Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Gramian Description of the Degree 4 Generalized Elliptope   One of the most widely studied convex relaxations in combinatorial
optimization is the relaxation of the cut polytope $\mathscr C^N$ to the
elliptope $\mathscr E^N$, which corresponds to the degree 2 sum-of-squares
(SOS) relaxation of optimizing a quadratic form over the hypercube $\{\pm
1\}^N$. We study the extension of this classical idea to degree 4 SOS, which
gives an intermediate relaxation we call the degree 4 generalized elliptope
$\mathscr E_4^N$. Our main result is a necessary and sufficient condition for
the Gram matrix of a collection of vectors to belong to $\mathscr E_4^N$.
Consequences include a tight rank inequality between degree 2 and degree 4
pseudomoment matrices, and a guarantee that the only extreme points of
$\mathscr E^N$ also in $\mathscr E_4^N$ are the cut matrices; that is,
$\mathscr E^N$ and $\mathscr E_4^N$ share no "spurious" extreme point.
  For Gram matrices of equiangular tight frames, we give a simple criterion for
membership in $\mathscr{E}_4^N$. This yields new inequalities satisfied in
$\mathscr{E}_4^N$ but not $\mathscr{E}^N$ whose structure is related to the
Schl\"{a}fli graph and which cannot be obtained as linear combinations of
triangle inequalities. We also give a new proof of the restriction to degree 4
of a result of Laurent showing that $\mathscr{E}_4^N$ does not satisfy certain
cut polytope inequalities capturing parity constraints. Though limited to this
special case, our proof of the positive semidefiniteness of Laurent's
pseudomoment matrix is short and elementary.
  Our techniques also suggest that membership in $\mathscr{E}_4^N$ is closely
related to the partial transpose operation on block matrices, which has
previously played an important role in the study of quantum entanglement. To
illustrate, we present a correspondence between certain entangled bipartite
quantum states and the matrices of $\mathscr{E}_4^N\setminus\mathscr{C}^N$.

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The categories of this paper are: hep-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: math-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: solv-int
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The categories of this paper are: supr-con
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The categories of this paper are: physics
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
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The categories of this paper are: q-alg
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The categories of this paper are: patt-sol