Generalized Effect Algebras of Positive Operators Densely Defined on Hilbert Spaces

Axioms of quantum structures, motivated by properties of some sets of linear operators in Hilbert spaces are studied. Namely, we consider examples of sets of positive linear operators defined on a dense linear subspace D in a (complex) Hilbert space ℋ. Some of these operators may have a physical meaning in quantum mechanics. We prove that the set of all positive linear operators with fixed such D and ℋ form a generalized effect algebra with respect to the usual addition of operators. Some sub-algebras are also mentioned. Moreover, on a set of all positive linear operators densely defined in an infinite dimensional complex Hilbert space, the partial binary operation is defined making this set a generalized effect algebra.     

Arrow of Time in Rigged Hilbert Space Quantum Mechanics

Arno Bohm and Ilya Prigogine's Brussels–Austin Group have been working on the quantum mechanical arrow of time and irreversibility in rigged Hilbert space quantum mechanics. A crucial notion in Bohm's approach is the so-called preparation/registration arrow. An analysis of this arrow and its role in Bohm's theory of scattering is given. Similarly, the Brussels–Austin Group uses an excitation/de-excitation arrow for ordering events, which is also analyzed. The relationship between the two approaches is initially discussed focusing on their semi-group operators and time arrows. Finally a possible realist interpretation of the rigged Hilbert space formulation of quantum mechanics is considered.     

The Representation Theory of Decoherence Functionals in History Quantum Theories

In the first part of this paper the general perspective of history quantum theoriesis reviewed. History quantum theories provide a conceptual and mathematicalframework for formulating quantum theories without a globally definedHamiltonian time evolution and for introducing the concept of space-time eventinto quantum theory. On a mathematical level a history quantum theory ischaracterized by the space of histories, which represent the space-time events, andby the space of decoherence functionals, which represent the quantum mechanicalstates in the history approach. The second part of this paper is devoted to thestudy of the structure of the space of decoherence functionals for some physicallyreasonable spaces of histories in some detail. The temporal reformulation ofstandard Hamiltonian quantum theories suggests to consider the case that thespace of histories is given by (i) the lattice of projection operators on someHilbert space or, slightly more generally, (ii) the set of projection operators insome von Neumann algebra. In the case (i) the conditions are identified underwhich decoherence functionals can be represented by, respectively, trace classoperators, bounded operators, or families of trace class operators on the tensorproduct of the underlying Hilbert space by itself. Moreover, we discuss thenaturally arising representations of decoherence functionals as sesquilinear forms.The paper ends with a discussion of the consequences of the results for thegeneral axiomatic framework of history theories.     

Properties of Quasi-Hermitian Operators Inherited from Self-Adjoint Operators

We study a generalized effect algebra of unbounded linear operators in an infinite-dimensional complex Hilbert space. This algebra equipped with a certain kind of topology allows us to show that unbounded quasi-Hermitian operators can be expressed as a difference of two infinite sums of bounded quasi-Hermitian operators.     

Nonideal quantum measurements

A partial ordering in the class of observables ( positive operator-valued measures, introduced by Davies and by Ludwig) is explored. The ordering is interpreted as a form of nonideality, and it allows one to compare ideal and nonideal versions of the same observable. Optimality is defined as maximality in the sense of the ordering. The framework gives a generalization of the usual (implicit) definition of self-adjoint operators as optimal observables (von Neumann), but it can, in contrast to this latter definition, be justified operationally. The nonideality notion is compared to other quantum estimation theoretic methods. Measures for the amount of nonideality are derived from information theory.     

Groupoids,von Neumann Algebras and the Integrated Density of States

We study spectral properties of random operators in the general setting of groupoids and von Neumann algebras. In particular, we establish an explicit formula for the canonical trace of the von Neumann algebra of random operators and define an abstract density of states. While the treatment applies to a general framework we lay special emphasis on three particular examples: random Schrödinger operators on manifolds, quantum percolation and quasi–crystal Hamiltonians. For these examples we show that the distribution function of the abstract density of states coincides with the integrated density of states defined via an exhaustion procedure.     

Local nets and self-adjoint extensions of quantum field operators

We consider Wightman fields having the property that some closed extensions of the field operators generate locally commuting von Neumann algebras. We show that for such fields the hermitian field operators have self-adjoint extensions, possibly in an enlarged Hilbert space, such that bounded functions of the self-adjoint operators commute locally.     

Convergence of observables on quantum logics

We define two types of convergence for observables on a quantum logic which we call M-weak and uniform M-weak convergence. These convergence modes correspond to weak convergence of probability measures. They are motivated by the idea that two (in general unbounded) observables are close if bounded functions of them are close. We show that M-weak and uniform M-weak convergence generalize strong resolvent and norm resolvent convergence for self-adjoint operators on a Hilbert space. Also, these types of convergence strengthen the weak operator convergence and operator norm convergence of bounded self-adjoint operators on a Hilbert space. Finally, we consider spectral perturbation by showing that the spectra of approximating observables approach the spectrum of the limit in a certain sense.     

The logic of quantum systems with diagonal singularities

The work of the Brussels-Austin groups on irreversibility over the last years has shown that Quantum Large Poincaré systems with diagonal singularity lead to an extension of the conventional formulation of dynamics at the level of mixtures which is manifestly time asymmetric. States with diagonal singularity acquire meaning as linear fractionals over the involutive Banach algebra of operators with diagonal singularity. We show in this paper that the logic of quantum systems with diagonal singularity is not the conventional logic of Hilbert space, because only finite combinations of prepositions are allowed.     

Brouwer-Zadeh (Fuzzy-intuitionistic) posets for unsharp quantum mechanics

The partial ordered structure which plays for unsharp quantum mechanics the same role of orthomodular lattices for ordinary quantum mechanics is introduced. Differently from the unsharp case, in which one can identify quantum propositions (i.e., Hilbert space subspaces) with yes-no devices (i.e., orthogonal projections) they are tested by, in the unsharp case this identification is broken down: every quantum generalized proposition (i.e., pair of mutually orthogonal subspaces) is tested by many different yes-no devices (i.e., Hilbert space effects). The set of all quantum effects has a structure of Brouwer-Zadeh poset, canonically embeddable in a (minimal) Brouwer-Zadeh lattice, whereas the set of all quantum generalized propositions has a structure of Brouwer-Zadeh complete lattice.A Brouwer-Zadeh poset is defined as a partially ordered structure equipped with two nonusual orthocomplementations: a regular degenerate (Zadeh or fuzzy-like) one and a weak (Brouwer or intuitionistic-like) one linked by an interconnection rule. Using these two orthocomplementations it is possible to introduce the two modal-like operators of necessity and possibility.     

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.     

On Realization of Partially Ordered Abelian Groups

The paper is devoted to algebraic structures connected with the logic of quantum mechanics. Since every (generalized) effect algebra with an order determining set of (generalized) states can be represented by means of an abelian partially ordered group and events in quantum mechanics can be described by positive operators in a suitable Hilbert space, we are focused in a representation of partially ordered abelian groups by means of sets of suitable linear operators. We show that there is a set of points separating ?-maps on a given partially ordered abelian group G if and only if there is an injective non-trivial homomorphism of G to the symmetric operators on a dense set in a complex Hilbert space $\mathcal{H}$ which is equivalent to an existence of an injective non-trivial homomorphism of G into a certain power of ?. A similar characterization is derived for an order determining set of ?-maps and symmetric operators on a dense set in a complex Hilbert space $\mathcal{H}$ . We also characterize effect algebras with an order determining set of states as interval operator effect algebras in groups of self-adjoint bounded linear operators.     

Perturbation Formulas for Traces on Normed Ideals

We prove perturbation results for traces on normed ideals in semifinite von Neumann algebra factors. This includes the case of Dixmier traces. In particular, we establish existence of spectral shift measures with initial operators being dissipative or bounded, and show that these measures can have singular components in the case of Dixmier traces. We also establish a linearization formula for a Dixmier trace applied to perturbed operator functions, a result that does not typically hold for normal traces.     

Canonical quantization of classical mechanics in curvilinear coordinates. Invariant quantization procedure

In the paper is presented an invariant quantization procedure of classical mechanics on the phase space over flat configuration space. Then, the passage to an operator representation of quantum mechanics in a Hilbert space over configuration space is derived. An explicit form of position and momentum operators as well as their appropriate ordering in arbitrary curvilinear coordinates is demonstrated. Finally, the extension of presented formalism onto non-flat case and related ambiguities of the process of quantization are discussed.     

Analysis of the role of von Neumann’s projection postulate in the canonical scheme of quantum teleportation

Modern development of quantum technologies based on quantum information theory (in particular, laser-based quantum-information technologies) stimulated the analysis of proposed computational, cryptographic, and teleportational schemes from the viewpoint of quantum foundations. It is evident that not all mathematical calculations performed in the complex Hilbert space can directly be realized in the physical space. Recently, analyzing the original EPR paper, we found that their argument was based on the misuse of von Neumann’s projection postulate. In contrast to von Neumann, Einstein, Podolsky, and Rosen (EPR) applied this postulate to the observables represented by the operators with degenerate spectra. It was completely forbidden by von Neumann’s axiomatics of quantum mechanics. It is impossible to repeat the EPR considerations within the von Neumann’s framework. We analyze here quantum teleportation by taking into account von Neumann’s projection postulate. Our analysis shows that the so-called quantum teleportation is impossible within the von Neumann’s framework.     

Joint distributions of observables on quantum logics

A joint distribution of a set of observables on a quantum logic in a statem is defined and its properties are derived. It is shown that if the joint distribution exists, then the observables can be represented in the statem by a set of commuting operators on a Hilbert space.     

Feynman quantization: An operator approach to path integration over commuting and anticommuting variables

We present an operator quantization scheme on a continuous direct product of Hilbert spaces over a time interval as an extension of the quantization using Feynman path integrals. We define the continuous direct product as a Hilbert space with two principal bases: the Fock and the Feynman ones. The Fock basis, defined by a complete set of commuting operators at different times, serves for a definition of the operator calculus. The Feynman basis, simultaneously diagonalizing the complete set of commuting operators, leads to path integrals constructed without time slicing as a spectral representation of certain operator functions. The construction of quantum theory and the corresponding path integrals for the harmonic oscillator is demonstrated both in the configuration and phase spaces. The extension of the theory to coherent states and anticommuting variables is performed.     

Do fuzzy quantum structures exist?

Recently, some fuzzy quantum structures were introduced. We focus on the fuzzy quantum logics arising from the isomorphism of some quantum logics and some systems of fuzzy subsets of the ordering sets of states. In general, a fuzzy quantum logic is equipped with the pointwise-defined fuzzy connectives generated by a common generator g. Stressing the pointwise nature of fuzzy structures and omitting the global properties of quantum elements, we find that only crisp values of elements of a fuzzy quantum logic are allowed. Consequently, fuzzy quantum structures do not exist! However, there exist quantum structures of fuzzy subsets.     

Measurement-Based Quantum Computation and Undecidable Logic

We establish a connection between measurement-based quantum computation and the field of mathematical logic. We show that the computational power of an important class of quantum states called graph states, representing resources for measurement-based quantum computation, is reflected in the expressive power of (classical) formal logic languages defined on the underlying mathematical graphs. In particular, we show that for all graph state resources which can yield a computational speed-up with respect to classical computation, the underlying graphs—describing the quantum correlations of the states—are associated with undecidable logic theories. Here undecidability is to be interpreted in a sense similar to Gödel’s incompleteness results, meaning that there exist propositions, expressible in the above classical formal logic, which cannot be proven or disproven.