A Link between Kraus operators and PVMs

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Question about how to find kraus operators from a pvm that generates a given POVM:
can you always find two subsystems and an observable such that tracing out the result of the measurement over one of those subsystems you get the action of the Kraus operators that generates a given POVM?
I'm sorry if this discussion may be dumb.

Is this a way to find a family of kraus operators that generates a given POVM from a PVM?

Assume that you have a POVM given by the family of operators ##\{E_i\}_i##. Then it is possible to find a PVM that generates the given POVM by this procedure:
It is possible to find two subsystems S1 and S2 with spaces respectively ##\mathcal{H}_1## and ##\mathcal{H}_2## and an observable ##M## with spectral projectors ##\{P_k\}_k## and eigenvalues ##\lambda_i ##(let ##\Omega:=\bigcup_i \lambda_i##, let ##S## be the state space). Suppose for example that S2 is ancillary. Suppose moreover that the joint state of the system is ##\rho## and the reduced state of S1 is ##\rho_1=Tr_2(\rho)##.

##\forall i## the Kraus operators will then be defined by this action:
$$A_i: \frac{A_i\rho_1A_i^*}{Tr(A_i\rho_1A_i^*)}=Tr_2(\frac{\sum_{j\in J(i)} P_j \rho P_j}{Tr(\sum_{j\in J(i)} P_j \rho)}) $$

This must hold ##\forall \rho \in S,\rho_1=Tr_2(\rho)##.
Where ##J(i): J(i)\cap J(j)=\emptyset## ##\forall i\neq j##, ##\bigcup_i J(i)=\Omega## is a partition. (I ask for the existence of this partition because the cardinality of the family ##\{E_i\}_i## and of ##\{P_k\}_k## may be different so that we need to cluster some outcome together. I don't know if this is necessary)
 
I read Hanbury Brown and Twiss's experiment is using one beam but split into two to test their correlation. It said the traditional correlation test were using two beams........ This confused me, sorry. All the correlation tests I learnt such as Stern-Gerlash are using one beam? (Sorry if I am wrong) I was also told traditional interferometers are concerning about amplitude but Hanbury Brown and Twiss were concerning about intensity? Isn't the square of amplitude is the intensity? Please...
I am not sure if this belongs in the biology section, but it appears more of a quantum physics question. Mike Wiest, Associate Professor of Neuroscience at Wellesley College in the US. In 2024 he published the results of an experiment on anaesthesia which purported to point to a role of quantum processes in consciousness; here is a popular exposition: https://neurosciencenews.com/quantum-process-consciousness-27624/ As my expertise in neuroscience doesn't reach up to an ant's ear...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA

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