Editing Bell state (section)

==Bell state measurement==
The '''Bell measurement''' is an important concept in [[quantum information science]]: It is a joint quantum-mechanical measurement of two [[qubit]]s that determines which of the four Bell states the two qubits are in.

[[File:Bell State Decoder.jpg|thumb|right|400px|Quantum circuit that performs Bell decoding. Bell states are sometimes called EPR pairs. Notice that the circuit that decodes the Bell state [[Quantum logic gate#Unitary inversion of gates|is the adjoint]] to the circuit that encodes, or creates, Bell states (described [[#Creating Bell states via quantum circuits|above]]).]]A helpful example of [[Measurement in quantum mechanics|quantum measurement]] in the Bell basis can be seen in quantum computing.  If a [[Controlled NOT gate|CNOT gate]] is applied to qubits A and B, followed by a [[Hadamard gate]] on qubit A, a measurement can be made in the computational basis. The CNOT gate performs the act of un-entangling the two previously entangled qubits. This allows the information to be converted from quantum information to a measurement of classical information.    

Quantum measurement obeys two key principles. The first, the principle of [[Deferred_Measurement_Principle|deferred measurement]], states that any measurement can be moved to the end of the circuit. The second principle, the principle of implicit measurement, states that at the end of a quantum circuit, measurement can be assumed for any unterminated wires.<ref name="Nielsen-2010" />

The following are applications of Bell state measurements:

Bell state measurement is the crucial step in [[quantum teleportation]]. The result of a Bell state measurement is used by one's co-conspirator to reconstruct the original state of a teleported particle from half of an entangled pair (the "quantum channel") that was previously shared between the two ends.

Experiments that utilize so-called "linear evolution, local measurement" techniques cannot realize a complete Bell state measurement. Linear evolution means that the detection apparatus acts on each particle independent of the state or evolution of the other, and local measurement means that each particle is localized at a particular detector registering a "click" to indicate that a particle has been detected. Such devices can be constructed from, for example: mirrors, beam splitters, and wave plates{{snd}}and are attractive from an experimental perspective because they are easy to use and have a high measurement [[cross section (physics)|cross-section]].

For entanglement in a single qubit variable, only three distinct classes out of four Bell states are distinguishable using such linear optical techniques. This means two Bell states cannot be distinguished from each other, limiting the efficiency of quantum communication protocols such as [[quantum teleportation|teleportation]]. If a Bell state is measured from this ambiguous class, the teleportation event fails.

Entangling particles in multiple qubit variables, such as (for photonic systems) [[polarization (waves)|polarization]] and a two-element subset of [[azimuthal quantum number|orbital angular momentum]] states, allows the experimenter to trace over one variable and achieve a complete Bell state measurement in the other.<ref>Kwiat, Weinfurter. [https://archive.today/20120712230327/http://pra.aps.org/abstract/PRA/v58/i4/pR2623_1 "Embedded Bell State Analysis"]</ref> Leveraging so-called hyper-entangled systems thus has an advantage for teleportation. It also has advantages for other protocols such as [[superdense coding]], in which hyper-entanglement increases the channel capacity.

In general, for hyper-entanglement in <math>n</math> variables, one can distinguish between at most <math>2^{n+1} - 1</math> classes out of <math>4^n</math> Bell states using linear optical techniques.<ref>Pisenti, Gaebler, Lynn. [http://www.opticsinfobase.org/abstract.cfm?uri=ICQI-2011-QMI25 "Distinguishability of Hyper-Entangled Bell States by Linear Evolution and Local Measurement"]</ref>