Accessible Information Without Disturbing Partially Known Quantum States on a von Neumann Algebra

This paper addresses the problem of how much information we can extract without disturbing a statistical experiment, which is a family of partially known normal states on a von Neumann algebra. We define the classical part of a statistical experiment as the restriction of the equivalent minimal sufficient statistical experiment to the center of the outcome space, which, in the case of density operators on a Hilbert space, corresponds to the classical probability distributions appearing in the maximal decomposition by Koashi and Imoto (Phys. Rev. A 66, 022,318 2002). We show that we can access by a Schwarz or completely positive channel at most the classical part of a statistical experiment if we do not disturb the states. We apply this result to the broadcasting problem of a statistical experiment. We also show that the classical part of the direct product of statistical experiments is the direct product of the classical parts of the statistical experiments. The proof of the latter result is based on the theorem that the direct product of minimal sufficient statistical experiments is also minimal sufficient.     

Hilbert Modules in Quantum Electro Dynamics and Quantum Probability

A physical system of the form with a distinguished state on may be described in a natural way on a Hilbert -module. Following the ideas of Accardi and Lu [1], we apply this possibility to a concrete system consisting of a boson field in the vacuum state coupled to a free electron. We show that the physical system is described adequately on a new type of Fock module: the symmetric Fock module. It turns out that a module has to fulfill an algebraic condition in order to allow for the construction of a symmetric Fock module. We prove in a central limit theorem that in the stochastic limit the moments of the collective operators (i.e. more or less the time-integrated interaction Hamiltonian) converge to the moments of free creators and annihilators on a full Fock module. In the sense of Voiculescu [22] and Speicher [20] these operators form a free white noise over the algebra . Received: 28 October 1996 / Accepted: 21 July 1997     

Factorization and Dilation Problems for Completely Positive Maps on von Neumann Algebras

We study factorization and dilation properties of Markov maps between von Neumann algebras equipped with normal faithful states, i.e., completely positive unital maps which preserve the given states and also intertwine their automorphism groups. The starting point for our investigation has been the question of existence of non-factorizable Markov maps, as formulated by C. Anantharaman-Delaroche. We provide simple examples of non-factorizable Markov maps on M_n(\mathbbC){M_n(\mathbb{C})} for all n ≥ 3, as well as an example of a one-parameter semigroup (T(t)) _t≥0 of Markov maps on M₄(\mathbbC){M_4(\mathbb{C})} such that T(t) fails to be factorizable for all small values of t > 0. As applications, we solve in the negative an open problem in quantum information theory concerning an asymptotic version of the quantum Birkhoff conjecture, as well as we sharpen the existing lower bound estimate for the best constant in the noncommutative little Grothendieck inequality.     

Dynamical entropy ofC* algebras and von Neumann algebras

The definition of the dynamical entropy is extended for automorphism groups ofC* algebras. As an example, the dynamical entropy of the shift of a lattice algebra is studied, and it is shown that in some cases it coincides with the entropy density.     

A New Dynamical Reflection Algebra and Related Quantum Integrable Systems

We propose a new dynamical reflection algebra, distinct from the previous dynamical boundary algebra and semi-dynamical reflection algebra. The associated Yang?CBaxter equations, coactions, fusions, and commuting traces are derived. Explicit examples are given and quantum integrable Hamiltonians are constructed. They exhibit features similar to the Ruijsenaars?CSchneider Hamiltonians.     

Quantum theory and geometry: sixty years after von Neumann

This paper is a brief review of some of the developments in the mathematical foundations of quantum mechanics that have taken place since the publication in 1932 of John von Neumann's celebrated treatiseMathematische Grundlagen der Quantenmechanik.     

Quantum phase transition and von Neumann entropy of quasiperiodic Hubbard chains

The half-filled Hubbard chains with the Fibonacci and Harper modulating site potentials are studied in a selfconsistent mean-field approximation. A new order parameter is introduced to describe a charge density order. We also calculate the von Neumann entropy of the ground state. The results show that the von Neumann entropy can identify a CDW/SDW （charge density wave/spin density wave） transition for quasiperiodic models.     

Modular Intersections of von Neumann Algebras in Quantum Field Theory

We show that modular intersections of von Neumann algebras occur naturally in quantum field theory. An example are local observable algebras associated with wedge regions, which have a lightray in common, see also [Bo 2, Wi 3]. Conversely, starting from a set of four algebras lying in a specified modular position relative to each other we construct a net of local observables of a 2+1 dimensional quantum field theory. Received: 26 February 1996/ Accepted: 30 August 1997     

Orthomodular Lattices and a Quantum Algebra

We show that one can formulate an algebra with lattice ordering so as to contain one quantum and five classical operations as opposed to the standard formulation of the Hilbert space subspace algebra. The standard orthomodular lattice is embeddable into the algebra. To obtain this result we devised algorithms and computer programs for obtaining expressions of all quantum and classical operations within an orthomodular lattice in terms of each other, many of which are presented in the paper. For quantum disjunction and conjunction we prove their associativity in an orthomodular lattice for any triple in which one of the elements commutes with the other two and their distributivity for any triple in which a particular element commutes with the other two. We also prove that the distributivity of symmetric identity holds in Hilbert space, although whether or not it holds in all orthomodular lattices remains an open problem, as it does not fail in any of over 50 million Greechie diagrams we tested.     

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.     

Decoherence Functionals for von Neumann Quantum Histories: Boundedness and Countable Additivity

Gell–Mann and Hartle have proposed a significant generalisation of quantum theory in which decoherence functionals perform a key role. Verifying a conjecture of Isham–Linden–Schreckenberg, the author analysed the structure of bounded, finitely additive, decoherence functionals for a general von Neumann algebra A (where A has no Type I₂ direct summand). Isham et al. had already given a penetrating analysis for the situation where A is finite dimensional. The assumption of countable additivity for a decoherence functional may seem more plausible, physically, than that of boundedness. The results of this note are obtained much more generally but, when specialised to L(H), the algebra of all bounded linear operators on a separable Hilbert space H, give: Let d be a countably additive decoherence functional defined on all pairs of projections in L(H). If H is infinite dimensional then d must be bounded. By contrast, when H is finite dimensional, unbounded (countably additive) decoherence functionals always exist for L(A). Received: 6 December 1996 / Accepted: 18 May 1997     

On the Relationships Between the Moments of a POVM and the Generator of the von Neumann Algebra It Generates

In the present paper, we review some recent results about commutative positive operator valued measures (POVMs) and single out some open problems. We introduce a conjecture about the extension of some recent results and prove some important consequences of such conjecture. In particular, we prove that it implies the universal character of some of the mathematical objects we introduce, i.e., the fact that they do not depend on the POV measure we are considering. We analyze the relevance of this result. Finally, we point out that some of the results we present admit a constructive proof and we show the relevance of this fact to the theory of commutative POV measures.     

Quantum Enveloping Algebras with von Neumann Regular Cartan-like Generators and the Pierce Decomposition

Quantum bialgebras derivable from U _q (sl ₂) which contain idempotents and von Neumann regular Cartan-like generators are introduced and investigated. Various types of antipodes (invertible and von Neumann regular) on these bialgebras are constructed, which leads to a Hopf algebra structure and a von Neumann-Hopf algebra structure, respectively. For them, explicit forms of some particular R-matrices (also, invertible and von Neumann regular) are presented, and the latter respects the Pierce decomposition. Dedicated to the memory of our colleague Leonid L. Vaksman (1951–2007) On leave of absence from: TheoryGroup, Nuclear Physics Laboratory,V.N.Karazin Kharkov National University, Svoboda Sq. 4, Kharkov 61077, Ukraine. E-mail: sduplij@gmail.com;     

Continuity of a Quantum Stochastic Process

We investigate a quantum counterpart of the classical notion of a stochastic process continuous with probability one, and prove that the L ²-limit of quantum martingales ‘continuous with probability one’ is a quantum martingale ‘continuous with probability one’. Applications of this result to a number of concrete situations is presented.     

The Quantum Commutator Algebra of a Perfect Fluid

A perfect fluid is quantized by the canonical method. The constraints are found and this allows the Dirac brackets to be calculated. Replacing the Dirac brackets with quantum commutators formally quantizes the system. There is a momentum operator in the denominator of some coordinate quantum commutators. It is shown that it is possible to multiply throughout by this momentum operator. Factor ordering differences can result in a viscosity term. The resulting quantum commutator algebra is unusual.     

Different Types of Conditional Expectation and the Lüders—von Neumann Quantum Measurement

In operator algebra theory, a conditional expectation is usually assumed to be a projection map onto a sub-algebra. In the paper, a further type of conditional expectation and an extension of the Lüders—von Neumann measurement to observables with continuous spectra are considered; both are defined for a single operator and become a projection map only if they exist for all operators. Criteria for the existence of the different types of conditional expectation and of the extension of the Lüders—von Neumann measurement are presented, and the question whether they coincide is studied. All this is done in the general framework of Jordan operator algebras. The examples considered include the type I and type II operator algebras, the standard Hilbert space model of quantum mechanics, and a no-go result concerning the conditional expectation of observables that satisfy the canonical commutator relation.     

Note on Entropies of Quantum Dynamical Systems

We review some techniques and notions for quantum information theory. It is shown that the dynamical entropies is discussed and some numerical computations of these entropies are carried for several states.