Continuity of the von Neumann Entropy

A general method for proving continuity of the von Neumann entropy on subsets of positive trace-class operators is considered. This makes it possible to re-derive the known conditions for continuity of the entropy in more general forms and to obtain several new conditions. The method is based on a particular approximation of the von Neumann entropy by an increasing sequence of concave continuous unitary invariant functions defined using decompositions into finite rank operators. The existence of this approximation is a corollary of a general property of the set of quantum states as a convex topological space called the strong stability property. This is considered in the first part of the paper.     

A Measure of Quantum Correlation Based on von Neumann Entropy and Positive Operator-Valued Measurement

Quantum correlations in composite quantum systems are at the origin of the most peculiar features of quantum mechanics such as the violation of Bells inequalities and non-locality. In quantum information theory, they are viewed as quantum resources used by quantum algorithms and communication protocols to outperform their classical analogs. In this paper, we define a new measure of quantum correlation based on von Neumann entropy and positive operator-valued measurement,which has clear physical meaning and we can prove that it satisfying many good property for a measure of quantumness.     

Generalizations of Talagrand Inequality for Sinkhorn Distance Using Entropy Power Inequality

The distance that compares the difference between two probability distributions plays a fundamental role in statistics and machine learning. Optimal transport (OT) theory provides a theoretical framework to study such distances. Recent advances in OT theory include a generalization of classical OT with an extra entropic constraint or regularization, called entropic OT. Despite its convenience in computation, entropic OT still lacks sufficient theoretical support. In this paper, we show that the quadratic cost in entropic OT can be upper-bounded using entropy power inequality (EPI)-type bounds. First, we prove an HWI-type inequality by making use of the infinitesimal displacement convexity of the OT map. Second, we derive two Talagrand-type inequalities using the saturation of EPI that corresponds to a numerical term in our expressions. These two new inequalities are shown to generalize two previous results obtained by Bolley et al. and Bai et al. Using the new Talagrand-type inequalities, we also show that the geometry observed by Sinkhorn distance is smoothed in the sense of measure concentration. Finally, we corroborate our results with various simulation studies.     

A new axiomatic characterization of the von Neumann entropy

We lay down a series of postulates which we consider indespensable for any measure of the intrinsic dispersion of quantum states. Then we show that the v. Neumann entropy S(W)=–kTr(WlnW) is uniquely determined by these postulates up to a scaling factor. Thus the v. Neumann entropy can be characterized as the only reasonable measure of the intrinsic dispersion of quantum states. Our postulates have no direct bearing on phenomenological thermodynamics or information theory.     

A prototype of quantum von Neumann architecture

A modern computer system, based on the von Neumann architecture, is a complicated system with several interactive modular parts. It requires a thorough understanding of the physics of information storage, processing, protection, readout, etc. Quantum computing, as the most generic usage of quantum information, follows a hybrid architecture so far, namely, quantum algorithms are stored and controlled classically, and mainly the executions of them are quantum, leading to the so-called quantum processing units. Such a quantum–classical hybrid is constrained by its classical ingredients, and cannot reveal the computational power of a fully quantum computer system as conceived from the beginning of the field. Recently, the nature of quantum information has been further recognized, such as the no-programming and no-control theorems, and the unifying understandings of quantum algorithms and computing models. As a result, in this work, we propose a model of a universal quantum computer system, the quantum version of the von Neumann architecture. It uses ebits (i.e. Bell states) as elements of the quantum memory unit, and qubits as elements of the quantum control unit and processing unit. As a digital quantum system, its global configurations can be viewed as tensor-network states. Its universality is proved by the capability to execute quantum algorithms based on a program composition scheme via a universal quantum gate teleportation. It is also protected by the uncertainty principle, the fundamental law of quantum information, making it quantum-secure and distinct from the classical case. In particular, we introduce a few variants of quantum circuits, including the tailed, nested, and topological ones, to characterize the roles of quantum memory and control, which could also be of independent interest in other contexts. In all, our primary study demonstrates the manifold power of quantum information and paves the way for the creation of quantum computer systems in the near future.     

Inequalities for traces on von Neumann algebras

A number of useful inequalities, which are known for the trace on a separable Hilbert space, are extended to traces on von Neumann algebras. In particular, we prove the Golden rule, Hölder inequality, and some convexity statements.Battelle Fellow, 1970–1971.     

von Neumann measurement-related matrices

<正>Quantum mechanics is the theory for small physical systems.It does provide a mathematical and conceptual framework for the development of laws that a physical system must obey. It has several basic postulates. The first is about state space, in which quantum mechanics takes place. The second is about the evolution of quantum system. The third is about quantum measurement, which describes what an experimentalist will obtain when he/she observes the system. These three postulates provide a connection between the physical world and the     

Dissipations on von Neumann algebras

It is shown that for certain classes of Euclidean fermion-boson systems on a lattice vacuum expectation values of scalar fields increase if a Yukawa-interaction is turned on. Applicability and possible extensions of this result in the framework of constructive quantum-field-theory are discussed.Supported by Studienstiftung des deutschen Volkes     

Can Sobolev Inequality Be Written for Sharma-Mittal Entropy?

In this paper, we focus on Sobolev inequality in the context of Sharma-Mittal entropy. Using this new inequality, generalized entropic uncertainty relation in accordance with Sharma-Mittal entropy is derived and the pseudoadditivity relation has been obtained. This new entropic uncertainty relation has then been applied to physical examples such as one dimensional harmonic oscillator and Pösch-Teller potential. Finally, it has been shown that for certain values of the parameters of Sharma-Mittal measure, the present results reduce to the corresponding results of Shannon, Renyi and Tsallis measures.     

On the Embedding of von Neumann Subalgebras

For a von Neumann algebra with a cyclic and separating vector it will be shown that the von Neumann subalgebras with the same cyclic vector can uniquely be characterized by one-parametric operator-valued functions obeying a set of conditions. Since the properties contain no reference to the subalgebra these operator-valued functions will be called characteristic functions. On the set of characteristic functions there exists a natural topology under which this set is complete. Received: 3 December 1998 /Accepted: 15 February 1999     

Logically independent von Neumann lattices

Three definitions of logical independence of two von Neumann latticesP₁,P₂ of two sub-von Neumann algebras ₁, ₂ of a von Neumann algebra are given and the relations of the definitions clarified. It is shown that under weak assumptions the following notion, called logical independence is the strongest:A B 0 for any 0 A P₁, 0 B P₂. Propositions relating logical independence ofP₁,P₂ toC ^*-independence,W ^* independence, and strict locality of ₁, ₂ are presented.     

Ruelle Inequality Relating Entropy,Folding Entropy and Negative Lyapunov Exponents

In this paper we prove an inequality conjectured by Ruelle relating the entropy, folding entropy and negative Lyapunov exponents of a differentiable map on a compact manifold, under a set of conditions on degenerate points of the map.This work is supported by SFMSBRP and NSFDYS     

Frobenius theory for positive maps of von Neumann algebras

Frobenius theory about the cyclic structure of eigenvalues of irreducible non negative matrices is extended to the case of positive linear maps of von Neumann algebras. Semigroups of such maps and ergodic properties are also considered.     

Conditional expectations on von Neumann algebras: A new approach

In the framework of non-commutative probability theory on von Neumann algebras the concept of conditional expectation is redefined in such a way that it exists relative to arbitrary a priori distributions and subalgebras. We derive explicit expressions for the expectations relative to Abelian subalgebras and establish their connection with the concept of coarse-graining.     

Inner automorphisms of von Neumann algebras

It is first shown that a *-automorphism of a factor is inner if and only if it is asymptotically equal to the identity automorphism. Then it is shown that a periodic *-automorphism of a von Neumann algebra is inner if and only if its fixed point algebra is a normal subalgebra of .     

von Neumann measurement-related matrices and the nullity condition for quantum correlation

We study von Neumann measurement-related matrices, and the nullity condition of quantum correlation. We investigate the properties of these matrices that are related to a von Neumann measurement. It is shown that these(m~2-1)×(m~2-1) matrices are idempotent, and have rank m-1. These properties give rise to necessary conditions for the nullity of quantum correlations in bipartite systems. Finally, as an example we discuss quantum correlation in Bell diagonal states.     

Perturbation theory of von Neumann entropy

In quantum information theory, von Neumann entropy plays an important role; it is related to quantum channel capacities. Only for a few states can one obtain their entropies. In a continuous variable system, numeric evaluation of entropy is not easy due to infinite dimensions. We develop the perturbation theory for systematically calculating von Neumann entropy of a non-degenerate system as well as a degenerate system.     

A New Integrable Symplectic Map of Neumann Type

By resorting to the nonlinearization approach, a Neumann constraint associated with a discrete 3 × 3 matrix eigenvalue problem is considered. A new symplectic map of the Neumann type is obtained through nonlinearization of the discrete eigenvalue problem and its adjoint one. The generating function of integrals of motion is presented, by which the symplectic reap＇is further proved to be completely integrable in the Liouville sense.     

Von Neumann Entropy of an Electron in One-Dimensional Determined Potentials

By using the measure of von Neumann entropy, we numerically investigate quantum entanglement of an electron moving in the one-dimensional Harper model and in the one-dimensional slowly varying potential model. The delocalized and localized eigenstates can be distinguished by von Neumann entropy of the individual eigenstates.There are drastic decreases in von Neumann entropy of the individual eigenstates at mobility edges. In the curve of the spectrum averaged von Neumann entropy as a function of potential parameter λ, a sharp transition exists at the metal-insulator transition point λc = 2. It is found that the von Neumann entropy is a good quantity to reflect localization and metal-insulator transition.