Mladen Pavicic

Kochen-Specker Vectors

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Bibliography on Quantum Logics and Related Structures

The bibliography contains 1851 references on axiomatic structures underlying quantum mechanics, with stress on varieties of algebraico-logical, probabilistic, and operational structures for which the term quantum logics is adopted. An index of about 250 keywords picked out from the titles is included and statistics about papers, journals, and authors are presented. Monographs and proceedings on the subject are noted.

Algorithms for Greechie Diagrams

We give a new algorithm for generating Greechie diagrams with arbitrary chosennumber of atoms or blocks (with 2, 3, 4, . . . atoms) and provide a computerprogram for generating the diagrams. The results show that the previous algorithmdoes not produce every diagram and that it is atleast 105 times slower. We alsoprovide an algorithm and programs for checking Greechie diagram passage byequations defining varieties of orthomodular lattices and give examples fromHilbert lattices. We also discuss some additional characteristics of Greechiediagrams.

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.

Quantum and Classical Implication Algebras with Primitive Implications

Join in an orthomodular lattice is obtained inthe same form for all five quantum implications. Theform holds for the classical implication in adistributive lattice as well. Even more, the definition added to an ortholattice makes it orthomodularfor quantum implications and distributive for theclassical one. Based on this result a quantumimplication algebra with a single primitive — andin this sense unique — implication is formulated. Acorresponding classical implication algebra is alsoformulated. The algebras are shown to be special casesof a universal implication algebra.

Nonordered quantum logic and its YES-NO representation

It is shown that an orthomodular lattice is an ortholattice in which aunique operation of bi-implication corresponds to the equality relation and that the ordering relation in the binary formulation of quantum logic as well as the operation of implication (conditional) in quantum logic are completely irrelevant for their axiomatization. The soundness and completeness theorems for the corresponding algebraic unified quantum logic are proved. A proper semantics, i.e., a representation of quantum logic, is given by means of a new YES-NO relation which might enable a proof of the finite model property and the decidability of quantum logic. A statistical YES-NO physical interpretation of the quantum logical propositions is provided.

Minimal quantum logic with merged implications

It is shown that the property of orthomodularity can be interpreted as a particular reduction of the operations of implication to the relation of implication, and a physical interpretation of the result is given.

Quantum logic and quantum computation

Publisher Summary Quantum logic is defined as algebra related to a complete description of quantum systems and its role in quantum computation. A complete description of a quantum system, such as a molecule, includes not only spins as with qubits, but also positions, momenta, and potentials of nucleons and electrons, and this, in the standard approach, requires infinite-dimensional Hilbert spaces. However, the Hilbert lattice has been proved isomorphic to the set of subspaces of an infinite-dimensional Hilbert space. This chapter briefly explores the translation from a classical problem to be computed on a quantum computer. It is noted that the way of handling operations in a computer is what differentiates a classical from a quantum computer and what enables the latter one to speed up exponentially given computations. One of the most successful quantum computing algorithms is Shors algorithm for the classical problem of factoring numbers. This algorithm makes use of a superposition of qubit states so as to reduce the problem of searching for factors to searching for a period of the wave function representing the superposition.

Identity Rule for Classical and Quantum Theories

It is shown that the identity rule — arule of inference which has the form of modus ponens butwith the operation of identity substituted for theoperation of implication — turns any ortholatticeinto either an orthomodular lattice (a model of a quantumtheory) or a distributive lattice (a model of aclassical theory). It is also shown that — asopposed to the implication algebras — one cannotconstruct an identity algebra although the identity rule contains theoperation of identity as the only operation.

Graph approach to quantum systems

Using a graph approach to quantum systems, we show that descriptions of 3-dim Kochen–Specker (KS) setups as well as descriptions of 3-dim spin systems by means of Greechie diagrams (a kind of lattice) that we find in the literature are wrong. Correct lattices generated by McKay-Megill-Pavicic (MMP) hypergraphs and Hilbert subspace equations are given. To enable future exhaustive generation of 3-dim KS setups by means of our recently found stripping technique, bipartite graph generation is used to provide us with lattices with equal numbers of elements and blocks (orthogonal triples of elements)—up to 41 of them. We obtain several new results on such lattices and hypergraphs, in particular, on properties such as superposition and orthoraguesian equations.