Hilbert lattices: New results and unsolved problems

The class of Hilbert lattices that derive from orthomodular spaces containing infinite orthonormal sets (normal Hilbert lattices) is investigated. Relevant open problems are listed. Comments on form-topological orthomodular spaces and results on arbitrary orthomodular spaces are appended.Deceased (October 29, 1989).     

Equations Holding in Hilbert Lattices

We produce and study several sequences of equations, in the language of orthomodular lattices, which hold in the ortholattice of closed subspaces of any classical Hilbert space, but not in all orthomodular lattices. Most of these equations hold in any orthomodular lattice admitting a strong set of states whose values are in a real Hilbert space. For some of these equations, we give conditions under which they hold in the ortholattice of closed subspaces of a generalised Hilbert space. These conditions are relative to the dimension of the Hilbert space and to the characteristic of its division ring of scalars. In some cases, we show that these equations cannot be deduced from the already known equations, and we study their mutual independence. To conclude, we suggest a new method for obtaining such equations, using the tensorial product. PACS numbers: 02.10, 03.65, 03.67     

Deduction, Ordering, and Operations in Quantum Logic

We show that in quantum logic of closed subspaces of Hilbert space one cannot substitute quantum operations for classical (standard Hilbert space) ones and treat them as primitive operations. We consider two possible ways of such a substitution and arrive at operation algebras that are not lattices what proves the claim. We devise algorithms and programs which write down any two-variable expression in an orthomodular lattice by means of classical and quantum operations in an identical form. Our results show that lattice structure and classical operations uniquely determine quantum logic underlying Hilbert space. As a consequence of our result, recent proposals for a deduction theorem with quantum operations in an orthomodular lattice as well as a, substitution of quantum operations for the usual standard Hilbert space ones in quantum logic prove to be misleading. Quantum computer quantum logic is also discussed.     

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.     

Spaces of Abstract Events

We generalize the concept of a space of numerical events in such a way that this generalization corresponds to arbitrary orthomodular posets whereas spaces of numerical events correspond to orthomodular posets having a full set of states. Moreover, we show that there is a natural one-to-one correspondence between orthomodular posets and certain posets with sectionally antitone involutions. Finally, we characterize orthomodular lattices among orthomodular posets.     

Testing for Classicality of a Physical System

Often quantum logics are algebraically modelled by orthomodular posets. The physical system described by such a quantum logic is classical if and only if the corresponding orthomodular poset is a Boolean algebra. We provide an easy testing procedure for this case. Moreover, we characterize orthomodular posets which are lattices and consider orthomodular posets which admit a full set of states and hence represent so-called spaces of numerical events. This way further test procedures are obtained.     

Complementarity in Categorical Quantum Mechanics

We relate notions of complementarity in three layers of quantum mechanics: (i) von Neumann algebras, (ii) Hilbert spaces, and (iii) orthomodular lattices. Taking a more general categorical perspective of which the above are instances, we consider dagger monoidal kernel categories for (ii), so that (i) become (sub)endohomsets and (iii) become subobject lattices. By developing a ‘point-free’ definition of copyability we link (i) commutative von Neumann subalgebras, (ii) classical structures, and (iii) Boolean subalgebras.     

The Covering Law in Orthomodular Lattices Generated by Graphs of Functions

In the paper [2] we introduced and investigated complete orthomodular lattices generated by graphs of continuous functions. A natural question arises: can such a lattice be represented by the lattice of projectors in a Hilbert space (the standard quantum logic)? The answer is no, because the covering law is not satisfied in this case.     

Quantum logic revisited

An adequate conjunction-implication pair is given for complete orthomodular lattices. The resulting conjunction is noncommutative in nature. We use the well-known lattice of closed subspaces of a Hilbert space, to give physical meaning to the given lattice operation.To the memory of Thomas A. Brody.     

Characterizations of Spectral Automorphisms and a Stone-Type Theorem in Orthomodular Lattices

The notion of spectral automorphism of an orthomodular lattice was introduced by Ivanov and Caragheorgheopol (Int. J. Theor. Phys. 49(12):3146–3152, 2010) to create an analogue of the Hilbert space spectral theory in the abstract framework of orthomodular lattices. We develop the theory of spectral automorphisms finding previously missing characterizations of spectral automorphisms, discussing several examples and the possibility to construct such automorphisms in direct products or horizontal sums of lattices. A factorization of the spectrum of a spectral automorphism is found. The last part of the paper addresses the problem of the unitary time evolution of a system from the point of view of the spectral automorphisms theory. An analogue of the Stone theorem concerning strongly continuous one-parameter unitary groups is given.     

Characterization of projection lattices of hilbert spaces

The classical lattices of projections of Hubert spaces over the real, the complex, or the quaternion number field are characterized among the totality of irreducible, complete, orthomodular, atomic lattices satisfying the covering property. To this end, so-called paratopological lattices are introduced, i.e., lattices carrying a topology that renders the lattice operations restrictedly continuous.     

Generalized Hermitian Algebras

We refer to the real Jordan Banach algebra of bounded Hermitian operators on a Hilbert space as a Hermitian algebra. In this paper we define and launch a study of a class of generalized Hermitian (GH) algebras. Among the examples of GH-algebras are ordered special Jordan algebras, JW-algebras, and AJW-algebras, but unlike these more restricted cases, a GH-algebra is not necessarily a Banach space and its lattice of projections is not necessarily complete. In this paper we develop the basic theory of GH-algebras, identify their unit intervals as effect algebras, and observe that their projection lattices are sigma-complete orthomodular lattices. We show that GH-algebras are spectral order-unit spaces and that they admit a substantial spectral theory. The second author was supported by Research and Development Support Agency under the contract No. APVV-0071-06, grant VEGA 2/0032/09 and Center of Excellence SAS, CEPI I/2/2005.     

Lattices and Quantum Logics with Separated Intervals, Atomicity

It is well known that a Boolean algebra B isatomic (atomistic) iff the interval topology on B isHausdorff. But this no longer holds for orthomodularlattices (quantum logics). There exist (even complete) atomic orthomodular lattices the intervaltopology of which is not Hausdorff. We show that anothercharacterization of atomicity for Boolean algebras isthe following: A Boolean algebra B is atomic iff B has separated intervals. Furthermore, we showthat the interval topology on a complete orthomodularlattice L is Hausdorff iff L has separated intervals iffL is atomic and it has separated intervals. An orthomodular lattice L with orthomodularMacNeille completion has separatedintervals iff L is atomic and it has separated intervalsiff the interval topology on isHausdorff.     

Noncommutativity and transition probability in quantum mechanics

Using an operation that behaves as a noncommutative conjuction in orthomodular lattices, a way to define the transition probability for arbitrary quantum logics is given.     

Small Quantum Structures with Small State Spaces

We summarize and extend results about “small” quantum structures with small dimensions of state spaces. These constructions have contributed to the theory of orthomodular lattices. More general quantum structures (orthomodular posets, orthoalgebras, and effect algebras) admit sometimes simplifications, but there are problems where no progress has been achieved.     

Pre-ideals of Basic Algebras

Basic algebras are a generalization of MV-algebras, also including orthomodular lattices and lattice effect algebras. A pre-ideal of a basic algebra is a non-empty subset that is closed under the addition ⊕ and downwards closed with respect to the underlying order. In this paper, we study the pre-ideal lattices of algebras in a particular subclass of basic algebras which are closer to MV-algebras than basic algebras in general. We also prove that finite members of this subclass are exactly finite MV-algebras.     

Type-Decomposition of an Effect Algebra

Effect algebras (EAs), play a significant role in quantum logic, are featured in the theory of partially ordered Abelian groups, and generalize orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras. We study centrally orthocomplete effect algebras (COEAs), i.e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. For COEAs, we introduce a general notion of decomposition into types; prove that a COEA factors uniquely as a direct sum of types I, II, and III; and obtain a generalization for COEAs of Ramsay’s fourfold decomposition of a complete orthomodular lattice.     

Logical Approach for Two-Valued States on Quantum Systems

In this paper we develop a logical system associated to two-valued states on orthomodular lattices. An completeness theorem with respect to a variety of orthomodular lattices enriched with an unary operation that represents two-valued states is given.     

Embedding quantum logics in Hilbert space

We show that an orthomodular lattice is embeddable in a Hilbert space if and only if states of a certain kind exist. A physical motivation for the existence of such states is given and a connection is provided between the quantum logic, algebraic, and operational approaches to quantum mechanics.     

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.