Editing Quantum logic gate (section)

== Measurement ==
{{Further|Measurement in quantum mechanics|Deferred measurement principle}}
[[Image:Qcircuit measure-arrow.svg|upright=0.6|thumb|Circuit representation of measurement. The two lines on the right hand side represent a classical bit, and the single line on the left hand side represents a qubit.]]
Measurement (sometimes called ''observation'') is irreversible and therefore not a quantum gate, because it assigns the observed quantum state to a single value. Measurement takes a quantum state and projects it to one of the [[basis vector]]s, with a likelihood equal to the square of the vector's length (in the [[Norm (mathematics)#p-norm|2-norm]]{{r|Nielsen-Chuang|page=66}}{{r|Yanofsky-Mannucci|pages=56,65}}) along that basis vector.{{r|Williams|pages=15–17}}<ref>{{cite book|author=Griffiths, D.J. |author-link=David J. Griffiths |pages=115–121, 126 |year=2008 |title=Introduction to Elementary Particles (2nd ed.) |publisher=[[John Wiley & Sons]] |isbn=978-3-527-40601-2}}</ref><ref>{{cite book|author=David Albert |pages=35 |year=1994 |title=Quantum mechanics and experience |publisher=[[Harvard University Press]] |isbn=0-674-74113-7}}</ref><ref>{{cite book|author=[[Sean M. Carroll]] |pages=376–394 |year=2019 |title=Spacetime and geometry: An introduction to general relativity |publisher=[[Cambridge University Press]] |isbn=978-1-108-48839-6}}</ref> This is known as the [[Born rule]] and appears{{efn|name="stochastic-interpretations"}} as a [[stochastic]] non-reversible operation as it probabilistically sets the quantum state equal to the basis vector that represents the measured state. At the instant of measurement, the state is said to "[[Wave function collapse|collapse]]" to the definite single value that was measured. Why and how, or even if<ref>{{cite book|author=[[David Wallace (physicist)|David Wallace]] |year=2012 |title=The emergent multiverse: Quantum theory according to the Everett Interpretation |publisher=[[Oxford University Press]] |isbn=9780199546961}}</ref><ref>{{cite book|author=[[Sean M. Carroll]] |year=2019 |title=Something deeply hidden: Quantum worlds and the emergence of spacetime |publisher=[[Penguin Random House]] |isbn=9781524743017}}</ref> the quantum state collapses at measurement, is called the [[measurement problem]].

The probability of measuring a value with [[probability amplitude]] <math>\phi</math> is {{nowrap|<math>1 \ge |\phi|^2 \ge 0</math>,}} where <math>|\cdot|</math> is the [[Absolute value#Complex numbers|modulus]].

Measuring a single qubit, whose quantum state is represented by the vector {{nowrap|<math>a|0\rangle + b|1\rangle = \begin{bmatrix} a \\ b \end{bmatrix}</math>,}} will result in <math>|0\rangle</math> with probability {{nowrap|<math>|a|^2</math>,}} and in {{nowrap|<math>|1\rangle</math> with probability <math>|b|^2</math>.}}

For example, measuring a qubit with the quantum state <math>\frac{|0\rangle -i|1\rangle }{\sqrt{2}} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ -i \end{bmatrix}</math> will yield with equal probability either <math>|0\rangle</math> or {{nowrap|<math>|1\rangle</math>.}}

[[Image:Qubit state with sin and cos.png|upright=0.8|thumb |For a single qubit, we have a unit sphere in <math>\mathbb C^2</math> with the quantum state <math>a|0\rangle + b|1\rangle</math> such that {{nowrap|<math>|a|^2 + |b|^2 = 1</math>.}} The state can be re-written as {{nowrap|<math>|\cos \theta|^2 + |\sin \theta|^2 = 1</math>,}} [[Pythagorean trigonometric identity|or]] <math>|a|^2 = \cos^2 \theta</math> and {{nowrap|<math>|b|^2 = \sin^2 \theta</math>.}}<br>Note: <math>|a|^2</math> is the probability of measuring <math>|0\rangle</math> and <math>|b|^2</math> is the probability of measuring {{nowrap|<math>|1\rangle</math>.}}]]
A quantum state <math>|\Psi\rangle</math> that spans {{mvar|n}} qubits can be written as a vector in <math>2^n</math> [[Complex number|complex]] dimensions: {{nowrap|<math>|\Psi\rangle \in \mathbb C^{2^n}</math>.}} This is because the tensor product of {{mvar|n}} qubits is a vector in <math>2^n</math> dimensions. This way, a [[quantum register|register]] of {{mvar|n}} qubits can be measured to <math>2^n</math> distinct states, similar to how a register of {{mvar|n}} classical [[bit]]s can hold <math>2^n</math> distinct states. Unlike with the bits of classical computers, quantum states can have non-zero probability amplitudes in multiple measurable values simultaneously. This is called ''superposition''.

The sum of all probabilities for all outcomes must always be equal to {{val|1}}.{{efn|See [[Probability axioms#Second axiom|Probability axioms § Second axiom]]}} Another way to say this is that the [[Pythagorean theorem]] generalized to <math>\mathbb C^{2^n}</math> has that all quantum states <math>|\Psi\rangle</math> with {{mvar|n}} qubits must satisfy <math display="inline">1 = \sum_{x=0}^{2^n-1}|a_x|^2,</math>{{efn|The [[hypotenuse]] has length 1 because the probabilities sum to 1, so the quantum state vector is a [[unit vector]].}} where <math>a_x</math> is the probability amplitude for measurable state {{nowrap|<math>|x\rangle</math>.}} A geometric interpretation of this is that the possible [[State space|value-space]] of a quantum state <math>|\Psi\rangle</math> with {{mvar|n}} qubits is the surface of the [[unit sphere]] in <math>\mathbb C^{2^n}</math> and that the [[Unitary transformation|unitary transform]]s (i.e. quantum logic gates) applied to it are rotations on the sphere. The rotations that the gates perform form the [[symmetry group]] [[unitary group|U(2<sup>n</sup>)]]. Measurement is then a probabilistic projection of the points at the surface of this [[complex number|complex]] sphere onto the [[basis vector]]s that span the space (and labels the outcomes).

In many cases the space is represented as a [[Hilbert space]] <math>\mathcal{H}</math> rather than some specific {{nowrap|<math>2^n</math>-dimensional}} complex space. The number of dimensions (defined by the basis vectors, and thus also the possible outcomes from measurement) is then often implied by the operands, for example as the required [[state space]] for solving a [[Computational problem|problem]]. In [[Grover's algorithm#Applications|Grover's algorithm]], [[Lov Grover|Grover]] named this generic basis vector set ''"the database"''.

The selection of basis vectors against which to measure a quantum state will influence the outcome of the measurement.{{r|Williams|pages=30–35}}{{r|Nielsen-Chuang|pages=22,84–85,185–188}}<ref>[https://docs.microsoft.com/en-us/quantum/concepts/pauli-measurements Q# Online manual: Measurement]</ref> See [[Change of basis#Endomorphisms|change of basis]] and [[Von Neumann entropy]] for details. In this article, we always use the ''computational [[Basis (linear algebra)|basis]]'', which means that we have labeled the <math>2^n</math> basis vectors of an {{mvar|n}}-qubit [[quantum register|register]] {{nowrap|<math>|0\rangle, |1\rangle, |2\rangle, \cdots, |2^n-1\rangle</math>,}} or use the [[Binary number#Counting in binary|binary representation]] {{nowrap|<math>|0_{10}\rangle = |0\dots 00_{2}\rangle, |1_{10}\rangle = |0\dots01_{2}\rangle, |2_{10}\rangle = |0\dots10_{2}\rangle, \cdots, |2^n-1\rangle = |111\dots1_{2}\rangle</math>.}}

In [[quantum mechanics]], the basis vectors constitute an [[orthonormal basis]].

An example of usage of an alternative measurement basis is in the [[BB84]] cipher.

=== The effect of measurement on entangled states ===
[[File:The_Hadamard-CNOT_transform_on_the_zero-state.png|thumb|right|upright=1.3|The [[#Hadamard|Hadamard]]-[[#CNOT|CNOT]] gate, which when given the input <math>|00\rangle</math> produces a [[Bell state]]]]

If two [[quantum state]]s (i.e. [[qubit]]s, or [[quantum register|register]]s) are [[quantum entanglement|entangled]] (meaning that their combined state cannot be expressed as a [[tensor product]]), measurement of one register affects or reveals the state of the other register by partially or entirely collapsing its state too. This effect can be used for computation, and is used in many algorithms.

The Hadamard-CNOT combination acts on the zero-state as follows: 
:<math>\operatorname{CNOT}(H \otimes I)|00\rangle
= \left(
    \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}
    \left(
        \frac{1}{\sqrt{2}}
        \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}
        \otimes
        \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
    \right)
  \right)
  \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}
= \frac{1}{\sqrt{2}} \begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix}
= \frac{|00\rangle + |11\rangle}{\sqrt{2}}</math>

[[Image:Used for geometric description of the Bell state.png|upright=0.8|thumb|The Bell state in the text is <math>|\Psi\rangle = a|00\rangle + b|01\rangle + c|10\rangle + d|11\rangle</math> where <math>a = d = \frac{1}{\sqrt{2}}</math> and {{nowrap|<math>b = c = 0</math>.}} Therefore, it can be described by the [[Plane (geometry)|plane]] spanned by the [[basis vector]]s <math>|00\rangle</math> and {{nowrap|<math>|11\rangle</math>,}} as in the picture. The [[unit sphere]] {{nowrap|(in <math>\mathbb C^{4}</math>)}} that represent the possible [[state space|value-space]] of the 2-qubit system intersects the plane and <math>|\Psi\rangle</math> lies on the unit spheres surface. Because {{nowrap|<math>|a|^2 = |d|^2 = 1/2</math>,}} there is equal probability of measuring this state to <math>|00\rangle</math> or {{nowrap|<math>|11\rangle</math>,}} and because <math>b=c=0</math> there is zero probability of measuring it to <math>|01\rangle</math> or {{nowrap|<math>|10\rangle</math>.}}]]
This resulting state is the [[Bell state]] {{nowrap|<math>\frac{|00\rangle + |11\rangle}{\sqrt{2}} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 0 \\ 0 \\ 1\end{bmatrix}</math>.}} It cannot be described as a tensor product of two qubits. There is no solution for

:<math>\begin{bmatrix} x \\ y \end{bmatrix} \otimes \begin{bmatrix} w \\ z \end{bmatrix} = \begin{bmatrix} xw \\ xz \\ yw \\ yz \end{bmatrix} = \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 0 \\ 0 \\ 1 \end{bmatrix},</math>

because for example {{mvar|w}} needs to be both non-zero and zero in the case of {{mvar|xw}} and {{mvar|yw}}.

The quantum state ''spans'' the two qubits. This is called ''entanglement''. Measuring one of the two qubits that make up this Bell state will result in that the other qubit logically must have the same value, both must be the same: Either it will be found in the state {{nowrap|<math>|00\rangle</math>,}} or in the state {{nowrap|<math>|11\rangle</math>.}} If we measure one of the qubits to be for example {{nowrap|<math>|1\rangle</math>,}} then the other qubit must also be {{nowrap|<math>|1\rangle</math>,}} because their combined state ''became'' {{nowrap|<math>|11\rangle</math>.}} Measurement of one of the qubits collapses the entire quantum state, that span the two qubits.

The [[Greenberger–Horne–Zeilinger state|GHZ state]] is a similar entangled quantum state that spans three or more qubits.

This type of value-assignment occurs ''instantaneously over any distance'' and this has as of 2018 been experimentally verified by [[Quantum Experiments at Space Scale|QUESS]] for distances of up to 1200 kilometers.<ref>{{cite journal |title=Satellite-based entanglement distribution over 1200 kilometers |author1=Juan Yin |author2=Yuan Cao |author3=Yu-Huai Li |author4=Sheng-Kai Liao |author5=Liang Zhang |author6=Ji-Gang Ren |author7=Wen-Qi Cai |author8=Wei-Yue Liu |author9=Bo Li |author10=Hui Dai |author11=Guang-Bing Li |author12=Qi-Ming Lu |author13=Yun-Hong Gong |author14=Yu Xu |author15=Shuang-Lin Li |author16=Feng-Zhi Li |author17=Ya-Yun Yin |author18=Zi-Qing Jiang |author19=Ming Li |author20=Jian-Jun Jia |author21=Ge Ren |author22=Dong He |author23=Yi-Lin Zhou |author24=Xiao-Xiang Zhang |author25=Na Wang |author26=Xiang Chang |author27=Zhen-Cai Zhu |author28=Nai-Le Liu |author29=Yu-Ao Chen |author30=Chao-Yang Lu |author31=Rong Shu |author32=Cheng-Zhi Peng |author33=Jian-Yu Wang |author34=[[Jian-Wei Pan]] |journal=Quantum Optics |year=2017 |volume=356 |issue=6343 |pages=1140–1144 |pmid=28619937 |arxiv=1707.01339 |doi=10.1126/science.aan3211 |s2cid=5206894}}</ref><ref>{{cite web |url=https://www.scientificamerican.com/article/china-shatters-ldquo-spooky-action-at-a-distance-rdquo-record-preps-for-quantum-internet/ |title=China Shatters "Spooky Action at a Distance" Record, Preps for Quantum Internet |first=Lee |last=Billings |website=Scientific American|date=23 April 2020 }}</ref><ref>{{cite web |last1=Popkin |first1=Gabriel |date=15 June 2017 |title=China's quantum satellite achieves 'spooky action' at record distance |url=https://www.science.org/content/article/china-s-quantum-satellite-achieves-spooky-action-record-distance |website=Science – AAAS}}</ref> That the phenomena appears to happen instantaneously as opposed to the time it would take to traverse the distance separating the qubits at the speed of light is called the [[EPR paradox]], and it is an open question in physics how to resolve this. Originally it was solved by giving up the assumption of [[local realism]], but other [[Interpretations of quantum mechanics|interpretations]] have also emerged. For more information see the [[Bell test experiments]]. The [[no-communication theorem]] proves that this phenomenon cannot be used for faster-than-light communication of [[Entropy (information theory)|classical information]].

=== Measurement on registers with pairwise entangled qubits ===
[[File:The_effect_of_unitary_transforms_on_registers_with_pairwise_entangled_qubits.png|thumb|upright=1.8|right|The effect of a unitary transform F on a register A that is in a superposition of <math>2^n</math> states and pairwise entangled with the register B. Here, {{mvar|n}} is 3 (each register has 3 qubits).]]
Take a [[Quantum register|register]] A with {{mvar|n}} qubits all initialized to {{nowrap|<math>|0\rangle</math>,}} and feed it through a [[#Hadamard transform|parallel Hadamard gate]] {{nowrap|<math display="inline">H^{\otimes n}</math>.}} Register A will then enter the state <math display="inline">\frac{1}{\sqrt{2^n}} \sum_{k=0}^{2^{n}-1} |k\rangle</math> that have equal probability of when measured to be in any of its <math>2^n</math> possible states; <math>|0\rangle</math> to {{nowrap|<math>|2^n-1\rangle</math>.}} Take a second register B, also with {{mvar|n}} qubits initialized to <math>|0\rangle</math> and pairwise [[#CNOT|CNOT]] its qubits with the qubits in register A, such that for each {{mvar|p}} the qubits <math>A_{p}</math> and <math>B_{p}</math> forms the state {{nowrap|<math>|A_{p}B_{p}\rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}</math>.}}

If we now measure the qubits in register A, then register B will be found to contain the same value as A. If we however instead apply a quantum logic gate {{mvar|F}} on A and then measure, then {{nowrap|<math>|A\rangle = F|B\rangle \iff F^\dagger|A\rangle = |B\rangle</math>,}} where <math>F^\dagger</math> is the [[Unitary matrix|unitary inverse]] of {{mvar|F}}.

Because of how [[#Unitary inversion of gates|unitary inverses of gates]] act, {{nowrap|<math>F^\dagger |A\rangle = F^{-1}(|A\rangle) = |B\rangle</math>.}} For example, say <math>F(x)=x+3 \pmod{2^n}</math>, then {{nowrap|<math>|B\rangle = |A - 3 \pmod{2^n}\rangle</math>.}}

The equality will hold no matter in which order measurement is performed (on the registers A or B), assuming that {{mvar|F}} has run to completion. Measurement can even be randomly and concurrently interleaved qubit by qubit, since the measurements assignment of one qubit will limit the possible value-space from the other entangled qubits.

Even though the equalities holds, the probabilities for measuring the possible outcomes may change as a result of applying {{mvar|F}}, as may be the intent in a quantum search algorithm.

This effect of value-sharing via entanglement is used in [[Shor's algorithm]], [[Quantum phase estimation|phase estimation]] and in [[Quantum counting algorithm|quantum counting]]. Using the [[Quantum Fourier transform|Fourier transform]] to amplify the probability amplitudes of the solution states for some [[Computational problem|problem]] is a generic method known as "[[Quantum algorithm#Fourier fishing and Fourier checking|Fourier fishing]]".<ref>{{Cite arXiv|last = Aaronson|first = Scott|year = 2009|title=BQP and the Polynomial Hierarchy|class=quant-ph|eprint=0910.4698}}</ref>