Editing Qubit (section)

==Qubit states==
[[File:Qubit represented by linear polarization of light.png|thumb|right|[[Polarization_(waves)|Polarization of light]] offers a straightforward way to present orthogonal states. With a typical mapping <math>|H\rangle=|0\rangle</math> and <math>|V\rangle=|1\rangle</math>, quantum states <math>(|0\rangle \pm |1\rangle)/\sqrt{2}</math> have a direct physical representation, both easily demonstrable experimentally in a class with [[Polarizer|linear polarizers]] and, for real <math>\alpha</math> and <math>\beta</math>, matching the high-school definition of [[orthogonality]].<ref>{{cite journal|last1=Seskir|first1=Zeki C.|last2=Migdał|first2=Piotr|last3=Weidner|first3=Carrie|last4=Anupam|first4=Aditya|last5=Case|first5=Nicky|last6=Davis|first6=Noah|last7=Decaroli|first7=Chiara|last8=Ercan|first8=İlke|last9=Foti|first9=Caterina|last10=Gora|first10=Paweł|last11=Jankiewicz|first11=Klementyna|last12=La Cour|first12=Brian R.|last13=Malo|first13=Jorge Yago|last14=Maniscalco|first14=Sabrina|last15=Naeemi|first15=Azad|last16=Nita|first16=Laurentiu|last17=Parvin|first17=Nassim|last18=Scafirimuto|first18=Fabio|last19=Sherson|first19=Jacob F.|last20=Surer|first20=Elif|last21=Wootton|first21=James|last22=Yeh|first22=Lia|last23=Zabello|first23=Olga|last24=Chiofalo|first24=Marilù|title=Quantum games and interactive tools for quantum technologies outreach and education|journal=Optical Engineering|volume=61|issue=8|pages=081809|year=2022|arxiv=2202.07756|doi=10.1117/1.OE.61.8.081809|bibcode=2022OptEn..61h1809S }}{{Creative Commons text attribution notice|cc=by4|from this source=yes}}</ref>]]

A pure qubit state is a [[quantum coherence|coherent]] [[quantum superposition|superposition]] of the basis states. This means that a single qubit (<math>\psi</math>) can be described by a [[linear combination]] of <math>|0 \rangle </math> and <math>|1 \rangle </math>:

: <math>| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle  </math>

where <var>α</var> and <var>β</var> are the [[probability amplitude]]s, and are both [[complex number]]s. When we measure this qubit in the standard basis, according to the [[Born rule]], the probability of outcome <math>|0 \rangle </math> with value "0" is <math>| \alpha |^2</math> and the probability of outcome <math>|1 \rangle </math> with value "1" is <math>| \beta |^2</math>. Because the absolute squares of the amplitudes equate to probabilities, it follows that <math>\alpha</math> and <math>\beta</math> must be constrained according to the [[Probability axioms#Second axiom|second axiom of probability theory]] by the equation<ref name="Williams">{{cite book |author=Williams |first=Colin P. |title=Explorations in Quantum Computing |publisher=[[Springer Science+Business Media|Springer]] |year=2011 |isbn=978-1-84628-887-6 |pages=9–13}}</ref>

: <math>| \alpha |^2 + | \beta |^2 = 1.</math>

The probability amplitudes, <math>\alpha</math> and <math>\beta</math>, encode more than just the probabilities of the outcomes of a measurement; the ''relative phase'' between <math>\alpha</math> and <math>\beta</math> is for example responsible for [[wave interference|quantum interference]], as seen in the [[double-slit experiment]].

===Bloch sphere representation===
[[File:Bloch sphere.svg|thumb|[[Bloch sphere]] representation of a qubit. The [[probability amplitude]]s for the superposition state, <math>| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle,\,</math> are given by <math> \alpha = \cos\left(\frac{\theta}{2}\right) </math> and <math> \beta = e^{i \varphi} \sin\left(\frac{\theta}{2}\right)</math>]]

It might, at first sight, seem that there should be four [[Degrees of freedom (physics and chemistry)|degrees of freedom]] in <math>| \psi \rangle = \alpha |0 \rangle + \beta |1 \rangle\,</math>, as <math>\alpha</math> and <math>\beta</math> are [[complex number]]s with two degrees of freedom each. However, one degree of freedom is removed by the normalization constraint {{math|{{!}}''α''{{!}}<sup>2</sup> + {{!}}''β''{{!}}<sup>2</sup> {{=}} 1}}. This means, with a suitable change of coordinates, one can eliminate one of the degrees of freedom. One possible choice is that of [[3-sphere#Hopf coordinates|Hopf coordinates]]:
:<math>\begin{align}
\alpha &= e^{i \delta} \cos\frac{\theta}{2}, \\
\beta &= e^{i (\delta + \varphi)} \sin\frac{\theta}{2}.
\end{align}</math>
Additionally, for a single qubit the ''global [[phase factor|phase]]'' of the state <math>e^{i\delta}</math> has no physically observable consequences,{{efn|This is because of the [[Born rule]]. The probability to observe an outcome upon [[Quantum measurement|measurement]] is the [[modulus squared]] of the [[probability amplitude]] for that outcome (or basis state, [[eigenstate]]). The ''global phase'' factor <math>e^{i\delta}</math> is not measurable, because it applies to both basis states, and is on the complex [[unit circle]] so <math>|e^{i\delta}|^2 = 1.</math><br>Note that by removing <math>e^{i\delta}</math> it means that [[quantum state]]s with global phase can not be represented as points on the surface of the Bloch sphere.}} so we can arbitrarily choose {{math|''α''}} to be real (or {{math|''β''}} in the case that {{math|''α''}} is zero), leaving just two degrees of freedom:
:<math>\begin{align}
\alpha &= \cos\frac{\theta}{2}, \\
\beta &= e^{i \varphi} \sin\frac{\theta}{2},
\end{align}</math>
where <math> e^{i \varphi} </math> is the physically significant ''relative phase''.<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|pages=13–16}}</ref>{{efn|The Pauli Z basis is usually called the ''computational basis'', where the relative phase have no effect on measurement. [[Quantum measurement|Measuring]] instead in the X or Y Pauli basis depends on the relative phase. For example, <math>(|0\rangle + e^{i\pi/2}|1\rangle)/{\sqrt{2}}</math> will (because this state lies on the positive pole of the Y-axis) in the Y-basis always measure to the same value, while in the Z-basis results in equal probability of being measured to <math>|0\rangle</math> or <math>|1\rangle</math>.<br/>Because measurement [[Wave function collapse|collapses]] the quantum state, measuring the state in one basis hides some of the values that would have been measurable the other basis; See the [[uncertainty principle]].}}

The possible quantum states for a single qubit can be visualised using a [[Bloch sphere]] (see picture). Represented on such a [[2-sphere]], a classical bit could only be at the "North Pole" or the "South Pole", in the locations where <math>|0 \rangle</math> and <math>|1 \rangle</math> are respectively. This particular choice of the polar axis is arbitrary, however. The rest of the surface of the Bloch sphere is inaccessible to a classical bit, but a pure qubit state can be represented by any point on the surface. For example, the pure qubit state <math>(|0 \rangle + |1 \rangle)/{\sqrt{2}}</math> would lie on the equator of the sphere at the positive X-axis. In the [[classical limit]], a qubit, which can have quantum states anywhere on the Bloch sphere, reduces to the classical bit, which can be found only at either poles.

The surface of the Bloch sphere is a [[Plane (mathematics)|two-dimensional space]], which represents the observable [[state space (physics)|state space]] of the pure qubit states. This state space has two local degrees of freedom, which can be represented by the two angles <math>\varphi</math> and <math>\theta</math>.

===Mixed state===
{{Main|Density matrix}} A pure state is fully specified by a single ket, <math>|\psi\rangle = \alpha |0\rangle + \beta |1\rangle,\,</math> a coherent superposition, represented by a point on the surface of the Bloch sphere as described above. Coherence is essential for a qubit to be in a superposition state. With interactions, [[quantum noise]] and [[decoherence]], it is possible to put the qubit in a [[Mixed state (physics)|mixed state]], a statistical combination or "incoherent mixture" of different pure states. Mixed states can be represented by points ''inside'' the Bloch sphere (or in the Bloch ball). A mixed qubit state has three degrees of freedom: the angles <math>\varphi</math> and <math>\theta </math>, as well as the length <math>r</math> of the vector that represents the mixed state.

[[Quantum error correction]] can be used to maintain the purity of qubits.