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Canonical divergence for measuring classical and quantum complexity

Domenico Felice, Stefano Mancini, Nihat Ay

Published Apr 24, 2019• 7 citations
Publisher

Abstract

A new canonical divergence is put forward for generalizing an information-geometric measure of complexity for both, classical and quantum systems. On the simplex of probability measures it is proved that the new divergence coincides with the Kullback-Leibler divergence, which is used to quantify how much a probability measure deviates from the non-interacting states that are modeled by exponential families of probabilities. On the space of positive density operators, we prove that the same divergence reduces to the quantum relative entropy, which quantifies many-party correlations of a quantum state from a Gibbs family.

DOI
10.15480/882.3736

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A new canonical divergence is put forward for generalizing an information-geometric measure of complexity for both, classical and quantum systems. On the simplex of probability measures it is proved that the new divergence coincides with the Kullback-Leibler divergence, which is used to quantify how much a probability measure deviates from the non-interacting states that are modeled by exponential families of probabilities. On the space of positive density operators, we prove that the same divergence reduces to the quantum relative entropy, which quantifies many-party correlations of a quantum state from a Gibbs family.
A new canonical divergence is put forward for generalizing an information-geometric measure of complexity for both classical and quantum systems. On the simplex of probability measures, it is proved that the new divergence coincides with the Kullback-Leibler divergence, which is used to quantify how much a probability measure deviates from the non-interacting states that are modeled by exponential families of probabilities. On the space of positive density operators, we prove that the same divergence reduces to the quantum relative entropy, which quantifies many-party correlations of a quantum state from a Gibbs family.
We give a new characterization of relative entropy, also known as the Kullback-Leibler divergence.We use a number of interesting categories related to probability theory.In particular, we consider a category FinStat where an object is a finite set equipped with a probability distribution, while a morphism is a measure-preserving function f : X → Y together with a stochastic right inverse s : Y → X.The function f can be thought of as a measurement process, while s provides a hypothesis about the state of the measured system given the result of a measurement.Given this data we can define the entropy of the probability distribution on X relative to the 'prior' given by pushing the probability distribution on Y forwards along s.We say that s is 'optimal' if these distributions agree.We show that any convex linear, lower semicontinuous functor from FinStat to the additive monoid [0, ∞] which vanishes when s is optimal must be a scalar multiple of this relative entropy.Our proof is independent of all earlier characterizations, but inspired by the work of Petz.We thank Ryszard Kostecki and Rob Spekkens for discussions
The MAXENT (maximum entropy) principle gives a natural correspondance between (classical and quantum) statistical mechanics and information theory. Its general formulation leverages on log-concavity (or Klein's inequality) and the celebrated information inequality, which is equivalent to Gibbs' inequality. The resulting convex optimization allows one to relate entropy to the log-canonical partition function by Legendre-Fenchel duality. This can be further generalized to dual variational characterization of relative entropy or Kullback-Leibler divergence, relating Gibbs variationnal principle to Donsker-Vardhan variational representations.From an information theoretical point of view, the Gibbs principle and its associated evidence lower bound serves as the basis of modern machine learning algorithms, such as the expectation-maximization (EM) algorithm and its restriction to parameters output by a neural network. Interestingly, a large class of alternating optimization algorithms including EM and Blahut-Arimoto for computing channel capacity and source rate-distortion functions can be cast in a generic information geometric alternating projections on convex sets of measures. On the other hand, Donsker-Vardhan representations, when restricted to the output of a neural network, allows one to efficiently estimate information measures, and when applied to parametric estimators with quadratic risk, provide HCR-like and CRB-like statistical lower bounds. An alternative derivation on such lower bounds in the Bayesian setting makes use of the Weyl-Heisenberg uncertainty principle, which opens up new perspectives.In this keynote, I will select and review several of the aforementioned topics, and discuss the interactions between classical and quantum interpretations.
In this PhD thesis, we take an axiomatic approach to studying multivariate divergences, which are measures of the information content of a tuple of classical or quantum states. These divergences are studied in the context of large-sample and catalytic convertibility of tuples of states. This type of convertibility is characterized by the ordering of an axiomatically defined family of divergences. In this thesis, we identify this family of divergences explicitly for two scenarios, one classical and one quantum. In the classical setting, we show that the relevant divergences are certain generalizations of the bivariate Rényi divergences. Unlike previous studies, we do not require the distributions to have the same support, yielding interesting new divergences. In the quantum setting, we consider pairs of classical-quantum states with pure component states, and find that majorization is characterized by all ααα-zzz relative entropies for α<1α<1α<1. This demonstrates the first operational interpretation of these relative entropies.
We study many-party correlations quantified in terms of the Umegaki relative entropy (divergence) from a Gibbs family known as a hierarchical model. We derive these quantities from the maximum-entropy principle which was used earlier to define the closely related irreducible correlation. We point out the differences between quantum states and probability vectors which exist in hierarchical models, in the divergence from a hierarchical model and in local maximizers of this divergence. The differences are, respectively, missing factorization, discontinuity and reduction of uncertainty. We discuss global maximizers of the mutual information of separable qubit states.
In the past decades, all of the efforts at quantifying systems complexity with a general tool has usually relied on using Shannon's classical information framework to address the disorder of the system through the Boltzmann–Gibbs–Shannon entropy, or one of its extensions. However, in recent years, there were some attempts to tackle the quantification of algorithmic complexities in quantum systems based on the Kolmogorov algorithmic complexity, obtaining some discrepant results against the classical approach. Therefore, an approach to the complexity measure is proposed here, using the quantum information formalism, taking advantage of the generality of the classical-based complexities, and being capable of expressing these systems' complexity on other framework than its algorithmic counterparts. To do so, the Shiner–Davison–Landsberg (SDL) complexity framework is considered jointly with linear entropy for the density operators representing the analyzed systems formalism along with the tangle for the entanglement measure. The proposed measure is then applied in a family of maximally entangled mixed state.