Qubit measurement probabilities with the Born rule

P. K. Majumdar

Published Aug 11, 2026• 0 citations

Abstract

This Jupyter Notebook provides a reproducible numerical demonstration of the Born rule for projective measurements of a single qubit. It explains how a complex inner product is converted into a physically meaningful measurement probability. Using Python and NumPy, the notebook defines a validated function that calculates the probability amplitude between an initial quantum state and a measurement state. It then calculates the transition probability by taking the squared absolute value of that amplitude. The implementation checks that both state vectors are finite, one dimensional, equal in size and correctly normalised. It uses the NumPy vdot function to apply complex conjugation to the measurement state correctly. In the worked example, the qubit is prepared in the computational zero state and measured in the Pauli X basis. The calculation shows that the probabilities of obtaining the plus and minus outcomes are both 0.5. It also confirms that the two probabilities sum to 1. The numerical result is compared with the corresponding Bloch sphere prediction. The notebook verifies that the vector calculation and the geometric calculation both produce a transition probability of 0.5. Numerical consistency checks confirm agreement between the two calculation methods and verify probability conservation. The notebook requires Python 3, Jupyter Notebook or JupyterLab, and NumPy. It does not require a quantum computing framework or access to physical quantum hardware.

DOI: 10.5281/zenodo.21892667

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