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0 A Quantum Logical and Geometrical Approach to the Study of Improper Mixtures

Graciela Domenech, Federico Holik, César Massri

Published Jan 1, 2009• 19 citations
OpenAlex

Abstract

In this work we present a study of improper mixtures from a quantum logical and geometrical point of view. We do not follow the standard approach which considers improper mixtures as measures over the algebra of projections. Instead of it, we consider the possibility of extending the lattice of projections in order to include improper mixtures as atoms of a new lattice. This lattice structure is constructed using the convex set of states instead of the traditional set of rays in the Hilbert space. We study the relationship between these two lattices and argue that this approach is suitable for the study of compound systems. Finally, we study the maps which can be defined between the extended lattices of a compound system and its subsystems. Key words: convex set of states, quantum logic, entanglement. Fellow of the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)

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In this work we present a study of improper mixtures from a quantum logical and geometrical point of view. We do not follow the standard approach which considers improper mixtures as measures over the algebra of projections. Instead of it, we consider the possibility of extending the lattice of projections in order to include improper mixtures as atoms of a new lattice. This lattice structure is constructed using the convex set of states instead of the traditional set of rays in the Hilbert space. We study the relationship between these two lattices and argue that this approach is suitable for the study of compound systems. Finally, we study the maps which can be defined between the extended lattices of a compound system and its subsystems. Key words: convex set of states, quantum logic, entanglement. Fellow of the Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET)
We study improper mixtures from a quantum logical and geometrical point of view. Taking into account the fact that improper mixtures do not admit an ignorance interpretation and must be considered as states in their own right, we do not follow the standard approach which considers improper mixtures as measures over the algebra of projections. Instead of it, we use the convex set of states in order to construct a new lattice whose atoms are all physical states: pure states and improper mixtures. This is done in order to overcome one of the problems which appear in the standard quantum logical formalism, namely, that for a subsystem of a larger system in an entangled state, the conjunction of all actual properties of the subsystem does not yield its actual state. In fact, its state is an improper mixture and cannot be represented in the von Neumann lattice as a minimal property which determines all other properties as is the case for pure states or classical systems. The new lattice also contains all propositions of the von Neumann lattice. We argue that this extension expresses in an algebraic form the fact that—alike the classical case—quantum interactions produce nontrivial correlations between the systems. Finally, we study the maps which can be defined between the extended lattice of a compound system and the lattices of its subsystems.
In this work we study the convex set of quantum states from a quantum logical point of view. We consider an algebraic structure based on the convex subsets of this set. The relationship of this algebraic structure with the lattice of propositions of quantum logic is shown. This new structure is suitable for the study of compound systems and shows new differences between quantum and classical mechanics. This differences are linked to the nontrivial correlations which appear when quantum systems interact. They are reflected in the new propositional structure, and do not have a classical analogue. This approach is also suitable for an algebraic characterization of entanglement.
It is well - known that the set of states of a given quantum mechanical system is to be closed from the point of view of the operational approach if we want to make mixed states or convex combinations. That is, s1s_1s1​ and s2s_2s2​ are states, then the same is to be true for λs1+(1−λ)s2,\lambda s_1 +(1-\lambda) s_2,λs1​+(1−λ)s2​, where 0<λ<1.0 < \lambda < 1.0<λ<1. We can define a convex combination of elements in a linear space, but unfortunately, in the general case the linear space is artificial for the set of states and has no physical meaning, but the procedure of forming the mixtures of states has a natural meaning. This is why we provide an abstract definition of the mixtures, which is independent of the linearity notion. We call this space a convex structure. In the work we consider state spaces, generalized state spaces, in which we select pure states, define operations and effects associated with the operations. We also consider ultraproducts of the sequences of these structures, operations and effects.
What picture does one see, looking at a physical theory from a distance, so that the details disappear? Since quantum mechanics is a statistical theory, the most universal picture which remains after the details are forgotten is that of a convex set. Bogdan Mielnik Convex sets Our object is to understand the geometry of the set of all possible states of a quantum system that can occur in nature. This is a very general question, especially since we are not trying to define ‘state’ or ‘system’ very precisely. Indeed we will not even discuss whether the state is a property of a thing, or of the preparation of a thing, or of a belief about a thing. Nevertheless we can ask what kind of restrictions are needed on a set if it is going to serve as a space of states in the first place. There is a restriction that arises naturally both in quantum mechanics and in classical statistics: the set must be a convex set . The idea is that a convex set is a set such that one can form ‘mixtures’ of any pair of points in the set. This is, as we will see, how probability enters (although we are not trying to define ‘probability’ either). From a geometrical point of view a mixture of two states can be defined as a point on the segment of the straight line between the two points that represent the states that we want to mix. We insist that given two points belonging to the set of states, the straight line segment between them must belong to the set too. This is certainly not true of any set.
According to quantum mechanics, the pure states of a microsystem are represented by vectors in a Hilbert Space. Sentences of the form, "x є L" (where x is the state vector for a system, L a subspace of the appropriate Hilbert space), may be called Q-propositions: such sentences serve to summarise our information about the results of possible experiments on the system. Quantum logic investigates the relations which hold among the Q-propositions about a given physical sys tem. These logical relations correspond to algebraic relations among the subspaces of Hilbert space. The algebra of this set of subspaces is non-Boolean, and may be regarded either as an orthomodular lattice or as a partial Boolean algebra. With each type of structure we can associate a logic. A general approach to the semantics for such a logic is provided in terms of interpretations of a formal language within an algebraic structure; an interpretation maps sentences of the language homomorphically onto elements of the structure. When the structure in question is a Boolean algera, the resulting logic is classical; here we develop a semantics for the logic associated with partial Boolean algebras. Two systems of proof, based on the natural deduction systems of Gentzen, are shown for this logic. With respect to the given sematics, these calculi are sound and weakly complete. Strong completeness is conjectured. Quantum logic deals with the logical relations between sentences, and so is properly called a logic. However, it is the logic appropriate to a limited class of sentences: proposals that it should replace classical logic wherever the latter is used should be viewed with suspicion.
A set of statements about the properties of a quantum system is looked at as at a partially ordered set of subspaces of finite or infinite dimensional Hilbert space. The operation of ordering is performed on a set of propositions comparing the truth values of these propositions and on the set of subspaces as the opera- tion of inclusion. Based on the required properties these structures are translated into operations on the lattice. The correspondence with Heisenberg uncertainty principle is shown there. Furthermore, it is shown that the lattices correspond- ing to the subspaces of infinite dimensional Hilbert space are not modular. This property is replaced with weaker property of orthomodularity, when operation of the negation is added. Following the work of G. Birkhoff and J. von Neumann, the structure of quantum logic is looked for in projective spaces, which are in- troduced either arithmetically or axiomatically. The examples of quantum logic, their physical implementation and eventual implementation in projective spaces are analysed. 1