Non-adjoint Conjugate of Pure State Vector

[Pythagorean]

If I have a pure state vector of a system (let's call it A):

-0.4431 + 0.2317i
-0.4431 + 0.2317i
0.5000
0.5000

A particularly interesting symmetry in the system allows a similar pure state (B):

-0.4431 - 0.2317i
-0.4431 - 0.2317i
0.5000
0.5000

the absolute value of the inner product is
|B'*A| = .8861
(as opposed to 1, since these are only conjugates of each other and not adjoints, or
hermitian conjugates. For instance, A'*A = 1 where A' is the adjoint of A).

question:
is there a specific name for the relationship between A and B besides just conjugate? In the literature, and especially with state vecto.rs, conjugate and adjoint are used interchangeably.

This is a classical case of misleading intuition with the complex space. I imagined that as you rotated the vector around, there would be a second position (B) that would return 1 if you took the inner product of it with A, by symmetry. I didn't suspect that the inner product could detect an "anti-parallel" vector.

Of course, note that as you rotate the vector B around, it has two clear maximum peaks. The largest peak is when B = A so your inner product is A'*A = 1 exactly, but the second largest peak occurs when B is "anti-parallel" (the conjugate non-adjoint) at the .8861 value mentioned.

In this respect, it reminds me of harmonics. I expected the two peaks opposite each other to both go to 1, but they did not and now I want to better understand what exactly I'm doing. It looks like maybe I'm only rotating it around the imaginary plane, not the whole complex plane, and the projection in the complex plane gets distorted (shortened) somehow.

 

[fresh_42]

When solving quadratic equations one usually gets conjugates as zeroes. E.g. the formula for ##x^2+px+q=0# is solved by conjugates. This is also true for higher degrees, just that the automorphisms are not of order 2 then. So it is very likely that your observation is due to the specific underlying equations.