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{{short description|Limitation on the minimum time for a quantum system to evolve between two states}}

{{Quantum book backlink|Quantum dynamics and evolution}}
In [[Physics:Quantum mechanics|quantum mechanics]], a '''quantum speed limit''' ('''QSL''') is a limitation on the minimum time for a quantum system to evolve between two distinguishable (orthogonal) states.<ref name="DEF">{{cite journal |last1=Deffner |first1=S. |last2=Campbell |first2=S. |title=Quantum speed limits: from Heisenberg's uncertainty principle to optimal quantum control |journal=J. Phys. A: Math. Theor. |date=10 October 2017 |volume=50 |issue=45 |page=453001 |doi=10.1088/1751-8121/aa86c6|arxiv=1705.08023 |bibcode=2017JPhA...50S3001D |s2cid=3477317 }}</ref> QSL theorems are closely related to time-energy uncertainty relations. In 1945, [[Biography:Leonid Mandelstam|Leonid Mandelstam]] and [[Biography:Igor Tamm|Igor Tamm]] derived a time-energy uncertainty relation that bounds the speed of evolution in terms of the energy dispersion.<ref name="MTT">{{cite journal |last1=Mandelshtam |first1=L. I. |last2=Tamm |first2=I. E. |title=The uncertainty relation between energy and time in nonrelativistic quantum mechanics |journal=J. Phys. (USSR) |date=1945 |volume=9 |pages=249–254}} Reprinted as {{cite book |chapter=The Uncertainty Relation Between Energy and Time in Non-relativistic Quantum Mechanics |date=1991 |title=Selected Papers |pages=115–123 |editor-last=Bolotovskii |editor-first=Boris M. |url=http://link.springer.com/10.1007/978-3-642-74626-0_8 |access-date=2024-04-06 |place=Berlin, Heidelberg |publisher=Springer |language=en |doi=10.1007/978-3-642-74626-0_8 |isbn=978-3-642-74628-4 |last1=Mandelstam |first1=L. |last2=Tamm |first2=Ig. |editor2-last=Frenkel |editor2-first=Victor Ya. |editor3-last=Peierls |editor3-first=Rudolf |editor-link3=Rudolf Peierls}}</ref> Over half a century later, [[Biography:Norman Margolus|Norman Margolus]] and [[Biography:Lev Levitin|Lev Levitin]] showed that the speed of evolution cannot exceed the mean energy,<ref name="Margolus1998">{{cite journal |last1=Margolus |first1=Norman |last2=Levitin |first2=Lev B. |title=The maximum speed of dynamical evolution |journal=Physica D: Nonlinear Phenomena |date=September 1998 |volume=120 |issue=1–2 |pages=188–195 |doi=10.1016/S0167-2789(98)00054-2 |arxiv=quant-ph/9710043 |bibcode=1998PhyD..120..188M |s2cid=468290 }}</ref> a result known as the Margolus–Levitin theorem. Realistic physical systems in contact with an environment are known as [[Physics:Open quantum system|open quantum system]]s and their evolution is also subject to QSL.<ref>{{cite journal |last1=Taddei |first1=M. M. |last2=Escher |first2=B. M. |last3=Davidovich |first3=L. |last4=de Matos Filho |first4=R. L. |title=Quantum Speed Limit for Physical Processes |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |date=30 January 2013 |volume=110 |issue=5 |article-number=050402 |doi=10.1103/PhysRevLett.110.050402 |pmid=23414007 |arxiv=1209.0362 |bibcode=2013PhRvL.110e0402T |s2cid=38373815 }}</ref><ref>{{cite journal |last1=del Campo |first1=A. |last2=Egusquiza |first2=I. L. |last3=Plenio |first3=M. B. |last4=Huelga |first4=S. F. |date=30 January 2013 |title=Quantum Speed Limits in Open System Dynamics |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |volume=110 |issue=5 |article-number=050403 |arxiv=1209.1737 |bibcode=2013PhRvL.110e0403D |doi=10.1103/PhysRevLett.110.050403 |pmid=23414008 |s2cid=8362503}}</ref> Quite remarkably it was shown that environmental effects, such as non-Markovian dynamics can speed up quantum processes,<ref>{{cite journal |last1=Deffner |first1=S. |last2=Lutz |first2=E. |title=Quantum speed limit for non-Markovian dynamics |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |date=3 July 2013 |volume=111 |issue=1 |article-number=010402 |doi=10.1103/PhysRevLett.111.010402|pmid=23862985 |arxiv=1302.5069 |bibcode=2013PhRvL.111a0402D |s2cid=36711861 }}</ref> which was verified in a cavity QED experiment.<ref>{{cite journal |last1=Cimmarusti |first1=A. D. |last2=Yan |first2=Z. |last3=Patterson |first3=B. D. |last4=Corcos |first4=L. P. |last5=Orozco |first5= L. A. |last6=Deffner |first6=S. |title=Environment-Assisted Speed-up of the Field Evolution in Cavity Quantum Electrodynamics |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |date=11 June 2015 |volume=114 |issue=23 |article-number=233602 |doi=10.1103/PhysRevLett.114.233602|pmid=26196802 |arxiv=1503.02591 |bibcode=2015PhRvL.114w3602C |s2cid=14904633 }}</ref>

QSL have been used to explore the [[Limits of computation|limits of computation]]<ref>{{cite journal |last1=Lloyd |first1=Seth |title=Ultimate physical limits to computation |journal=Nature |date=31 August 2000 |volume=406 |issue=6799 |pages=1047–1054 |doi=10.1038/35023282 |pmid=10984064 |language=en |issn=1476-4687|arxiv=quant-ph/9908043 |bibcode=2000Natur.406.1047L |s2cid=75923 }}</ref><ref>{{cite journal |last1=Lloyd |first1=Seth |title=Computational Capacity of the Universe |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |date=24 May 2002 |volume=88 |issue=23 |article-number=237901 |doi=10.1103/PhysRevLett.88.237901 |pmid=12059399 |arxiv=quant-ph/0110141 |bibcode=2002PhRvL..88w7901L |s2cid=6341263 }}</ref> and complexity. In 2017, QSLs were studied in a quantum oscillator at high temperature.<ref>{{cite journal |last1=Deffner |first1=S. |title=Geometric quantum speed limits: a case for Wigner phase space |journal=[[Physics:New Journal of Physics|New Journal of Physics]] |date=20 October 2017 |volume=19 |issue=10 |page=103018 |doi=10.1088/1367-2630/aa83dc|doi-access=free |arxiv=1704.03357 |bibcode=2017NJPh...19j3018D |hdl=11603/19409 |hdl-access=free }}</ref> In 2018, it was shown that QSL are not restricted to the quantum domain and that similar bounds hold in classical systems.<ref>{{cite journal |last1=Shanahan |first1=B. |last2=Chenu |first2=A. |last3=Margolus |first3=N. |last4=del Campo |first4=A. |title=Quantum Speed Limits across the Quantum-to-Classical Transition |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |date=12 February 2018 |volume=120 |issue=7 |article-number=070401 |doi=10.1103/PhysRevLett.120.070401 |pmid=29542956 |doi-access=free |arxiv=1710.07335 |bibcode=2018PhRvL.120g0401S }}</ref><ref>{{cite journal |last1=Okuyama |first1=Manaka |last2=Ohzeki |first2=Masayuki |title=Quantum Speed Limit is Not Quantum |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |date=12 February 2018 |volume=120 |issue=7 |article-number=070402 |doi=10.1103/PhysRevLett.120.070402 |pmid=29542975 |arxiv=1710.03498 |bibcode=2018PhRvL.120g0402O |s2cid=4027745 }}</ref> In 2021, both the Mandelstam-Tamm and the Margolus–Levitin QSL bounds were concurrently tested in a single experiment<ref>{{cite journal |last1=Ness |first1=Gal |last2=Lam |first2=Manolo R. |last3=Alt |first3=Wolfgang |last4=Meschede |first4=Dieter |last5=Sagi |first5=Yoav |last6=Alberti |first6=Andrea |title=Observing crossover between quantum speed limits |journal=Science Advances |date=22 December 2021 |volume=7 |issue=52 |article-number=eabj9119 |doi=10.1126/sciadv.abj9119 |doi-access=free |pmid=34936463 |pmc=8694601 }}</ref> which indicated there are "two different regimes: one where the Mandelstam-Tamm limit constrains the evolution at all times, and a second where a crossover to the Margolus-Levitin limit occurs at longer times."

In [[Quantum sensor|quantum sensing]], QSLs impose fundamental constraints on the maximum achievable time resolution of quantum sensors. These limits stem from the requirement that quantum states must evolve to orthogonal states to extract precise information. For example, in applications like [[Physics:Ramsey interferometry|Ramsey interferometry]], the QSL determines the minimum time required for phase accumulation during control sequences, directly impacting the sensor's [[Engineering:Temporal resolution|temporal resolution]] and sensitivity.<ref>{{cite journal |last1=Herb |first1=Konstantin | last2=Degen | first2=Christian L. |title=Quantum speed limit in quantum sensing |journal=[[Physics:Physical Review Letters|Physical Review Letters]] | date=19 November 2024 | volume=133 |issue=21 |article-number=210802|doi=10.1103/PhysRevLett.133.210802 |arxiv=2406.18348 }}</ref>

== Preliminary definitions ==
The speed limit theorems can be stated for pure states, and for mixed states; they take a simpler form for pure states. An arbitrary pure state can be written as a [[Physics:Quantum superposition|linear combination]] of energy eigenstates:

:<math>|\psi\rangle = \sum_n c_n |E_n\rangle.</math>

The task is to provide a lower bound for the time interval <math>t_\perp</math> required for the initial state <math>|\psi\rangle</math> to evolve into a state orthogonal to <math>|\psi\rangle</math>. The time evolution of a pure state is given by the [[Schrödinger equation]]:

:<math>|\psi_t\rangle = \sum_n c_n e^{itE_n/\hbar}|E_n\rangle.</math>

Orthogonality is obtained when

:<math>\langle\psi_0|\psi_t\rangle=0</math>

and the minimum time interval <math>t=t_\perp</math> required to achieve this condition is called the orthogonalization interval<ref name="MTT"/> or orthogonalization time.<ref name="LTT">
{{citation |author1=Lev B. Levitin |title=Fundamental Limit on the Rate of Quantum Dynamics: The Unified Bound Is Tight |url=https://link.aps.org/doi/10.1103/PhysRevLett.103.160502 |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |volume=103 |number=16 |article-number=160502 |year=2009 |arxiv=0905.3417 |doi=10.1103/PhysRevLett.103.160502 |issn=0031-9007 |author2=Tommaso Toffoli|pmid=19905679 |bibcode=2009PhRvL.103p0502L |s2cid=36320152 }}</ref>

== Mandelstam–Tamm limit ==
For pure states, the Mandelstam–Tamm theorem states that the minimum time <math>t_{\perp}</math> required for a state to evolve into an orthogonal state is bounded below:
:<math>t_{\perp} \ge \frac{\pi\hbar}{2\,\delta E}= \frac{h}{4\,\delta E}</math>,
where
:<math>(\delta E)^2 = \left\langle \psi|H^2|\psi\right\rangle - (\left\langle \psi|H|\psi\right\rangle)^2
=\frac{1}{2}\sum_{n,m} |c_n|^2 |c_m|^2 (E_n-E_m)^2
</math>,
is the [[Variance|variance]] of the system's energy and <math>H</math> is the [[Physics:Hamiltonian operator|Hamiltonian operator]]. The quantum evolution is independent of the particular Hamiltonian used to transport the quantum system along a given curve in the [[Projective Hilbert space|projective Hilbert space]]; the distance along this curve is measured by the [[Fubini–Study metric]].<ref name="GQE">{{cite journal
|first1=Yakir |last1=Aharonov
|first2=Jeeva |last2=Anandan
|title=Geometry of quantum evolution
|journal=[[Physics:Physical Review Letters|Physical Review Letters]]
|volume=65
|pages=1697–1700
|year=1990
|issue=14
|doi=10.1103/PhysRevLett.65.1697
|pmid=10042340
|bibcode=1990PhRvL..65.1697A
}}</ref> This is sometimes called the quantum angle, as it can be understood as the arccos of the inner product of the initial and final states.

===For mixed states===
The Mandelstam–Tamm limit can also be stated for mixed states and for time-varying Hamiltonians. In this case, the [[Bures metric]] must be employed in place of the Fubini–Study metric. A mixed state can be understood as a sum over pure states, weighted by [[Probability measure|classical probabilities]]; likewise, the Bures metric is a weighted sum of the Fubini–Study metric. For a time-varying Hamiltonian <math>H_t</math> and time-varying [[Density matrix|density matrix]] <math>\rho_t,</math> the variance of the energy is given by

:<math>\sigma^2_H(t)=|\text{tr}(\rho_t H^2_{t})|-|\text{tr}(\rho_t H_{t})|^2</math>

The Mandelstam–Tamm limit then takes the form
:<math>\int_0^{\tau} \sigma_H(t) dt \geq \hbar D_B(\rho_0, \rho_{\tau})</math>,
where <math>D_B</math> is the Bures distance between the starting and ending states. The Bures distance is [[Geodesic|geodesic]], giving the shortest possible distance of any continuous curve connecting two points, with <math>\sigma_H(t)</math> understood as an infinitessimal path length along a curve parametrized by <math>t.</math> Equivalently, the time <math>\tau</math> taken to evolve from <math>\rho</math> to <math>\rho'</math> is bounded as
:<math>\tau \geq \frac{\hbar}{\overline\sigma_H}D_B(\rho, \rho')</math>
where
:<math>\overline \sigma_H = \frac{1}{\tau}\int_0^\tau \sigma_H(t)dt</math>
is the time-averaged uncertainty in energy. For a pure state evolving under a time-varying Hamiltonian, the time <math>\tau</math> taken to evolve from one pure state to another pure state orthogonal to it is bounded as<ref name="Deffner2013"/>
:<math>\tau \geq \frac{\hbar}{\overline\sigma_H} \frac{\pi}{2}</math>
This follows, as for a pure state, one has the density matrix <math>\rho_t=|\psi_t\rangle\langle\psi_t|.</math> The quantum angle (Fubini–Study distance) is then <math>D_B(\rho_0,\rho_t)=\arccos| \langle\psi_0|\psi_t\rangle|</math> and so one concludes <math>D_B=\arccos 0=\pi/2</math> when the initial and final states are orthogonal.

== Margolus–Levitin limit ==
For the case of a pure state, Margolus and Levitin<ref name="Margolus1998"/> obtain a different limit, that

:<math>\tau_\perp \geq \frac{h}{4\langle E\rangle},</math>

where <math>\langle E\rangle</math> is the average energy,
<math display="block">\langle E \rangle = E_\text{avg} = \langle \psi |H | \psi \rangle =\sum_n |c_n|^2 E_n.</math>
This form applies when the Hamiltonian is not time-dependent, and the ground-state energy is defined to be zero.

===For time-varying states===
The Margolus–Levitin theorem can also be generalized to the case where the Hamiltonian varies with time, and the system is described by a mixed state.<ref name="Deffner2013">
{{Cite journal |last1=Deffner |first1=Sebastian |last2=Lutz |first2=Eric |date=2013-08-23 |title=Energy–time uncertainty relation for driven quantum systems |url=https://iopscience.iop.org/article/10.1088/1751-8113/46/33/335302 |journal=Journal of Physics A: Mathematical and Theoretical |volume=46 |issue=33 |article-number=335302 |arxiv=1104.5104 |doi=10.1088/1751-8113/46/33/335302 |bibcode=2013JPhA...46G5302D |hdl=11603/19394 |s2cid=119313370 |issn=1751-8113}}</ref> In this form, it is given by
:<math>\int_0^{\tau}|\text{tr}(\rho_t H_{t})| dt \geq \hbar D_B(\rho_0, \rho_{\tau})</math>
with the ground-state defined so that it has energy zero at all times.

This provides a result for time varying states. Although it also provides a bound for mixed states, the bound (for mixed states) can be so loose as to be uninformative.<ref>{{Cite journal |last1=Marvian |first1=Iman |last2=Spekkens |first2=Robert W. |last3=Zanardi |first3=Paolo |date=2016-05-24 |title=Quantum speed limits, coherence, and asymmetry |url=https://link.aps.org/doi/10.1103/PhysRevA.93.052331 |journal=[[Physical Review A]] |language=en |volume=93 |issue=5 |article-number=052331 |arxiv=1510.06474 |bibcode=2016PhRvA..93e2331M |doi=10.1103/PhysRevA.93.052331 |issn=2469-9926}}</ref> The Margolus–Levitin theorem has not yet been experimentally established in time-dependent quantum systems, whose Hamiltonians <math>H_t</math> are driven by arbitrary time-dependent parameters, except for the adiabatic case.<ref name="CETURDQS">
{{Cite journal
|last1=Okuyama
|first1=Manaka
|last2=Ohzeki
|first2=Masayuki
|year=2018
|title=Comment on 'Energy-time uncertainty relation for driven quantum systems'
|url=https://iopscience.iop.org/article/10.1088/1751-8121/aacb90
|journal=Journal of Physics A: Mathematical and Theoretical
|page=318001
|volume = 51
|issue=31
|arxiv=1802.00995
|doi=10.1088/1751-8121/aacb90
|bibcode=2018JPhA...51E8001O
|issn=1751-8113}}</ref>

=== Dual Margolus–Levitin limit ===
In addition to the original Margolus–Levitin limit, a dual bound exists for quantum systems with a bounded energy spectrum. This dual bound, also known as the Ness–Alberti–Sagi limit or the Ness limit, depends on the difference between the state's mean energy and the energy of the highest occupied eigenstate. In bounded systems, the minimum time <math>\tau_{\perp}</math> required for a state to evolve to an orthogonal state is bounded by

<math>\tau_{\perp} \geq \frac{h}{4(E_{\max}-\langle E\rangle)},</math>

where <math>E_{\max}</math> is the energy of the highest occupied eigenstate and <math>\langle E\rangle</math> is the mean energy of the state. This bound complements the original Margolus–Levitin limit and the Mandelstam–Tamm limit, forming a trio of constraints on quantum evolution speed.<ref>{{Cite journal |last=Ness |first=Gal |last2=Alberti |first2=Andrea |last3=Sagi |first3=Yoav |date=2022-09-29 |title=Quantum Speed Limit for States with a Bounded Energy Spectrum |url=https://link.aps.org/doi/10.1103/PhysRevLett.129.140403 |journal=Physical Review Letters |language=en |volume=129 |issue=14 |doi=10.1103/PhysRevLett.129.140403 |issn=0031-9007|arxiv=2206.14803 }}</ref>

== Levitin–Toffoli limit ==
A 2009 result by Lev B. Levitin and [[Biography:Tommaso Toffoli|Tommaso Toffoli]] states that the precise bound for the Mandelstam–Tamm theorem is attained only for a [[Qubit|qubit]] state.<ref name="LTT"/> This is a two-level state in an [[Physics:Quantum superposition|equal superposition]]

:<math>\left|\psi_q\right\rangle = \frac{1}{\sqrt{2}}\left(\left|E_0\right\rangle + e^{i \varphi}\left|E_1\right\rangle \right)</math>

for energy eigenstates <math>E_0=0</math> and <math>E_1=\pm \pi\hbar /\Delta t</math>. The states <math>\left|E_0\right\rangle</math> and <math>\left|E_1\right\rangle</math> are unique up to [[Physics:Degenerate energy levels|degeneracy]] of the energy level <math>E_1</math> and an arbitrary [[Phase factor|phase factor]] <math>\varphi.</math> This result is sharp, in that this state also satisfies the Margolus–Levitin bound, in that <math>E_\text{avg}=\delta E</math> and so <math>t_{\perp}=\hbar\pi/2E_\text{avg}=\hbar\pi/2\delta E.</math> This result establishes that the combined limits are strict:

:<math>t_\perp\ge\max\left(\frac{\pi\hbar}{2\,\delta E}\;,\; \frac{\pi\hbar}{2\,E_\text{avg}}\right)</math>

Levitin and Toffoli also provide a bound for the average energy in terms of the maximum. For any pure state <math>\left|\psi\right\rangle,</math> the average energy is bounded as
:<math>\frac{E_\text{max}}{4} \le E_\text{avg} \le \frac{E_\text{max}}{2}</math>
where <math>E_\text{max}</math> is the maximum energy [[Eigenvalues and eigenvectors|eigenvalue]] appearing in <math>\left|\psi\right\rangle.</math> (This is the quarter-pinched sphere theorem in disguise, transported to [[Complex projective space|complex projective space]].) Thus, one has the bound
:<math>\frac{\pi \hbar}{E_\text{max}} \le t_{\perp} \le \frac{2 \pi \hbar}{E_\text{max}}</math>

The strict lower bound <math>E_\text{max} t_{\perp} = \pi \hbar</math> is again attained for the qubit state
<math>\left|\psi_q\right\rangle</math> with <math>E_\text{max} = E_1</math>.

== Bremermann's limit ==
{{main|Bremermann's limit}}

The quantum speed limit bounds establish an upper bound at which [[Computation|computation]] can be performed. Computational machinery is constructed out of physical matter that follows quantum mechanics, and each operation, if it is to be unambiguous, must be a transition of the system from one state to an orthogonal state. Suppose the computing machinery is a [[Physics:Physical system|physical system]] evolving under Hamiltonian that does not change with time. Then, according to the Margolus–Levitin theorem, the number of operations per unit time per unit energy is bounded above by

:<math>\frac{2}{\hbar \pi} = 6 \times 10^{33} \mathrm{s}^{-1}\cdot \mathrm{J}^{-1} </math>

This establishes a strict upper limit on the number of calculations that can be performed by physical matter. The processing rate of ''all'' forms of computation cannot be higher than about 6 × 10<sup>33</sup> operations per second per [[Joule|joule]] of energy. This is including "classical" computers, since even classical computers are still made of matter that follows quantum mechanics.<ref>Bremermann, H.J. (1962) [http://holtz.org/Library/Natural%20Science/Physics/Optimization%20Through%20Evolution%20and%20Recombination%20-%20Bremermann%201962.htm ''Optimization through evolution and recombination''] {{Webarchive|url=https://web.archive.org/web/20191218210942/http://holtz.org/Library/Natural%20Science/Physics/Optimization%20Through%20Evolution%20and%20Recombination%20-%20Bremermann%201962.htm |date=2019-12-18 }} In: Self-Organizing systems 1962, edited M.C. Yovits et al., Spartan Books, Washington, D.C. pp. 93–106.</ref><ref>Bremermann, H.J. (1965) [http://projecteuclid.org/euclid.bsmsp/1200513783 Quantum noise and information]. 5th Berkeley Symposium on Mathematical Statistics and Probability; Univ. of California Press, Berkeley, California.</ref>

This bound is not merely a fanciful limit: it has practical ramifications for quantum-resistant cryptography. Imagining a computer operating at this limit, a [[Brute-force search|brute-force search]] to break a 128-bit encryption key requires only modest resources. Brute-forcing a 256-bit key requires planetary-scale computers, while a brute-force search of 512-bit keys is effectively unattainable within the lifetime of the universe, even if galactic-sized computers were applied to the problem.

The [[Astronomy:Bekenstein bound|Bekenstein bound]] limits the amount of information that can be stored within a volume of space. The maximal rate of change of information within that volume of space is given by the quantum speed limit. This product of limits is sometimes called the '''Bremermann–Bekenstein limit'''; it is saturated by [[Physics:Hawking radiation|Hawking radiation]].<ref name="DEF"/> That is, Hawking radiation is emitted at the maximal allowed rate set by these bounds.

== References ==
{{reflist}}

== Further reading ==
* {{cite journal | last=Jordan|first=Stephen P.| title=Fast quantum computation at arbitrarily low energy|journal = [[Physical Review A]] | volume = 95 | article-number=032305 | year=2017 |issue=3|arxiv = 1701.01175|bibcode=2017PhRvA..95c2305J|doi=10.1103/PhysRevA.95.032305|s2cid=118953874}}
* {{cite journal|first=Nikolai A.|last=Sinitsyn|title=Is there a quantum limit on speed of computation?|journal=Physics Letters A |year=2018|volume=382|issue=7 |pages=477–481|arxiv = 1701.05550|bibcode=2018PhLA..382..477S|doi=10.1016/j.physleta.2017.12.042|s2cid=55887738}}

[[Category:Quantum mechanics]]
[[Category:Mathematical physics]]

{{Sourceattribution|Quantum speed limit}}

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