The Boltzmann--Gibbs entropy is a functional on the space of probability measures. When a state space is countable, one characterization of the Boltzmann--Gibbs entropy is given by the Shannon--Khinchin axioms, which consist of continuity, maximality, expandability and extensivity. Among these four properties, the extensivity is generalized in various ways. The extensivity of a functional is interpreted as the property that, for any random variables $(X,Y)$ taking finitely many values in $\mathbb{N}$, the difference between the value of the functional at the joint law of $(X,Y)$ and that at the law of $X$ coincides with the linear combinations of the values at the conditional laws of $Y$ given $X=n$ with coefficients given by the probabilities of each event $X=n$. A generalization of the extensivity obtained by replacing the coefficients with a power of the probabilities of the events $X=n$ provides a characterization of the Tsallis entropy.
In this paper, we first prove the impossibility to replace the coefficients with a non-power function of the probabilities of the events $X=n$. Then we estimate the difference between the value at the joint law of $(X,Y)$ and that at the law of $X$ for a general functional.
We consider a family of quantum conditional entropies based on the concept of quantum $f$-divergences. First, we explicitly formulate conditions under which the notion of quantum conditional entropy is well defined in this way. In particular, we demand that the value of conditional entropy be independent of any extension of the principal Hilbert space. Using fundamental properties of the quantum $f$-divergences, several interesting relations for such conditional entropies are formulated. We separately examine an especially important case of quantum conditional entropies related to the Tsallis …
The Renyi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies or mutual information, and have found many applications in information theory and beyond. Various generalizations of Renyi entropies to the quantum setting have been proposed, most notably Petz's quasi-entropies and Renner's conditional min-, max- and collision entropy. Here, we argue that previous quantum extensions are incompatible and thus unsatisfactory. We p…
We investigate the possibility of discrete groups furnishing a kinematic framework for systems whose thermodynamic behaviour may be given by non-additive entropies. Relying on the well-known result of the growth rate of balls of nilpotent groups, we see that maintaining extensivity of the entropy of a nilpotent group requires using a non-Boltzmann/Gibbs/Shannon (BGS) entropic form. We use the Tsallis entropy as an indicative alternative. Using basic results from hyperbolic and random groups, we investigate the genericity and possible range of applicability of the BGS entropy in this context. W…
We prove the quantum conditional Entropy Power Inequality for quantum additive noise channels. This inequality lower bounds the quantum conditional entropy of the output of an additive noise channel in terms of the quantum conditional entropies of the input state and the noise when they are conditionally independent given the memory. We also show that this conditional Entropy Power Inequality is optimal in the sense that we can achieve equality asymptotically by choosing a suitable sequence of Gaussian input states. We apply the conditional Entropy Power Inequality to find an array of informat…
We present a bouquet of continuity bounds for quantum entropies, falling broadly into two classes: First, a tight analysis of the Alicki-Fannes continuity bounds for the conditional von Neumann entropy, reaching almost the best possible form that depends only on the system dimension and the trace distance of the states. Almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators. As applications we give new proofs, with tighter bounds, of the asymptotic continuity of the relative entropy of entangleme…
We prove the conditional Entropy Power Inequality for Gaussian quantum systems. This fundamental inequality determines the minimum quantum conditional von Neumann entropy of the output of the beam-splitter or of the squeezing among all the input states where the two inputs are conditionally independent given the memory and have given quantum conditional entropies. We also prove that, for any couple of values of the quantum conditional entropies of the two inputs, the minimum of the quantum conditional entropy of the output given by the conditional Entropy Power Inequality is asymptotically ach…
We discuss the idea that the Tsallis-type (q-additive) entropic chain rule allows for a wider class of entropic functionals than previously thought. In particular, we point out that the ensuing entropy solutions (e.g., Tsallis entropy) can be determined uniquely only when one fixes the prescription for handling conditional entropies. Our point is substantiated with the Darótzy's mapping theorem and DeFinetti-Kolmogorov theorem for escort distributions and illustrated with a number of examples. Connection with Landsberg's classification of non-extensive thermodynamical systems is also briefly d…
Quantum information measures such as the entropy and the mutual information find applications in physics, e.g., as correlation measures. Generalizing such measures based on the Rényi entropies is expected to enhance their scope in applications. We prescribe Rényi generalizations for any quantum information measure which consists of a linear combination of von Neumann entropies with coefficients chosen from the set {-1,0,1}. As examples, we describe Rényi generalizations of the conditional quantum mutual information, some quantum multipartite information measures, and the topological entangleme…
This paper introduces "swiveled Renyi entropies" as an alternative to the Renyi entropic quantities put forward in [Berta et al., Phys. Rev. A 91, 022333 (2015)]. What distinguishes the swiveled Renyi entropies from the prior proposal of Berta et al. is that there is an extra degree of freedom: an optimization over unitary rotations with respect to particular fixed bases (swivels). A consequence of this extra degree of freedom is that the swiveled Renyi entropies are ordered, which is an important property of the Renyi family of entropies. The swiveled Renyi entropies are however generally dis…
We develop a semidefinite programming method for the optimization of quantum networks, including both causal networks and networks with indefinite causal structure. Our method applies to a broad class of performance measures, defined operationally in terms of interactive tests set up by a verifier. We show that the optimal performance is equal to a max relative entropy, which quantifies the informativeness of the test. Building on this result, we extend the notion of conditional min-entropy from quantum states to quantum causal networks. The optimization method is illustrated in a number of ap…
Distance measures between quantum states like the trace distance and the fidelity can naturally be defined by optimizing a classical distance measure over all measurement statistics that can be obtained from the respective quantum states. In contrast, Petz showed that the measured relative entropy, defined as a maximization of the Kullback-Leibler divergence over projective measurement statistics, is strictly smaller than Umegaki's quantum relative entropy whenever the states do not commute. We extend this result in two ways. First, we show that Petz' conclusion remains true if we allow genera…
This thesis consolidates, improves and extends the smooth entropy framework for non-asymptotic information theory and cryptography. We investigate the conditional min- and max-entropy for quantum states, generalizations of classical Rényi entropies. We introduce the purified distance, a novel metric for unnormalized quantum states and use it to define smooth entropies as optimizations of the min- and max-entropies over a ball of close states. We explore various properties of these entropies, including data-processing inequalities, chain rules and their classical limits. The most important prop…
We define a new divergence of von Neumann algebras using a variational expression that is similar in nature to Kosaki's formula for the relative entropy. Our divergence satisfies the usual desirable properties, upper bounds the sandwiched Renyi entropy and reduces to the fidelity in a limit. As an illustration, we use the formula in quantum field theory to compute our divergence between the vacuum in a bipartite system and an "orbifolded" -- in the sense of conditional expectation -- system in terms of the Jones index. We take the opportunity to point out entropic certainty relation for arbitr…
The quantum relative entropy is a measure of the distinguishability of two quantum states, and it is a unifying concept in quantum information theory: many information measures such as entropy, conditional entropy, mutual information, and entanglement measures can be realized from it. As such, there has been broad interest in generalizing the notion to further understand its most basic properties, one of which is the data processing inequality. The quantum f-divergence of Petz is one generalization of the quantum relative entropy, and it also leads to other relative entropies, such as the Petz…
Quantum states can be subjected to classical measurements, whose incompatibility, or uncertainty, can be quantified by a comparison of certain entropies. There is a long history of such entropy inequalities between position and momentum. Recently these inequalities have been generalized to the tensor product of several Hilbert spaces and we show here how their derivations can be shortened to a few lines and how they can be generalized. All the recently derived uncertainty relations utilize the strong subadditivity (SSA) theorem; our contribution relies on directly utilizing the proof technique…
We introduce an axiomatic approach for characterizing quantum conditional entropy. Our approach relies on two physically motivated axioms: monotonicity under conditional majorization and additivity. We show that these two axioms provide sufficient structure that enable us to derive several key properties applicable to all quantum conditional entropies studied in the literature. Specifically, we prove that any quantum conditional entropy must be negative on certain entangled states and must equal -log(d) on dxd maximally entangled states. We also prove the non-negativity of conditional entropy …
In [Berta et al., J. Math. Phys. 56, 022205 (2015)], we recently proposed Renyi generalizations of the conditional quantum mutual information of a tripartite state on $ABC$ (with $C$ being the conditioning system), which were shown to satisfy some properties that hold for the original quantity, such as non-negativity, duality, and monotonicity with respect to local operations on the system $B$ (with it being left open to show that the Renyi quantity is monotone with respect to local operations on system $A$). Here we define a Renyi squashed entanglement and a Renyi quantum discord based on a R…
Conditioning on Subalgebras - Entropy Duality and Generalized Quantum Stein's Lemma in von Neumann Algebras
Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；Department of Mathematical Sciences, Chalmers University of Technology（查尔姆斯理工大学数学科学系）
Additivity and chain rules for quantum entropies via multi-index Schatten norms
Inria（法国国家信息与自动化研究所）；ENS Lyon（里昂高等师范学院）；UCBL（里昂第一大学）；LIP（计算机科学实验室）；Télécom Paris - LTCI（巴黎电信学院-通信与图像实验室）；Institut Polytechnique de Paris（巴黎理工学院）；Inria Paris（法国国家信息与自动化研究所巴黎分部）
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 co…
