Why the global phase does not matter?
In books on quantum computing, sometimes it is said that the global phase of a state does not matter. Because?
Making an analogy, let’s say I want to leave my house and walk to the bakery.
What matters is the relative position between my position and the destination.
However, strictly speaking, we can say that the planet Earth is rotating, and at every moment, my position is modified in relation to the center of the Earth. The same happens with the bakery, which has its position modified by the rotation of the Earth at every moment.
Therefore, the global phase affects both terms of the equation equally and can be eliminated. I don’t need to map the absolute position in relation to the universe (and, according to Einstein, there’s no absolute position).
Mathematically, be the quantum state:
We can isolate the global phase using complex numbers in polar notation:
We can isolate the global phase factor and leave only the relative phase:
We can call the difference phi_a — phi_b just phi, to facilitate the notation, and throw out the global phase, as explained earlier.
Plus the normalization constraint, these three variables become two, the two in the Bloch sphere.
Join my quantum computing study group in Facebook: