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Sep 20, 2021 at 12:02 comment added ZeroTheHero this is true of every system: two states differing by an overall phase are equivalent.
Sep 20, 2021 at 11:49 comment added Marion Ok, but one might not have angular momentum or such as a parameter. Depends on what is the system considering, right? For a superconducting qubit I guess there is not much point to talk about angular momentum. But point is that in the Bloch sphere, am I correct that any two states related by a $U(1)$ rotation (that is a phase) are then physically equivalent?
Sep 20, 2021 at 11:21 comment added ZeroTheHero $\ell,m$ are angular momentum quantum number… $i$ has magnitude 1 so it’s hardly a scaling, and since all states differing by an overall phase are equivalent, this isn’t an issue unless you have grounds to keep track of this overall phase.
Sep 20, 2021 at 7:50 comment added Marion I am not sure what is $\ell,m$ here but let's say I don't allow piecewise Hamiltonians. It seems from your answer that the evolution of the state is unique up to a complex number so one can say it is "conformally unique" since $i$ scales the result in the complex plane (actually rotates it). And this is the result of a toric action. But is the state $|n\rangle$ and $i|n\rangle$ indistinguishable?
Sep 19, 2021 at 2:14 comment added ZeroTheHero still not quite precise. Do you allow piecewise Hamiltonian evolution? Do you specify input AND output? Do you have restrictions on your Hamiltonians? (for instance, you cannot go from $\ell=2,m=2$ to $\ell=2,m=1$ states using a single Hamiltonian constructed from angular momentum operators alone...
Sep 18, 2021 at 17:29 comment added Marion Thanks a lot for the answer. I guess that I would like to cut phases down since given an experiment the output is "agnostic" about the overall phase so we would only read/estimate the state as it is given: In this case the complex $i$ makes a difference as to the numerical (not physical) output. But even if we allow phases, then there must be a family of Hamiltonians parametrized by some variable right? So I guess I am asking the equivalence classes classifications of such Hamiltonians.
Sep 18, 2021 at 13:54 history answered ZeroTheHero CC BY-SA 4.0