Abstract In quantum information theory, the quantum relative Rényi functional is a generalisation of quantum relative entropy that quantifies the difference between two quantum states. We study a new type of quantum relative Rényi functional, motivated by the concepts of the cumulant-generating function and quantum surprisal difference, and propose an information-theoretic interpretation of the Rényi parameter. In addition, we demonstrate that this functional can be naturally expressed in a path-integral-like formulation, providing a distinct perspective compared to existing approaches. To support the viability of our proposal, we conduct a preliminary numerical study of its key properties for qubits and qutrits. These properties include the monotonicity in the Rényi parameter (α) and the quantum data-processing inequality (QDPI).

Information Inequalities prove to be a powerful tool to quantify uncertainties and data dependencies. They provide useful insights into the relations between information-theoretic probability distributions. The entropy and divergence measure makes the data processing inequalities more effective for communication and informatic concepts. This study provides the constructive perspective of Ostrowski type inequality for functions of bounded variation. Explored approximation of new f divergence measure by applying principles of numerical integration theory. Discussed function f and its first derivatives exhibit bounded variation characteristics. Furthermore, by applying the bounded variation properties of the function f and its derivatives, accurate and efficient approximations can be achieved. Additionally, we have found applications of the obtained information inequalities related to the Relative Arithmetic-Geometric Divergence (AGD) that quantify the difference between probability distributions. Some identified means are also used to specify the results.

Quantum information quantities, such as mutual information and entropies, are essential for characterizing quantum systems and protocols in quantum information science. In this contribution, we identify types of information measures based on generalized divergences and prove their invariance under local isometric or unitary transformations. Leveraging the reversal channel for local isometries together with the data-processing inequality, we establish invariance for information quantities used in both asymptotic and one-shot regimes without relying on the specific functional form of the underlying divergence. These invariances can be applied to improve the computation of such information quantities or optimize protocols and their output states, whose performance is determined by some invariant measure. Our results improve the capability to characterize and compute many operationally relevant information measures with application across the field of quantum information processing.

In this paper, inspired by the idea due to Takahashi and Fujiwara, for density matrices ρ , σ , we define the rescaled ♮ α -Rényi divergence D ˆ α ( ρ | | σ ) := 1 α ( α − 1 ) log ⁡ Tr [ σ ♮ α ρ ] of all real orders α ∈ R of ρ from σ, where σ ♮ α ρ := σ 1 / 2 ( σ − 1 / 2 ρ σ − 1 / 2 ) α σ 1 / 2 for α ∈ R , and study its properties such as the positivity, the data processing inequality, continuity on R and jointly convexity. Moreover, we determine the order relation among D ˆ α ( ρ | | σ ) , the rescaled sandwiched α-Rényi divergence D ~ α ( ρ | | σ ) and the rescaled α-Rényi divergence D α ( ρ | | σ ) .

The ability to quantify the directional flow of information is vital to understanding natural systems and designing engineered information-processing systems. A widely used measure to quantify this information flow is the transfer entropy. However, until now, this quantity could only be obtained in dynamical models using approximations that are typically uncontrolled. Here we introduce a computational algorithm called "transfer entropy-path weight sampling" (TE-PWS), which makes it possible, for the first time, to quantify the transfer entropy and its variants exactly for any stochastic model, including those with multiple hidden variables, nonlinearity, transient conditions, and feedback. By leveraging techniques from polymer and path sampling, TE-PWS efficiently computes the transfer entropy as a MonteCarlo average over signal trajectory space. We use our exact technique to demonstrate that commonly used approximate methods to compute transfer entropies incur large systematic errors and high computational costs. As an application, we use TE-PWS in linear and nonlinear systems to reveal how transfer entropy can overcome naive applications of the data processing inequality in the presence of feedback.

Carrying the insights of conditional probability to the quantum realm is notoriously difficult due to the noncommutative nature of quantum observables. Nevertheless, conditional expectations on von Neumann algebras have played a significant role in the development of quantum information theory, and especially the study of quantum error correction. In quantum gravity, it has been suggested that conditional expectations may be used to implement the holographic map algebraically, with quantum error correction underlying the emergence of spacetime through the generalized entropy formula. However, the requirements for exact error correction are almost certainly too strong for realistic theories of quantum gravity. In this paper, we present a relaxed notion of quantum conditional expectation which implements approximate error correction. We introduce a generalization of Connes’s spatial theory adapted to completely positive maps, and derive a chain rule allowing for the noncommutative factorization of relative modular operators into a marginal and conditional part, constituting a quantum Bayes’s law. This allows for an exact quantification of the information gap occurring in the data processing inequality for arbitrary quantum channels. When applied to algebraic inclusions, this also provides an approach to factorizing the entropy of states into a sum of terms which, in the gravitational context, may be interpreted as a generalized entropy. We illustrate that the emergent area operator is fully noncommutative rather than central, except under the conditions of exact error correction. We provide some comments on how this result may be used to construct a fully algebraic quantum extremal surface prescription and to probe the quantum nature of black holes.

The Strong Data Processing Inequality Under the Heat Flow

The study of information revivals, witnessing the violation of certain data-processing inequalities, has provided an important paradigm in the study of non-Markovian quantum stochastic processes. Although often used interchangeably, we argue here that the notions of “revivals” and “backflows,” i.e., flows of information from the environment back into the system, are distinct: an information revival can occur without any backflow ever taking place. In this paper, we examine in detail the phenomenon of noncausal revivals and relate them to the theory of short Markov chains and squashed non-Markovianity. We also provide an operational condition, in terms of system-only degrees of freedom, to witness the presence of genuine backflow that cannot be explained by noncausal revivals. As a byproduct, we demonstrate that focusing on processes with genuine backflows, while excluding those with only noncausal revivals, resolves the issue of nonconvexity of Markovianity, thus enabling the construction of a convex resource theory of genuine quantum non-Markovianity.

Abstract The bivariate classical fidelity is a widely used measure of the similarity of two probability distributions. There exist a few extensions of the notion of the bivariate classical fidelity to more than two probability distributions; herein we call these extensions multivariate classical fidelities, with some examples being the Matusita multivariate fidelity and the average pairwise fidelity. Hitherto, quantum generalizations of multivariate classical fidelities have not been systematically explored, even though there are several well known generalizations of the bivariate classical fidelity to quantum states, such as the Uhlmann and Holevo fidelities. The main contribution of our paper is to introduce a number of multivariate quantum fidelities and show that they satisfy several desirable properties that are natural extensions of those of the Uhlmann and Holevo fidelities. We propose several variants that reduce to the average pairwise fidelity for commuting states, including the average pairwise z-fidelities, the multivariate semi-definite programming (SDP) fidelity, and a multivariate fidelity inspired by an existing secrecy measure. The second one is obtained by extending the SDP formulation of the Uhlmann fidelity to more than two states. All of these variants satisfy the following properties: (i) reduction to multivariate classical fidelities for commuting states, (ii) the data-processing inequality, (iii) invariance under permutations of the states, (iv) its values are in the interval [ 0 , 1 ] ; they are faithful, that is, their values are equal to one if and only if all the states are equal, and they satisfy orthogonality, that is their values are equal to zero if and only if the states are mutually orthogonal to each other, (v) direct-sum property, (vi) joint concavity, and (vii) uniform continuity bounds under certain conditions. Furthermore, we establish inequalities relating these different variants, indeed clarifying that all these definitions coincide with the average pairwise fidelity for commuting states. We also introduce another multivariate fidelity called multivariate log-Euclidean fidelity, which is a quantum generalization of the Matusita multivariate fidelity. We also show that it satisfies most of the desirable properties listed above, it is a function of a multivariate log-Euclidean divergence, and it has an operational interpretation in terms of quantum hypothesis testing with an arbitrarily varying null hypothesis. Lastly, we propose multivariate generalizations of Matsumoto’s geometric fidelity and establish several properties of them.

It is a challenging task to decipher the mechanisms of a complex system from observational data, especially in biology, where systems are sophisticated, measurements coarse, and multi-modality common. The typical approaches of inferring a network of relationships between a system's components struggle with the quality and feasibility of estimation, as well as with the interpretability of the results they yield. Said issues can be avoided, however, when dealing with a simpler problem of tracking only the influence paths, defined as circuits relying on the information of an experimental perturbation as it spreads through the system. Such an approach can be formalized with information theory and leads to a relatively streamlined, interpretable output, in contrast to the incomprehensibly dense 'haystack' networks produced by typical tools. Following this idea, the paper introduces Vistla, a novel method built around tri-variate mutual information and data processing inequality, combined with a higher-order generalization of the widest path problem. Vistla can be used standalone, in a machine learning pipeline to aid interpretability, or as a tool for mediation analysis; the paper demonstrates its efficiency both in synthetic and real-world problems. The R package implementing the method is available at https://gitlab.com/mbq/vistla, as well as on CRAN.

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Process tensors are quantum combs describing the evolution of open quantum systems through multiple steps of quantum dynamics. While there is more than one way to measure how different two processes are, special care must be taken to ensure quantifiers obey physically desirable conditions such as data-processing inequalities. Here, we analyze two classes of distinguishability measures commonly used in general applications of quantum combs. We show that the first class, called Choi divergences, does not satisfy an important data-processing inequality, while the second one, which we call generalized divergences, does. We also extend to quantum combs some other relevant results of generalized divergences of quantum channels. Finally, given the properties we proved, we argue that generalized divergences may be more adequate than Choi divergences for distinguishing quantum combs in most of their applications. Particularly, this is crucial for defining monotones for resource theories whose states have a comb structure, such as resource theories of quantum processes and resource theories of quantum strategies. Published by the American Physical Society 2024

Directional chemosensing is ubiquitous in cell biology, but some cells such as mating yeast paradoxically degrade the signal they aim to detect. While the data processing inequality suggests that such signal modification cannot increase the sensory information, we show using a reaction-diffusion model and an exactly solvable discrete-state reduction that it can. We identify a non-Markovian step in the information chain allowing the system to evade the data processing inequality, reflecting the nonlocal nature of diffusion. Our results apply to any sensory system in which degradation couples to diffusion. Experimental data suggest that mating yeast operate in the beneficial regime where degradation improves sensing.

Cells sense environmental signals and transmit information intracellularly through changes in the abundance of molecular components. Such molecular abundances can be measured in single cells and exhibit significant heterogeneity in clonal populations even in identical environments. Experimentally observed joint probability distributions can then be used to quantify the covariability and mutual information between molecular abundances along signaling cascades. However, because stationary state abundances along stochastic biochemical reaction cascades are not conditionally independent, their mutual information is not constrained by the data-processing inequality. Here, we report the conditions under which the mutual information between stationary state abundances increases along a cascade of biochemical reactions. This nonmonotonic behavior can be intuitively understood in terms of noise propagation and time-averaging stochastic fluctuations that are short-lived compared to an extrinsic signal. Our results reemphasize that mutual information measurements of stationary state distributions of cellular components may be of limited utility for characterizing cellular signaling processes because they do not measure information transfer.

The causal structure of a system imposes constraints on the joint probability distribution of variables that can be generated by the system. Archetypal constraints consist of conditional independencies between variables. However, particularly in the presence of hidden variables, many causal structures are compatible with the same set of independencies inferred from the marginal distributions of observed variables. Additional constraints allow further testing for the compatibility of data with specific causal structures. An existing family of causally informative inequalities compares the information about a set of target variables contained in a collection of variables, with a sum of the information contained in different groups defined as subsets of that collection. While procedures to identify the form of these groups-decomposition inequalities have been previously derived, we substantially enlarge the applicability of the framework. We derive groups-decomposition inequalities subject to weaker independence conditions, with weaker requirements in the configuration of the groups, and additionally allowing for conditioning sets. Furthermore, we show how constraints with higher inferential power may be derived with collections that include hidden variables, and then converted into testable constraints using data processing inequalities. For this purpose, we apply the standard data processing inequality of conditional mutual information and derive an analogous property for a measure of conditional unique information recently introduced to separate redundant, synergistic, and unique contributions to the information that a set of variables has about a target.

Information spreads in time. For example, correlations dissipate when the correlated system locally couples to a third party, such as the environment. This simple but important fact forms the known quantum data-processing inequality. Here we theoretically uncover the quantum fluctuation theorem behind the quantum informational inequality. The fluctuation theorem quantitatively predicts the statistics of the underlying stochastic quantum process. To fully capture the quantum nature, the fluctuation theorem established here is extended to the quasiprobability regime. We also experimentally apply an interference-based method to measure the amplitudes composing the quasiprobability and verify our established fluctuation theorem by the IBM quantum computer.

FAGnet: Family-aware-based android malware analysis using graph neural network

Mutual information (MI) quantifies the statistical dependency between a pair of random variables and plays a central role in signal processing and data analysis. Recent advances in machine learning have enabled the estimation of MI from a dataset using the expressive power of neural networks. In this study, we conducted a comparative experimental analysis of several existing neural estimators of MI between random vectors that model power spectrum features. We explored alternative models of power spectrum features by leveraging information-theoretic data processing inequality and bijective transformations. Empirical results demonstrated that each neural estimator of MI covered in this study has its limitations. In practical applications, we recommend the collective use of existing neural estimators in a complementary manner for the problem of estimating MI between power spectrum features.

We study hypothesis testing under communication constraints, where each sample is quantized before being revealed to a statistician. Without communication constraints, it is well known that the sample complexity of simple binary hypothesis testing is characterized by the Hellinger distance between the distributions. We show that the sample complexity of simple binary hypothesis testing under communication constraints is at most a logarithmic factor larger than in the unconstrained setting and this bound is tight. We develop a polynomial-time algorithm that achieves the aforementioned sample complexity. Our framework extends to robust hypothesis testing, where the distributions are corrupted in total variation distance. Our proofs rely on a new reverse data processing inequality and a reverse Markov inequality, which may be of independent interest. For simple M -ary hypothesis testing, the sample complexity in the absence of communication constraints has a logarithmic dependence on M . We show that communication constraints can cause an exponential blow-up, leading to Ω(M ) sample complexity even for adaptive algorithms.

Given M ≥ 2 distributions defined on a general measurable space, we introduce a nonparametric (kernel) measure of multi-sample dissimilarity (KMD)—a parameter that quantifies the difference between the M distributions. The population KMD, which takes values between 0 and 1, is 0 if and only if all the M distributions are the same, and 1 if and only if all the distributions are mutually singular. Moreover, KMD possesses many properties commonly associated with f-divergences such as the data processing inequality and invariance under bijective transformations. The sample estimate of KMD, based on independent observations from the M distributions, can be computed in near linear time (up to logarithmic factors) using k-nearest neighbor graphs (for k ≥ 1 fixed). We develop an easily implementable test for the equality of M distributions based on the sample KMD that is consistent against all alternatives where at least two distributions are not equal. We prove central limit theorems for the sample KMD, and provide a complete characterization of the asymptotic power of the test, as well as its detection threshold. The usefulness of our measure is demonstrated via real and synthetic data examples; our method is also implemented in an R package. Supplementary materials for this article are available online.