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Consider a tripartite quantum system with the three subsystems labeled $A, B,$ and $C$. Now take two states $\rho_{AB}$ on the joint system $AB$ and $\rho_{BC}$ on the joint system $BC$. Under what conditions are these compatible with the same global state of $ABC.$

In other words when does a state $\tau_{ABC}$ exist such that $Tr_{C}(\tau_{ABC}) = \rho_{AB}$ and $Tr_A(\tau_{ABC}) = \rho_{BC}.$

A necessary condition for the existence of such a state would be $Tr_A(\rho_{AB}) = Tr_C(\rho_{BC}).$

Is this condition sufficient? If not, are sufficient conditions known?

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  • $\begingroup$ All states $\rho_{ABC}$ have $Tr_{C}(\rho_{ABC})=\rho_{AB}$ and $Tr_{A}(\rho_{ABC})=\rho_{BC}$. I think you need to edit the second line to make your question clearer. $\endgroup$
    – GaussStrife
    Commented Nov 9, 2021 at 15:04
  • $\begingroup$ The notation is fixed now. Thanks $\endgroup$
    – biryani
    Commented Nov 9, 2021 at 16:17
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    $\begingroup$ We can show the condition is insufficient using an example: Let A, B and C be 2d systems, and both \rho_{AB} and \rho_{BC} equal the maximally entangled state |\phi^+>. Then, your condition is satisfied, however, due to the monogamy of entanglement, there cannot exist a state where both \rho_{AB} and \rho_{BC} is maximally entangled. $\endgroup$
    – Mahathi Vempati
    Commented Nov 19, 2021 at 11:21

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