Splitter Branch Floor Boundary Scan - Quantum Mechanics IV: Photons, Interferometers, and Optical Qubits Step by Step

Even input counts move each splitter branch across a floor. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Each splitter branch has a minimum count

A half splitter sends the same count to the upper and lower branches.

m=N_{\text{upper}}-N_{\min}

Three even inputs bracket the branch floor

Eight input photons are the boundary because each branch receives four.

\begin{array}{c|c|c|c}N_{\text{in}}&N_{\text{upper}}&m&\text{gate}\\4&2&-2&\text{fail}\\8&4&0&\text{pass}\\12&6&2&\text{pass}\\\end{array}

Twelve input photons clear the branch floor

The lower branch carries the same count, so the balance remains exact.

m=2\quad \text{pass}