaches in non-Hermitian quantum sensing, Nat. Commun. 9, 4320 (2018).
  9. Y.-H. Lai, Y.-K. Lu, M.-G. Suh, Z. Yuan, and K. Vahala, Observation of the exceptional-point-enhanced Sagnac effect, Nature (London) 576, 65 (2019).
  10. L. Bao, B. Qi, D. Dong, and F. Nori, Fundamental limits for reciprocal and nonreciprocal non-Hermitian quantum sensing, Phys. Rev. A 103, 042418 (2021).
  11. L. Bao, B. Qi, F. Nori, and D. Dong, Exponential sensitivity revival of noisy non-Hermitian quantum sensing with two-photon drives, Phys. Rev. Res. 6, 023216 (2024).
  12. R. Kononchuk, J. Cai, F. Ellis, R. Thevamaran, and T. Kottos, Exceptional-point-based accelerometers with enhanced signal-to-noise ratio, Nature (London) 607, 697 (2022).
  13. S. Yu, Y. Meng, J.-S. Tang, X.-Y. Xu, Y.-T. Wang, P. Yin, Z.-J. Ke, W. Liu, Z.-P. Li, Y.-Z. Yang, G. Chen, Y.-J. Han, C.-F. Li, and G.-C. Guo, Experimental investigation of quantum PT-enhanced sensor, Phys. Rev. Lett. 125, 240506 (2020).
  14. M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. D. Stone, and L. Jiang, Quantum noise theory of exceptional point amplifying sensors, Phys. Rev. Lett. 123, 180501 (2019).
  15. K. J. H. Peters and S. R. K. Rodriguez, Exceptional precision of a nonlinear optical sensor at a square-root singularity, Phys. Rev. Lett. 129, 013901 (2022).
  16. X. Yu, X. Zhao, L. Li, X.-M. Hu, X. Duan, H. Yuan, and C. Zhang, Toward Heisenberg scaling in non-Hermitian metrology at the quantum regime, Sci. Adv. 10, eadk7616 (2024).
  17. Q.-C. Wu, Y.-H. Zhou, T. Liu, Y.-H. Kang, Q.-P. Su, C.-P. Yang, and Z.-W. Zhou, Enhanced quantum sensing in time-modulated non-Hermitian systems, Chin. J. Phys. 98, 1116 (2025).
  18. J. Naikoo, R. W. Chhajlany, and A. Miranowicz, Enhanced quantum sensing with hybrid exceptional-diabolic singularities, New J. Phys. 27, 064505 (2025).
  19. Y. Chu, Y. Liu, H. Liu, and J. Cai, Quantum sensing with a single-qubit pseudo-Hermitian system, Phys. Rev. Lett. 124, 020501 (2020).
  20. L. Xiao, Y. Chu, Q. Lin, H. Lin, W. Yi, J. Cai, and P. Xue, Non-Hermitian sensing in the absence of exceptional points, Phys. Rev. Lett. 133, 180801 (2024).
  21. J. Zhang, B. Peng, Ş. K. Özdemir, K. Pichler, D. O. Krimer, G. Zhao, F. Nori, Y. X. Liu, S. Rotter, and L. Yang, A phonon laser operating at an exceptional point, Nat. Photonics 12, 479 (2018).
  22. W. Langbein, No exceptional precision of exceptional-point sensors, Phys. Rev. A 98, 023805 (2018).
  23. H. Wang, Y.-H. Lai, Z. Yuan, M.-G. Suh, and K. Vahala, Petermann-factor sensitivity limit near an exceptional point in a Brillouin ring laser gyroscope, Nat. Commun. 11, 1610 (2020).
  24. C. Chen, L. Jin, and R.-B. Liu, Sensitivity of parameter estimation near the exceptional point of a non-Hermitian system, New J. Phys. 21, 083002 (2019).
  25. H. Loughlin and V. Sudhir, Exceptional-point sensors offer no fundamental signal-to-noise ratio enhancement, Phys. Rev. Lett. 132, 243601 (2024).
  26. J. Naikoo, R. W. Chhajlany, and J. Kołodyński, Multiparameter estimation perspective on non-Hermitian singularity-enhanced sensing, Phys. Rev. Lett. 131, 220801 (2023).
  27. J.-X. Peng, B. Zhu, W. Zhang, and K. Zhang, Dissipative quantum Fisher information for a general Liouvillian parametrized process, Phys. Rev. A 109, 012432 (2024).
  28. X. Zheng and Y. D. Chong, Noise constraints for nonlinear exceptional point sensing, Phys. Rev. Lett. 134, 133801 (2025).
  29. J. Ma, X. Wang, C. Sun, and F. Nori, Quantum spin squeezing, Phys. Rep. 509, 89 (2011).
  30. W. Ding, X. Wang, and S. Chen, Fundamental sensitivity limits for non-Hermitian quantum sensors, Phys. Rev. Lett. 131, 160801 (2023).
  31. N. Zeng, T. Liu, K. Xia, Y.-R. Zhang, and F. Nori, Non-Hermitian sensing from the perspective of post-selected measurements, Phys. Rev. Res. 7, 043219 (2025).
  32. J. Naikoo, R. W. Chhajlany, J. Kołodyński, and A. Miranowicz, Metric formalism for precision assessment in non-Hermitian systems, arXiv:2506.22571.

  33. Pseudo-Hermitian Hamiltonians can be physically realized through several approaches. For instance, they can be implemented via the Naimark dilation method, where a pseudo-Hermitian system can be embedded into a higher-dimensional Hermitian system [34]. Alternatively, one can realize them through the postselection procedure, where by properly monitoring the environment of an open quantum system, one can effectively induce the continuous nonunitary dynamics [35, 36].

  34. U. Günther and B. F. Samsonov, Naimark-dilated PT-symmetric brachistochrone, Phys. Rev. Lett. 101, 230404 (2008).
  35. F. Minganti, A. Miranowicz, R. W. Chhajlany, I. I. Arkhipov, and F. Nori, Hybrid-Liouvillian formalism connecting exceptional points of non-Hermitian Hamiltonians and Liouvillians via postselection of quantum trajectories, Phys. Rev. A 101, 062112 (2020).
  36. I. I. Arkhipov, P. Lewalle, F. Nori, Ş. K. Özdemir, and K. B. Whaley, Topological state permutations in time-modulated non-Hermitian multiqubit systems with suppressed nonadiabatic transitions, Phys. Rev. Res. 7, 033242 (2025).
  37. A. Mostafazadeh, Pseudo-Hermitian representation of quantum mechanics, Int. J. Geom. Methods Mod. Phys. 07, 1191 (2010).
  38. C.-Y. Ju, A. Miranowicz, F. Minganti, C.-T. Chan, G.-Y. Chen, and F. Nori, Einstein’s quantum elevator: Hermitization of non-Hermitian Hamiltonians via a generalized vielbein formalism, Phys. Rev. Res. 4, 023070 (2022).

  39. Note that the associated Hermitian Hamiltonian HH is not unique given a metric η, since by redefining S→TS, where T is some unitary, the metric remains invariant, though the Hermitian Hamiltonian is transformed as HH→THHT†. Since this unitary freedom corresponds to a gauge choice and therefore does not modify the underlying physics, one may, without loss of generality, choose T to be parameter independent.

  40. See Supplemental Material at http://link.aps.org/supplemental/10.1103/qk5r-h851, which also include Ref. [41] for detailed derivations of the expressions presented in the main text.
  41. I. I. Arkhipov and F. Minganti, Emergent non-Hermitian localization phenomena in the synthetic space of zero-dimensional bosonic systems, Phys. Rev. A 107, 012202 (2023).
  42. M. Nakahara, Geometry, Topology and Physics, 2nd ed. (IOP Publishing, Bristol, 2003).

  43. The covariant derivative with the metric-compatible connection in Eq. (2) satisfies the following relation [42]: ∂θ⟨ψ|ψ⟩η=⟨Dθψ|ψ⟩η+⟨ψ|Dθψ⟩η. In particular, when η=I, implying Γ=0, the covariant derivative is reduced to the standard derivative Dθ=∂θ. As a result, the derivative of the scalar ∂θ⟨ψ|ψ⟩ simplifies to the ordinary derivative of the inner product of the two state-vectors on the standard flat Hilbert space.

  44. C. Helstrom, Minimum mean-squared error of estimates in quantum statistics, Phys. Lett. 25A, 101 (1967).
  45. S. L. Braunstein and C. M. Caves, Statistical distance and the geometry of quantum states, Phys. Rev. Lett. 72, 3439 (1994).
  46. V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Phys. Rev. Lett. 96, 010401 (2006).
  47. R. M. Wilcox, Exponential operators and parameter differentiation in quantum physics, J. Math. Phys. (N.Y.) 8, 962 (1967).
  48. S. Pang and T. A. Brun, Quantum metrology for a general Hamiltonian parameter, Phys. Rev. A 90, 022117 (2014).
  49. S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, Quantum metrology: Dynamics versus entanglement, Phys. Rev. Lett. 101, 040403 (2008).
  50. V. Giovannetti, S. Lloyd, and L. Maccone, Advances in quantum metrology, Nat. Photonics 5, 222 (2011).
  51. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic Press, New York, 1976).
  52. A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (Springer, Basel, 2011).
  53. M. G. A. Paris, Quantum estimation for quantum technology, Int. J. Quantum. Inform. 07, 125 (2009).
  54. H. Zhang, K. Wang, L. Xiao, and P. Xue, Self-normal and biorthogonal dynamical quantum phase transitions in non-Hermitian quantum walks, Light Sci. Appl. 14, 253 (2025).
  55. X. Wang and G. Ma, Experimental measurement of non-Hermitian left eigenvectors, Front. Phys. 20, 54202 (2025).
  56. Y.-T. Wang, Z.-A. Wang, Z.-P. Li et al., Experimental investigation of direct non-Hermitian measurement and uncertainty relation towards high-dimensional quantum domain, arXiv:2407.05332.
  57. S. Soleymani, Q. Zhong, M. Mokim, S. Rotter, R. El-Ganainy, and Ş. K. Özdemir, Chiral and degenerate perfect absorption on exceptional surfaces, Nat. Commun. 13, 599 (2022).
  58. Z. Zhang, Z. Xu, R. Huang et al., Chirality-induced quantum non-reciprocity, Nat. Photonics 19, 840 (2025).
  59. T.-L. Wang, L.-N. Wu, W. Yang, G.-R. Jin, N. Lambert, and F. Nori, Quantum Fisher information as a signature of the superradiant quantum phase transition, New J. Phys. 16, 063039 (2014).
  60. S. Yin, J. Song, Y. Zhang, and S. Liu, Quantum Fisher information in quantum critical systems with topological characterization, Phys. Rev. B 100, 184417 (2019).
  61. Q. Guan and R. J. Lewis-Swan, Identifying and harnessing dynamical phase transitions for quantum-enhanced sensing, Phys. Rev. Res. 3, 033199 (2021).
  62. V. Montenegro, C. Mukhopadhyay, R. Yousefjani, S. Sarkar, U. Mishra, M. G. Paris, and A. Bayat, Review: Quantum metrology and sensing with many-body systems, Phys. Rep. 1134, 1 (2025).
  63. Q. Liu, Z. Hu, H. Yuan, and Y. Yang, Fully-optimized quantum metrology: Framework, tools, and applications, Adv. Quantum Technol. 7, 2400094 (2024).

  64. We note that in evaluating the probabilities p±(θ)=⟨ψt(θ)|Π±(x0)|ψt(θ)⟩η, the inner product involves states defined, in general, at different points in parameter space. Consequently, the states |ν(x0)⟩ and |ψt⟩ may correspond to distinct local metrics. This ambiguity can be resolved either by parallel transporting one state to the parameter location of the other, or by flattening the η-deformed Hilbert space. Here we adopt the latter approach and discuss it in Supplemental Material [40].

  65. C.-Y. Ju, A. Miranowicz, G.-Y. Chen, and F. Nori, Non-Hermitian Hamiltonians and no-go theorems in quantum information, Phys. Rev. A 100, 062118 (2019).

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Phys. Rev. Lett. 136, 080802– Published 24 February, 2026
  • Received 12 June 2025
  • Revised 26 September 2025
  • Accepted 27 January 2026
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