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I'm trying to understand how to describe the quantum state after weak measurement using these two toy examples. Hopefully, these simple examples and your answer will help others who want to learn about weak measurement.

  1. Suppose photons are prepared in a superposition of horizontal and vertical polarization: $ \frac{1}{\sqrt{2}}\left(\left|H\right\rangle+\left|V\right\rangle\right) $. The weak measurement is implemented, let's say, by stacking uncoated glass plates oriented at the Brewster angle, so they would transmit the horizontal polarization state with the probability of 100% and reflect the vertical polarization state with the probability of 50%. For the reflected photons, the state collapses to $ \left|V\right\rangle $ because we know that only vertically polarized photons can be reflected (strong measurement). But what is the state of the transmitted photons? Does their state change to $ \frac{\sqrt{3}}{2}\left|H\right\rangle+\frac{1}{2}\left|V\right\rangle $ because 25% of photons have already been reflected, i.e., they have been measured as vertically polarized?

  2. Suppose photons are prepared in a maximally entangled state $ \frac{1}{\sqrt{2}}\left(\left|H_{s}H_{i}\right\rangle+\left|V_{s}V_{i}\right\rangle\right) $ instead. We conduct the same weak measurement as above on the s photon. If this photon is transmitted, what is the description of the combined system (both the transmitted s and its i pair)? And did the entanglement of the two photons weaken? If it did, how would you express it mathematically?

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