# Equivalence of mixed states

I have questions regarding equivalence of mixed states. Consider a single photon and its polarization as an example. If I know that some kind of process creates a single photon in an exact polarization state either horizontal or vertical, both with equal 50% probability, then I can describe its state with this density matrix: $$\rho=\frac{1}{2}\begin{pmatrix}1&0\\0&1\end{pmatrix}$$ On the other hand if a process creates a photon in a polarization state with probability density evenly distributed on a bloch sphere (this seems to be a totaly random state), then this photon state is also described by the same density matrix: $$\rho=\frac{1}{2}\begin{pmatrix}1&0\\0&1\end{pmatrix}$$ Now the questions:

1. Both cases have the same density matrix and their von Neumann entropy is the same, but it seems that in the first case I have more information about the state because I know that photon is in either of only two states. Is it correct here to assume that I have more information about the first state?
2. Why was density matrix invented the way that it makes it the same in both cases? I mean what is the physical reasoning to describe: totaly random state; and equal probability mixed state of all basis vectors by exactly the same density matrix?
• You device is claimed to make $H$ and $V$ equally. Are you sure it doesn't make $H\cos 2\theta + V\sin 2\theta$ and $V\cos 2\theta - H\sin 2\theta$ equally? How can you tell?
– JEB
Commented Mar 8, 2023 at 16:17