Suppose A,B,C are Hilbert space, and P,Q be density matrices defined on $A\times B$, and $B\times C$ respectively. Relation $Tr_A P=Tr_C Q$ holds(i.e., partial trace of the two matrix are the same in the common space.) Is there a density matrix $R$ on $A\times B\times C$ such that $Tr_{C}R=P$, and $Tr_{A}R=Q$?


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  • $\begingroup$ Cross-posted all over the place: physics.stackexchange.com/q/284414/24498, cstheory.stackexchange.com/q/36726/5038, math.stackexchange.com/q/1955870/14578, cs.stackexchange.com/q/64286/755. Please do not post the same question on multiple sites. Each community should have an honest shot at answering without anybody's time being wasted. $\endgroup$ – D.W. Oct 6 '16 at 17:07
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No. Consider e.g. $P$ and $Q$ to be the maximally entangled state $|\omega\rangle = (|00\rangle+|11\rangle)/\sqrt{2}$, $P=Q=|\omega\rangle\langle\omega|$. Then, the monogamy of entanglement tells us that there can be no such state $R$.

(Obviously, there are also cases where this possible, since you can always start from $R$ and derive $P$ and $Q$ from there.)


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