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Role of the extended Hilbert space in the attainability of the Quantum Cramér-Rao bound for multiparameter estimation (2404.01520v1)

Published 1 Apr 2024 in quant-ph

Abstract: The symmetric logarithmic derivative Cram\'er-Rao bound (SLDCRB) provides a fundamental limit to the minimum variance with which a set of unknown parameters can be estimated in an unbiased manner. It is known that the SLDCRB can be saturated provided the optimal measurements for the individual parameters commute with one another. However, when this is not the case the SLDCRB cannot be attained in general. In the experimentally relevant setting, where quantum states are measured individually, necessary and sufficient conditions for when the SLDCRB can be saturated are not known. In this setting the SLDCRB is attainable provided the SLD operators can be chosen to commute on an extended Hilbert space. However, beyond this relatively little is known about when the SLD operators can be chosen in this manner. In this paper we present explicit examples which demonstrate novel aspects of this condition. Our examples demonstrate that the SLD operators commuting on any two of the following three spaces: support space, support-kernel space and kernel space, is neither a necessary nor sufficient condition for commutativity on the extended space. We present a simple analytic example showing that the Nagaoka-Hayashi Cram\'er-Rao bound is not always attainable. Finally, we provide necessary and sufficient conditions for the attainability of the SLDCRB in the case when the kernel space is one-dimensional. These results provide new information on the necessary and sufficient conditions for the attainability of the SLDCRB.

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References (61)
  1. Enhanced sensitivity of the LIGO gravitational wave detector by using squeezed states of light. Nat. Photonics, 7(8):613–619, 2013.
  2. Quantum-enhanced nonlinear microscopy. Nature, 594(7862):201–206, 2021.
  3. Toward Heisenberg-limited spectroscopy with multiparticle entangled states. Science, 304(5676):1476–1478, 2004.
  4. Optimal metrology with programmable quantum sensors. Nature, 603(7902):604–609, 2022.
  5. Quantum metrology: Dynamics versus entanglement. Phys. Rev. Lett., 101(4):040403, 2008.
  6. Carlton M Caves. Quantum-mechanical noise in an interferometer. Phys. Rev. D, 23(8):1693, 1981.
  7. Distributed quantum sensing in a continuous-variable entangled network. Nat. Phys., 16(3):281–284, 2020.
  8. Measurement of damping and temperature: Precision bounds in gaussian dissipative channels. Phys. Rev. A, 83(1):012315, 2011.
  9. Quantum enhanced multiple phase estimation. Phys. Rev. Lett., 111(7):070403, 2013.
  10. Quantum enhanced estimation of a multidimensional field. Phys. Rev. Lett., 116(3):030801, 2016.
  11. Quantum sensing for dynamical tracking of chemical processes. Phys. Rev. A, 99(5):053817, 2019.
  12. Joint estimation of phase and phase diffusion for quantum metrology. Nat. Commun., 5(1):1–7, 2014.
  13. On super-resolution imaging as a multiparameter estimation problem. Int. J. Quantum Inf., 15(08):1740005, 2017.
  14. Reaching for the quantum limits in the simultaneous estimation of phase and phase diffusion. Quantum Sci.Technol., 2(4):044004, 2017.
  15. Minimal tradeoff and ultimate precision limit of multiparameter quantum magnetometry under the parallel scheme. Phys. Rev. Lett., 125(2):020501, 2020.
  16. Multiparameter estimation with two-qubit probes in noisy channels. Entropy, 25(8):1122, 2023.
  17. Verifying the security of a continuous variable quantum communication protocol via quantum metrology. arXiv preprint arXiv:2311.05389, 2023.
  18. Werner Heisenberg. Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik. In Original Scientific Papers Wissenschaftliche Originalarbeiten, pages 478–504. Springer, 1985.
  19. Howard Percy Robertson. The uncertainty principle. Phys. Rev., 34(1):163, 1929.
  20. E Arthurs and JL Kelly Jr. On the simultaneous measurement of a pair of conjugate observables. Bell Syst. Tech. J., 44(4):725–729, 1965.
  21. Multi-parameter quantum metrology. Adv. Phys. X, 1(4):621–639, 2016.
  22. Quantum fisher information matrix and multiparameter estimation. J. Phys. A Math. Theor., 53(2):023001, 2019.
  23. Geometric perspective on quantum parameter estimation. AVS Quantum Science, 2(1):014701, 2020.
  24. Multi-parameter estimation beyond quantum fisher information. J. Phys. A Math. Theor., 53(36):363001, 2020.
  25. Lorcan Conlon. Quantum Multiparameter Estimation: Exploring Fundamental Limits and Testing Quantum Mechanics. PhD thesis, The Australian National University, 2023.
  26. Carl W Helstrom. Minimum mean-squared error of estimates in quantum statistics. Phys. Lett. A, 25(2):101–102, 1967.
  27. Carl W Helstrom. The minimum variance of estimates in quantum signal detection. IEEE Trans. Inf. Theory, 14(2):234–242, 1968.
  28. H. Yuen and M. Lax. Multiple-parameter quantum estimation and measurement of nonselfadjoint observables. IEEE Trans. Inf. Theory, 19(6):740–750, 1973.
  29. R. D. Gill and S. Massar. State estimation for large ensembles. Phys. Rev. A, 61(4):042312, 2000.
  30. Alexander S Holevo. Statistical decision theory for quantum systems. J. Multivar. Anal., 3(4):337–394, 1973.
  31. Alexander S Holevo. Probabilistic and statistical aspects of quantum theory, volume 1. Springer Science & Business Media, 2011.
  32. Hiroshi Nagaoka. A new approach to Cramér-Rao bounds for quantum state estimation. In Asymptotic Theory Of Quantum Statistical Inference: Selected Papers, pages 100–112. 2005. Originally published as IEICE Technical Report, 89, 228, IT 89-42, 9-14, (1989).
  33. H. Nagaoka. A generalization of the simultaneous diagonalization of Hermitian matrices and its relation to quantum estimation theory. In Asymptotic Theory Of Quantum Statistical Inference: Selected Papers, pages 133–149. World Scientific, 2005. Originally published as Trans. Jap. Soc. Indust. Appl. Math., 1, 43-56, (1991) in Japanese. Translated to English by Y.Tsuda.
  34. Efficient computation of the nagaoka–hayashi bound for multi-parameter estimation with separable measurements. npj Quantum Inf., 7(110), 2020.
  35. Local asymptotic normality for finite dimensional quantum systems. Commun. Math. Phys., 289(2):597–652, 2009.
  36. Quantum local asymptotic normality based on a new quantum likelihood ratio. Ann. Stat., 41(4):2197–2217, 2013.
  37. Attaining the ultimate precision limit in quantum state estimation. Commun. Math. Phys., 368(1):223–293, 2019.
  38. Tight Cramér-Rao type bounds for multiparameter quantum metrology through conic programming. Quantum, 7:1094, 2023.
  39. Statistical distance and the geometry of quantum states. Phys. Rev. Lett., 72(22):3439, 1994.
  40. Compatibility in multiparameter quantum metrology. Phys. Rev. A, 94(5):052108, 2016.
  41. Five open problems in quantum information theory. Phys. Rev. X Quantum, 3:010101, 2022.
  42. Optimal measurements for simultaneous quantum estimation of multiple phases. Phys. Rev. Lett., 119(13):130504, 2017.
  43. Optimal measurements for quantum multiparameter estimation with general states. Phys. Rev. A, 100(3):032104, 2019.
  44. Information geometry under hierarchical quantum measurement. Phys. Rev. Lett., 128:250502, 2022.
  45. Incompatibility measures in multiparameter quantum estimation under hierarchical quantum measurements. Phys. Rev. A, 105:062442, 2022.
  46. Methods of information geometry, volume 191. American Mathematical Soc., 2000.
  47. Hendra I Nurdin. Saturability of the Quantum Cramér-Rao Bound in Multiparameter Quantum Estimation at the Single-Copy Level. arXiv preprint arXiv:2402.11567, 2024.
  48. Quantum state estimation with nuisance parameters. Journal of Physics A: Mathematical and Theoretical, 53(45):453001, 2020.
  49. Łukasz Rudnicki. Private Communication (Dec 2022).
  50. The gap persistence theorem for quantum multiparameter estimation. arXiv preprint arXiv:2208.07386, 2022.
  51. MA Neumark. On spectral functions of a symmetric operator. Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 7(6):285–296, 1943.
  52. Ancilla dimensions needed to carry out positive-operator-valued measurement. Physical Review A, 76(6):060303, 2007.
  53. Entangling measurements for multiparameter estimation with two qubits. Quantum Sci. Technol., 3(1):01LT01, 2017.
  54. Beating the Rayleigh limit using two-photon interference. Phys. Rev. Lett., 121(25):250503, 2018.
  55. Deterministic realization of collective measurements via photonic quantum walks. Nat. Commun., 9(1):1–7, 2018.
  56. Experimentally reducing the quantum measurement back action in work distributions by a collective measurement. Sci. Adv., 5(3):eaav4944, 2019.
  57. Direct estimation of quantum coherence by collective measurements. npj Quantum Inf., 6(1):1–5, 2020.
  58. Approaching optimal entangling collective measurements on quantum computing platforms. Nature Physics, 19(3):351–357, 2023.
  59. Discriminating mixed qubit states with collective measurements. Communications Physics, 6(1):337, 2023.
  60. Experimental optimal quantum state estimation with genuine three-copy collective measurements. arXiv preprint arXiv:2312.01651, 2023.
  61. Minimum-consumption discrimination of quantum states via globally optimal adaptive measurements. Physical Review Letters, 132(11):110801, 2024.

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