Direct-basis probability bars read squared magnitudes, so some phase changes leave the displayed row unchanged. Here up carries 9/25 and down carries 16/25 whether the down branch is positive, negative, or imaginary, but the ket still keeps the phase for later recombination or a different basis. Real phase claims need controlled preparation, stable references, interference checks, finite shots, and uncertainty.
Changing a branch sign or phase can preserve direct probabilities while keeping state information. 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 positive state sets the direct probability row
A direct-basis measurement only reads squared magnitudes. For this state, up carries 9/25 and down carries 16/25.
P↑=259,P↓=2516
Sign and imaginary phase keep this direct row
Changing the down branch from positive to negative or imaginary does not change these direct probabilities. It changes what can happen later when amplitudes recombine.
down amp+54−54i54P↑259259259P↓251625162516sum111
The phase is not discarded
The imaginary state has the same direct row, but the displayed ket keeps the phase. A later basis choice can use that phase, so it is not decoration.