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The four Bell states form a Bell basis. A two-qubit state that is described as a sum of Bell states describes a correlation between any two bases on the individual qubits. For example, <math>\frac{1}{\sqrt{2}}(|0+\rangle + |1-\rangle)</math> is maximally entangled; it represents a correlation between the basis <math>b_1</math> and <math>b_2 = H.b_1</math>. It can be rewritten as <math>\frac{1}{\sqrt{2}}(|\Phi^-\rangle + |\Psi^+\rangle)</math> using Bell basis states. |
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The four Bell states form a Bell basis. A two-qubit state that is described as a sum of Bell states describes a correlation between any two bases on the individual qubits. For example, <math>\frac{1}{\sqrt{2}}(|0+\rangle + |1-\rangle)</math> is maximally entangled; it represents a correlation between the basis <math>b_1</math> and <math>b_2 = H.b_1</math>. It can be rewritten as <math>\frac{1}{\sqrt{2}}(|\Phi^-\rangle + |\Psi^+\rangle)</math> using Bell basis states. |
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== Notes == |
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{{Notelist}} |
Separating the Bell State
When the CNOT gate acts upon two qubits that are perfectly correlated in the state, the outputs are the unentangled states and . The CNOT gate is its own inverse.
To demonstrate this, we show that in any chosen basis the perfect correlation and the operation of the CNOT gate combine to produce a constant output.
Selecting the computational basis we have:
Qubit A's effect on qubit B
Based on qubit B correlating exactly with qubit A and then qubit B being subjected to the CNOT X-rotation depending on qubit A:
correlates to which results in
correlates to which results in
Qubit B's effect on qubit A
The basis vectors that we've chosen, represented by Hadamard basis vectors are:
Separates into:
and
The other basis vector:
Separates into:
and
So the resulting state of summing the results of the basis transformations (and dividing by 2) is the constant:
Further worked example
Using an arbitrarily-selected basis of:
Qubit A's effect on qubit B
Based on qubit B correlating exactly with qubit A and then qubit B being subjected to the CNOT X-rotation depending on qubit A:
Separates into:
and which equals
The other basis vector:
Separates into:
and which equals
So the resulting state of summing the results of the basis transformations (and dividing by 2) is the constant:
Qubit B's effect on qubit A
The basis vectors that we've chosen, represented by Hadamard basis vectors are:
Separates into:
and which equals
The other basis vector:
Separates into:
and which equals
So the resulting state of summing the results of the basis transformations (and dividing by 2) is the constant:
Bell basis
The four Bell states form a Bell basis. A two-qubit state that is described as a sum of Bell states describes a correlation between any two bases on the individual qubits. For example, is maximally entangled; it represents a correlation between the basis and . It can be rewritten as using Bell basis states.
Notes