A physical phase shift can be modeled as an exact quantum gate. 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 physical phase shift can be a gate

The phase-control model applies a checked phase gate to an equal superposition.

S+x=+yS\lvert +x\rangle = \lvert +y\rangle
Phase control as a gateThe gate output is exact complex arithmetic.state1/sqrt(2) zero1/sqrt(2) onestate1/sqrt(2) zero1/sqrt(2)i oneSgate

Direct balance becomes a definite y-basis row

The phase gate keeps direct probabilities balanced, then the y-basis row becomes definite.

rowplusminustotaldirect12121y basis101\begin{array}{c|c|c|c}\text{row}&\text{plus}&\text{minus}&\text{total}\\\text{direct}&\frac{1}{2}&\frac{1}{2}&1\\\text{y basis}&1&0&1\\\end{array}
Phase-control readout scanThe later basis turns the phase into a definite row.state1/sqrt(2) up1/sqrt(2)i downy basis1plusy0minusy

The controlled phase changes a later basis result

In the y basis, the modeled result has probability 1 for plus y.

P+y=1P_{+y}=1
Phase control readoutThe y-basis bar is computed from the phase state.state1/sqrt(2) up1/sqrt(2)i downy basis1plusy0minusy