Timeline for Is Hamiltonian evolution unique given a fixed input and output state?
Current License: CC BY-SA 4.0
7 events
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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 |