# Some Ulam's reconstruction problems for quantum states. (arXiv:1802.00783v1 [quant-ph])

Provided by a complete set of putative $k$-body reductions of a multipartite

quantum state, can one determine if a joint state exists? We derive necessary

conditions for this to be true. In contrast to what is known as the quantum

marginal problem, we consider a setting where the labeling of the subsystems is

unknown. The problem can be seen in analogy to Ulam's reconstruction conjecture

in graph theory. The conjecture - still unsolved - claims that every graph can

uniquely be reconstructed from the set of its vertex-deleted subgraphs. When

considering quantum states, we demonstrate that the non-existence of joint

states can, in some cases, already be inferred from a set of marginals having

the size of just more than half of the parties. We apply these methods to graph

states, where many constraints can be evaluated by knowing the number of

stabilizer elements of certain weights that appear in the reductions. This

perspective links with constraints that were derived in the context of quantum

error-correcting codes and polynomial invariants. Some of these constraints can

be interpreted as monogamy-like relations that limit the correlations arising

from quantum states.