Stanley Gudder

Quantum Computational Logic

A quantum computational logic is constructed by employing density operators on spaces of qubits and quantum gates represented by unitary operators. It is shown that this quantum computational logic is isomorphic to the basic sequential effect algebra [0, 1].

On Hidden‐Variable Theories

An abstract definition of a general hidden‐variables theory is given, and it is shown that such a theory is always possible in the present framework of quantum mechanics and is, in fact, unique in a certain sense. It is noted that the Bohm‐Bub hidden‐variables example is contained in this theory and an attempt is made to clarify the position of this theory with respect to hidden‐variable impossibility proofs. The general definition is used in the consideration of quantum‐mechanical ordering and the measurement process.

Quantum Markov chains

A new approach to quantum Markov chains is presented. We first define a transition operation matrix (TOM) as a matrix whose entries are completely positive maps whose column sums form a quantum operation. A quantum Markov chain is defined to be a pair (G,E) where G is a directed graph and E=[Eij] is a TOM whose entry Eij labels the edge from vertex j to vertex i. We think of the vertices of G as sites that a quantum system can occupy and Eij is the transition operation from site j to site i in one time step. The discrete dynamics of the system is obtained by iterating the TOM E. We next consider a special type of TOM called a transition effect matrix. In this case, there are two types of dynamics, a state dynamics and an operator dynamics. Although these two types are not identical, they are statistically equivalent. We next give examples that illustrate various properties of quantum Markov chains. We conclude by showing that our formalism generalizes the usual framework for quantum random walks.

A general theory of convexity

SuntoSi sviluppa una generalizzazione della nozione usuale di convessità. Si mostra che per certe applicazioni alcuni dei postulati della teoria generalizzata devone essere indeboliti. Si discute l’indipendenza dei postulati e si stabiliscono teoremi di rappresentazione. Si costruisce una topologia compatibile con la struttura di convessità.AbstractA generalization of the usual notion of convexity is developed. It is shown that for certain applications some of the postulates of the generalized theory must be relaxed. The independence of the postulates is discussed and representation theorems are given. A topology which is compatible with the convexity structure is constructed.

Convex structures and operational quantum mechanics

A general mathematical framework called a convex structure is introduced. This framework generalizes the usual concept of a convex set in a real linear space. A metric is constructed on a convex structure and it is shown that mappings which preserve the structure are contractions. Convex structures which are isomomorphic to convex sets are characterized and for such convex structures it is shown that the metric is induced by a norm and that structure preserving mappings can be extended to bounded linear operators.Convex structures are shown to give an axiomatization of the states of a physical system and the metric is physically motivated. We demonstrate how convex structures give a generalizing and unifying formalism for convex set and operational methods in axiomatic quantum mechanics.

Convex and linear effect algebras

Abstract It is shown that convex effect algebras arise naturally in the description of a physical statistical system. An effect algebra that is a convex subset of a real linear space is called a linear effect algebra and it is demonstrated that any convex effect algebra is affinely isomorphic to a linear effect algebra. Convex effect algebras that possess separating and order determining state spaces are characterized. It is shown that an effect algebra P is imbeddable in an interval of an order unit space if and only if the state space of P is order determining. Sharp and extreme elements of convex effect algebras are studied and compared. An alternative definition of a convex effect algebra called a CE-algebra is considered and MV-algebras are discussed.

Lattice properties of quantum effects

Sufficient conditions for the existence of the infimum AΛB of two quantum effects A and B are given. The existence of AΛB is characterized for commuting A and B with pure point spectrum. Properties of a generalized infimum and supremum are studied. Some previous finite dimensional, commutative results are extended to the infinite dimensional and noncommutative case.

A Superposition Principle in Physics

A superposition principle is considered both in classical mechanics and in the quantum logic approach to quantum mechanics. It is shown, roughly speaking, that in classical mechanics the only type of superposition of states is a mixture and that no pure state is a nontrivial superposition of other pure states. In quantum mechanics it is shown that, if a superposition principle holds, then the quantum logic is a complete atomic lattice.

Sequential product of quantum effects

Unsharp quantum measurements can be modelled by means of the class e(H) of positive contractions on a Hilbert space H, in brief, quantum effects. For A, B ∈ e(H) the operation of sequential product AcB = A 1/2 BA 1/2 was proposed as a model for sequential quantum measurements. We continue these investigations on sequential product and answer positively the following question: the assumption A o B > B implies AB = BA = B. Then we propose a geometric approach of quantum effects and their sequential product by means of contractively contained Hilbert spaces and operator ranges. This framework leads us naturally to consider lattice properties of quantum effects, sums and intersections, and to prove that the sequential product is left distributive with respect to the intersection.

S-DOMINATING EFFECT ALGEBRAS

A special type of effect algebra called anS-dominating effect algebra is introduced. It is shownthat an S-dominating effect algebra P has a naturallydefined Brouwer-complementation that gives P thestructure of a Brouwer–Zadeh poset. This enables usto prove that the sharp elements of P form anorthomodular lattice. We then show that a standardHilbert space effect algebra is S-dominating. Weconclude that S-dominating effect algebras may be usefulabstract models for sets of quantum effects in physicalsystems.