Geometric quantum speed limits: a case for Wigner phase space
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The quantum speed limit is a fundamental upper bound on the speed of quantum evolution. However, the actual mathematical expression of this fundamental limit depends on the choice of a measure of distinguishability of quantum states. We show that quantum speed limits are qualitatively governed by the Schatten- p -norm of the generator of quantum dynamics. Since computing Schatten- p -norms can be mathematically involved, we then develop an alternative approach in Wigner phase space. We find that the quantum speed limit in Wigner space is fully equivalent to expressions in density operator space, but that the new bound is significantly easier to compute. Our results are illustrated for the parametric harmonic oscillator and for quantum Brownian motion.
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Reference [53] proposes a bound with a straightforward experimental interpretation, but it is orbit-dependent, valid only for nondegenerate states, and lacks an explicit relation between purity and minimal evolution time.
“Reference [53] proposes a bound that has a straightforward experimental interpretation, but that is obtained from an orbit-dependent distance, is valid only for nondegenerate states, and does not provide an explicit relation between the purity of the state and the minimal time of evolution.”
From Tightening Quantum Speed Limits for Almost All States (2018) · cites this paper as [53]
An equivalent and more tractable quantum speed limit can be derived using the Wigner representation, where the rate of Wasserstein-p-distance is bounded by the Wigner function norm, as in equation (119).
“It was shown in [188] that if one considers the rate of Wasserstein-p-distance D p (W t , W 0 ) = W t -W 0 p ≡ dΓ |W(Γ, t) -W 0 (Γ)| p 1/p ,(118) one can derive Ḋp (W t , W 0 ) Ẇt p = v W QSL . (119”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [188]
The Schatten-p and Wasserstein-p expressions (116) and (119) are fully equivalent up to normalization.
“The two expressions (116) and (119) are fully equivalent up to normalization [188].”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [188]
Independent of the choice of measure, quantum speed is universally characterised by the Schatten-p-norm of the generator of the dynamics, and this norm can be computed from the Wasserstein-p-norm of the corresponding Wigner function.
“In conclusion, it was shown in [188] that, independent of the choice of measure, the quantum speed is universally characterised by the Schatten-p-norm of the generator of the dynamics.Moreover, this Schatten-p-norm can be computed from the mathematically tractable Wasserstein-p-norm of the corresponding Wigner function.”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [188]
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Fan, J., Wu, S.-X., & Yu, C. (2021) · Quantum Information Processing · doi:10.1007/s11128-020-02964-3 · Cited by 5 · see citing works
Quantum speed limit for the maximum coherent state under the squeezed environment*
Du, K.-Y., Ma, Y.-J., Wu, S.-X., & Yu, C. (2021) · Chinese Physics B · doi:10.1088/1674-1056/ac0daf · Cited by 5 · see citing works
Hadipour, M., Haseli, S., & Haddadi, S. (2024) · Communications in Theoretical Physics · doi:10.1088/1572-9494/ad54ae · Cited by 4 · see citing works
Generalized entropic quantum speed limits
de Sousa, J. F., & Pires, D. P. (2025) · Physical Review A · doi:10.1103/ghr4-d2vb · Cited by 4 · see citing works
Quantum speed limits of quantum system in colored environments
Zheng, L., & Peng, Y. (2024) · Results in Physics · doi:10.1016/j.rinp.2023.107315 · Cited by 4 · see citing works
Quantum speed limit of a single atom in a squeezed optical cavity mode
Ma, Y.-J., Gao, X.-C., Wu, S.-X., & Yu, C. (2023) · Chinese Physics B · doi:10.1088/1674-1056/acbd2b · Cited by 4 · see citing works
Quantum speed limits in dephasing dynamics of a qubit system coupled to thermal environments
Cai, X.-J., Feng, Y., Ren, J., Lan, K., Sun, S., Meng, X., & Czerwiński, A. (2025) · APL Quantum · doi:10.1063/5.0255326 · Cited by 3 · see citing works
Non-Markovian dynamics control of spin-1/2 system interacting with magnets
Zhang, Y.-J., Han, W., Yan, W., Man, Z., Xia, Y., & Fan, H. (2021) · New Journal of Physics · doi:10.1088/1367-2630/ac2c2a · Cited by 3 · see citing works
Theory of Quantum Super Impulses
Jarzynski, C. (2024) · PRX Quantum · doi:10.1103/prxquantum.5.010322 · Cited by 2 · see citing works
Witnessing Quantum Speedup through Mutual Information
Wang, J., Wu, Y., & Shi-xue, S. (2019) · Annalen der Physik · doi:10.1002/andp.201900225 · Cited by 2 · see citing works
Energetic cost as a consequence of parallel transporting speed limit
Peng, Y., Ni, Z., & Zheng, Y. (2024) · Physica Scripta · doi:10.1088/1402-4896/ad6e2a · Cited by 2 · see citing works
Quantum-to-classical transition via single-shot generalized measurements
Xu, Z. (2026) · Physical Review A · doi:10.1103/2md2-nxz3 · Cited by 1 · see citing works
Quantum speed limits based on Jensen-Shannon and Jeffreys divergences for general physical processes
de Sousa, J. F., & Pires, D. P. (2026) · Physical Review A · doi:10.1103/qnd8-dww7 · Cited by 1 · see citing works
Quantum Speed Limit and Nonclassicality in Open Quantum System Models Using the Wigner Function
Gaharwar, A., Tiwari, D., & Banerjee, S. (2024) · Annalen der Physik · doi:10.1002/andp.202400194 · Cited by 1 · see citing works
Liu, T., Li, Z., Zhang, Y., & Lan, K. (2022) · Acta Physica Sinica · doi:10.7498/aps.72.20222159 · Cited by 1 · see citing works
Quantum Speed Limit Time of Topological Qubits Influenced by the Fermionic and Bosonic Environments
Ahmadi, F., Haseli, S., Hadipour, M., Heshmatian, S., Dolatkhah, H., & Salimi, S. (2022) · Brazilian Journal of Physics · doi:10.1007/s13538-022-01092-w
Quantum evolution speed and its limit in the driven double‐well system
Xu, F., Zhang, L., & Huang, W. (2023) · International Journal of Quantum Chemistry · doi:10.1002/qua.27263
Experimental investigation of the trade-off between quantum speed and energy cost
Dong, Y., Jiang, W., Liu, Z.-W., Liu, Y., Zhang, S., Pires, D. P., Soares-Pinto, D. O., Chen, X., Guo, G.-C., & Sun, F. (2025) · npj Quantum Information · doi:10.1038/s41534-025-01149-z
Chau, H. F., & Li, J. (2026) · Physical Review A · doi:10.1103/trss-qhc8
Saturable quantum speed limits for imaginary-time evolution
Kobayashi, K. (2026) · Physical Review A · doi:10.1103/bln8-8xzy
Information processing in quantum thermodynamic systems: An autonomous Hamiltonian approach
Anonymous, Doucet, E., Touil, A., Deffner, S., & Sone, A. (2026) · Physical review. E · doi:10.1103/rwkz-dc5z
Self-citation
- Authors
- Deffner, S.
- Affiliations
- University of Maryland, Baltimore County (US)
- Corresponding author
- Deffner, Sebastian
- ORCID
- Deffner, Sebastian
- Venue
- New Journal of Physics
- Publisher
- IOP Publishing
- Published
- 2017-10-23
- Volume
- 19
- Issue
- 10
- Pages
- 103018
- ISSN
- 1367-2630 (ISSN-L)
- OpenAlex
- W2607459592
- Cited by
- 78
- References
- 44
- Citations by year
- 2026: 5 · 2025: 6 · 2024: 11 · 2023: 8 · 2022: 14 · 2021: 11 · 2020: 9 · 2019: 5 · 2018: 8 · 2017: 1
- FWCI
- 3.09
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- New Journal of Physics (publishedVersion, cc-by)
- arXiv (Cornell University) (submittedVersion)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Directorate for Biological Sciences (CHE-1648973)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Keywords
- quantum speed limit, Wigner phase space, Schatten p-norm, quantum state distinguishability, quantum Brownian motion, quantum evolution
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