Phase Gate Three-Row Scan - Quantum Mechanics: Waves, Spin, and Tunneling Step by Step

Three gates act on the same balanced input; only one of them genuinely redistributes the z-basis populations. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

A phase gate can leave the direct populations fixed

The S gate applied to the plus-x state keeps the z-basis populations balanced at one half each, exactly as before the gate. Only the phase between the branches changed.

P_{\uparrow}=\frac{1}{2},\quad P_{\downarrow}=\frac{1}{2}

Three gates on the same input separate phase from population

S and Z only ever touch phase in this model: both leave the z-basis populations balanced. H is not a phase gate here; it genuinely moves population, collapsing the balanced input onto one definite branch.

\begin{array}{c|c|c}\text{gate}&P_{\uparrow}&P_{\downarrow}\\S&\frac{1}{2}&\frac{1}{2}\\Z&\frac{1}{2}&\frac{1}{2}\\H&1&0\\\end{array}

H genuinely redistributes population, not just phase

The H gate output is the definite up branch: probability one up, probability zero down. This is the one row in the scan where the population itself changed, not only the phase between branches.

P_{\uparrow}=1,\quad P_{\downarrow}=0