5 research outputs found
Approximate tensorization of the relative entropy for noncommuting conditional expectations
In this paper, we derive a new generalisation of the strong subadditivity of
the entropy to the setting of general conditional expectations onto arbitrary
finite-dimensional von Neumann algebras. The latter inequality, which we call
approximate tensorization of the relative entropy, can be expressed as a lower
bound for the sum of relative entropies between a given density and its
respective projections onto two intersecting von Neumann algebras in terms of
the relative entropy between the same density and its projection onto an
algebra in the intersection, up to multiplicative and additive constants. In
particular, our inequality reduces to the so-called quasi-factorization of the
entropy for commuting algebras, which is a key step in modern proofs of the
logarithmic Sobolev inequality for classical lattice spin systems. We also
provide estimates on the constants in terms of conditions of clustering of
correlations in the setting of quantum lattice spin systems. Along the way, we
show the equivalence between conditional expectations arising from Petz
recovery maps and those of general Davies semigroups.Comment: 31 page
Entropy Uncertainty Relations and Strong Sub-additivity of Quantum Channels
We prove an entropic uncertainty relation for two quantum channels, extending
the work of Frank and Lieb for quantum measurements. This is obtained via a
generalized strong super-additivity (SSA) of quantum entropy. Motivated by
Petz's algebraic SSA inequality, we also obtain a generalized SSA for quantum
relative entropy. As a special case, it gives an improved data processing
inequality.Comment: 33 pages. Comments are welcome
Quantum Brascamp-Lieb dualities
BrascampâLieb inequalities are entropy inequalities which have a dual formulation as generalized Young inequalities. In this work, we introduce a fully quantum version of this duality, relating quantum relative entropy inequalities to matrix exponential inequalities of Young type. We demonstrate this novel duality by means of examples from quantum information theoryâincluding entropic uncertainty relations, strong data-processing inequalities, super-additivity inequalities, and many more. As an application we find novel uncertainty relations for Gaussian quantum operations that can be interpreted as quantum duals of the well-known family of âgeometricâ BrascampâLieb inequalities
On the nature and decay of quantum relative entropy
Historically at the core of thermodynamics and information theory, entropy's use in quantum information extends to diverse topics including high-energy physics and operator algebras. Entropy can gauge the extent to which a quantum system departs from classicality, including by measuring entanglement and coherence, and in the form of entropic uncertainty relations between incompatible measurements. The theme of this dissertation is the quantum nature of entropy, and how exposure to a noisy environment limits and degrades non-classical features.
An especially useful and general form of entropy is the quantum relative entropy, of which special cases include the von Neumann and Shannon entropies, coherent and mutual information, and a broad range of resource-theoretic measures. We use mathematical results on relative entropy to connect and unify features that distinguish quantum from classical information. We present generalizations of the strong subadditivity inequality and uncertainty-like entropy inequalities to subalgebras of operators on quantum systems for which usual independence assumptions fail. We construct new measures of non-classicality that simultaneously quantify entanglement and uncertainty, leading to a new resource theory of operations under which these forms of non-classicalty become interchangeable. Physically, our results deepen our understanding of how quantum entanglement relates to quantum uncertainty.
We show how properties of entanglement limit the advantages of quantum superadditivity for information transmission through channels with high but detectable loss. Our method, based on the monogamy and faithfulness of the squashed entanglement, suggests a broader paradigm for bounding non-classical effects in lossy processes. We also propose an experiment to demonstrate superadditivity.
Finally, we estimate decay rates in the form of modified logarithmic Sobolev inequalities for a variety of quantum channels, and in many cases we obtain the stronger, tensor-stable form known as a complete logarithmic Sobolev inequality. We compare these with our earlier results that bound relative entropy of the outputs of a particular class of quantum channels