Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Perfect Matchings and Quantum Physics: Progress on Krenn's Conjecture   In 2017, Krenn reported that certain problems related to the perfect
matchings and colourings of graphs emerge out of studying the constructability
of general quantum states using modern photonic technologies. He realized that
if we can prove that the \emph{weighted matching index} of a graph, a parameter
defined in terms of perfect matchings and colourings of the graph is at most 2,
that could lead to exciting insights on the potential of resources of quantum
inference. Motivated by this, he conjectured that the {weighted matching index}
of any graph is at most 2. The first result on this conjecture was by Bogdanov,
who proved that the \emph{(unweighted) matching index} of graphs
(non-isomorphic to $K_4$) is at most 2, thus classifying graphs non-isomorphic
to $K_4$ into Type 0, Type 1 and Type 2. By definition, the weighted matching
index of Type 0 graphs is 0. We give a structural characterization for Type 2
graphs, using which we settle Krenn's conjecture for Type 2 graphs. Using this
characterization, we provide a simple $O(|V||E|)$ time algorithm to find the
unweighted matching index of any graph. In view of our work, Krenn's conjecture
remains to be proved only for Type 1 graphs. We give upper bounds for the
weighted matching index in terms of connectivity parameters for such graphs.
Using these bounds, for a slightly simplified version, we settle Krenn's
conjecture for the class of graphs with vertex connectivity at most 2 and the
class of graphs with maximum degree at most 4.
  Krenn has been publicizing his conjecture in various ways since 2017. He has
even declared a reward for a resolution of his conjecture. We hope that this
article will popularize the problem among computer scientists.

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The categories of this paper are: astro-ph
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The categories of this paper are: q-fin
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ex
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: eess
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: supr-con