How to write the density matrix for a state in the basis $\{ |H\rangle,|V\rangle \}$? [closed]

Question

Suppose we have a set of photons in a mixed state with probabilities $P_1=0.2$ and $P_2=0.8$, respectively, of being in the pure states;

$$|\phi_1\rangle=|H\rangle; \; |\phi_2\rangle=\frac{3}{5}|H\rangle+\frac{4i}{5}|V\rangle$$

where $|H\rangle$ and $|V\rangle$ are the orthonormal basis states representing horizontal and vertical polarization, respectively. We are asked to write the density matrix in the $\{|H\rangle,|V\rangle\}$ basis.

The density matrix is defined as:

$$\rho=\sum_jp_j|\phi_j\rangle\langle\phi_j|$$

Inserting the corresponding pure states, we have:

$$\rho=p_1|\phi_1\rangle\langle\phi_1| + p_2|\phi_2\rangle\langle\phi_2|\\ =0.2 \cdot |H\rangle\langle H| + 0.8 \cdot \left[ \frac{3}{5} |H\rangle + \frac{4i}{5} |V\rangle \right]\left[ \frac{3}{5} \langle H| - \frac{4i}{5} \langle V| \right]\\ =0.2 \cdot |H\rangle\langle H| + 0.8 \cdot \left[ \frac{9}{25} |H\rangle\langle H|-\frac{12i}{25}|H\rangle\langle V|+\frac{12i}{25} |V\rangle\langle H| + \frac{16}{25} |V\rangle\langle V| \right]\\ =0.2 \cdot |H\rangle\langle H| + 0.8 \cdot \left[ \frac{9}{25} |H\rangle\langle H|+\frac{12i}{25} \left(|V\rangle\langle H|-|H\rangle\langle V|\right) + \frac{16}{25} |V\rangle\langle V| \right]$$

Now; my question is: Can we somehow reduce this expression? For instance, can we use some relation to reduce $|V\rangle\langle H|-|H\rangle\langle V|$? Also, what does it mean that we have a an operator consisting of two different state-operators? Is my approach valid, or should I use another approach?

Answer 1

Besides combining the $|H\rangle\langle H|$ terms, there's nothing to reduce. If we use matrix notation such that $$a|H\rangle + b|V\rangle \leftrightarrow \pmatrix{a\\b}$$ then generic operator takes the matrix form

$$a|H\rangle\langle H| + b|H\rangle\langle V|+ c|V\rangle\langle H| + d|V\rangle\langle V| \leftrightarrow \pmatrix{a& b\\c& d}$$

Asking if we can reduce $|H\rangle\langle V | - |V\rangle\langle H|$ is asking if we can reduce the matrix $\pmatrix{0&1\\-1&0}$.