Symbolic computation using complex variables

Questions
1

I have a complex matrix and a vector of two complex numbers:

bs1 = ( {
   {1/Sqrt[2], u},
   {-Conjugate[u], 1/Sqrt[2]}
  } )

where u is a complex number.

and the vector

state = ( {
   {\[Alpha]},
   {\[Beta]}
  } )

where Alpha and Beta are complex numbers. The Norm of the vector should be 1

I wish to compute

Norm[bs1.state]^2

and then simplify. The result should be:

1/2 + Norm[u]^2 = 1

However this is predicated on the vector Norm being 1. How do I define the vector specifying that the norm is 1?

As an aside if the Conjugate is not defined say there was u and -u where u, Alpha and Beta are complex how would I specify that they are indeed complex variables for the symbolic evaluation?

2

Perhaps like this? (Note the change in the notation of state)

bs1 = ({{1/Sqrt[2], u}, {-Conjugate[u], 1/Sqrt[2]}});
state = {\[Alpha], \[Beta]}
xx = bs1.state

normxxsquared = Conjugate[xx].xx // FullSimplify
normxxsquared /. Abs[\[Alpha]]^2 -> 1 - Abs[\[Beta]]^2
3

Thanks. That seems to work. I was hoping for a global definition of the condition but it really makes little difference.

Any chance you know the answer to the second question ;)?

PS - it works for me with or without the change of notation of state.

4

I must admit that I don’t understand what you mean. I think you should use Conjugate at the appropriate places.