Editing Quantum logic gate (section)

=== 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]].