# How can we derive the form of POVMs on a subspace from a projective measurement on a larger space?

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.

• 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?
– glS
Apr 21, 2023 at 19:36
• 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 ? Apr 23, 2023 at 15:59
• 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
– glS
Apr 23, 2023 at 20:10