Quantum State Probability CalculatorĀ
Quantum State Probability Calculator computes the probability of measuring a qubit in a specific state using the Born rule. Enter the complex coefficients for the state \\( |\psi\rangle = a|0\rangle + b|1\rangle \\) (e.g., \\( a = x + yi \\), \\( b = w + zi \\)) and select a measurement basis. The state must be normalized (\\( |a|^2 + |b|^2 = 1 \\)).
Methodology Used in Quantum State Probability Calculator
In quantum mechanics, the probability of measuring a quantum system in a specific state is given by the Born rule. For a qubit in state \\( |\psi\rangle = a|0\rangle + b|1\rangle \\), where \\( a, b \\) are complex coefficients, the state is normalized if \\( |a|^2 + |b|^2 = 1 \\).
Born Rule: The probability of measuring the system in a basis state \\( |n\rangle \\) is:
\\[ P(n) = |\langle n | \psi \rangle|^2 \\]Z-Basis (\\( |0\rangle, |1\rangle \\)):
\\[ P(|0\rangle) = |a|^2, \quad P(|1\rangle) = |b|^2 \\]X-Basis (\\( |+\rangle, |-\rangle \\)): Where \\( |+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle) \\), \\( |-\rangle = \frac{1}{\sqrt{2}}(|0\rangle – |1\rangle) \\), the probabilities are:
\\[ P(|+\rangle) = \left| \frac{a + b}{\sqrt{2}} \right|^2, \quad P(|-\rangle) = \left| \frac{a – b}{\sqrt{2}} \right|^2 \\]Algorithm Steps:
1. Input complex coefficients \\( a = x + yi \\), \\( b = w + zi \\).
2. Check normalization: \\( |a|^2 + |b|^2 = x^2 + y^2 + w^2 + z^2 \approx 1 \\) (within \\( 10^{-4} \\)).
3. For Z-basis, compute \\( P(|0\rangle) = |a|^2 \\), \\( P(|1\rangle) = |b|^2 \\).
4. For X-basis, compute amplitudes \\( \langle + | \psi \rangle = \frac{a + b}{\sqrt{2}} \\), \\( \langle – | \psi \rangle = \frac{a – b}{\sqrt{2}} \\), then square their magnitudes.
5. Display probabilities with LaTeX formatting.
Example
For state \\( |\psi\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{i}{\sqrt{2}}|1\rangle \\):
Z-Basis:
\\[ P(|0\rangle) = \left| \frac{1}{\sqrt{2}} \right|^2 = \frac{1}{2}, \quad P(|1\rangle) = \left| \frac{i}{\sqrt{2}} \right|^2 = \frac{1}{2} \\]X-Basis:
\\[ \langle + | \psi \rangle = \frac{\frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}}}{\sqrt{2}} = \frac{1 + i}{2}, \quad P(|+\rangle) = \left| \frac{1 + i}{2} \right|^2 = \frac{1 + 1}{4} = \frac{1}{2} \\] \\[ \langle – | \psi \rangle = \frac{\frac{1}{\sqrt{2}} – \frac{i}{\sqrt{2}}}{\sqrt{2}} = \frac{1 – i}{2}, \quad P(|-\rangle) = \left| \frac{1 – i}{2} \right|^2 = \frac{1 + 1}{4} = \frac{1}{2} \\]