Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Matrix Monge-Kantorovich Problem   The classical Monge-Kantorovich (MK) problem as originally posed is concerned
with how best to move a pile of soil or rubble to an excavation or fill with
the least amount of work relative to some cost function. When the cost is given
by the square of the Euclidean distance, one can define a metric on densities
called the "Wasserstein distance." In this note, we formulate a natural matrix
counterpart of the MK problem for positive definite density matrices. We prove
a number of results about this metric including showing that it can be
formulated as a convex optimization problem, strong duality, an analogue of the
Poincare-Wirtinger inequality, and a Lax-Hopf-Oleinik type result.

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The categories of this paper are: hep-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: bayes-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: hep-lat
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The categories of this paper are: mtrl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-th
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The categories of this paper are: supr-con
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The categories of this paper are: solv-int
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The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph
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The categories of this paper are: dg-ga