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

The splitter scan checks whether both output branches meet a minimum count after a half splitter. Eight input photons sit on the boundary because each branch receives four, while twelve input photons clear the branch floor with balanced outputs. Real splitters add imbalance, insertion loss, alignment drift, detector efficiency limits, finite counts, and uncertainty.

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}