Editing Quantum superposition

{{Short description|Principle of quantum mechanics}}
{{broader|Superposition principle}}
{{Use dmy dates|date=April 2020}}
[[File:Quantum superposition of states and decoherence.ogv|thumb|upright=1.5|Quantum superposition of states and decoherence]]
{{Quantum mechanics|cTopic=Fundamental concepts}}

'''Quantum superposition''' is a fundamental principle of [[quantum mechanics]] that states that linear combinations of solutions to the [[Schrödinger equation]] are also solutions of the Schrödinger equation. This follows from the fact that the Schrödinger equation is a [[linear differential equation]] in time and position. More precisely, the state of a system is given by a [[linear combination]] of all the [[eigenfunction]]s of the Schrödinger equation governing that system.

An example is a [[qubit]] used in [[quantum information processing]]. A qubit state is most generally a superposition of the basis states <math>|0 \rangle</math> and <math>|1 \rangle</math>:

: <math>|\Psi \rangle = c_0|0\rangle + c_1|1\rangle,</math>

where <math>|\Psi \rangle</math> is the [[quantum state]] of the qubit, and <math>|0 \rangle</math>, <math>|1 \rangle</math> denote particular solutions to the Schrödinger equation in [[Bra–ket notation|Dirac notation]] weighted by the two [[probability amplitude]]s <math>c_0</math> and <math>c_1</math> that both are complex numbers. Here <math>|0 \rangle </math> corresponds to the classical 0 [[bit]], and <math>|1 \rangle </math> to the classical 1 bit. The probabilities of measuring the system in the <math>|0 \rangle</math> or <math>|1 \rangle</math> state are given by <math>|c_0|^2</math> and <math>|c_1|^2</math> respectively (see the [[Born rule]]). Before the measurement occurs the qubit is in a superposition of both states.

The interference fringes in the [[double-slit experiment]] provide another example of the superposition principle.

== Wave postulate ==
The theory of quantum mechanics postulates that a [[wave equation]] completely determines the state of a quantum system at all times. Furthermore, this differential equation is restricted to be [[linear differential equation|linear]] and [[homogeneous differential equation|homogeneous]]. These conditions mean that for any two solutions of the wave equation, <math>\Psi_1</math> and <math>\Psi_2</math>, a linear combination of those solutions also solve the wave equation:
<math display="block">\Psi = c_1\Psi_1 + c_2\Psi_2</math>
for arbitrary complex coefficients <math>c_1</math> and <math>c_2</math>.<ref name=Messiah>{{Cite book |last=Messiah |first=Albert |title=Quantum mechanics. 1 |date=1976 |publisher=North-Holland |isbn=978-0-471-59766-7 |edition=2|location=Amsterdam}}</ref>{{rp|61}} If the wave equation has more than two solutions, combinations of all such solutions are again valid solutions.

== Transformation ==
The quantum wave equation can be solved using functions of position, <math>\Psi(\vec{r})</math>, or using functions of momentum, <math>\Phi(\vec{p})</math> and consequently the superposition of momentum functions are also solutions:
<math display="block">\Phi(\vec{p}) = d_1\Phi_1(\vec{p}) + d_2\Phi_2(\vec{p})</math>
The position and momentum solutions are related by a [[linear transformation]], a [[Fourier transformation]].  This transformation is itself a quantum superposition and every position wave function can be represented as a superposition of momentum wave functions and vice versa. These superpositions involve an infinite number of component waves.<ref name=Messiah/>{{rp|244}}

== Generalization to basis states ==
Other transformations express a quantum solution as a superposition of [[eigenvectors]], each corresponding to a possible result of a measurement on the quantum system. An eigenvector <math>\psi_i</math> for a mathematical operator, <math>\hat{A}</math>, has the equation
<math display="block">\hat{A}\psi_i = \lambda_i\psi_i</math>
where <math>\lambda_i</math> is one possible measured quantum value for the observable <math>A</math>. A superposition of these eigenvectors can represent any solution:
<math display="block">\Psi = \sum_n a_i\psi_i.</math>
The states like <math>\psi_i</math> are called basis states. 

== Compact notation for superpositions ==
Important mathematical operations on quantum system solutions can be performed using only the coefficients of the superposition, suppressing the details of the superposed functions. This leads to quantum systems expressed in the [[bra-ket notation|Dirac bra-ket notation]]:<ref name=Messiah/>{{rp|245}}
<math display="block">|v\rangle = d_1|1\rangle + d_2|2\rangle </math>
This approach is especially effect for systems like quantum spin with no classical coordinate analog. Such shorthand notation is very common in textbooks and papers on quantum mechanics and superposition of basis states is a fundamental tool in quantum mechanics.

==Consequences==
[[Paul Dirac]] described the superposition principle as follows:
<blockquote>The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, ''A'' and ''B'', such that there exists an observation which, when made on the system in state ''A'', is certain to lead to one particular result, ''a'' say, and when made on the system in state ''B'' is certain to lead to some different result, ''b'' say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes ''a'' and sometimes ''b'', according to a probability law depending on the relative weights of ''A'' and ''B'' in the superposition process. It will never be different from both ''a'' and ''b'' [i.e., either ''a'' or ''b'']. ''The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.''<ref name="Dirac1947">{{cite book|author=P.A.M. Dirac|title=The Principles of Quantum Mechanics|year=1947|publisher=Clarendon Press|page=12|edition=2nd}}</ref></blockquote>

[[Anton Zeilinger]], referring to the prototypical example of the [[double-slit experiment]], has elaborated regarding the creation and destruction of quantum superposition:

<blockquote>"[T]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ The absence of any such information is ''the essential criterion'' for quantum interference to appear.<ref name=Zeilinger>{{cite journal | author = Zeilinger A | year = 1999 | title = Experiment and the foundations of quantum physics | journal = Rev. Mod. Phys. | volume = 71 | issue = 2 | pages = S288–S297 | doi=10.1103/revmodphys.71.s288|bibcode = 1999RvMPS..71..288Z }}</ref></blockquote>

==Theory==

=== General formalism ===
Any quantum state can be expanded as a sum or superposition of the eigenstates of an Hermitian operator, like the Hamiltonian, because the eigenstates form a complete basis:
:<math>
 |\alpha\rangle = \sum_n c_n |n\rangle,
</math>

where <math>|n\rangle</math> are the energy eigenstates of the Hamiltonian. For continuous variables like position eigenstates, <math>|x\rangle</math>:

:<math>
 |\alpha \rangle = \int dx' |x'\rangle \langle x'|\alpha \rangle,
</math>

where <math>\phi_\alpha(x) = \langle x| \alpha \rangle</math> is the projection of the state into the <math>|x\rangle</math> basis and is called the wave function of the particle. In both instances we notice that <math>|\alpha\rangle</math> can be expanded as a superposition of an infinite number of basis states.

===Example===
Given the Schrödinger equation

: <math>\hat H |n\rangle = E_n |n\rangle, </math>

where <math>|n\rangle</math> indexes the set of eigenstates of the Hamiltonian with energy eigenvalues <math>E_n,</math> we see immediately that

: <math>\hat H\big(|n\rangle + |n'\rangle\big) = E_n |n\rangle + E_{n'} |n'\rangle,</math>

where

: <math>|\Psi\rangle = |n\rangle + |n'\rangle</math>

is a solution of the Schrödinger equation but is not generally an eigenstate because <math>E_n</math> and <math>E_{n'}</math> are not generally equal. We say that <math>|\Psi\rangle</math> is made up of a superposition of energy eigenstates. Now consider the more concrete case of an [[electron]] that has either [[Spin (physics)|spin]] up or down. We now index the eigenstates with the [[spinor]]s in the <math>\hat z</math> basis:

: <math>|\Psi\rangle = c_1 |{\uparrow}\rangle + c_2 |{\downarrow}\rangle,</math>

where <math>|{\uparrow}\rangle</math> and <math>|{\downarrow}\rangle</math> denote spin-up and spin-down states respectively. As previously discussed, the magnitudes of the complex coefficients give the probability of finding the electron in either definite spin state:

: <math> P\big(|{\uparrow}\rangle\big) = |c_1|^2,</math>
: <math> P\big(|{\downarrow}\rangle\big) = |c_2|^2,</math>
: <math> P_\text{total} = P\big(|{\uparrow}\rangle\big) + P\big(|{\downarrow}\rangle\big) = |c_1|^2 + |c_2|^2 = 1,</math>

where the probability of finding the particle with either spin up or down is normalized to 1. Notice that <math>c_1</math> and <math>c_2</math> are complex numbers, so that

: <math>|\Psi\rangle = \frac{3}{5} i |{\uparrow}\rangle + \frac{4}{5} |{\downarrow}\rangle.</math>

is an example of an allowed state. We now get

: <math>P\big(|{\uparrow}\rangle\big) = \left|\frac{3i}{5}\right|^2 = \frac{9}{25},</math>
: <math>P\big(|{\downarrow}\rangle\big) = \left|\frac{4}{5}\right|^2 = \frac{16}{25},</math>
: <math>P_\text{total} = P\big(|{\uparrow}\rangle\big) + P\big(|{\downarrow}\rangle\big) = \frac{9}{25} + \frac{16}{25} = 1.</math>

If we consider a qubit with both position and spin, the state is a superposition of all possibilities for both:

:<math>
 \Psi = \psi_+(x) \otimes |{\uparrow}\rangle + \psi_-(x) \otimes |{\downarrow}\rangle,
</math>

where we have a general state <math>\Psi</math> is the sum of the [[tensor products]] of the position space wave functions and spinors.

==Experiments ==

Successful experiments involving superpositions of [[mesoscopic|relatively large]] (by the standards of quantum physics) objects have been performed.
* A [[beryllium]] [[ion]] has been trapped in a superposed state.<ref>{{Cite journal |last=Monroe |first=C. |last2=Meekhof |first2=D. M. |last3=King |first3=B. E. |last4=Wineland |first4=D. J. |date=1996-05-24 |title=A “Schrödinger Cat” Superposition State of an Atom |url=https://www.science.org/doi/10.1126/science.272.5265.1131 |journal=Science |language=en |volume=272 |issue=5265 |pages=1131–1136 |doi=10.1126/science.272.5265.1131 |issn=0036-8075}}</ref>
* A [[double slit experiment]] has been performed with molecules as large as [[Buckminsterfullerene|buckyballs]] and functionalized oligoporphyrins with up to 2000 atoms.<ref>{{cite web|url=http://www.quantum.at/research/molecule-interferometry-foundations/wave-particle-duality-of-c60.html |title=Wave-particle duality of C60 |date=31 March 2012 |url-status=bot: unknown |archive-url=https://web.archive.org/web/20120331115055/http://www.quantum.at/research/molecule-interferometry-foundations/wave-particle-duality-of-c60.html |archive-date=31 March 2012 }}</ref><ref>{{cite web|url=http://www.univie.ac.at/qfp/research/matterwave/stehwelle/standinglightwave.html|title=standinglightwave|first=Olaf|last=Nairz}}{{cite journal |title=Quantum superposition of molecules beyond 25 kDa |author=Yaakov Y. Fein |author2=Philipp Geyer |author3=Patrick Zwick |author4=Filip Kiałka |author5=Sebastian Pedalino |author6=Marcel Mayor |author7=Stefan Gerlich |author8=Markus Arndt |journal=Nature Physics |volume=15 |pages=1242–1245 |date=September 2019 |issue=12 |doi=10.1038/s41567-019-0663-9|bibcode=2019NatPh..15.1242F |s2cid=203638258 }}</ref>
* Molecules with masses exceeding 10,000 and composed of over 810 atoms have successfully been superposed<ref>Eibenberger, S., Gerlich, S., Arndt, M., Mayor, M., Tüxen, J. (2013). "Matter-wave interference with particles selected from a molecular library with masses exceeding 10 000 amu", ''Physical Chemistry Chemical Physics'', '''15''': 14696-14700. {{ArXiv|1310.8343}}</ref>
* Very sensitive magnetometers have been realized using [[SQUID|superconducting quantum interference devices]] (SQUIDS) that operate using quantum  interference effects in superconducting circuits.

* A [[piezoelectric]] "[[tuning fork]]" has been constructed, which can be placed into a superposition of vibrating and non-vibrating states. The resonator comprises about 10 trillion atoms.<ref>Scientific American: [http://www.scientificamerican.com/article.cfm?id=quantum-microphone ''Macro-Weirdness: "Quantum Microphone" Puts Naked-Eye Object in 2 Places at Once: A new device tests the limits of Schrödinger's cat'']</ref>
* Recent research indicates that [[chlorophyll]] within [[plants]] appears to exploit the feature of quantum superposition to achieve greater efficiency in transporting energy, allowing pigment proteins to be spaced further apart than would otherwise be possible.<ref name="doi:10.1038/nature08811">{{Cite journal|last=Scholes|first=Gregory |author2=Elisabetta Collini |author3=Cathy Y. Wong |author4=Krystyna E. Wilk |author5=Paul M. G. Curmi |author6=Paul Brumer |author7=Gregory D. Scholes|date=4 February 2010|title=Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature|journal=[[Nature (journal)|Nature]]|volume=463|issue= 7281|pages=644–647|doi=10.1038/nature08811|bibcode = 2010Natur.463..644C|pmid=20130647|s2cid=4369439 }}</ref><ref>{{Cite news|url=http://www.scientificamerican.com/article.cfm?id=quantum-entanglement-and-photo|title=Quantum Entanglement, Photosynthesis and Better Solar Cells|last=Moyer|first=Michael|date=September 2009|work=Scientific American|access-date=12 May 2010}}</ref>

== In quantum computers ==
In [[quantum computers]], a [[qubit]] is the analog of the classical information [[bit]] and qubits can be superposed.<ref name="Nielsen-Chuang"/>{{rp|13}} Unlike classical bits, a superposition of qubits represents information about two states in parallel.<ref name="Nielsen-Chuang"/>{{rp|31}} Controlling the superposition of qubits is a central challenge in quantum computation. Qubit systems like [[nuclear spin]]s with small coupling strength are robust to outside disturbances but the same small coupling makes it difficult to readout results.<ref name="Nielsen-Chuang">{{Cite book |title=Quantum Computation and Quantum Information |last1=Nielsen |first1=Michael A. |last2=Chuang |first2=Isaac |date=2010 |publisher=[[Cambridge University Press]] |isbn=978-1-10700-217-3 |location=Cambridge |oclc=43641333 |author-link=Michael Nielsen |author-link2=Isaac Chuang |url=https://www.cambridge.org/9781107002173}}</ref>{{rp|278}}

==See also==
* {{annotated link|Eigenstate}}
* {{annotated link|Mach–Zehnder interferometer}}
* {{annotated link|Penrose interpretation}}
* {{annotated link|Pure qubit state}}
* {{annotated link|Quantum computation}}
* {{annotated link|Schrödinger's cat}}
* {{annotated link|Superposition principle}}
* {{annotated link|Wave packet}}

==References==
{{Reflist|30em}}

==Further reading==
* [[Niels Bohr|Bohr, N.]] (1927/1928). The quantum postulate and the recent development of atomic theory, [http://www.nature.com/nature/journal/v121/n3050/abs/121580a0.html ''Nature'' Supplement 14 April 1928, '''121''': 580–590].
* [[Claude Cohen-Tannoudji|Cohen-Tannoudji, C.]], Diu, B., Laloë, F. (1973/1977). ''Quantum Mechanics'', translated from the French by S. R. Hemley, N. Ostrowsky, D. Ostrowsky, second edition, volume 1, Wiley, New York, {{ISBN|0471164321}}.
* [[Albert Einstein|Einstein, A.]] (1949). Remarks concerning the essays brought together in this co-operative volume, translated from the original German by the editor, pp.&nbsp;665–688 in [[Paul Arthur Schilpp|Schilpp, P. A.]] editor (1949), [https://www.worldcat.org/oclc/311439 ''Albert Einstein: Philosopher-Scientist''], volume {{math|II}}, Open Court, La Salle IL.
* [[Richard Feynman|Feynman, R. P.]], Leighton, R.B., Sands, M. (1965). ''The Feynman Lectures on Physics'', [https://feynmanlectures.caltech.edu/III_toc.html volume 3], Addison-Wesley, Reading, MA.
* [[Eugen Merzbacher|Merzbacher, E.]] (1961/1970). ''Quantum Mechanics'', second edition, Wiley, New York.
* [[Albert Messiah|Messiah, A.]] (1961). ''Quantum Mechanics'', volume 1, translated by G.M. Temmer from the French ''Mécanique Quantique'', North-Holland, Amsterdam.
* {{cite book|first1=J. A.|last1=Wheeler|author1-link=John Archibald Wheeler|first2=W.H.|last2=Zurek|author2-link=Wojciech H. Zurek|year=1983|title=Quantum Theory and Measurement|publisher=Princeton University Press|location=Princeton NJ}}
* {{Cite book|title=Quantum Computation and Quantum Information|last1=Nielsen|first1=Michael A.|last2=Chuang|first2=Isaac|date=2000|publisher=[[Cambridge University Press]]|isbn=0521632358|location=Cambridge|oclc=43641333|author-link=Michael Nielsen|author-link2=Isaac Chuang}}
* {{cite book |author=Williams |first=Colin P. |year=2011 |title=Explorations in Quantum Computing |publisher=[[Springer Science+Business Media|Springer]] |isbn=978-1-84628-887-6 |language=en}}
* {{Cite book|title=Quantum computing for computer scientists|last1=Yanofsky|first1=Noson S.|last2=Mannucci|first2=Mirco|date=2013|publisher=[[Cambridge University Press]]|isbn=978-0-521-87996-5}}

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