Quantum Information: Why |<φ|exp(iα)|ψ>|^2 Differs

[brydustin]

Why is |<{\phi}|exp(i{\alpha}|{\psi}>|^2 for any alpha not dependent on choice of alpha, but |<phi | exp(i\alpha) \psi >|^2 is dependent on choice of alpha. Also, is there a list somewhere for how to type this garbage on physics forums; I'm used to using TeX maker for writing math, but I always type horrendously at p.f.

 

[Ken G]

Is alpha supposed to be an operator in the first case and a constant in the second case? Because if I regard it as a constant both times, then I cannot see any difference in those expressions. Also, did you mean to invert their order? Because the second term clearly does not depend on alpha, just carry out the multiplication, and if alpha is an operator in the first case, then it is not true that the outcome doesn't depend on what alpha is.

 

[brydustin]

Typo, yet another reason why I need to learn how to type the math here (so I can see it properly).
The second one is supposed to be: |<phi|psi_1 +exp(iα)psi_2 >|^2

 

[Mark M]

For LaTeX, I use this. Make sure you select phpBB as the output.

 

[brydustin]

For LaTeX, I use this. Make sure you select phpBB as the output.

Thanks Mark!

But here is my REAL question:
Given
$$|<{\phi}|e^{i\alpha}{\psi}>|^2 \\ |<\phi|\psi_1+e^{i\alpha}\psi_2>|^2$$
why does the first one NOT depend on choice of alpha but the second one does (in terms of "states" of the system)?

 

[Ken G]

Ah, OK that makes sense. It is the difference between "absolute phase" and "relative phase" in QM. If two states that are being superimposed with each other or interacting in some way have a relative phase difference, that can cause real effects. If a single state is attributed a different phase in some absolute sense, and this new phase is never referenced to the phase of any other state it is superimposed with or interacting with, then it cannot cause real effects. Phase is a relative concept. You can think of it like a clock with no numbers on it floating in deep space-- if the clock has only a single hand, then a photograph of it cannot carry a meaningful concept of phase, but it can if there are at least two hands on the clock (like your psi_1 and psi_2).