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.

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