Imaginary Phase - Quantum Mechanics: Waves, Spin, and Tunneling Step by Step

An imaginary phase changes the relation between branches while leaving direct squared-magnitude probabilities unchanged. The equal direct rows are exact only for the declared amplitudes and basis; real phase tests add coherence loss, phase drift, pulse error, alignment limits, and finite readout statistics.

An imaginary phase changes the state without changing direct measurement probabilities. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

An imaginary phase preserves direct probabilities

The phase gate turns the down branch into an imaginary branch. The direct probabilities remain 1/2 and 1/2.

P_{\text{zero}}=\frac{1}{2},\quad P_{\text{one}}=\frac{1}{2}

Both direct rows keep one half

The phase changes the branch relation, while the direct probability rows stay equal.

\begin{array}{c|c|c}\text{branch}&P_{\text{before}}&P_{\text{after}}\\\text{zero}&\frac{1}{2}&\frac{1}{2}\\\text{one}&\frac{1}{2}&\frac{1}{2}\\\text{total}&1&1\\\end{array}

The phase is stored for later

The diagram does not show a new direct chance. It shows a state that can behave differently when measured in another basis.

S\lvert +x\rangle = \lvert +y\rangle