QMetro++ – Python optimization package for large scale quantum metrology with customized strategy structures
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AbstractQMetro++ is a Python package that provides a set of tools for identifying optimal estimation protocols that maximize quantum Fisher information (QFI). Optimization can be performed for arbitrary configurations of input states, parameter-encoding channels, noise correlations, control operations, and measurements. The use of tensor networks and an iterative see-saw algorithm allows for an efficient optimization even in the regime of a large number of channel uses ($N\approx100$). Additionally, the package includes implementations of the recently developed methods for computing fundamental upper bounds on QFI, which serve as benchmarks for assessing the optimality of numerical optimization results. All functionalities are wrapped up in a user-friendly interface which enables the definition of strategies at various levels of detail.QMetro++ – Python optimization package for large scale quantum metrology with customized strategy structures at Github ► BibTeX data@article{Dulian2026qmetropython, doi = {10.22331/q-2026-01-29-1991}, url = {https://doi.org/10.22331/q-2026-01-29-1991}, title = {{QM}etro++ – {P}ython optimization package for large scale quantum metrology with customized strategy structures}, author = {Dulian, Piotr and Kurdzia{\l{}}ek, Stanis{\l{}}aw and Demkowicz-Dobrz{\'{a}}ski, Rafa{\l{}}}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1991}, month = jan, year = {2026} }► References [1] Piotr Dulian and Stanisław Kurdziałek. ``Python optimization package for large scale quantum metrology with customized strategy structures''. GitHub repository (2025). url: https://github.com/pdulian/qmetro. https://github.com/pdulian/qmetro [2] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. ``Quantum metrology''. Phys. Rev. Lett. 96, 010401 (2006). https://doi.org/10.1103/PhysRevLett.96.010401 [3] Geza Toth and Iagoba Apellaniz. ``Quantum metrology from a quantum information science perspective''. J. Phys. A: Math. Theor. 47, 424006 (2014). https://doi.org/10.1088/1751-8113/47/42/424006 [4] Luca Pezzè, Augusto Smerzi, Markus K. Oberthaler, Roman Schmied, and Philipp Treutlein. ``Quantum metrology with nonclassical states of atomic ensembles''. Rev. Mod. Phys. 90, 035005 (2018). https://doi.org/10.1103/RevModPhys.90.035005 [5] Emanuele Polino, Mauro Valeri, Nicolò Spagnolo, and Fabio Sciarrino. ``Photonic quantum metrology''. AVS Quantum Science 2, 024703 (2020). https://doi.org/10.1116/5.0007577 [6] David S. Simon. ``Introduction to quantum science and technology''.
Springer Nature Switzerland. (2025). https://doi.org/10.1007/978-3-031-81315-3 [7] R. Schnabel. ``Squeezed states of light and their applications in laser interferometers''. Physics Reports 684, 1 – 51 (2017). https://doi.org/10.1016/j.physrep.2017.04.001 [8] D. Ganapathy and et al. ``Broadband quantum enhancement of the ligo detectors with frequency-dependent squeezing''. Phys. Rev. X 13, 041021 (2023). https://doi.org/10.1103/PhysRevX.13.041021 [9] Charikleia Troullinou, Vito Giovanni Lucivero, and Morgan W. Mitchell. ``Quantum-enhanced magnetometry at optimal number density''. Phys. Rev. Lett. 131, 133602 (2023). https://doi.org/10.1103/PhysRevLett.131.133602 [10] Christophe Cassens, Bernd Meyer-Hoppe, Ernst Rasel, and Carsten Klempt. ``Entanglement-enhanced atomic gravimeter''. Phys. Rev. X 15, 011029 (2025). https://doi.org/10.1103/PhysRevX.15.011029 [11] Edwin Pedrozo-Peñafiel, Simone Colombo, Chi Shu, Albert F. Adiyatullin, Zeyang Li, Enrique Mendez, Boris Braverman, Akio Kawasaki, Daisuke Akamatsu, Yanhong Xiao, and Vladan Vuletić. ``Entanglement on an optical atomic-clock transition''. Nature 588, 414–418 (2020). https://doi.org/10.1038/s41586-020-3006-1 [12] B. M. Escher, R. L. de Matos Filho, and L. Davidovich. ``General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology''. Nature Phys. 7, 406–411 (2011). https://doi.org/10.1038/nphys1958 [13] Rafał Demkowicz-Dobrzański, Jan Kołodyński, and Mădălin Guţă. ``The elusive Heisenberg limit in quantum-enhanced metrology''. Nat. Commun. 3, 1063 (2012). arXiv:1201.3940. https://doi.org/10.1038/ncomms2067 arXiv:1201.3940 [14] Rafal Demkowicz-Dobrzański and Lorenzo Maccone. ``Using entanglement against noise in quantum metrology''. Phys. Rev. Lett. 113, 250801 (2014). https://doi.org/10.1103/PhysRevLett.113.250801 [15] Rafal Demkowicz-Dobrzański, Marcin Jarzyna, and Jan Kołodyński. ``Chapter four - quantum limits in optical interferometry''. In E. Wolf, editor, Progress in Optics. Volume 60 of Progress in Optics, pages 345–435. Elsevier (2015). https://doi.org/10.1016/bs.po.2015.02.003 [16] Rafał Demkowicz-Dobrzański, Jan Czajkowski, and Pavel Sekatski. ``Adaptive quantum metrology under general markovian noise''. Phys. Rev. X 7, 041009 (2017). https://doi.org/10.1103/PhysRevX.7.041009 [17] Sisi Zhou, Mengzhen Zhang, John Preskill, and Liang Jiang. ``Achieving the heisenberg limit in quantum metrology using quantum error correction''. Nature Communications 9, 78 (2018). https://doi.org/10.1038/s41467-017-02510-3 [18] Krzysztof Chabuda, Jacek Dziarmaga, Tobias J. Osborne, and Rafal Demkowicz-Dobrzanski. ``Tensor-network approach for quantum metrology in many-body quantum systems''. Nature Communications 11, 250 (2020). https://doi.org/10.1038/s41467-019-13735-9 [19] Sisi Zhou and Liang Jiang. ``Asymptotic theory of quantum channel estimation''. PRX Quantum 2, 010343 (2021). https://doi.org/10.1103/PRXQuantum.2.010343 [20] Anian Altherr and Yuxiang Yang. ``Quantum metrology for non-markovian processes''. Phys. Rev. Lett. 127, 060501 (2021). https://doi.org/10.1103/PhysRevLett.127.060501 [21] Qiushi Liu, Zihao Hu, Haidong Yuan, and Yuxiang Yang. ``Optimal strategies of quantum metrology with a strict hierarchy''. Phys. Rev. Lett. 130, 070803 (2023). https://doi.org/10.1103/PhysRevLett.130.070803 [22] Stanisław Kurdziałek, Wojciech Górecki, Francesco Albarelli, and Rafał Demkowicz-Dobrzański. ``Using adaptiveness and causal superpositions against noise in quantum metrology''. Phys. Rev. Lett. 131, 090801 (2023). https://doi.org/10.1103/PhysRevLett.131.090801 [23] Stanisław Kurdziałek, Piotr Dulian, Joanna Majsak, Sagnik Chakraborty, and Rafał Demkowicz-Dobrzański. ``Quantum metrology using quantum combs and tensor network formalism''. New Journal of Physics 27, 013019 (2025). https://doi.org/10.1088/1367-2630/ada8d1 [24] Qiushi Liu and Yuxiang Yang. ``Efficient tensor networks for control-enhanced quantum metrology''. Quantum 8, 1571 (2024). https://doi.org/10.22331/q-2024-12-18-1571 [25] C. W. Helstrom. ``Quantum detection and estimation theory''. Academic press. (1976). https://doi.org/10.1007/BF01007479 [26] Samuel L. Braunstein and Carlton M. Caves. ``Statistical distance and the geometry of quantum states''. Phys. Rev. Lett. 72, 3439–3443 (1994). https://doi.org/10.1103/PhysRevLett.72.3439 [27] Mattteo G. A. Paris. ``Quantum estimation for quantum technologies''. International Journal of Quantum Information 07, 125–137 (2009). https://doi.org/10.1142/S0219749909004839 [28] Krzysztof Chabuda and Rafał Demkowicz-Dobrzański. ``Tnqmetro: Tensor-network based package for efficient quantum metrology computations''.
Computer Physics Communications 274, 108282 (2022). https://doi.org/10.1016/j.cpc.2021.108282 [29] Stanislaw Kurdzialek, Francesco Albarelli, and Rafal Demkowicz-Dobrzanski. ``Universal bounds for quantum metrology in the presence of correlated noise'' (2024). arXiv:2410.01881. https://doi.org/10.1103/jy3v-wkcb arXiv:2410.01881 [30] Akio Fujiwara. ``Quantum channel identification problem''. Phys. Rev. A 63, 042304 (2001). https://doi.org/10.1103/PhysRevA.63.042304 [31] Jan Kołodyński and Rafał Demkowicz-Dobrzański. ``Efficient tools for quantum metrology with uncorrelated noise''. New Journal of Physics 15, 073043 (2013). https://doi.org/10.1088/1367-2630/15/7/073043 [32] Masahito Hayashi. ``Comparison between the cramer-rao and the mini-max approaches in quantum channel estimation''. Communications in Mathematical Physics 304, 689–709 (2011). https://doi.org/10.1007/s00220-011-1239-4 [33] Johannes Jakob Meyer, Sumeet Khatri, Daniel Stilck França, Jens Eisert, and Philippe Faist. ``Quantum metrology in the finite-sample regime'' (2023). arXiv:2307.06370. https://doi.org/10.1103/qbn1-p6bq arXiv:2307.06370 [34] Rafał Demkowicz-Dobrzański, Wojciech Górecki, and Mădălin Guţă. ``Multi-parameter estimation beyond quantum fisher information''. Journal of Physics A: Mathematical and Theoretical 53, 363001 (2020). https://doi.org/10.1088/1751-8121/ab8ef3 [35] S. M. Kay. ``Fundamentals of statistical signal processing: Estimation theory''. Signal processing series. Prentice Hall. (1993). [36] Giulio Chiribella, Giacomo Mauro D'Ariano, and Paolo Perinotti. ``Theoretical framework for quantum networks''. Phys. Rev. A 80, 022339 (2009). https://doi.org/10.1103/PhysRevA.80.022339 [37] Akio Fujiwara and Hiroshi Imai. ``A fibre bundle over manifolds of quantum channels and its application to quantum statistics''. J. Phys. A 41, 255304 (2008). https://doi.org/10.1088/1751-8113/41/25/255304 [38] Ingemar Bengtsson and Karol Życzkowski. ``Geometry of quantum states: an introduction to quantum entanglement''.
Cambridge Univeristy Press. (2006). [39] MOSEK ApS. ``The mosek optimization toolbox for python manual. version 10.2.''. (2024). url: http://docs.mosek.com/latest/pythonapi/index.html. http://docs.mosek.com/latest/pythonapi/index.html [40] Rafał Demkowicz-Dobrzański. ``Optimal phase estimation with arbitrary a priori knowledge''. Phys. Rev. A 83, 061802 (2011). https://doi.org/10.1103/PhysRevA.83.061802 [41] Katarzyna Macieszczak. ``Quantum Fisher Information: Variational principle and simple iterative algorithm for its efficient computation'' (2013). arXiv:1312.1356. arXiv:1312.1356 [42] Katarzyna Macieszczak, Martin Fraas, and Rafał Demkowicz-Dobrzański. ``Bayesian quantum frequency estimation in presence of collective dephasing''. New Journal of Physics 16, 113002 (2014). https://doi.org/10.1088/1367-2630/16/11/113002 [43] Géza Tóth and Tamás Vértesi. ``Quantum states with a positive partial transpose are useful for metrology''. Phys. Rev. Lett. 120, 020506 (2018). https://doi.org/10.1103/PhysRevLett.120.020506 [44] Árpád Lukács, Róbert Trényi, Tamás Vértesi, and Géza Tóth. ``Optimization in quantum metrology and entanglement theory using semidefinite programming'' (2022). arXiv:2206.02820. https://doi.org/10.1088/2058-9565/ae24a6 arXiv:2206.02820 [45] Jacob C Bridgeman and Christopher T Chubb. ``Hand-waving and interpretive dance: an introductory course on tensor networks''. Journal of Physics A: Mathematical and Theoretical 50, 223001 (2017). https://doi.org/10.1088/1751-8121/aa6dc3 [46] Ulrich Schollwöck. ``The density-matrix renormalization group in the age of matrix product states''. Annals of Physics 326, 96 – 192 (2011). https://doi.org/10.1016/j.aop.2010.09.012 [47] Joshua Bloch. ``Effective java: Programming language guide''. Addison-Wesley. (2018). [48] Mao Zhang, Huai-Ming Yu, Haidong Yuan, Xiaoguang Wang, Rafał Demkowicz-Dobrzański, and Jing Liu. ``Quanestimation: An open-source toolkit for quantum parameter estimation''. Phys. Rev. Res. 4, 043057 (2022). https://doi.org/10.1103/PhysRevResearch.4.043057 [49] Francesco Albarelli and Rafał Demkowicz-Dobrzański. ``Probe incompatibility in multiparameter noisy quantum metrology''. Phys. Rev. X 12, 011039 (2022). https://doi.org/10.1103/PhysRevX.12.011039 [50] Masahito Hayashi and Yingkai Ouyang. ``Finding the optimal probe state for multiparameter quantum metrology using conic programming''. npj Quantum Information 10 (2024). https://doi.org/10.1038/s41534-024-00905-x [51] Jessica Bavaresco, Mio Murao, and Marco Túlio Quintino. ``Strict hierarchy between parallel, sequential, and indefinite-causal-order strategies for channel discrimination''.
Physical Review Letters 127 (2021). https://doi.org/10.1103/physrevlett.127.200504Cited byCould not fetch Crossref cited-by data during last attempt 2026-01-29 12:52:02: Could not fetch cited-by data for 10.22331/q-2026-01-29-1991 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-01-29 12:52:08: No response from ADS or unable to decode the received json data when getting the list of citing works.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. AbstractQMetro++ is a Python package that provides a set of tools for identifying optimal estimation protocols that maximize quantum Fisher information (QFI). Optimization can be performed for arbitrary configurations of input states, parameter-encoding channels, noise correlations, control operations, and measurements. The use of tensor networks and an iterative see-saw algorithm allows for an efficient optimization even in the regime of a large number of channel uses ($N\approx100$). Additionally, the package includes implementations of the recently developed methods for computing fundamental upper bounds on QFI, which serve as benchmarks for assessing the optimality of numerical optimization results. All functionalities are wrapped up in a user-friendly interface which enables the definition of strategies at various levels of detail.QMetro++ – Python optimization package for large scale quantum metrology with customized strategy structures at Github ► BibTeX data@article{Dulian2026qmetropython, doi = {10.22331/q-2026-01-29-1991}, url = {https://doi.org/10.22331/q-2026-01-29-1991}, title = {{QM}etro++ – {P}ython optimization package for large scale quantum metrology with customized strategy structures}, author = {Dulian, Piotr and Kurdzia{\l{}}ek, Stanis{\l{}}aw and Demkowicz-Dobrz{\'{a}}ski, Rafa{\l{}}}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {1991}, month = jan, year = {2026} }► References [1] Piotr Dulian and Stanisław Kurdziałek. ``Python optimization package for large scale quantum metrology with customized strategy structures''. GitHub repository (2025). url: https://github.com/pdulian/qmetro. https://github.com/pdulian/qmetro [2] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. ``Quantum metrology''. Phys. Rev. Lett. 96, 010401 (2006). https://doi.org/10.1103/PhysRevLett.96.010401 [3] Geza Toth and Iagoba Apellaniz. ``Quantum metrology from a quantum information science perspective''. J. Phys. A: Math. Theor. 47, 424006 (2014). https://doi.org/10.1088/1751-8113/47/42/424006 [4] Luca Pezzè, Augusto Smerzi, Markus K. Oberthaler, Roman Schmied, and Philipp Treutlein. ``Quantum metrology with nonclassical states of atomic ensembles''. Rev. Mod. Phys. 90, 035005 (2018). https://doi.org/10.1103/RevModPhys.90.035005 [5] Emanuele Polino, Mauro Valeri, Nicolò Spagnolo, and Fabio Sciarrino. ``Photonic quantum metrology''. AVS Quantum Science 2, 024703 (2020). https://doi.org/10.1116/5.0007577 [6] David S. Simon. ``Introduction to quantum science and technology''.
Springer Nature Switzerland. (2025). https://doi.org/10.1007/978-3-031-81315-3 [7] R. Schnabel. ``Squeezed states of light and their applications in laser interferometers''. Physics Reports 684, 1 – 51 (2017). https://doi.org/10.1016/j.physrep.2017.04.001 [8] D. Ganapathy and et al. ``Broadband quantum enhancement of the ligo detectors with frequency-dependent squeezing''. Phys. Rev. X 13, 041021 (2023). https://doi.org/10.1103/PhysRevX.13.041021 [9] Charikleia Troullinou, Vito Giovanni Lucivero, and Morgan W. Mitchell. ``Quantum-enhanced magnetometry at optimal number density''. Phys. Rev. Lett. 131, 133602 (2023). https://doi.org/10.1103/PhysRevLett.131.133602 [10] Christophe Cassens, Bernd Meyer-Hoppe, Ernst Rasel, and Carsten Klempt. ``Entanglement-enhanced atomic gravimeter''. Phys. Rev. X 15, 011029 (2025). https://doi.org/10.1103/PhysRevX.15.011029 [11] Edwin Pedrozo-Peñafiel, Simone Colombo, Chi Shu, Albert F. Adiyatullin, Zeyang Li, Enrique Mendez, Boris Braverman, Akio Kawasaki, Daisuke Akamatsu, Yanhong Xiao, and Vladan Vuletić. ``Entanglement on an optical atomic-clock transition''. Nature 588, 414–418 (2020). https://doi.org/10.1038/s41586-020-3006-1 [12] B. M. Escher, R. L. de Matos Filho, and L. Davidovich. ``General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology''. Nature Phys. 7, 406–411 (2011). https://doi.org/10.1038/nphys1958 [13] Rafał Demkowicz-Dobrzański, Jan Kołodyński, and Mădălin Guţă. ``The elusive Heisenberg limit in quantum-enhanced metrology''. Nat. Commun. 3, 1063 (2012). arXiv:1201.3940. https://doi.org/10.1038/ncomms2067 arXiv:1201.3940 [14] Rafal Demkowicz-Dobrzański and Lorenzo Maccone. ``Using entanglement against noise in quantum metrology''. Phys. Rev. Lett. 113, 250801 (2014). https://doi.org/10.1103/PhysRevLett.113.250801 [15] Rafal Demkowicz-Dobrzański, Marcin Jarzyna, and Jan Kołodyński. ``Chapter four - quantum limits in optical interferometry''. In E. Wolf, editor, Progress in Optics. Volume 60 of Progress in Optics, pages 345–435. Elsevier (2015). https://doi.org/10.1016/bs.po.2015.02.003 [16] Rafał Demkowicz-Dobrzański, Jan Czajkowski, and Pavel Sekatski. ``Adaptive quantum metrology under general markovian noise''. Phys. Rev. X 7, 041009 (2017). https://doi.org/10.1103/PhysRevX.7.041009 [17] Sisi Zhou, Mengzhen Zhang, John Preskill, and Liang Jiang. ``Achieving the heisenberg limit in quantum metrology using quantum error correction''. Nature Communications 9, 78 (2018). https://doi.org/10.1038/s41467-017-02510-3 [18] Krzysztof Chabuda, Jacek Dziarmaga, Tobias J. Osborne, and Rafal Demkowicz-Dobrzanski. ``Tensor-network approach for quantum metrology in many-body quantum systems''. Nature Communications 11, 250 (2020). https://doi.org/10.1038/s41467-019-13735-9 [19] Sisi Zhou and Liang Jiang. ``Asymptotic theory of quantum channel estimation''. PRX Quantum 2, 010343 (2021). https://doi.org/10.1103/PRXQuantum.2.010343 [20] Anian Altherr and Yuxiang Yang. ``Quantum metrology for non-markovian processes''. Phys. Rev. Lett. 127, 060501 (2021). https://doi.org/10.1103/PhysRevLett.127.060501 [21] Qiushi Liu, Zihao Hu, Haidong Yuan, and Yuxiang Yang. ``Optimal strategies of quantum metrology with a strict hierarchy''. Phys. Rev. Lett. 130, 070803 (2023). https://doi.org/10.1103/PhysRevLett.130.070803 [22] Stanisław Kurdziałek, Wojciech Górecki, Francesco Albarelli, and Rafał Demkowicz-Dobrzański. ``Using adaptiveness and causal superpositions against noise in quantum metrology''. Phys. Rev. Lett. 131, 090801 (2023). https://doi.org/10.1103/PhysRevLett.131.090801 [23] Stanisław Kurdziałek, Piotr Dulian, Joanna Majsak, Sagnik Chakraborty, and Rafał Demkowicz-Dobrzański. ``Quantum metrology using quantum combs and tensor network formalism''. New Journal of Physics 27, 013019 (2025). https://doi.org/10.1088/1367-2630/ada8d1 [24] Qiushi Liu and Yuxiang Yang. ``Efficient tensor networks for control-enhanced quantum metrology''. Quantum 8, 1571 (2024). https://doi.org/10.22331/q-2024-12-18-1571 [25] C. W. Helstrom. ``Quantum detection and estimation theory''. Academic press. (1976). https://doi.org/10.1007/BF01007479 [26] Samuel L. Braunstein and Carlton M. Caves. ``Statistical distance and the geometry of quantum states''. Phys. Rev. Lett. 72, 3439–3443 (1994). https://doi.org/10.1103/PhysRevLett.72.3439 [27] Mattteo G. A. Paris. ``Quantum estimation for quantum technologies''. International Journal of Quantum Information 07, 125–137 (2009). https://doi.org/10.1142/S0219749909004839 [28] Krzysztof Chabuda and Rafał Demkowicz-Dobrzański. ``Tnqmetro: Tensor-network based package for efficient quantum metrology computations''.
Computer Physics Communications 274, 108282 (2022). https://doi.org/10.1016/j.cpc.2021.108282 [29] Stanislaw Kurdzialek, Francesco Albarelli, and Rafal Demkowicz-Dobrzanski. ``Universal bounds for quantum metrology in the presence of correlated noise'' (2024). arXiv:2410.01881. https://doi.org/10.1103/jy3v-wkcb arXiv:2410.01881 [30] Akio Fujiwara. ``Quantum channel identification problem''. Phys. Rev. A 63, 042304 (2001). https://doi.org/10.1103/PhysRevA.63.042304 [31] Jan Kołodyński and Rafał Demkowicz-Dobrzański. ``Efficient tools for quantum metrology with uncorrelated noise''. New Journal of Physics 15, 073043 (2013). https://doi.org/10.1088/1367-2630/15/7/073043 [32] Masahito Hayashi. ``Comparison between the cramer-rao and the mini-max approaches in quantum channel estimation''. Communications in Mathematical Physics 304, 689–709 (2011). https://doi.org/10.1007/s00220-011-1239-4 [33] Johannes Jakob Meyer, Sumeet Khatri, Daniel Stilck França, Jens Eisert, and Philippe Faist. ``Quantum metrology in the finite-sample regime'' (2023). arXiv:2307.06370. https://doi.org/10.1103/qbn1-p6bq arXiv:2307.06370 [34] Rafał Demkowicz-Dobrzański, Wojciech Górecki, and Mădălin Guţă. ``Multi-parameter estimation beyond quantum fisher information''. Journal of Physics A: Mathematical and Theoretical 53, 363001 (2020). https://doi.org/10.1088/1751-8121/ab8ef3 [35] S. M. Kay. ``Fundamentals of statistical signal processing: Estimation theory''. Signal processing series. Prentice Hall. (1993). [36] Giulio Chiribella, Giacomo Mauro D'Ariano, and Paolo Perinotti. ``Theoretical framework for quantum networks''. Phys. Rev. A 80, 022339 (2009). https://doi.org/10.1103/PhysRevA.80.022339 [37] Akio Fujiwara and Hiroshi Imai. ``A fibre bundle over manifolds of quantum channels and its application to quantum statistics''. J. Phys. A 41, 255304 (2008). https://doi.org/10.1088/1751-8113/41/25/255304 [38] Ingemar Bengtsson and Karol Życzkowski. ``Geometry of quantum states: an introduction to quantum entanglement''.
Cambridge Univeristy Press. (2006). [39] MOSEK ApS. ``The mosek optimization toolbox for python manual. version 10.2.''. (2024). url: http://docs.mosek.com/latest/pythonapi/index.html. http://docs.mosek.com/latest/pythonapi/index.html [40] Rafał Demkowicz-Dobrzański. ``Optimal phase estimation with arbitrary a priori knowledge''. Phys. Rev. A 83, 061802 (2011). https://doi.org/10.1103/PhysRevA.83.061802 [41] Katarzyna Macieszczak. ``Quantum Fisher Information: Variational principle and simple iterative algorithm for its efficient computation'' (2013). arXiv:1312.1356. arXiv:1312.1356 [42] Katarzyna Macieszczak, Martin Fraas, and Rafał Demkowicz-Dobrzański. ``Bayesian quantum frequency estimation in presence of collective dephasing''. New Journal of Physics 16, 113002 (2014). https://doi.org/10.1088/1367-2630/16/11/113002 [43] Géza Tóth and Tamás Vértesi. ``Quantum states with a positive partial transpose are useful for metrology''. Phys. Rev. Lett. 120, 020506 (2018). https://doi.org/10.1103/PhysRevLett.120.020506 [44] Árpád Lukács, Róbert Trényi, Tamás Vértesi, and Géza Tóth. ``Optimization in quantum metrology and entanglement theory using semidefinite programming'' (2022). arXiv:2206.02820. https://doi.org/10.1088/2058-9565/ae24a6 arXiv:2206.02820 [45] Jacob C Bridgeman and Christopher T Chubb. ``Hand-waving and interpretive dance: an introductory course on tensor networks''. Journal of Physics A: Mathematical and Theoretical 50, 223001 (2017). https://doi.org/10.1088/1751-8121/aa6dc3 [46] Ulrich Schollwöck. ``The density-matrix renormalization group in the age of matrix product states''. Annals of Physics 326, 96 – 192 (2011). https://doi.org/10.1016/j.aop.2010.09.012 [47] Joshua Bloch. ``Effective java: Programming language guide''. Addison-Wesley. (2018). [48] Mao Zhang, Huai-Ming Yu, Haidong Yuan, Xiaoguang Wang, Rafał Demkowicz-Dobrzański, and Jing Liu. ``Quanestimation: An open-source toolkit for quantum parameter estimation''. Phys. Rev. Res. 4, 043057 (2022). https://doi.org/10.1103/PhysRevResearch.4.043057 [49] Francesco Albarelli and Rafał Demkowicz-Dobrzański. ``Probe incompatibility in multiparameter noisy quantum metrology''. Phys. Rev. X 12, 011039 (2022). https://doi.org/10.1103/PhysRevX.12.011039 [50] Masahito Hayashi and Yingkai Ouyang. ``Finding the optimal probe state for multiparameter quantum metrology using conic programming''. npj Quantum Information 10 (2024). https://doi.org/10.1038/s41534-024-00905-x [51] Jessica Bavaresco, Mio Murao, and Marco Túlio Quintino. ``Strict hierarchy between parallel, sequential, and indefinite-causal-order strategies for channel discrimination''.
Physical Review Letters 127 (2021). https://doi.org/10.1103/physrevlett.127.200504Cited byCould not fetch Crossref cited-by data during last attempt 2026-01-29 12:52:02: Could not fetch cited-by data for 10.22331/q-2026-01-29-1991 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2026-01-29 12:52:08: No response from ADS or unable to decode the received json data when getting the list of citing works.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.