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     

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.     

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 .     

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.     

States and Structure of von Neumann Algebras

We summarize and deepen recent results on the interplay between properties of states and the structure of von Neumann algebras. We treat Jauch–Piron states and the concept of independence in noncommutative probability theory.     

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.     

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     

Representations of von Neumann Algebras and Ultraproducts

International Journal of Theoretical Physics - We will introduce the concept of ergodicity of states with respect to some group of transformations on a von Neumann algebra and its properties are...     

States and Structure of von Neumann Algebras

We summarize and deepen recent results on the interplay between properties of states and the structure of von Neumann algebras. We treat Jauch–Piron states and the concept of independence in noncommutative probability theory.     

Unbounded generalizations of standard von Neumann algebras

The primary purpose of this paper is to study unbounded operator algebras called standard EW^#-algebras which are generalizations of standard von Neumann algebras to the unbounded case.The first part is an investigation of the weak, σ-weak, strong and σ-strong topologies on a standard EW^#-algebra. It is showed that the weak and σ-weak topologies on a standard EW^#-algebra coincide.The second part is a study of a locally convex topology called _∞L^ω₂-topology on a standard EW^#-algebra U. It is proved that U is a GB¹-algebra under the topology.The third part is an investigation of isomorphisms of standard EW^#-algebras. It is showed that if standard EW^#-algebras U and B are isomorphic then U and B are spatially isomorphic.     

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.     

Quantum Correlations Induced by Local von Neumann Measurement

We study the total quantum correlation, semiquantum correlation and joint quantum correlation induced by local von Neumann measurement in bipartite system. We analyze the properties of these quantum correlations and obtain analytical formula for pure states. The experimental witness for these quantum correlations is further provided and the significance of these quantum correlations is discussed in the context of local distinguishability of quantum states.     

Extension of covariant POV-measures in von Neumann algebras

We discuss the extension of POV-measures transforming covariantly with respect to automorphic group representations on von Neumann algebras.     

States on projection logics of von Neumann algebras

We summarize recent results concerning states on projection lattices of von Neumann algebras. In particular, we present an analysis of the Jauch-Piron property in the von Neumann algebra setting.     

Conditional expectation and commutativity in von Neumann algebras

LetH be a Hilbert space,A the von Neumann algebra of all bounded operators onH,B a von Neumann subalgebra ofA, andw a bounded linear functional onA. The functionalw is said to commute withBA ifw(AB)=w(BA) for allAA. It is shown that the mapBw (BAB) is a complex measure on the orthocomplemented partially ordered set of all orthogonal projections inB for everyAA if and only ifw commutes with all members ofB. For anyAA, the conditional expectation ofA with respect toB andw is defined and it is shown that this expectation exists for an Abelian separableB ifw commutes with all members ofB. Using Gleason's theorem it is shown thatw commutes withB if and only if the density operator ofw commutes withB.     

A New Inequality for the von Neumann Entropy

Strong subadditivity of von Neumann entropy, proved in 1973 by Lieb and Ruskai, is a cornerstone of quantum coding theory. All other known inequalities for entropies of quantum systems may be derived from it. Here we prove a new inequality for the von Neumann entropy which we prove is independent of strong subadditivity: it is an inequality which is true for any four party quantum state, provided that it satisfies three linear relations (constraints) on the entropies of certain reduced states.