Editing Butterfly effect (section)

===In quantum mechanics===
The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of cases in [[semiclassical physics|semiclassical]] and [[quantum mechanics|quantum physics]], including atoms in strong fields and the anisotropic [[Kepler problem]].<ref>{{cite journal |title=Postmodern Quantum Mechanics |first1=E. J. |last1=Heller |first2=S. |last2=Tomsovic |journal=[[Physics Today]] |date=July 1993 |doi=10.1063/1.881358 |volume=46 |issue=7 |pages=38–46 |bibcode=1993PhT....46g..38H}}</ref><ref>{{cite book |last=Gutzwiller |first=Martin C. |title=Chaos in Classical and Quantum Mechanics |year=1990 |publisher=Springer-Verlag |location=New York |isbn=0-387-97173-4}}</ref> Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure quantum treatments;<ref name="What is... Quantum Chaos">{{cite web |url=https://www.ams.org/notices/200801/tx080100032p.pdf |title=What is... Quantum Chaos? |last=Rudnick |first=Ze'ev |date=January 2008 |work=Notices of the American Mathematical Society |url-status=live |archive-url=https://web.archive.org/web/20091002000354/http://www.ams.org/notices/200801/tx080100032p.pdf |archive-date=2009-10-02}}</ref><ref>{{cite journal |last1=Berry |first1=Michael |title=Quantum chaology, not quantum chaos |journal=Physica Scripta |volume=40 |pages=335–336 |year=1989 |doi=10.1088/0031-8949/40/3/013 |bibcode=1989PhyS...40..335B |issue=3 |s2cid=250776260}}</ref> however, the sensitive dependence on initial conditions demonstrated in classical motion is included in the semiclassical treatments developed by [[Martin Gutzwiller]]<ref>{{cite journal |last=Gutzwiller |first=Martin C. |title=Periodic Orbits and Classical Quantization Conditions |journal=[[Journal of Mathematical Physics]] |year=1971 |volume=12 |issue=3 |page=343 |doi=10.1063/1.1665596 |bibcode=1971JMP....12..343G}}</ref> and John B. Delos and co-workers.<ref>{{cite journal |title=Closed-orbit theory of oscillations in atomic photoabsorption cross sections in a strong electric field. II. Derivation of formulas |last1=Gao |first1=J. |first2=J. B. |last2=Delos |name-list-style=amp |journal=[[Physical Review A]] |volume=46 |issue=3 |pages=1455–1467 |year=1992 |doi=10.1103/PhysRevA.46.1455 |pmid=9908268 |bibcode=1992PhRvA..46.1455G |s2cid=7877923 |url=https://scholarworks.wm.edu/cgi/viewcontent.cgi?article=2818&context=aspubs}}</ref> The random matrix theory and simulations with quantum computers prove that some versions of the butterfly effect in quantum mechanics do not exist.<ref>{{cite journal |last1=Yan |first1=Bin |last2=Sinitsyn |first2=Nikolai A. |title=Recovery of Damaged Information and the Out-of-Time-Ordered Correlators |journal=Physical Review Letters |volume=125 |pages=040605 |year=2020 |issue=4 |doi=10.1103/PhysRevLett.125.040605 |pmid=32794812 |arxiv=2003.07267 |bibcode=2020PhRvL.125d0605Y |s2cid=212725801}}</ref>

Other authors suggest that the butterfly effect can be observed in quantum systems. Zbyszek P. Karkuszewski et al. consider the time evolution of quantum systems which have slightly different [[Hamiltonian (quantum mechanics)|Hamiltonians]]. They investigate the level of sensitivity of quantum systems to small changes in their given Hamiltonians.<ref>{{cite journal |title=Quantum Chaotic Environments, the Butterfly Effect, and Decoherence |last1=Karkuszewski |first1=Zbyszek P. |last2=Jarzynski |first2=Christopher |last3=Zurek |first3=Wojciech H. |journal=[[Physical Review Letters]] |volume=89 |issue=17 |year=2002 |page=170405 |doi=10.1103/PhysRevLett.89.170405 |bibcode=2002PhRvL..89q0405K |arxiv=quant-ph/0111002 |pmid=12398653 |s2cid=33363344}}</ref> David Poulin et al. presented a quantum algorithm to measure fidelity decay, which "measures the rate at which identical initial states diverge when subjected to slightly different dynamics". They consider fidelity decay to be "the closest quantum analog to the (purely classical) butterfly effect".<ref>{{cite journal |last1=Poulin |first1=David |last2=Blume-Kohout |first2=Robin |last3=Laflamme |first3=Raymond |name-list-style=amp |last4=Ollivier |first4=Harold |year=2004 |title=Exponential Speedup with a Single Bit of Quantum Information: Measuring the Average Fidelity Decay |journal=Physical Review Letters |volume=92 |issue=17 |page=177906 |doi=10.1103/PhysRevLett.92.177906 |bibcode=2004PhRvL..92q7906P |arxiv=quant-ph/0310038 |pmid=15169196 |s2cid=6218604}}</ref> Whereas the classical butterfly effect considers the effect of a small change in the position and/or velocity of an object in a given [[Hamiltonian system]], the quantum butterfly effect considers the effect of a small change in the Hamiltonian system with a given initial position and velocity.<ref name="iqc.ca">{{cite web |title=A Rough Guide to Quantum Chaos |first=David |last=Poulin |url=http://www.iqc.ca/publications/tutorials/chaos.pdf |url-status=dead |archive-url=https://web.archive.org/web/20101104132156/http://www.iqc.ca/publications/tutorials/chaos.pdf |archive-date=2010-11-04}}</ref><ref>{{cite book |last=Peres |first=A. |title=[[Quantum Theory: Concepts and Methods]] |publisher=Kluwer Academic |location=Dordrecht |year=1995}}</ref> This quantum butterfly effect has been demonstrated experimentally.<ref>{{cite journal |title=Quantum amplifier: Measurement with entangled spins |last1=Lee |first1=Jae-Seung |last2=Khitrin |first2=A. K. |name-list-style=amp |journal=[[Journal of Chemical Physics]] |volume=121 |issue=9 |pages=3949–51 |year=2004 |doi=10.1063/1.1788661 |pmid=15332940 |bibcode=2004JChPh.121.3949L|doi-access=free }}</ref> Quantum and semiclassical treatments of system sensitivity to initial conditions are known as [[quantum chaos]].<ref name="What is... Quantum Chaos"/><ref name="iqc.ca"/>