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## Homework Statement

Is the following matrix a state operator ? and if it is a state operator is it a pure state ? and if it is so then find the state vectors for the pure state.

If you dont see image here is the matrix which is 2X2 in matlab code:

[9/25 12/25; 12/25 16/25]

## Homework Equations

To be a state operator, if we have a operator ρ we know :

Tr(ρ)=1

ρ=ρ

^{t}(self-adjoint)

<u|ρ|u> >= 0 for all vectors |u>

and these means :

the sum of eigenvalues must be 1 and eigenvalues must be greater or equal to zero

For pure state what do I know are these:

ρ=|ψ><ψ| where |ψ> is the unit-normed vector called state vector.

The average value of an observable R in this pure state is:

<R> = Tr(|ψ><ψ|R) = <ψ|R|ψ>

The other condition is :

ρ

^{2}=ρ (which is possible for 1 or 0 but the sum of eigenvalues must be 1)

The third condition is :

Tr(ρ

^{2})=1

## The Attempt at a Solution

This matrix has eigenvalues 1 and 0. And this means it is a state operator. In my solution I do see that this matrix is a pure state and it has the vector state : (3/5 4/5). But I dont know how I can use conditions for pure state to see that if a matrix or an operator is a pure state and I can not either get the state vectors.

What do I know is that :

WWith eigenvalue 1 we get vector (-(4/3) 1). I do see that (3/5 4/5) is the norm of the diagonal of the matrix, [9/25 12/25; 12/25 16/25], that is in the first place in the matrix we have 9/25 and √(9/25)= 3/5.

In the last place of this matrix we have 16/25 and √(16/25) = 4/5

ofcourse 16/25 + 9/25 =1

But is that correct to think so ?