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Suppose we have the Hilbert space $\mathcal{H}_{0}$ describing the states of the system $s$, and the Hilbert space $\mathcal{H}_{e}$ describing the states of the environment. I have seen that [1][2], if the system is initially in the product state $|\psi_{0}\rangle|\chi_{e}\rangle$ we can write the time evolution as $$\hat{U}|\psi_{0}\rangle|\chi_{\mathrm{e}}\rangle=\sum_{k}\left(\hat{E}_{k}|\psi_{0}\rangle\right)\otimes|\chi_{k}\rangle$$ where $$\sum_{k}\hat{E}^{\dagger}_{k}\hat{E}_{k}=\mathbb{1}$$ Taking the partial trace over the environment, we see that the density operator of state $s$ evolves as $$|\psi_{0}\rangle\langle\psi_{0}|=\sum_{k}\hat{E}_{k}|\psi_{0}\rangle\langle\psi_{0}|\hat{E}^{\dagger}_{k}$$ So, from unitary evolution of a system, the Kraus representation of quantum channels on the subsystem can be derived. According to Neumark's theorem, any POVM measurement in a subsystem can be achieved by a POVM measurement in a larger system, a result which is (to my understanding) analogous to the relationship between channels in a subsystem and unitary operations in a larger system. My question is then, is it possible to similarly derive the form of POVMs by considering a projective measurement in the larger system?

I am asking this, because I currently don't fully understand the point of POVMs, and I feel like seeing this relationship may help to enlighten me a bit. Thanks for any help you may be able to provide.

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  • $\begingroup$ I don't understand the question. Yes, Neumark 's theorem tells you that any POVM can be represented as a projective measurement in an enlarged space. Is that what you are asking? $\endgroup$
    – glS
    Apr 21, 2023 at 19:36
  • $\begingroup$ What I’m asking is, if we start with a general projective measurement on a space can we derive the form of resulting POVMs on a sub space ? $\endgroup$ Apr 23, 2023 at 15:59
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    $\begingroup$ depends on what you mean. What is the "resulting POVM on a sub space"? What "subspace" are you referring to? Simply ignoring some outcomes doesn't result in a POVM, if that's what you're asking $\endgroup$
    – glS
    Apr 23, 2023 at 20:10

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