Editing Quantum teleportation (section)

== Alternative notations ==
{{wide image|Teleport.png|780px| Quantum teleportation in its diagrammatic form.<ref name=coecke>{{Cite journal |last1=Coecke |first1=Bob|title=Quantum Picturalism |journal=Contemporary Physics|volume=51|issue=2010 |pages=59–83|year=2009 |doi=10.1080/00107510903257624 |bibcode=2010ConPh..51...59C|s2cid=752173|arxiv=0908.1787}}</ref> employing [[Penrose graphical notation]].<ref name=Penrose>R. Penrose, Applications of negative dimensional tensors, In: Combinatorial Mathematics and its Applications, D.~Welsh (Ed), pages 221–244. Academic Press (1971).</ref> Formally, such a computation takes place in a [[dagger compact category]]. This results in the abstract description of quantum teleportation as employed in [[categorical quantum mechanics]].}}

[[File:Quantum teleportation circuit.svg|upright=1.5|thumb |right| [[Quantum circuit]] representation for teleportation of a quantum state,<ref>{{cite book |author=Williams |first=Colin P. |title=Explorations in Quantum Computing |publisher=[[Springer Science+Business Media|Springer]] |year=2010 |isbn=978-1-4471-6801-0 |pages=496–499}}</ref><ref>{{Cite book|title=Quantum Computation and Quantum Information |last1=Nielsen |first1=Michael A.|last2=Chuang|first2=Isaac|year=2010 |publisher=[[Cambridge University Press]] |isbn=978-1-10700-217-3 |location=Cambridge|oclc=43641333|author-link=Michael Nielsen|author-link2=Isaac Chuang |pages=26–28}}</ref> as [[#Formal presentation|described above]]. The circuit consumes the <math>|\Phi^{+}\rangle</math> [[Bell state]] and the qubit to teleport as input, and consists of [[quantum logic gate#CNOT|CNOT]], [[quantum logic gate#Hadamard|Hadamard]], two [[Measurement in quantum mechanics#Quantum circuits|measurements]] of two qubits, and finally, two gates with [[Quantum logic gate#Classical control|classical control]]: a [[quantum logic gate#X|Pauli X]], and a [[quantum logic gate#Z|Pauli Z]], meaning that if the result from the measurement was <math>|1\rangle</math>, then the classically controlled Pauli gate is executed. After the circuit has run to completion, the value of <math>|\psi\rangle_C</math> will have moved to, or ''teleported'' to <math>|\psi\rangle_B</math>, and <math>|\psi\rangle_C</math> will have its value set to either <math>|0\rangle</math> or <math>|1\rangle</math>, depending on the result from the measurement on that qubit.<br>This circuit can also be used for ''entanglement swapping'', if <math>|\psi\rangle_C</math> is one of the qubits that make up an entangled state, as described in [[#Entanglement swapping|the text]].]]

There are a variety of different notations in use that describe the teleportation protocol. One common one is by using the notation of [[quantum gate]]s.

In the above derivation, the unitary transformation that is the change of basis (from the standard product basis into the Bell basis) can be written using quantum gates. Direct calculation shows that this gate is given by

:<math>G = (H \otimes I) \operatorname{CNOT}</math>

where ''H'' is the one qubit [[Hadamard gate|Walsh-Hadamard gate]] and <math>\operatorname{CNOT}</math> is the Controlled NOT gate.