Quantum – Theory – Superposition

Definition:

Superposition represents any state of the qubit that is a linear combination of basis states |j^{^}> and |j^v> where both components are non-zero.

States that are not in superposition:

|j^{^}> = $\begin {bmatrix} 1^h \\ 0 \end{bmatrix}$

|j^v> = $\begin {bmatrix} 0 \\ 1^h \end{bmatrix}$

^–|j^{^}> = $\begin {bmatrix} 1^v \\ 0 \end{bmatrix}$

^–|j^v> = $\begin {bmatrix} 0 \\ 1^v \end{bmatrix}$

$\begin {bmatrix} 0 \\ i^h \end{bmatrix}$

$\begin {bmatrix} i^h \\ 0 \end{bmatrix}$

$\begin {bmatrix} 0 \\ i^v\end{bmatrix}$

$\begin {bmatrix} i^v \\ 0 \end{bmatrix}$

Note that although the state $\begin {bmatrix} 0 \\ 0 \end{bmatrix}$ is mathematically possible, a qubit cannot physically exist in this state. Also, this state breaches Born’s rule which states that:

P(x) = |<x|ψ>|² and ∑P(xi) = 1.

In this case ∑P(xi) = 0.

Some States that are in Superposition:

|^{^}> = $\begin {bmatrix} 1^h/\sqrt{2} \\ 1^h/\sqrt{2} \end{bmatrix}$ = 1/√2(|j^{^}> + |j^v>)

|^v> = $\begin {bmatrix} 1^h/\sqrt{2} \\ 1^v/\sqrt{2} \end{bmatrix}$ = 1/√2(|j^{^}> ^–|j^v>)

|i^{^}> = $\begin {bmatrix} 1^h/\sqrt{2} \\ i^h/\sqrt{2} \end{bmatrix}$ = 1/√2(|j^{^}> + i|j^v>)

|i^v> = $\begin {bmatrix} 1^h/\sqrt{2} \\ i^v/\sqrt{2} \end{bmatrix}$ = 1/√2(|j^{^}> + ^–i|j^v>)

$\begin {bmatrix} 1^h/\sqrt{3} \\ 2^h/\sqrt{3} \end{bmatrix}$ = 1/√3|j^{^}> + 2/√3|j^v>

Conclusion

In summary, any state where either of the matrix column entries equal 0 is not in superposition. Conversely, any state, where both of the column entries are not 0, are in superposition.

Check out the Bella g io Circuit where states are changed into superposition using Haadamard circuits.

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Quantum – Theory – Superposition

Definition:

States that are not in superposition:

Some States that are in Superposition:

Conclusion

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