Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Triangle inequalities in coherence measures and entanglement concurrence   We provide detailed proofs of triangle inequalities in coherence measures and
entanglement concurrence. If a rank-$2$ state $\varrho$ can be expressed as a
convex combination of two pure states, i.e.,
$\varrho=p_{1}|\psi_{1}\rangle\langle\psi_{1}|+p_{2}|\psi_{2}\rangle\langle\psi_{2}|$,
a triangle inequality can be established as
$\big{|}E(|\Psi_{1}\rangle)-E(|\Psi_{2}\rangle)\big{|}\leq E(\varrho)\leq
E(|\Psi_{1}\rangle)+E(|\Psi_{2}\rangle)$, where
$|\Psi_{1}\rangle=\sqrt{p_{1}}|\psi_{1}\rangle$ and
$|\Psi_{2}\rangle=\sqrt{p_{2}}|\psi_{2}\rangle$, $E$ can be considered either
coherence measures or entanglement concurrence. This inequality displays
mathematical beauty for its similarity to the triangle inequality in plane
geometry. An illustrative example is given after the proof.

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The categories of this paper are: physics
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The categories of this paper are: hep-ex
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The categories of this paper are: gr-qc
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The categories of this paper are: adap-org
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: patt-sol
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The categories of this paper are: ao-sci
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The categories of this paper are: alg-geom
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The categories of this paper are: nlin
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The categories of this paper are: nucl-ex
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The categories of this paper are: stat
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The categories of this paper are: math-ph
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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: hep-lat