Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $4\times4$ unextendible product basis and genuinely entangled space   We show that there are six inequivalent $4\times4$ unextendible product bases
(UPBs) of size eight, when we consider only 4-qubit product vectors. We apply
our results to construct Positive-Partial-Transpose entangled states of rank
nine. They are at the same 4-qubit, $2\times2\times4$ and $4\times4$ states,
and their ranges have product vectors. One of the six UPBs turns out to be
orthogonal to an almost genuinely entangled space, in the sense that the latter
does not contain $4\times4$ product vector in any bipartition of 4-qubit
systems. We also show that the multipartite UPB orthogonal to a genuinely
entangled space exists if and only if the $n\times n\times n$ UPB orthogonal to
a genuinely entangled space exists for some $n$. These results help understand
an open problem in [Phys. Rev. A 98, 012313, 2018].

18
The categories of this paper are: econ
37
The categories of this paper are: supr-con
14
The categories of this paper are: math-ph
33
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: atom-ph
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The categories of this paper are: nucl-ex
2
The categories of this paper are: math
1
The categories of this paper are: cs
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The categories of this paper are: q-fin
5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
12
The categories of this paper are: nucl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: cond-mat
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The categories of this paper are: funct-an