It is the purpose of this paper to show that, in any separable Hilbert space of dimension at least three, whether real or complex, every measure on the closed subspaces is derived in this fashion. For example, some mathematical aspects of the notion of probability involved by the density operator have been studied by Veeravalli Varadarajan .
But it was the representation theorem of Constantin Piron  which clarified the field. The theorem states that if L is a complete orthocomplemented atomic lattice which is weakly modular and satisfies the covering law, then each irreducible component of the lattice L can be represented as the lattice of all biorthogonal subspaces of a vector space V over a division ring K. In the sixties, Jauch and Piron [, ] also aimed at reconstructing the formalism of QM from first principles with special interest in the relation between concepts and real physical operations that can be performed in the laboratory.
The distinction between the system and its states cannot be maintained under all circumstances with the precision implied by this definition. The reason is that systems which we regard under normal circumstances as different may be considered as two different states of the same system. An example is a positronium and a system of two photons. Moreover, its purpose is to attempt to give an independent motivation to the general program to understand QM .
One of the main results of the operational line of research is due to Aerts in Orthodox QL faces a deep problem for treating composite systems. In fact, when considering two classical systems, it is meaningful to organize the whole set of propositions about them in the corresponding Boolean lattice built up as the Cartesian product of the individual lattices. Informally one may say that each factor lattice corresponds to the properties of each physical system. But the quantum case is completely different. When two or more systems are considered together, the state space of their pure states is taken to be the tensor product of their Hilbert spaces.
But it is not true, as a naive classical analogy would suggest, that any pure state of the compound system factorizes after the interaction in pure states of the subsystems, and that they evolve with their own Hamiltonian operators. It was shown, in a non-separability theorem by Aerts , that when trying to repeat the classical procedure of taking the tensor product of the lattices of the properties of two systems, to obtain the lattice of the properties of the composite, the procedure fails [5, 6, 8, 57, , ]. Attempts to vary the conditions that define the product of lattices have been made but in all cases it results that the Hilbert lattice factorizes only in the case in which one of the factors is a Boolean lattice, or when the systems have never interacted.
During the late sixties and beginning of the seventies there was a radical philosophical view initiated by David Finkelstein [, ] and Hilary Putnam [, ] arguing that logic is in a certain sense empirical.
Finkelstein highlighted the abstractions we make in passing from mechanics to geometry to logic, and suggested that the dynamical processes of fracture and flow already observed at the first two levels should also arise at the third. Putnam, on the other hand, argued that the metaphysical pathologies of superposition and complementarity are nothing more than artifacts of logical contradictions generated by an indiscriminate use of the distributive law.
We live in a world with a non-classical logic. Inasmuch as this picture of physical properties is confirmed by the empirical success of QM, this view means we must accept that the way in which physical properties actually hang together is not Boolean. Since logic is, for Putnam, very much the study of how physical properties actually hang together, he concludes that classical logic is simply mistaken: the distributive law is not universally valid.
The study of the modal character of QM was explicitly formalized in the seventies and eighties by a group of physicists and philosophers of science. Bas van Fraassen was the first to formally include the reasoning of modal logic in QM. He presented a modal interpretation MI of QL in terms of its semantical analysis [, , , ].
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The purpose of which was to clarify which properties among those of the complete set structured in the lattice of subspaces of Hilbert space pertain to the system. In , Simon Kochen presented his own modal version  at one of the famous conferences on the foundations of QM organized by Kalervo Laurikainen in Finland. This interpretation of QM also has a direct link to the discussions between the founding fathers of the theory. We consider it is an illuminating clarification of the mathematical structure of the theory, especially apt to describe the measuring process.
We would, however feel that it means not an alternative but a continuation to the Copenhagen interpretation Bohr and, to some extent, Heisenberg. Taking as a standpoint the work done by van Fraassen, Dieks went further in relation to the metaphysical presuppositions involved, making explicit the idea that MIs [94, 95, 96, 97] could be also considered from a realist stance as describing systems with properties. Of course, the way in which MIs attack the problem rests on the distinction between the realms of possibility and actuality.
As noted by Dirac in the first chapter of his famous book , the existence of superpositions is responsible for the striking difference between quantum and classical behavior. Superpositions are also central when dealing with the measurement process, where the various terms associated with the possible outcomes of a measurement must be assumed to be present together in the description. This fact leads van Fraassen to the distinction between value-attributing propositions and state-attributing propositions , between value-states and dynamic-states :.
In other words, the state delimits what can and cannot occur, and how likely it is—it delimits possibility, impossibility, and probability of occurrence—but does not say what actually occurs. So, van Fraassen distinguishes propositions about events and propositions about states. Value-states are specified by stating which observables have values and what these values are. Dynamic-states state how the system will develop.
This is endowed with the following interpretation:. This interpretation informs the consideration of possibility in the realm of QL [, chapter 9]. The logic operations among value-attribution propositions are defined as:. It may be enriched to approach the lattice of subspaces of Hilbert space. One may recognize a modal relation between both kind of propositions.
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For example, one starts denying the collapse in the measurement process and recognizing that the observable has one of the possible eigenvalues. Then it may be asked what may be inferred with respect to those values when one knows the dynamic state. Thus, the logic of V is that of P , that is, QL. Endowed with these tools, van Fraassen gives an interpretation of the probabilities of the measurement outcomes which is in agreement with the Born rule.
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The MI proposed by Kochen and Dieks K-D, for short , proposes to use the so called biorthogonal decomposition theorem also called Schmidt theorem in order to describe the correlations between the quantum system and the apparatus in the measurement process. From a realistic perspective, an interpretational issue which MIs need to take into account is the assignment of definite values to properties.
But if we try to interpret eigenvalues which pertain to different sets of observables as the actual pre-existent values of the physical properties of a system, we are faced with all kind of no-go theorems that preclude this possibility. Regarding the specific scheme of the MI, Bacciagaluppi and Clifton were able to derive KS-type contradictions in the K-D interpretation which showed that one cannot extend the set of definite valued properties to non-disjoint sub-systems [26, 56].
In this way one can avoid KS contradictions and maintain a consistent discourse about statements which pertain to the sublattice determined by the preferred observable R. It is this distinction between property states and dynamical states which according to Bub provides the modal character to the interpretation:. In precise terms, as L H does not admit a global family of compatible valuations, and thus not all propositions about the system are determinately true or false, probabilities defined by the pure state cannot be interpreted epistemically  p.
Quantum Logic in Historical and Philosophical Perspective
So, dynamical states do not coincide with property states. The determinate sublattice, which changes with the dynamics of the system, is a partial Boolean algebra, that is, the union of a family of Boolean algebras pasted together in such a way that the maximum and minimum elements of each one, and eventually other elements, are identified and, for every n -tuple of pair-wise compatible elements, there exists a Boolean algebra in the family containing the n elements.
Thus constructed, the structure avoids KS-type theorems. Then, given a system S and a measuring apparatus M ,. Moreover, the quantum state can be interpreted as assigning probabilities to the different possible ways in which the set of determinate quantities can have values, where one particular set of values represents the actual but unknown values of these quantities. The problem with this interpretation is that, in the case of an isolated system, there is no single element in the formalism of QM that allows us to choose an observable R , rather than another.
This is why the move seems flagrantly ad hoc. Were we dealing with an apparatus, there would be a preferred observable, namely the pointer position, but the quantum wave function contains in itself mutually incompatible representations choices of apparatuses each of which provides non-trivial information about the state of affairs. The authors of this work have also contributed to the understanding of modality in the context of orthodox QL [, , , ]. From our investigation there are several conclusions which can be drawn.
We started our analysis with a question regarding the contextual aspect of possibility. As it is well known, the KS theorem does not talk about probabilities, but rather about the constraints of the formalism to actual definite valued properties considered from multiple contexts. What we found via the analysis of possible families of valuations is that a theorem which we called, for obvious reasons, the Modal KS MKS theorem can be derived which proves that quantum possibility, contrary to classical possibility, is also contextually constrained .
This means that, regardless of its use in the literature, quantum possibility is not classical possibility. In a paper written in , we concentrated on the analysis of actualization within the orthodox frame and interpreted, following the structure, the logical realm of possibility in terms of ontological potentiality. The study of the structure of tensor products [57, , , , ] motivated a fruitful development of different algebraic structures that could represent quantum propositions, which in turn became a line of investigation by itself.
Beginning with the proposal of test spaces by Foulis and Randall [, , , , , , ], which are related to orthoalgebras, the theory of structures as orthomodular lattices, partial Boolean algebras, orthomodular posets, effect algebras, quantum MV-algebras and the like became widely discussed.
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The weakened structures allow consideration of unsharp propositions related, not to projections, but to the elements of the more general set of linear bounded operators—called effects —over which the probability measure given by the Born rule may be defined. An important line of research in the subject of quantum structures is the application of QL methods to languages of information processing and, more specifically, to quantum computational logic QCL [53, 80, , 82, , , , , , , ].
In this way several logical systems associated to quantum computation were developed. They provide a new form of quantum logic strongly connected with the fuzzy logic of continuous t -norms . The groups in Firenze directed by Dalla Chiara, and Cagliari directed by Giuntini, have also developed different languages for quantum computation.
A sentence in QL may be interpreted as a closed subspace of H. Instead, the meaning of an elementary sentence in QCL is a quantum information quantity encoded in a collection of qbits —unit vectors pertaining to the tensorial product of two dimensional complex Hilbert spaces—or qmixes —positive semi-definite Hermitian operators of trace one over Hilbert space. Conjunction and disjunction are not associated to the join and meet lattice operations.
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On the one hand, NRL is, in a wide sense, a logic in which the relation of identity or equality is restricted, eliminated, replaced, at least in part, by a weaker relation, or employed together with a new non-reflexive implication or equivalence relation. There are other versions in higher-order logic, in which higher order variables appear. Some of the above principles are not in general valid in non-reflexive logics. They are total or partially eliminated, restricted, or not applied to the relation that is employed instead of identity.
Several of these principles are the motivations for the development of non-reflexive logics. In the Congress, Manin proposed as one of the new set of problems for the next century:. New quantum physics has shown us models of entities with quite different behaviour. Within this context, the weakening of the concept of identity—substituted by that of indiscernibility—allows the development of non-reflexive logics which, in a wide sense, are logics in which the relation of identity or equality is restricted, eliminated, replaced, at least in part, by a weaker relation, or employed together with a new non-reflexive implication or equivalence relation [68, 73, , 75].
There are also different approaches to the logic related to quantum set theories. Gaisi Takeuti proposed a quantum set theory developed in the lattice of projections-valued universe [, ] and Satoko Titani formulated a lattice valued logic corresponding to general complete lattices developed in the classical set theory based on the classical logic . On the other hand, PL are the logics of inconsistent but non-trivial theories.
The origins of PL go back to the first systematic studies dealing with the possibility of rejecting the PNC. PL was elaborated, independently, by Stanislaw Jaskowski in Poland, and by Newton da Costa in Brazil, around the middle of the last century on PL, see, for example: . T is called trivial if any sentence of its language is also a theorem of T ; otherwise, T is said to be non-trivial. In classical logics and in most usual logics, a theory is inconsistent if, and only if, it is trivial.
L is paraconsistent when it can be the underlying logic of inconsistent but non-trivial theories. Clearly, no classical logic is paraconsistent. The notion of complementarity was developed by Bohr in order to consider the contradictory representations of wave representation and corpuscular representation found in the double-slit experiment see for example .
There is a great amount of work in progress in QL from new quantum structures, to the use of non-reflexive logics, paraconsistent logics, dynamical logics, etc.