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We consider the problem of characterizing states of a multipartite quantumsystem from restricted, quasi-local information, with emphasis on uniquelydetermined pure states. By leveraging tools from dissipative quantum controltheory, we show how the search for states consistent with an assigned list ofreduced density matrices may be restricted to a proper subspace, which isdetermined solely by their supports. The existence of a quasi-local observablewhich attains its unique minimum over such a subspace further provides asufficient criterion for a pure state to be uniquely determined by its reducedstates. While the condition that a pure state is uniquely determined isnecessary for it to arise as a non-degenerate ground state of a quasi-localHamiltonian, we prove the opposite implication to be false in general, byexhibiting an explicit analytic counterexample. We show how the problem ofdetermining whether a quasi-local parent Hamiltonian admitting a given purestate as its unique ground state is dual, in the sense of semidefiniteprogramming, to the one of determining whether such a state is uniquelydetermined by the quasi-local information. Failure of this dual program toattain its optimal value is what prevents these two classes of states tocoincide.

Uniquely determined pure quantum states need not be unique ground states of quasi-local Hamiltonians