Partial and unsharp quantum logics

The total and the sharp character of orthodox quantum logic has been put in question in different contexts. This paper presents the basic ideas for a unified approach to partial and unsharp forms of quantum logic. We prove a completeness theorem for some partial logics based on orthoalgebras and orthomodular posets. We introduce the notion of unsharp orthoalgebra and of generalized MV algebra. The class of all effects of any Hilbert space gives rise to particular examples of these structures. Finally, we investigate the relationship between unsharp orthoalgebras, generalized MV algebras, and orthomodular lattices.     

Effect algebras and unsharp quantum logics

The effects in a quantum-mechanical system form a partial algebra and a partially ordered set which is the prototypical example of the effect algebras discussed in this paper. The relationships among effect algebras and such structures as orthoalgebras and orthomodular posets are investigated, as are morphisms and group- valued measures (or charges) on effect algebras. It is proved that there is a universal group for every effect algebra, as well as a universal vector space over an arbitrary field.     

Toward a formal language for unsharp properties

Some algebraic structures of the set of all effects are investigated and summarized in the notion of a(weak) orthoalgebra. It is shown that these structures can be embedded in a natural way in lattices, via the so-calledMacNeille completion. These structures serve as a model ofparaconsistent quantum logic, orthologic, andorthomodular quantum logic.     

Some New Quantum BCH Codes over Finite Fields

Quantum error correcting codes (QECCs) play an important role in preventing quantum information decoherence. Good quantum stabilizer codes were constructed by classical error correcting codes. In this paper, Bose–Chaudhuri–Hocquenghem (BCH) codes over finite fields are used to construct quantum codes. First, we try to find such classical BCH codes, which contain their dual codes, by studying the suitable cyclotomic cosets. Then, we construct nonbinary quantum BCH codes with given parameter sets. Finally, a new family of quantum BCH codes can be realized by Steane’s enlargement of nonbinary Calderbank-Shor-Steane (CSS) construction and Hermitian construction. We have proven that the cyclotomic cosets are good tools to study quantum BCH codes. The defining sets contain the highest numbers of consecutive integers. Compared with the results in the references, the new quantum BCH codes have better code parameters without restrictions and better lower bounds on minimum distances. What is more, the new quantum codes can be constructed over any finite fields, which enlarges the range of quantum BCH codes.     

Evaluation of two-center electric field gradient integrals over STOs using elliptical coordinates

In this study, analytical expressions are obtained for two different types of two-center EFG integrals and the integrals are calculated using these expressions. Analytical expressions for the integrals are given in terms of auxiliary J and K functions defined by Özmen et al. [Int. J. Quantum Chem. 91, 13 (2003)] for the evaluation of two-center Coulomb integrals using elliptical coordinates. Results are compared and found to be in good agreement for small quantum numbers with the existing results from the literature. Numerical stability of the proposed method is very good and therefore we also present results for higher quantum numbers, however there is no previous results found in the literature to compare.     

Scalable Network Coding for Heterogeneous Devices over Embedded Fields

In complex network environments, there always exist heterogeneous devices with different computational powers. In this work, we propose a novel scalable random linear network coding (RLNC) framework based on embedded fields, so as to endow heterogeneous receivers with different decoding capabilities. In this framework, the source linearly combines the original packets over embedded fields based on a precoding matrix and then encodes the precoded packets over GF(2) before transmission to the network. After justifying the arithmetic compatibility over different finite fields in the encoding process, we derive a sufficient and necessary condition for decodability over different fields. Moreover, we theoretically study the construction of an optimal precoding matrix in terms of decodability. The numerical analysis in classical wireless broadcast networks illustrates that the proposed scalable RLNC not only guarantees a better decoding compatibility over different fields compared with classical RLNC over a single field, but also outperforms Fulcrum RLNC in terms of a better decoding performance over GF(2). Moreover, we take the sparsity of the received binary coding vector into consideration, and demonstrate that for a large enough batch size, this sparsity does not affect the completion delay performance much in a wireless broadcast network.     

Minimal Orthomodular Lattices from Quadratic Spaces over Finite Fields

If T is a finite, nonmodular, orthomodular lattice (OML), T is called minimal ifall its proper subOMLs are modular. In this paper we give a new infinite list ofminimal OMLs. They are obtained from quadratic spaces over a finite field Kof cardinality q 3 (mod 4). Their Greechie diagrams for q = 7 and q = 11are presented in a new way.     

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.     

Some remarks on effects,operations, and unsharp measurements

A brief survey is given of some important features of a generalized quantum mechanics and its measurement theory, together with an overview of papers and monographs representative of the rather new research field dealing with these subjects.     

Some results on BZ structures from Hilbertian unsharp quantum physics

Some algebraic structures determined by the class (þ) of all effects of a Hilbert space þ and by some subclasses of (þ) are investigated, in particular de Morgan-Brouwer-Zadeh posets [it is proved that (þ ⁿ )(n<) has such a structure], Brouwer-Zadeh ^* posets (a quite trivial example consisting of suitable effects is given), and Brouwer-Zadeh ³ posets which are both de Morgan and ^*.It is shown that a nontrivial class of effects of a Hilbert space exists which is a BZ ³ poset. An -preclusivity relation on the set of all vectors of þ is introduced, and it is shown that it satisfies the regularity condition also for [1/2, 1].     

A unified framework for the algebra of unsharp quantum mechanics

On the basis of the concrete operations definable on the set of effect operators on a Hilbert space, an abstract algebraic structure of sum Brouwer-Zadeh (SBZ)-algebra is introduced. This structure consists of a partial sum operation and two mappings which turn out to be Kleene and Brouwer unusual orthocomplementations. The Foulis-Bennett effect algebra substructure induced by any SBZ-algebra, allows one to introduce the notions of unsharp “state” and “observable” in such a way that any “state-observable” composition is a standad probability measure (classical state). The Cattaneo-Nisticò BZ substructure induced by any SBZ-algebra permits one to distinguish, in an equational and simple way, the sharp elements from the really unsharp ones. The family of all sharp elements turns out to be a Foulis-Randall orthoalgebra. Any unsharp element can be “roughly” approximated by a pair of sharp elements representing the best sharp approximation from the bottom and from the top respectively, according to an abstract generalization introduced by Cattaneo of Pawlack “rough set” theory (a generalization of set theory, complementary to fuzzy set theory, which describes approximate knowledge with applications in computer sciences). In both the concrete examples of fuzzy sets and effect operators the “algebra” of rough elements shows a weak SBZ structure (weak effect algebra plus BZ standard poset) whose investigation is set as an interesting open problem.     

On the Geometry of Orthomodular Spaces over Fields of Power Series

Orthomodular spaces are generalizations ofHilbert spaces with which they share the basic propertyexpressed by the Projection Theorem. We study twoinfinite-dimensional orthomodular spaces, bothconstructed over the same field of power series, but withdifferent inner products. On the first space everybounded, self-adjoint operator decomposes into anorthogonal sum of operators of rank 1 or 2; on thesecond space, in contrast, there exist self-adjointoperators that are undecomposable. These differencesreflect the fact that the underlying geometries aredissimiliar.     

Fuzzy quantum logic II. The logics of unsharp quantum mechanics

A survey of the main results of the Italian group about the logics of unsharp quantum mechanics is presented. In particular partial ordered structures playing with respect to effect operators (linear bounded operatorsF on a Hilbert space such that, 0¦F²) the role played by orthomodular posets with respect to orthogonal projections (corresponding to sharp effects) are analyzed. These structures are generally characterized by the splitting of standard orthocomplementation on projectors into two nonusual orthocomplementations (afuzzy-like and anintuitionistic-like) giving rise to different kinds of Brouwer-Zadeh (BZ) posets: de Morgan BZ posets, BZ^* posets, and BZ³ posets. Physically relevant generalizations of ortho-pair semantics (paraconsistent, regular paraconsistent, and minimal quantum logics) are introduced and their relevance with respect to the logic of unsharp quantum mechanics are discussed.     

Axiomatic unsharp quantum theory (From Mackey to Ludwig and Piron)

On the basis of Mackey's axiomatic approach to quantum physics or, equivalently, of a state-event-probability (SEVP) structure, using a quite standard fuzzification procedure, a set of unsharp events (or effects) is constructed and the corresponding state-effect-probability (SEFP) structure is introduced. The introduction of some suitable axioms gives rise to a partially ordered structure of quantum Brouwer-Zadeh (BZ) poset; i.e., a poset endowed with two nonusual orthocomplementation mappings, a fuzzy-like orthocomplementation, and an intuitionistic-like orthocomplementation, whose set of sharp elements is an orthomodular complete lattice. As customary, by these orthocomplementations the two modal-like necessity and possibility operators are introduced, and it is shown that Ludwig's and Jauch-Piron's approaches to quantum physics are interpreted in complete SEFP. As a marginal result, a standard procedure to construct a lot of unsharp realizations starting from any sharp realization of a fixed observable is given, and the relationship among sharp and corresponding unsharp realizations is studied.     

Extended Fermi coordinates

We extend the notion of Fermi coordinates to a generalized definition in which the highest orders are described by arbitrary functions. From this definition rises a formalism that naturally gives coordinate transformation formulae. Some examples are developed in order to discuss the physical meaning of Fermi coordinates.     

Canonical normal coordinates

We examine the advantages and disadvantages of fixing the coordinate gauge to be that of the Riemann normal coordinates by applying the Dirac formalism for constrained canonical systems in general relativity.     

Adapted harmonic coordinates

We obtain the necessary and sufficient conditions that a timelike congruence has to satisfy to admit three independent adapted harmonic coordinates of space, proving in the process that if it does then these coordinates are unique up to a linear transformation with constant coefficients. As a particular example we prove that irrotational pure Born (i.e. not Killing) congruences never admit a system of adapted harmonic coordinates of space.     

Isotropic spherical coordinates

The method of using regularization to analyze singularities is applied to the Schwarzschild metric in isotropic spherical coordinates (3). It is found that there arises a singularity at the gravitational radius which does not satisfy the conditions of physical realizability (T =T ,T ₀ ⁰ =T =0). Consequently, this metric cannot be considered as corresponding to pure vacuum.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 84–87, April, 1987.     

On identifying coordinates

Various techniques are reviewed for identifying the coordinates employed in a given line element. In particular the questions of coordinate range, signature, character of coordinates, isometries, and linearization are considered. The consideration of isometries leads to the introduction of functional form-invariant (FFI) transformations, which may in particular be of help in obtaining Killing solutions. Finally the general question of coordinate identification is discussed.     

Bethe Approximation for Pure and Diluted Magnets as Averaging over Local Exchange Fields

Russian Physics Journal - An interpretation of the Bethe approximation based on the method of averaging over local exchange fields taking into account the correlation of neighboring spins is...