Editing Qubit (section)

==Quantum entanglement==
{{Main|Quantum entanglement|Bell state}} An important distinguishing feature between qubits and classical bits is that multiple qubits can exhibit [[quantum entanglement]]; the qubit itself is an exhibition of quantum entanglement. In this case, quantum entanglement is a local or [[quantum nonlocality|nonlocal]] property of two or more qubits that allows a set of qubits to express higher correlation than is possible in classical systems.

The simplest system to display quantum entanglement is the system of two qubits. Consider, for example, two entangled qubits in the <math>|\Phi^+\rangle</math> [[Bell state]]:

:<math>\frac{1}{\sqrt{2}} (|00\rangle + |11\rangle).</math>

In this state, called an ''equal superposition'', there are equal probabilities of measuring either product state <math>|00\rangle</math> or <math>|11\rangle</math>, as <math>|1/\sqrt{2}|^2 = 1/2</math>. In other words, there is no way to tell if the first qubit has value "0" or "1" and likewise for the second qubit.

Imagine that these two entangled qubits are separated, with one each given to Alice and Bob. Alice makes a measurement of her qubit, obtaining—with equal probabilities—either <math>|0\rangle</math> or <math>|1\rangle</math>, i.e., she can now tell if her qubit has value "0" or "1". Because of the qubits' entanglement, Bob must now get exactly the same measurement as Alice. For example, if she measures a <math>|0\rangle</math>, Bob must measure the same, as <math>|00\rangle</math> is the only state where Alice's qubit is a <math>|0\rangle</math>. In short, for these two entangled qubits, whatever Alice measures, so would Bob, with perfect correlation, in any basis, however far apart they may be and even though both can not tell if their qubit has value "0" or "1"—a most surprising circumstance that cannot be explained by classical physics.

===Controlled gate to construct the Bell state===
[[Quantum logic gate#Controlled gates|Controlled gates]] act on 2 or more qubits, where one or more qubits act as a control for some specified operation. In particular, the [[controlled NOT gate]] (CNOT or CX) acts on 2 qubits, and performs the NOT operation on the second qubit only when the first qubit is <math>|1\rangle</math>, and otherwise leaves it unchanged. With respect to the unentangled product basis <math>\{|00\rangle</math>, <math>|01\rangle</math>, <math>|10\rangle</math>, <math>|11\rangle\}</math>, it maps the basis states as follows:
:<math> | 0 0 \rangle \mapsto | 0 0 \rangle </math>
:<math> | 0 1 \rangle \mapsto | 0 1 \rangle </math>
:<math> | 1 0 \rangle \mapsto | 1 1 \rangle </math>
:<math> | 1 1 \rangle \mapsto | 1 0 \rangle </math>.

A common application of the CNOT gate is to maximally entangle two qubits into the <math>|\Phi^+\rangle</math> [[Bell state]]. To construct <math>|\Phi^+\rangle</math>, the inputs A (control) and B (target) to the CNOT gate are:

<math>\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)_A</math> <math>\otimes</math> <math>|0\rangle_B</math> = <math>\frac{1}{\sqrt{2}}</math> <math>(|00\rangle + |10\rangle)</math>.

After applying CNOT, the output is the <math>|\Phi^+\rangle</math> Bell State: <math>\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)</math>.

===Applications===
The <math>|\Phi^+\rangle</math> Bell state forms part of the setup of the [[superdense coding]], [[quantum teleportation]], and entangled [[quantum cryptography]] algorithms.

Quantum entanglement also allows multiple states (such as the Bell state mentioned above) to be acted on simultaneously, unlike classical bits that can only have one value at a time. Entanglement is a necessary ingredient of any quantum computation that cannot be done efficiently on a classical computer. Many of the successes of quantum computation and communication, such as quantum teleportation and superdense coding, make use of entanglement, suggesting that entanglement is a [[Computational resource|resource]] that is unique to quantum computation.<ref>{{Cite journal |last=Horodecki |first=Ryszard |display-authors=et al |date=2009 |title=Quantum entanglement |journal=Reviews of Modern Physics |volume=81|issue=2 |pages=865–942 |doi=10.1103/RevModPhys.81.865 |arxiv=quant-ph/0702225 |bibcode=2009RvMP...81..865H |s2cid=59577352}}</ref> A major hurdle facing quantum computing, as of 2018, in its quest to surpass classical digital computing, is noise in quantum gates that limits the size of [[quantum circuit]]s that can be executed reliably.<ref name="preskill2018">{{cite journal |last1=Preskill |first1=John |date=2018 |title=Quantum Computing in the NISQ era and beyond |journal=Quantum |volume=2 |pages=79 |arxiv=1801.00862 |doi=10.22331/q-2018-08-06-79 |bibcode=2018Quant...2...79P |s2cid=44098998 }}</ref>