Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An Analog of the 2-Wasserstein Metric in Non-commutative Probability
  under which the Fermionic Fokker-Planck Equation is Gradient Flow for the
  Entropy   Let $\Cl$ denote the Clifford algebra over $\R^n$, which is the von Neumann
algebra generated by $n$ self-adjoint operators $Q_j$, $j=1,...,n$ satisfying
the canonical anticommutation relations, $Q_iQ_j+Q_jQ_i = 2\delta_{ij}I$, and
let $\tau$ denote the normalized trace on $\Cl$. This algebra arises in quantum
mechanics as the algebra of observables generated by $n$ Fermionic degrees of
freedom. Let $\Dens$ denote the set of all positive operators $\rho\in\Cl$ such
that $\tau(\rho) =1$; these are the non-commutative analogs of probability
densities in the non-commutative probability space $(\Cl,\tau)$. The Fermionic
Fokker-Planck equation is a quantum-mechanical analog of the classical
Fokker-Planck equation with which it has much in common, such as the same
optimal hypercontractivity properties. In this paper we construct a Riemannian
metric on $\Dens$ that we show to be a natural analog of the classical
2-Wasserstein metric, and we show that, in analogy with the classical case, the
Fermionic Fokker-Planck equation is gradient flow in this metric for the
relative entropy with respect to the ground state. We derive a number of
consequences of this, such as a sharp Talagrand inequality for this metric, and
we prove a number of results pertaining to this metric. Several open problems
are raised.

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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
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The categories of this paper are: q-bio
15
The categories of this paper are: nucl-ex
34
The categories of this paper are: chem-ph
26
The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-th
30
The categories of this paper are: q-alg
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn
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The categories of this paper are: adap-org
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The categories of this paper are: plasm-ph
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The categories of this paper are: eess