# How to determine the projectors of a given basis?

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 $$|\phi_1\rangle$$, $$|\phi_2\rangle$$?

• I have considered your answer with an update to my post. It is correct that the kronecker delta is 'alive' when considering the terms $|e_1\rangle\langle e_2$, so only the non-perpendicular projections survive? Jul 3 at 8:12