On the von Neumann and Shannon Entropies for Quantum Walks on Z^2

Comments on "Generalization of Shannon-Khinchin axioms to nonextensive systems and the uniqueness theorem for the nonextensive entropy"

Recently, Suyari has proposed a generalization of Shannon-Khinchin axioms, which determines a class of entropies containing the well-known Tsalis and Havrda-Charvat entropies [H. Suyari, IEEE Trans. Inf. Theory, vol. 50, pp. 1783-1787, Aug. 2004]. In this comment we show that the class of entropy functions determined by Suyari's axioms is wider than the one proposed by Suyari and give a counterexample. Additionally, we generalize Suyari's axioms characterizing recently introduced class of entropies obtained by averaging pseudoadditive information content introduced in [V. Ilic and M. Stankovic…

Maximum Entropy Principle in statistical inference: case for non-Shannonian entropies

In this letter we show that the Shore--Johnson axioms for Maximum Entropy Principle in statistical estimation theory account for a considerably wider class of entropic functional than previously thought. Apart from a formal side of the proof, we substantiate our point by analyzing the effect of weak correlations and discuss two pertinent examples: $2$-qubit quantum system and strongly interacting nuclear systems.

Mappings preserving quantum Renyi's entropies in von Neumann algebras

We investigate the situation when a normal positive linear unital map on a semifinite von Neumann algebra leaving the trace invariant does not change fixed quantum Renyi's entropy of the density of a normal state. It is also shown that such a map does not change the entropy of any density if and only if it is a Jordan *-isomorphism on the algebra.

Detecting dimensional crossover and finite Hilbert space through entanglement entropies

The information content of the two-particle one- and two-dimensional Calogero model is studied using the von Neumann and Rényi entropies. The one-dimensional model is shown to have non-monotonic entropies with finite values in the large interaction strength limit. On the other hand, the von Neumann entropy of the two-dimensional model with isotropic confinement is a monotone increasing function of the interaction strength which diverges logarithmically. By considering an anisotropic confinement in the two-dimensional case we show that the one-dimensional behavior is eventually reached when the…

Chain Rules for Smooth Min- and Max-Entropies

The chain rule for the Shannon and von Neumann entropy, which relates the total entropy of a system to the entropies of its parts, is of central importance to information theory. Here we consider the chain rule for the more general smooth min- and max-entropy, used in one-shot information theory. For these entropy measures, the chain rule no longer holds as an equality, but manifests itself as a set of inequalities that reduce to the chain rule for the von Neumann entropy in the i.i.d. case.

Comments on "A two-parameter generalization of Shannon-Khinchin Axioms and the uniqueness theorem"

Wada and Suyari proposed a two-parameter generalization of Shannon-Khinchin axioms (TGSK axioms) [T. Wada and H. Suyari, Physics Letters A, 368(3)]. We derive a new class of entropies which differs from Wada-Suyari's class by fixing the incorrectness which occurs in the mentioned paper. Also, we consider a two-parameter class of entropies derived from the maxent principle proposed in [Kaniadakis, G. and Lissia, M. and Scarfone, AM, Physica A: Statistical Mechanics and its Applications, 340(1)]. We rederived this class by changing initial condition, obtaining the same class as our class derived…

Entropies in General Probabilistic Theories and its Application to Holevo Bound

General probabilistic theories are designed to provide operationally the most general probabilistic models including both classical and quantum theories. In this letter, we introduce a systematic method to construct a series of entropies, all of which generalize Shannon entropy in classical system and von Neumann entropy in quantum system. Using these entropies, the Holevo bound, an upper bound of the accessible information from a quantum system, is generalized to hold in any general probabilistic theory.

When Shannon and Khinchin meet Shore and Johnson: equivalence of information theory and statistical inference axiomatics

We propose a unified framework for both Shannon-Khinchin and Shore-Johnson axiomatic systems. We do it by rephrasing Shannon-Khinchine axioms in terms of generalized arithmetics of Kolmogorov and Nagumo. We prove that the two axiomatic schemes yield identical classes of entropic functionals -- Uffink class of entropies. This allows to re-establish the entropic parallelism between information theory and statistical inference that has seemed to be "broken" by the use of non-Shannonian entropies.

Generalization of Shannon-Khinchin Axioms to Nonextensive Systems and the Uniqueness Theorem

The Shannon-Khinchin axioms are generalized to nonextensive systems and the uniqueness theorem for the nonextensive entropy is proved rigorously. In the present axioms, Shannon additivity is used as additivity in contrast to pseudoadditivity in Abe's axioms. The results reveal that Tsallis entropy is the simplest among all nonextensive entropies which can be obtained from the generalized Shannon-Khinchin axioms.

Infinite Shannon entropy

Even if a probability distribution is properly normalizable, its associated Shannon (or von Neumann) entropy can easily be infinite. We carefully analyze conditions under which this phenomenon can occur. Roughly speaking, this happens when arbitrarily small amounts of probability are dispersed into an infinite number of states; we shall quantify this observation and make it precise. We develop several particularly simple, elementary, and useful bounds, and also provide some asymptotic estimates, leading to necessary and sufficient conditions for the occurrence of infinite Shannon entropy. We g…

Some new Karamata type inequalities and their applications to some entropies

Some new inequalities of Karamata type are established with a convex function in this paper. The methods of our proof allow us to obtain an extended version of the reverse of Jensen inequality given by Pe{\v} carić and Mićić. Applying the obtained results, we give reverses for information inequality (Shannon inequality) in different types, namely ratio type and difference type, under some conditions. Also, we provide interesting inequalities for von Neumann entropy and quantum Tsallis entropy which is a parametric extension of von Neumann entropy. The inequality for von Neumann entropy recover…

Beyond the Shannon-Khinchin Formulation: The Composability Axiom and the Universal Group Entropy

The notion of entropy is ubiquitous both in natural and social sciences. In the last two decades, a considerable effort has been devoted to the study of new entropic forms, which generalize the standard Boltzmann-Gibbs (BG) entropy and are widely applicable in thermodynamics, quantum mechanics and information theory. In [23], by extending previous ideas of Shannon [38], [39], Khinchin proposed an axiomatic definition of the BG entropy, based on four requirements, nowadays known as the Shannon-Khinchin (SK) axioms. The purpose of this paper is twofold. First, we show that there exists an intrin…

具有多个通道的量子图的散射熵

Scattering entropies of quantum graphs with several channels

本研究探讨了具有多个通道的量子图在多种情况下的散射熵，分析了 Shannon、Rényi 和 Tsallis 熵，并描述了多顶点、边和引线存在时的相关结果，指出其可作为量子图输运模型的量化指标。

源于基本熵不等式的量子熵连续性界

Continuity bounds for quantum entropies arising from a fundamental entropic inequality

该研究针对两个量子态的冯·诺依曼熵差建立紧上界，提出蕴含并细化Audenaert-Fannes不等式的新型熵不等式，据此得到量子条件熵和双变量相对熵的连续性界，且不等式在无限维中有效。

熵与自由能的半经典极限

Semiclassical limit of entropies and free energies

本文从半经典分析角度研究熵与自由能，证明量子吉布斯态的冯·诺依曼熵经重整化后收敛到韦尔熵，且韦尔熵自由能泛函Γ-收敛到经典形式，从而保证极小值与极小化子的收敛。

Convergence Conditions of Mixed States and their von Neumann Entropy in Continuous Quantum Measurements

By carrying out appropriate continuous quantum measurements with a family of projection operators, a unitary channel can be approximated in an arbitrary precision in the trace norm sense. In particular, the quantum Zeno effect is described as an application. In the case of an infinite dimension, although the von Neumann entropy is not necessarily continuous, the difference of the entropies between the states, as mentioned above, can be arbitrarily made small under some conditions.

The Shannon-entropy-based uncertainty relation for $D$-dimensional central potentials

The uncertainty relation based on the Shannon entropies of the probability densities in position and momentum spaces is improved for quantum systems in arbitrary $D$-dimensional spherically symmetric potentials. To find it, we have used the $L^p$ -- $L^q$ norm inequality of L. De Carli and the logarithmic uncertainty relation for the Hankel transform of S. Omri. Applications to some relevant three-dimensional central potentials are shown.

Equivalence of information production and generalized entropies in complex processes

Complex systems that are characterized by strong correlations and fat-tailed distribution functions have been argued to be incompatible within the framework of Boltzmann-Gibbs entropy. As an alternative, so-called generalized entropies were proposed and intensively studied. Here we show that this incompatibility is a misconception. For a broad class of processes, Boltzmann entropy the log multiplicity remains the valid entropy concept, however, for non-i.i.d., non-multinomial, and non-ergodic processes, Boltzmann entropy is not of Shannon form. The correct form of Boltzmann entropy can be show…

玻色量子系统中冯·诺依曼熵的广义强次可加性

The generalized strong subadditivity of the von Neumann entropy for bosonic quantum systems

本文推广了玻色量子高斯系统中冯·诺依曼熵的强次可加性，给出任意线性函数关联子系统熵组合的最小值，并据此证明新的熵不确定性关系、量子熵功率不等式推广及纠缠熵线性时间标度。

Variational approach to relative entropies (with application to QFT)

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…

Rényi relative entropies and noncommutative $L_p$-spaces

We propose an extension of the sandwiched Rényi relative $α$-entropy to normal positive functionals on arbitrary von Neumann algebras, for the values $α>1$. For this, we use Kosaki's definition of noncommutative $L_p$-spaces with respect to a state. We show that these extensions coincide with the previously defined Araki-Masuda divergences [M. Berta et al., Annales Henri Poincaré, 19:1843--1867, 2018] and prove some of their properties, in particular the data processing inequality with respect to positive normal unital maps. As a consequence, we obtain monotonicity of the Araki relative entrop…

Generalized Additive Entropies in Fully Developed Turbulence

We explore a possible application of the additive generalization of the Boltzmann-Gibbs-Shannon entropy proposed in [A.N. Gorban, I.V. Karlin, Phys. Rev. E, 67:016104 (2003)] to fully developed turbulence. The predicted probability distribution functions are compared with those obtained from Tsallis' entropy and with experimental data. Consequences of the existence of hidden subsystems are illustrated.

关于子代数的条件作用——冯·诺依曼代数中的熵对偶性与广义量子斯坦引理

Conditioning on Subalgebras - Entropy Duality and Generalized Quantum Stein's Lemma in von Neumann Algebras

该论文将条件熵对偶性推广至任意冯·诺依曼代数子代数，并建立无限维量子斯坦引理，为子代数相对熵提供复合假设检验中的操作解释。

Bures-Hall Ensemble: Spectral Densities and Average Entropies

We consider an ensemble of random density matrices distributed according to the Bures measure. The corresponding joint probability density of eigenvalues is described by the fixed trace Bures-Hall ensemble of random matrices which, in turn, is related to its unrestricted trace counterpart via a Laplace transform. We investigate the spectral statistics of both these ensembles and, in particular, focus on the level density, for which we obtain exact closed-form results involving Pfaffians. In the fixed trace case, the level density expression is used to obtain an exact result for the average Hav…

Fractal von Neumann entropy

We consider the {\it fractal von Neumann entropy} associated with the {\it fractal distribution function} and we obtain for some {\it universal classes h of fractons} their entropies. We obtain also for each of these classes a {\it fractal-deformed Heisenberg algebra}. This one takes into account the braid group structure of these objects which live in two-dimensional multiply connected space.

Relative modular operator in semifinite von Neumann algebras and its use

We present some results concerning the relative modular operator in semifinite von Neumann algebras. These results allow one to prove some basic formula for trace, to obtain equivalence between Araki's relative entropy and Umegaki's information as well as to derive some formulae for quasi-entropies, and Rényi's relative entropy known in finite dimension.

On quantum Renyi entropies: a new generalization and some properties

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…

对数与幂律熵的凸性来源

Logarithmic and power-law entropies from convexity

通过几何方法探讨熵泛函的起源，利用凸体相关定理对比玻尔兹曼熵与幂律熵的适用性，提出幂律熵可能适用于某些系统。

von Neumann 代数中态熵型泛函的变分公式

Variational formulae for entropy-like functionals for states in von Neumann algebras

本文针对半有限 von Neumann 代数上的正规态，建立了 Segal 熵与 Rényi 熵等熵型泛函的变分公式，推广了相对熵的已知结果，并揭示了量子熵与交换子代数结构的联系，为量子信息论提供新视角。

Hypergraph min-cuts from quantum entropies

The min-cut function of weighted hypergraphs and the von Neumann entropy of pure quantum states are both symmetric submodular functions. In this note, we explain this coincidence by proving that the min-cut function of any weighted hypergraph can be approximated (up to an overall rescaling) by the entropies of quantum states known as stabilizer states. This implies that the min-cuts of hypergraphs are constrained by quantum entropy inequalities, and it shows that the recently defined hypergraph cones are contained in the quantum stabilizer entropy cones, as has been conjectured in the recent l…

Comments on "On q-non-extensive statistics with non-Tsallisian entropy"

Recently, in [P. Jizba and J. Korbel, Physica A 444, 2016, 808-827], four generalized Shannon-Khinchin [GSK] axioms have been proposed and a generalized entropy which uniquely satisfies the GSK axioms has been derived. In this comment, we show that the unique class of the entropies derived in the aforementioned paper is not correct, as it violates the fourth GSK axiom, and we derive the correct one. Nevertheless, the class of entropies proposed in the commented paper still can serve as a basis for generalized statistical mechanics. We propose a new axiomatic system which characterizes the clas…

从经典到量子：无限维中熵的一致连续性界

From Classical to Quantum: Uniform Continuity Bounds on Entropies in Infinite Dimensions

本文证明了无限维经典与量子系统熵的一致连续性界，利用Fano型不等式和极大耦合获得Shannon与von Neumann熵的紧致估计，并提出基于算子Hölder连续性的新近似方案处理Rényi等熵，解决了有限维近似性质的开放问题。

The Shannon-McMillan Theorem for Ergodic Quantum Lattice Systems

We formulate and prove a quantum Shannon-McMillan theorem. The theorem demonstrates the significance of the von Neumann entropy for translation invariant ergodic quantum spin systems on n-dimensional lattices: the entropy gives the logarithm of the essential number of eigenvectors of the system on large boxes. The one-dimensional case covers quantum information sources and is basic for coding theorems.

Affine connections, duality and divergences for a von Neumann algebra

On the predual of a von Neumann algebra, we define a differentiable manifold structure and affine connections by embeddings into non-commutative L_p-spaces. Using the geometry of uniformly convex Banach spaces and duality of the L_p and L_q spaces for 1/p+1/q=1, we show that we can introduce the α-divergence, for αin (-1,1), in a similar manner as Amari in the classical case. If restricted to the positive cone, the α-divergence belongs to the class of quasi-entropies, defined by Petz.