A phase error position scan moves a single minus sign across the three carriers of a phase-basis codeword and reads off the resulting syndrome: minus on the left gives syndrome 1-0, minus in the middle gives syndrome 1-1, and minus on the right gives syndrome 0-1 exactly for the stated toy phase-parity rule. Every earlier audit in this book fixed the phase flip at the right carrier; moving it to each position in turn builds the same complete single-error table chapter two already built in the computational basis, now built in the phase basis instead.
Every earlier audit fixed the phase flip at the right carrier; moving it to each position builds the phase-basis syndrome table. 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 left phase flip gives syndrome one-zero
Chapter three's basis-bridge lesson always put the minus sign on the right carrier. Here it sits on the left instead: 011. The phase syndrome is 10 and the correction index is 0.
s=10c=0
A middle phase flip fires both phase parity checks
Moving the minus sign to the middle carrier, 101, makes both phase parity checks fire, giving syndrome 11 and correction index 1.
s=11c=1
The three phase positions give three distinct syndromes
The right position is the one every earlier audit in this book used: 110, syndrome 01. Together the three rows form the same complete single-error table chapter two already built in the computational basis, now built in the phase basis instead.