Editing Bell state (section)

== Applications ==

=== Superdense coding ===
[[Superdense coding]] allows two individuals to communicate two bits of classical information by only sending a single qubit. The basis of this phenomenon is the entangled states or Bell states of a two qubit system. In this example, Alice and Bob are very far from each other, and have each been given one qubit of the entangled state.

<math>|\psi \rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}</math>.

In this example, Alice is trying to communicate two bits of classical information, one of four two bit strings: <math>'00', '01', '10',</math>or <math>'11'</math>. If Alice chooses to send the two bit message <math>'01'</math>, she would perform the <math>X</math> gate to her qubit. Similarly, if Alice wants to send <math>'10'</math>, she would apply the phase flip <math>Z</math>; if she wanted to send <math>'11'</math>, she would apply the <math>iY</math>gate to her qubit; and finally, if Alice wanted to send the two bit message <math>'00'</math>, she would do nothing to her qubit. Alice performs these [[quantum gate]] transformations locally, transforming the initial entangled state <math>|\psi\rangle</math> into one of the four Bell states.

The steps below show the necessary quantum gate transformations, and resulting Bell states, that Alice needs to apply to her qubit for each possible two bit message she desires to send to Bob.

<math>00: I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \longrightarrow |\psi \rangle = \frac{|00\rangle + |11\rangle}{\sqrt2}\equiv |{\Phi^+}\rangle</math>

<math>01: X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\longrightarrow |\psi \rangle = \frac{|01\rangle + |10\rangle}{\sqrt2}\equiv |{\Psi^+}\rangle</math>

<math>10: Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\longrightarrow |\psi \rangle = \frac{|00\rangle - |11\rangle}{\sqrt2}\equiv |{\Phi^-}\rangle</math>                                                     

<math>11: iY = ZX = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\longrightarrow |\psi \rangle = \frac{|01\rangle -  |10\rangle}{\sqrt2}\equiv |{\Psi^-}\rangle</math>.

After Alice applies the desired transformations to her qubit, she sends it to Bob. Bob then performs a measurement on the Bell state, which projects the entangled state onto one of the four two-qubit basis vectors, one of which will coincide with the original two bit message Alice was trying to send.

=== Quantum teleportation ===
{{Main|Quantum teleportation}}
[[Quantum teleportation]] is the transfer of a quantum state over a distance. It is facilitated by entanglement between A, the giver, and B, the receiver of this quantum state. This process has become a fundamental research topic for quantum communication and computing.  More recently, scientists have been testing its applications in information transfer through optical fibers.<ref>{{Cite journal|last=Huo|first=Meiru|date=19 October 2018|title=Deterministic Quantum Teleportation through Fiber Channels|journal=Science Advances|volume=4|issue=10|pages=eaas9401|doi=10.1126/sciadv.aas9401|pmid=30345350|pmc=6195333|bibcode=2018SciA....4.9401H|doi-access=free}}</ref> The process of quantum teleportation is defined as the following:

Alice and Bob share an EPR pair and each took one qubit before they became separated. Alice must deliver a qubit of information to Bob, but she does not know the state of this qubit and can only send classical information to Bob.

It is performed step by step as the following:

# Alice sends her qubits through a [[Controlled NOT gate|CNOT gate]].
# Alice then sends the first qubit through a [[Hadamard gate]].
# Alice measures her qubits, obtaining one of four results, and sends this information to Bob.
# Given Alice's measurements, Bob performs one of four operations on his half of the EPR pair and recovers the original quantum state.<ref name="Nielsen-2010" />

The following quantum circuit describes teleportation:
[[File:Telep.jpg|center|thumb|Quantum circuit for teleporting a qubit]]

=== Quantum cryptography ===
[[Quantum cryptography]] is the use of quantum mechanical properties in order to encode and send information safely.  The theory behind this process is the fact that it is impossible to measure a quantum state of a system without disturbing the system.  This can be used to detect eavesdropping within a system.

The most common form of [[quantum cryptography]] is [[quantum key distribution]].  It enables two parties to produce a shared random secret key that can be used to encrypt messages.  Its private key is created between the two parties through a public channel.<ref name="Nielsen-2010" />

Quantum cryptography can be considered a state of entanglement between two multi-dimensional systems, also known as two-[[qudit]] (quantum digit) entanglement.<ref name="Sych-2009" />