Editing Bell state (section)

==Bell states==
The Bell states are four specific maximally entangled [[quantum states]] of two [[qubit]]s.  They are in a superposition of 0 and 1{{snd}}a linear combination of the two states.  Their entanglement means the following:

The qubit held by Alice (subscript "A") can be in a superposition of 0 and 1. If Alice measured her qubit in the standard basis, the outcome would be either 0 or 1, each with probability 1/2; if Bob (subscript "B") also measured his qubit, the outcome would be the same as for Alice. Thus, Alice and Bob would each seemingly have random outcome. Through communication they would discover that, although their outcomes separately seemed random, these were perfectly correlated.

This perfect correlation at a distance is special: maybe the two particles "agreed" in advance, when the pair was created (before the qubits were separated), which outcome they would show in case of a measurement.

Hence, following [[Albert Einstein]], [[Boris Podolsky]], and [[Nathan Rosen]] in their famous 1935 "[[EPR paradox|EPR]] paper", there is something missing in the description of the qubit pair given above{{snd}}namely this "agreement", called more formally a [[hidden-variable theory|hidden variable]]. In his famous paper of 1964, [[John S.&nbsp;Bell]] showed by simple [[probability theory]] arguments that these correlations (the one for the 0,&nbsp;1 basis and the one for the +,&nbsp;− basis) cannot ''both'' be made perfect by the use of any "pre-agreement" stored in some hidden variables{{snd}}but that quantum mechanics predicts perfect correlations. In a more refined formulation known as the [[CHSH Bell test|Bell–CHSH inequality]], it is shown that a certain correlation measure cannot exceed the value 2 if one assumes that physics respects the constraints of [[local hidden-variable theory|local "hidden-variable" theory]] (a sort of common-sense formulation of how information is conveyed), but certain systems permitted in quantum mechanics can attain values as high as <math>2\sqrt{2}</math>.  Thus, quantum theory violates the Bell inequality and the idea of local "hidden variables".

=== Bell basis ===
Four specific two-qubit states with the maximal value of <math>2\sqrt{2}</math> are designated as "Bell states". They are known as the four ''maximally entangled two-qubit Bell states'' and form a maximally entangled basis, known as the Bell basis, of the four-dimensional [[Hilbert space]] for two qubits:<ref name="Nielsen-2010">{{Cite book |last=Nielsen |first=Michael A. |title=Quantum computation and quantum information |last2=Chuang |first2=Isaac L. |publisher=Cambridge Univ. Press |year=2010 |isbn=978-0-521-63503-5 |edition=10 |location=Cambridge}}</ref>
: <math>|\Phi^+\rangle = \frac{1}{\sqrt{2}} \big(|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B\big) \qquad (1)</math>
: <math>|\Phi^-\rangle = \frac{1}{\sqrt{2}} \big(|0\rangle_A \otimes |0\rangle_B - |1\rangle_A \otimes |1\rangle_B\big) \qquad (2)</math>
: <math>|\Psi^+\rangle = \frac{1}{\sqrt{2}} \big(|0\rangle_A \otimes |1\rangle_B + |1\rangle_A \otimes |0\rangle_B\big) \qquad (3)</math>
: <math>|\Psi^-\rangle = \frac{1}{\sqrt{2}} \big(|0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B\big) \qquad (4)</math>

=== Creating Bell states via quantum circuits ===
[[File:The Hadamard-CNOT transform on the zero-state.png|thumb|right|400px|Quantum circuit to create Bell state <math>|\Phi^+\rangle</math>.]]Although there are many possible ways to create entangled Bell states through [[Quantum circuit|quantum circuits]], the simplest takes a computational basis as the input, and contains a [[Hadamard gate]] and a [[Cnot gate|CNOT gate]] (see picture). As an example, the pictured quantum circuit takes the two qubit input <math>|00\rangle</math> and transforms it to the first Bell state <math>|\Phi^+\rangle.</math> Explicitly, the Hadamard gate transforms <math>|00\rangle</math> into a [[Quantum superposition|superposition]] of <math>(|0\rangle|0\rangle + |1\rangle|0\rangle) \over \sqrt{2}</math>. This will then act as a control input to the CNOT gate, which only inverts the target (the second qubit) when the control (the first qubit) is 1. Thus, the CNOT gate transforms the second qubit as follows <math>\frac{(|00\rangle + |11\rangle)}{\sqrt{2} } = |\Phi^+\rangle</math>.

For the four basic two-qubit inputs, <math>|00\rangle, |01\rangle, |10\rangle, |11\rangle</math>, the circuit outputs the four Bell states ([[#Bell basis|listed above]]). More generally, the circuit transforms the input in accordance with the equation

<math display="block">|\beta(x,y)\rangle = \left ( \frac{|0,y\rangle + (-1)^x|1,\bar{y}\rangle}{\sqrt{2}} \right ),</math>

where <math>\bar{y}</math> is the negation of <math>y</math>.<ref name="Nielsen-2010" />

=== Properties of Bell states ===
The result of a measurement of a single qubit in a Bell state is indeterminate, but upon measuring the first qubit in the ''z''-basis, the result of measuring the second qubit is guaranteed to yield the same value (for the <math>\Phi</math> Bell states) or the opposite value (for the <math>\Psi</math> Bell states). This implies that the measurement outcomes are correlated.  [[John Stewart Bell|John Bell]] was the first to prove that the measurement correlations in the Bell State are stronger than could ever exist between classical systems. This hints that quantum mechanics allows information processing beyond what is possible with classical mechanics.  In addition, the Bell states form an orthonormal basis and can therefore be defined with an appropriate measurement. Because Bell states are entangled states, information on the entire system may be known, while withholding information on the individual subsystems.  For example, the Bell state is a [[Quantum state|pure state]], but the reduced density operator of the first qubit is a [[Quantum state|mixed state]]. The mixed state implies that not all the information on this first qubit is known.<ref name="Nielsen-2010" /> Bell States are either symmetric or antisymmetric with respect to the subsystems.<ref name="Sych-2009" /> Bell states are maximally entangled in the sense that its reduced density operators are maximally mixed, the multipartite generalization of Bell states in this spirit is called the [[Absolutely maximally entangled state|absolutely maximally entangled (AME) state]].