scholarly journals Non-commutative Calculus, Optimal Transport and Functional Inequalities in Dissipative Quantum Systems

2019 ◽  
Vol 178 (2) ◽  
pp. 319-378 ◽  
Author(s):  
Eric A. Carlen ◽  
Jan Maas
Keyword(s):  
Spectral Gap ◽  
Optimal Transport ◽  
Quantum Systems ◽  
Sobolev Inequalities ◽  
Unified Framework ◽  
Logarithmic Sobolev Inequalities ◽  
Curvature Bounds ◽  
Finite Dimensional ◽  
Dissipative Quantum Systems ◽  
Entropy Inequalities

AbstractWe study dynamical optimal transport metrics between density matrices associated to symmetric Dirichlet forms on finite-dimensional $$C^*$$ C ∗ -algebras. Our setting covers arbitrary skew-derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for Markov chains, Lindblad equations, and the Fermi Ornstein–Uhlenbeck semigroup. We develop a non-nommutative differential calculus that allows us to obtain non-commutative Ricci curvature bounds, logarithmic Sobolev inequalities, transport-entropy inequalities, and spectral gap estimates.

scholarly journals Some drawbacks of finite modified logarithmic Sobolev inequalities

2018 ◽  
Vol 123 (1) ◽  
pp. 147-159
Author(s):  
Laurent Miclo
Keyword(s):  
Invariant Measure ◽  
State Space ◽  
Differential Inequality ◽  
Spectral Gap ◽  
Umbrella Sampling ◽  
Sampling Procedure ◽  
The Other ◽  
Sobolev Inequalities ◽  
Logarithmic Sobolev Inequalities ◽  
Markov Generator

Classically, finite modified logarithmic Sobolev inequalities are used to deduce a differential inequality for the evolution of the relative entropy with respect to the invariant measure. We will check that these inequalities are ill-behaved with respect, on one hand, to the symmetrization procedure, and on the other hand, to the umbrella sampling procedure for Poincaré inequalities. A short spectral proof of the latter method is given to estimate the spectral gap of a finite reversible Markov generator $L$ in terms of the spectral gap of the restrictions of $L$ on two subsets whose union is the whole state space and whose intersection is not empty.

Logarithmic Sobolev inequalities in non-commutative algebras

2015 ◽  
Vol 18 (02) ◽  
pp. 1550011 ◽  
Author(s):  
Raffaella Carbone ◽  
Andrea Martinelli
Keyword(s):  
Markov Processes ◽  
Relative Entropy ◽  
Spectral Gap ◽  
Von Neumann Algebras ◽  
Regularity Conditions ◽  
Sobolev Inequalities ◽  
Von Neumann ◽  
Logarithmic Sobolev Inequalities ◽  
Commutative Algebras ◽  
Positive Identity

We study the relations between (tight) logarithmic Sobolev inequalities, entropy decay and spectral gap inequalities for Markov evolutions on von Neumann algebras. We prove that log-Sobolev inequalities (in the non-commutative form defined by Olkiewicz and Zegarlinski in Ref. 25) imply spectral gap inequalities, with optimal relation between the constants. Furthermore, we show that a uniform exponential decay of a proper relative entropy is equivalent to a modified version of log-Sobolev inequalities. The relations among the mentioned inequalities are investigated and often depend on some regularity conditions, which are also discussed. With regard to this aspect, we provide an example of a positive identity-preserving semigroup not verifying the usually requested regularity conditions (which are always fulfilled for reversible classical Markov processes).

scholarly journals LineageOT is a unified framework for lineage tracing and trajectory inference

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Aden Forrow ◽  
Geoffrey Schiebinger
Keyword(s):  
Graphical Models ◽  
Optimal Transport ◽  
Developmental Trajectories ◽  
Lineage Tracing ◽  
Developmental Pathways ◽  
State Transitions ◽  
Unified Framework ◽  
Measured Time ◽  
Mathematical Problems ◽  
Time Courses

AbstractUnderstanding the genetic and epigenetic programs that control differentiation during development is a fundamental challenge, with broad impacts across biology and medicine. Measurement technologies like single-cell RNA-sequencing and CRISPR-based lineage tracing have opened new windows on these processes, through computational trajectory inference and lineage reconstruction. While these two mathematical problems are deeply related, methods for trajectory inference are not typically designed to leverage information from lineage tracing and vice versa. Here, we present LineageOT, a unified framework for lineage tracing and trajectory inference. Specifically, we leverage mathematical tools from graphical models and optimal transport to reconstruct developmental trajectories from time courses with snapshots of both cell states and lineages. We find that lineage data helps disentangle complex state transitions with increased accuracy using fewer measured time points. Moreover, integrating lineage tracing with trajectory inference in this way could enable accurate reconstruction of developmental pathways that are impossible to recover with state-based methods alone.

BEYOND STRONG SUBADDITIVITY? IMPROVED BOUNDS ON THE CONTRACTION OF GENERALIZED RELATIVE ENTROPY

1994 ◽  
Vol 06 (05a) ◽  
pp. 1147-1161 ◽  
Author(s):  
MARY BETH RUSKAI
Keyword(s):  
Relative Entropy ◽  
Discrete Case ◽  
Sobolev Inequalities ◽  
Completely Positive ◽  
Logarithmic Sobolev Inequalities ◽  
Strong Subadditivity

New bounds are given on the contraction of certain generalized forms of the relative entropy of two positive semi-definite operators under completely positive mappings. In addition, several conjectures are presented, one of which would give a strengthening of strong subadditivity. As an application of these bounds in the classical discrete case, a new proof of 2-point logarithmic Sobolev inequalities is presented in an Appendix.

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