Complexifying Quantum States

Abstract

This chapter explains why complex numbers are important for more general quantum mechanical situations. The complex exponential function, aka Euler’s formula, plays a critical role in expressing phase differences between components of a quantum state vector. These phase differences affect the outcome of quantum operations. The Bloch sphere and Bloch vector provide a visual representation of these more complex quantum states with the real and imaginary parts on different axes. The complex exponential is important in the quantum Fourier transform, which is then used to prove an important result for finding the repetition period in the Shor algorithm.