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- TL;DR Summary
- 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)
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)