Cross-Checks of Quantum Budgets

Spin Basis Probability Scan

same state, different question

A three-row basis scan separates the state from the measurement question. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

The plus-x state is balanced in the z basis

The same physical state can be asked a z question first. For the plus-x state the answer row is balanced.

P↑=12,P↓=12P_{\uparrow}=\frac{1}{2},\quad P_{\downarrow}=\frac{1}{2}
Plus-x read as zThe z-basis bars are computed from the plus-x state.state1/sqrt(2) up1/sqrt(2) downz basis1/2up1/2down

A basis scan changes the question, not the state by caption

The rows compare three checked questions. The x and y rows become definite only when the state and basis match.

state/questionP+P−sum+x asked z12121+x asked x101+y asked y101\begin{array}{c|c|c|c}\text{state/question}&P_+&P_-&\text{sum}\\+x\text{ asked }z&\frac{1}{2}&\frac{1}{2}&1\\+x\text{ asked }x&1&0&1\\+y\text{ asked }y&1&0&1\\\end{array}
Spin basis scanThe x-basis row is definite for plus-x.state1/sqrt(2) up1/sqrt(2) downx basis1plusx0minusx

A y-basis state gives a different definite row

The plus-y state also has balanced direct z probabilities, but its y-basis row is definite. The phase in the state decides which basis can read it cleanly.

P+y=1,P−y=0P_{+y}=1,\quad P_{-y}=0
Plus-y read as yThe y-basis bars are computed from the phase state.state1/sqrt(2) up1/sqrt(2)i downy basis1plusy0minusy