Faster than Hermitian quantum mechanics
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Given an initial quantum state $|{\ensuremath{\psi}}_{I}⟩$ and a final quantum state $|{\ensuremath{\psi}}_{F}⟩$, there exist Hamiltonians $H$ under which $|{\ensuremath{\psi}}_{I}⟩$ evolves into $|{\ensuremath{\psi}}_{F}⟩$. Consider the following quantum brachistochrone problem: subject to the constraint that the difference between the largest and smallest eigenvalues of $H$ is held fixed, which $H$ achieves this transformation in the least time $\ensuremath{\tau}$? For Hermitian Hamiltonians $\ensuremath{\tau}$ has a nonzero lower bound. However, among non-Hermitian $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians satisfying the same energy constraint, $\ensuremath{\tau}$ can be made arbitrarily small without violating the time-energy uncertainty principle. This is because for such Hamiltonians the path from $|{\ensuremath{\psi}}_{I}⟩$ to $|{\ensuremath{\psi}}_{F}⟩$ can be made short. The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole. This result may have applications in quantum computing.
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Bender et al argued that PT-symmetric Hamiltonians provide means to accelerate quantum mechanics.
“Bender et al [249] then argued that PT -symmetric Hamiltonians provide means to accelerate quantum mechanics.”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [249]
Under PT-symmetric Hamiltonians, states that appear to evolve arbitrarily fast are no longer orthogonal because the inner product and Hilbert-space geometry are modified.
“However, Bender et al [249] also note that such |ψ 0 and |ψ T are no longer orthogonal, since under PT -symmetric Hamiltonians the inner product, and hence the geometry of Hilbert space has to be modified (137).”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [249]
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Interactions of Hermitian and non-Hermitian Hamiltonians
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- Authors
- Bender, C. M., Brody, D. C., Jones, H. F., & Meister, B. K.
- Affiliations
- Washington University in St. Louis (US)
- Imperial College London (GB)
- Renmin University of China (CN)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2007-01-24
- Volume
- 98
- Issue
- 4
- Pages
- 040403-040403
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2010599738
- Cited by
- 417
- References
- 20
- Citations by year
- 2026: 23 · 2025: 25 · 2024: 26 · 2023: 23 · 2022: 29 · 2021: 26 · 2020: 28 · 2019: 28 · 2018: 28 · 2017: 24 · 2016: 18 · 2015: 18 · 2014: 16 · 2013: 16 · 2012: 24
- FWCI
- 10.16
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Non-Hermitian Physics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Quantum chaos and dynamical systems
- Keywords
- quantum brachistochrone, non-Hermitian Hamiltonian, PT-symmetric Hamiltonians, time-energy uncertainty relation, quantum state evolution, energy constraints, Hamiltonian spectrum, quantum computing
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