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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?
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    $\begingroup$ 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? $\endgroup$
    – JEB
    Commented Mar 8, 2023 at 16:17

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There is no experiment which can distinguish the two cases. The "more information" you have is solely about the past of the system. You have no more information in the first as compared to the second case when it comes to predict anything about the future (that is, outcomes of experiments).

That's precisely what the density matrix is meant to do: It can be used to predict the future (that is, outcomes of any experiment), but has no information about the past (unless it is relevant for the future).

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