Editing Bell state (section)

==Bell state correlations==

Independent measurements made on two qubits that are entangled in Bell states positively correlate perfectly if each qubit is measured in the relevant basis. For the <math>|\Phi^+\rangle</math> state, this means selecting the same basis for both qubits. If an experimenter chose to measure both qubits in a <math>|\Phi^-\rangle</math> Bell state using the same basis, the qubits would appear positively correlated when measuring in the <math>\{|0\rangle,|1\rangle\}</math> basis, anti-correlated in the <math>\{|+\rangle,|-\rangle\}</math> basis{{Efn|<math>|\Phi^-\rangle = \frac{1}{\sqrt{2}} (|00\rangle - |11\rangle)</math>

<math>= \frac{1}{2\sqrt{2}} ((|+\rangle_A + |-\rangle_A)(|+\rangle_B + |-\rangle_B) - (|+\rangle_A - |-\rangle_A)(|+\rangle_B - |-\rangle_B))</math>

<math>= \frac{1}{2\sqrt{2}} (|++\rangle + |+-\rangle + |-+\rangle + |--\rangle - |++\rangle + |+-\rangle + |-+\rangle - |--\rangle)
</math>

<math>= \frac{1}{\sqrt{2}} (|+-\rangle + |-+\rangle)
</math>
}}, and partially (probabilistically) correlated in other bases.

The <math>|\Psi^+\rangle</math> correlations can be understood by measuring both qubits in the same basis and observing perfectly anti-correlated results. More generally, <math>|\Psi^+\rangle</math> can be understood by measuring the first qubit in basis <math>b_1</math>, the second qubit in basis <math>b_2 = X.b_1</math>, and observing perfectly positively correlated results.

[[File:Representation of the two-qubit Phi-minus entangled state.svg|thumb|Relationship between the correlated bases of two qubits in the <math>|\Phi^-\rangle</math> state.]]

{| class="wikitable"
|-
! Bell state !! Basis <math>b_2</math>
|-
| <math>|\Phi^+\rangle</math>|| <math>b_1</math>
|-
| <math>|\Phi^-\rangle</math>|| <math>Z.b_1</math>
|-
| <math>|\Psi^+\rangle</math> || <math>X.b_1</math>
|-
| <math>|\Psi^-\rangle</math> || <math>X.Z.b_1</math>
|}