I am currently working with an exercise set about discriminating between two different quantum states. We consider a 2-dimensional Hilbert space expanded by $|\phi_1\rangle$, $|\phi_2\rangle$. The system is secretly prepared into one of the following states at random:
$|\psi_1\rangle=|{\phi_1}\rangle \tag{1}\\$
$|\psi_2\rangle=\cos\theta|\phi_1\rangle+\sin\theta |\phi_2\rangle \tag{2}$
Where angle $\theta$ is a real number between $0$ and $\frac{\pi}{2}$.
We are handed one of these states and want to identify if we receive $|\psi_1\rangle$ or $|\psi_2\rangle$.
To start with we consider another basis $|e_1\rangle$, $|e_2\rangle$ which is rotated with respect to $|\phi_1\rangle$ and $|\phi_2\rangle$:
$$|e_1\rangle=\cos\alpha|\phi_1\rangle+\sin\alpha|\phi_2\rangle\\$$
$$|e_2\rangle=-\sin\alpha|\phi_1\rangle+\cos\alpha|\phi_2\rangle$$
Where angle $\alpha$ originates from the fact that basis $|e_1\rangle$, $|e_2\rangle$ is rotated relative to $|\phi_1\rangle$, $|\phi_2\rangle$.
I am asked to write down general expressions using $|\phi_1\rangle$ and $|\phi_2\rangle$ for the projectors corresponding to a measurement in the $|e_1\rangle$, $|e_2\rangle$ basis.
Attempted solution:
I have previously constructed the basis $|\phi_1\rangle$ and $|\phi_2\rangle$ in terms of $|e_1\rangle$ and $|e_2\rangle$ as:
$$|\phi_1\rangle=-\sin\alpha |e_2\rangle$$
$$|\phi_2\rangle=\sin\alpha |e_1\rangle+\cos\alpha |e_2\rangle$$
But I see now that they want me to construct the projectors instead of the vectors themselves. What do they mean exactly, and how do I determine these projectors?
Update:
Following the first answer, i get:
$P=\sum_i|\phi_i\rangle\langle\phi_i|\delta_{a,a_i}=|e_1\rangle\langle e_1|+|e_2\rangle\langle e_2|$
$|e_1\rangle\langle e_1| = [\text{cos}\alpha | \phi_1 \rangle + \text{sin}\alpha |\phi_2 \rangle]\;[\text{cos}\alpha \langle \phi_1 |+\text{sin}\alpha \langle \phi_2 |]=\text{cos}^2\alpha |\phi_1\rangle \langle \phi_1 | + \text{sin}^2\alpha |\phi_2\rangle\langle \phi_2 |$
$|e_2\rangle\langle e_2| = [-\text{sin}\alpha | \phi_1 \rangle + \text{cos}\alpha |\phi_2 \rangle]\;[-\text{sin}\alpha \langle \phi_1 |+\text{cos}\alpha \langle \phi_2 |]=\text{sin}^2\alpha |\phi_1\rangle \langle \phi_1 | + \text{cos}^2\alpha |\phi_2\rangle\langle \psi_2 |$
So $P=|\phi_1\rangle\langle \phi_1|+|\phi_2\rangle\langle \phi_2|$
Is it correct to only look at the non-perpencular states for $|e_1\rangle\langle e_1|$, i.e. does the kronecker delta 'survive' for this expression when considering the states $|\phi_1\rangle$, $|\phi_2\rangle$?
Best Answer
If "the projectors corresponding to a measurement in the |e1⟩, |e2⟩ basis" means operators that project out the |e1> and |e2> components from a state, aren't they just |e1><e1| and |e2><e2|? You just need to plug in the expression of |e1> and |e2> in terms of |ϕ1⟩ and |ϕ2⟩.