Error Bounds for Open Quantum Systems with Harmonic Bosonic Bath
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AbstractWe investigate the dependence of physical observable of open quantum systems with Bosonic bath on the bath correlation function. We provide an error estimate of the difference of physical observable induced by the variation of bath correlation function, based on diagrammatic and combinatorial arguments. This gives a mathematically rigorous justification of the result in [6].► BibTeX data@article{Liu2025errorboundsopen, doi = {10.22331/q-2025-10-28-1896}, url = {https://doi.org/10.22331/q-2025-10-28-1896}, title = {Error {B}ounds for {O}pen {Q}uantum {S}ystems with {H}armonic {B}osonic {B}ath}, author = {Liu, Kaizhao and Lu, Jianfeng}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1896}, month = oct, year = {2025} }► References [1] Massimiliano Esposito, Upendra Harbola, and Shaul Mukamel. ``Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems''. Reviews of Modern Physics 81, 1665–1702 (2009). https://doi.org/10.1103/revmodphys.81.1665 [2] Peter W. Shor. ``Scheme for reducing decoherence in quantum computer memory''. Physical Review A 52, R2493–R2496 (1995). https://doi.org/10.1103/physreva.52.r2493 [3] G. Lindblad. ``On the generators of quantum dynamical semigroups''. Communications in Mathematical Physics 48, 119–130 (1976). https://doi.org/10.1007/bf01608499 [4] Heinz-Peter Breuer and Francesco Petruccione. ``Open quantum electrodynamics''. Pages 568–618.
Oxford University Press. (2007). https://doi.org/10.1093/acprof:oso/9780199213900.003.12 [5] R.P Feynman and F.L Vernon. ``The theory of a general quantum system interacting with a linear dissipative system''. Annals of Physics 24, 118–173 (1963). https://doi.org/10.1016/0003-4916(63)90068-x [6] F. Mascherpa, A. Smirne, S. F. Huelga, and M. B. Plenio. ``Open systems with error bounds: Spin-boson model with spectral density variations''.
Physical Review Letters 118, 100401 (2017). https://doi.org/10.1103/physrevlett.118.100401 [7] Yoshitaka Tanimura and Ryogo Kubo. ``Time evolution of a quantum system in contact with a nearly gaussian-markoffian noise bath''. Journal of the Physical Society of Japan 58, 101–114 (1989). https://doi.org/10.1143/jpsj.58.101 [8] Christoph Meier and David J. Tannor. ``Non-Markovian evolution of the density operator in the presence of strong laser fields''. The Journal of Chemical Physics 111, 3365–3376 (1999). https://doi.org/10.1063/1.479669 [9] Evgenii Kochetov. ``Comment on “Coherent-state path integrals in the continuum”''. Physical Review A 99, 026101 (2019). https://doi.org/10.1103/physreva.99.026101 [10] Justin H. Wilson and Victor Galitski. ``Breakdown of the coherent state path integral: Two simple examples''.
Physical Review Letters 106, 110401 (2011). https://doi.org/10.1103/physrevlett.106.110401 [11] G. Kordas, S. I. Mistakidis, and A. I. Karanikas. ``Coherent-state path integrals in the continuum''. Physical Review A 90, 032104 (2014). https://doi.org/10.1103/physreva.90.032104 [12] Zhenning Cai, Jianfeng Lu, and Siyao Yang. ``Inchworm Monte Carlo method for open quantum systems''. Communications on Pure and Applied Mathematics 73, 2430–2472 (2020). https://doi.org/10.1002/cpa.21888 [13] J. J. Sakurai and Jim Napolitano. ``Modern quantum mechanics''.
Cambridge University Press. (2017). https://doi.org/10.1017/9781108499996 [14] Alex Kamenev. ``Field theory of non-equilibrium systems''.
Cambridge University Press. (2011). https://doi.org/10.1017/cbo9781139003667 [15] Henrik Bruus and Karsten Flensberg. ``Many–body quantum theory in condensed matter physics''.
Oxford University Press. (2004). https://doi.org/10.1093/oso/9780198566335.001.0001 [16] Zhenning Cai, Jianfeng Lu, and Siyao Yang. ``Numerical analysis for inchworm Monte Carlo method: Sign problem and error growth''. Mathematics of Computation 92, 1141–1209 (2022). https://doi.org/10.1090/mcom/3785 [17] Hans C. Fogedby. ``Field-theoretical approach to open quantum systems and the lindblad equation''. Physical Review A 106, 022205 (2022). https://doi.org/10.1103/physreva.106.022205 [18] Erik Aurell, Ryochi Kawai, and Ketan Goyal. ``An operator derivation of the Feynman–Vernon theory, with applications to the generating function of bath energy changes and to an-harmonic baths''. Journal of Physics A: Mathematical and Theoretical 53, 275303 (2020). https://doi.org/10.1088/1751-8121/ab9274 [19] Mauro Cirio, Pengfei Liang, and Neill Lambert. ``Input-output hierarchical equations of motion''. Physical Review A 112, 012211 (2025). https://doi.org/10.1103/4q4p-z14x [20] Qiongtao Xie, Honghua Zhong, Murray T Batchelor, and Chaohong Lee. ``The quantum Rabi model: solution and dynamics''. Journal of Physics A: Mathematical and Theoretical 50, 113001 (2017). https://doi.org/10.1088/1751-8121/aa5a65 [21] Klaus Hepp and Elliott H Lieb. ``On the superradiant phase transition for molecules in a quantized radiation field: the dicke maser model''. Annals of Physics 76, 360–404 (1973). https://doi.org/10.1016/0003-4916(73)90039-0 [22] A C Hewson and D Meyer. ``Numerical renormalization group study of the Anderson-Holstein impurity model''. Journal of Physics: Condensed Matter 14, 427–445 (2001). https://doi.org/10.1088/0953-8984/14/3/312 [23] Si Luo, Neill Lambert, Pengfei Liang, and Mauro Cirio. ``Quantum-classical decomposition of gaussian quantum environments: A stochastic pseudomode model''. PRX Quantum 4, 030316 (2023). https://doi.org/10.1103/prxquantum.4.030316 [24] Gunhee Park, Zhen Huang, Yuanran Zhu, Chao Yang, Garnet Kin-Lic Chan, and Lin Lin. ``Quasi-lindblad pseudomode theory for open quantum systems''. Physical Review B 110, 195148 (2024). https://doi.org/10.1103/physrevb.110.195148 [25] D. Tamascelli, A. Smirne, S. F. Huelga, and M. B. Plenio. ``Nonperturbative treatment of non-Markovian dynamics of open quantum systems''.
Physical Review Letters 120, 030402 (2018). https://doi.org/10.1103/physrevlett.120.030402 [26] F. Mascherpa, A. Smirne, A. D. Somoza, P. Fernández-Acebal, S. Donadi, D. Tamascelli, S. F. Huelga, and M. B. Plenio. ``Optimized auxiliary oscillators for the simulation of general open quantum systems''. Physical Review A 101, 052108 (2020). https://doi.org/10.1103/physreva.101.052108 [27] Leonardo A. Pachón and Paul Brumer. ``Direct experimental determination of spectral densities of molecular complexes''. The Journal of Chemical Physics 141 (2014). https://doi.org/10.1063/1.4900512Cited byCould not fetch Crossref cited-by data during last attempt 2025-10-28 15:21:33: Could not fetch cited-by data for 10.22331/q-2025-10-28-1896 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-10-28 15:21:34: 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. AbstractWe investigate the dependence of physical observable of open quantum systems with Bosonic bath on the bath correlation function. We provide an error estimate of the difference of physical observable induced by the variation of bath correlation function, based on diagrammatic and combinatorial arguments. This gives a mathematically rigorous justification of the result in [6].► BibTeX data@article{Liu2025errorboundsopen, doi = {10.22331/q-2025-10-28-1896}, url = {https://doi.org/10.22331/q-2025-10-28-1896}, title = {Error {B}ounds for {O}pen {Q}uantum {S}ystems with {H}armonic {B}osonic {B}ath}, author = {Liu, Kaizhao and Lu, Jianfeng}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {9}, pages = {1896}, month = oct, year = {2025} }► References [1] Massimiliano Esposito, Upendra Harbola, and Shaul Mukamel. ``Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems''. Reviews of Modern Physics 81, 1665–1702 (2009). https://doi.org/10.1103/revmodphys.81.1665 [2] Peter W. Shor. ``Scheme for reducing decoherence in quantum computer memory''. Physical Review A 52, R2493–R2496 (1995). https://doi.org/10.1103/physreva.52.r2493 [3] G. Lindblad. ``On the generators of quantum dynamical semigroups''. Communications in Mathematical Physics 48, 119–130 (1976). https://doi.org/10.1007/bf01608499 [4] Heinz-Peter Breuer and Francesco Petruccione. ``Open quantum electrodynamics''. Pages 568–618.
Oxford University Press. (2007). https://doi.org/10.1093/acprof:oso/9780199213900.003.12 [5] R.P Feynman and F.L Vernon. ``The theory of a general quantum system interacting with a linear dissipative system''. Annals of Physics 24, 118–173 (1963). https://doi.org/10.1016/0003-4916(63)90068-x [6] F. Mascherpa, A. Smirne, S. F. Huelga, and M. B. Plenio. ``Open systems with error bounds: Spin-boson model with spectral density variations''.
Physical Review Letters 118, 100401 (2017). https://doi.org/10.1103/physrevlett.118.100401 [7] Yoshitaka Tanimura and Ryogo Kubo. ``Time evolution of a quantum system in contact with a nearly gaussian-markoffian noise bath''. Journal of the Physical Society of Japan 58, 101–114 (1989). https://doi.org/10.1143/jpsj.58.101 [8] Christoph Meier and David J. Tannor. ``Non-Markovian evolution of the density operator in the presence of strong laser fields''. The Journal of Chemical Physics 111, 3365–3376 (1999). https://doi.org/10.1063/1.479669 [9] Evgenii Kochetov. ``Comment on “Coherent-state path integrals in the continuum”''. Physical Review A 99, 026101 (2019). https://doi.org/10.1103/physreva.99.026101 [10] Justin H. Wilson and Victor Galitski. ``Breakdown of the coherent state path integral: Two simple examples''.
Physical Review Letters 106, 110401 (2011). https://doi.org/10.1103/physrevlett.106.110401 [11] G. Kordas, S. I. Mistakidis, and A. I. Karanikas. ``Coherent-state path integrals in the continuum''. Physical Review A 90, 032104 (2014). https://doi.org/10.1103/physreva.90.032104 [12] Zhenning Cai, Jianfeng Lu, and Siyao Yang. ``Inchworm Monte Carlo method for open quantum systems''. Communications on Pure and Applied Mathematics 73, 2430–2472 (2020). https://doi.org/10.1002/cpa.21888 [13] J. J. Sakurai and Jim Napolitano. ``Modern quantum mechanics''.
Cambridge University Press. (2017). https://doi.org/10.1017/9781108499996 [14] Alex Kamenev. ``Field theory of non-equilibrium systems''.
Cambridge University Press. (2011). https://doi.org/10.1017/cbo9781139003667 [15] Henrik Bruus and Karsten Flensberg. ``Many–body quantum theory in condensed matter physics''.
Oxford University Press. (2004). https://doi.org/10.1093/oso/9780198566335.001.0001 [16] Zhenning Cai, Jianfeng Lu, and Siyao Yang. ``Numerical analysis for inchworm Monte Carlo method: Sign problem and error growth''. Mathematics of Computation 92, 1141–1209 (2022). https://doi.org/10.1090/mcom/3785 [17] Hans C. Fogedby. ``Field-theoretical approach to open quantum systems and the lindblad equation''. Physical Review A 106, 022205 (2022). https://doi.org/10.1103/physreva.106.022205 [18] Erik Aurell, Ryochi Kawai, and Ketan Goyal. ``An operator derivation of the Feynman–Vernon theory, with applications to the generating function of bath energy changes and to an-harmonic baths''. Journal of Physics A: Mathematical and Theoretical 53, 275303 (2020). https://doi.org/10.1088/1751-8121/ab9274 [19] Mauro Cirio, Pengfei Liang, and Neill Lambert. ``Input-output hierarchical equations of motion''. Physical Review A 112, 012211 (2025). https://doi.org/10.1103/4q4p-z14x [20] Qiongtao Xie, Honghua Zhong, Murray T Batchelor, and Chaohong Lee. ``The quantum Rabi model: solution and dynamics''. Journal of Physics A: Mathematical and Theoretical 50, 113001 (2017). https://doi.org/10.1088/1751-8121/aa5a65 [21] Klaus Hepp and Elliott H Lieb. ``On the superradiant phase transition for molecules in a quantized radiation field: the dicke maser model''. Annals of Physics 76, 360–404 (1973). https://doi.org/10.1016/0003-4916(73)90039-0 [22] A C Hewson and D Meyer. ``Numerical renormalization group study of the Anderson-Holstein impurity model''. Journal of Physics: Condensed Matter 14, 427–445 (2001). https://doi.org/10.1088/0953-8984/14/3/312 [23] Si Luo, Neill Lambert, Pengfei Liang, and Mauro Cirio. ``Quantum-classical decomposition of gaussian quantum environments: A stochastic pseudomode model''. PRX Quantum 4, 030316 (2023). https://doi.org/10.1103/prxquantum.4.030316 [24] Gunhee Park, Zhen Huang, Yuanran Zhu, Chao Yang, Garnet Kin-Lic Chan, and Lin Lin. ``Quasi-lindblad pseudomode theory for open quantum systems''. Physical Review B 110, 195148 (2024). https://doi.org/10.1103/physrevb.110.195148 [25] D. Tamascelli, A. Smirne, S. F. Huelga, and M. B. Plenio. ``Nonperturbative treatment of non-Markovian dynamics of open quantum systems''.
Physical Review Letters 120, 030402 (2018). https://doi.org/10.1103/physrevlett.120.030402 [26] F. Mascherpa, A. Smirne, A. D. Somoza, P. Fernández-Acebal, S. Donadi, D. Tamascelli, S. F. Huelga, and M. B. Plenio. ``Optimized auxiliary oscillators for the simulation of general open quantum systems''. Physical Review A 101, 052108 (2020). https://doi.org/10.1103/physreva.101.052108 [27] Leonardo A. Pachón and Paul Brumer. ``Direct experimental determination of spectral densities of molecular complexes''. The Journal of Chemical Physics 141 (2014). https://doi.org/10.1063/1.4900512Cited byCould not fetch Crossref cited-by data during last attempt 2025-10-28 15:21:33: Could not fetch cited-by data for 10.22331/q-2025-10-28-1896 from Crossref. This is normal if the DOI was registered recently. Could not fetch ADS cited-by data during last attempt 2025-10-28 15:21:34: 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.
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