8,299 research outputs found
Some Ulam's reconstruction problems for quantum states
Provided a complete set of putative -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 on
at least three vertices 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. Lastly, we provide an answer to
Ulam's reconstruction problem for generic quantum states.Comment: 22 pages, 3 figures, v2: significantly revised final versio
Explicit reconstruction in quantum cohomology and K-theory
Cohomological genus-0 Gromov-Witten invariants of a given target space can be
encoded by the "descendant potential," a generating function defined on the
space of power series in one variable with coefficients in the cohomology space
of the target. Replacing the coefficient space with the subspace
multiplicatively generated by degree-2 classes, we explicitly reconstruct the
graph of the differential of the restricted generating function from one point
on it. Using the Quantum Hirzebruch--Riemann--Roch Theorem from our joint work
with Valentin Tonita, we derive a similar reconstruction formula in genus-0
quantum K-theory. The results amplify the role of the divisor equations, and
the structures of -modules and -modules in quantum cohomology and
quantum K-theory with respect to Novikov's variables.Comment: 13 page
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