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Nonseparability of inferribility and measurability in quantum mechanics as a Systema Magnum

Lars Löfgren

Published Jan 1, 1998• 4 citations
OpenAlex

Abstract

Quantum measurement theory has a peculiar nature with its TWO processes of assertibility, by proof and by measurement, which is beyond formal logic. This two-sortedness with respect to assertibility is discussed in terms of developments of many--sorted logics, sometimes referred to as Logica Magna. Although useful insights are obtained this way, we argue that there is a nonseparability of inferribility and measurability that is understandable first with a shift from logic to language in its systemic conception. That is, from a Logica Magna to a Systema Magnum understood as a truly systemic whole of complementary description and interpretation processes. In introspective quantum mechanics as a Systema Magnum, the nonseparability sheds light on interconnections between formalism and experimentalism which, for example, helps a critical understanding of Deutsch’s hypothesis of a programmable experimentability ("quantum computation").

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•Feb 17, 2015•0 citations
Quantum measurement theory has a peculiar nature with its TWO processes of assertibility, by proof and by measurement, which is beyond formal logic. This two-sortedness with respect to assertibility is discussed in terms of developments of many--sorted logics, sometimes referred to as Logica Magna. Although useful insights are obtained this way, we argue that there is a nonseparability of inferribility and measurability that is understandable first with a shift from logic to language in its systemic conception. That is, from a Logica Magna to a Systema Magnum understood as a truly systemic whole of complementary description and interpretation processes. In introspective quantum mechanics as a Systema Magnum, the nonseparability sheds light on interconnections between formalism and experimentalism which, for example, helps a critical understanding of Deutsch’s hypothesis of a programmable experimentability ("quantum computation").
Quantum measurement theories are interesting mainly because of the fundamental questions. Since quantum mechanics is primarily a theory of small, simple systems, it could a priori be expected that a full description of this formalism would be possible in a language that would refer only to the objective properties of such small systems. Unless quantum physics is objectively false in some of its predictions, limitations of such a kind are unavoidable. The chapter argues that such limitations extend to systems of arbitrary finite size and of arbitrary complexity. As a matter of fact, this is the first and perhaps the most significant conclusion in investigation of the conceptual foundations of quantum physics. An equally tenable standpoint is to assert that truly infinite systems have special features of their own that make them fall outside the realm of elementary quantum mechanics. Implicitly or explicitly, nonseparability is indeed present in all the various interpretations of the quantum rules which have been set forth.
Scholars have wondered for a long time whether quantum mechanics (QM) subtends a quantum concept of truth which originates quantum logic (QL) and is radically different from the classical (Tarskian) concept of truth. We show in this paper that QL can be interpreted as a pragmatic language of pragmatically decidable assertive formulas, which formalize statements about physical systems that are empirically justified or unjustified in the framework of QM. According to this interpretation, QL formalizes properties of the metalinguistic concept of empirical justification within QM rather than properties of a quantum concept of truth. This conclusion agrees with a general integrationist perspective, according to which nonstandard logics can be interpreted as theories of metalinguistic concepts different from truth, avoiding competition with classical notions and preserving the globality of logic. By the way, some elucidations of the standard concept of quantum truth are also obtained.
In quantum mechanics, quantum nonseparability is at the core of philosophical debates regarding its meaning. In the context of the process matrix formalism, causal nonseparability characterises quantum processes (connecting the inputs and outputs of different local quantum operations) that are incompatible with any definite causal structure among interacting parties. One talks about indefinite causal orders. A famous example of causally nonseparable processes is called the quantum switch (QS). It is extensively studied in the literature in virtue of its simple architecture and its various implementations in laboratories.The present work will discuss the possible interpretations of the QS's causal nonseparability under the assumption that causal nonseparability is seen as pointing towards novel objective features of nature.It was first defended that a scientific realist approach towards the process matrix formalism, which is an operational theory generalising quantum mechanics, was as much legitimate as any possible antirealist reading, contrary to certain views found in the literature. The reason is that operational formalisms are ontologically and epistemically neutral.From there, the theoretical concepts of interest, namely causal nonseparability and its model-independent counterpart called noncausality, were analysed in more details, in order to highlight in what sense they are distinct from the standard notions of quantum nonseparability and nonlocality in quantum mechanics.The discussion then focused on noncausality. It was argued that noncausality has a strong connection with a notion of temporal nonlocality. In the same way that Bell nonlocality is given different underlying explanations depending on the details of the chosen quantum mechanics' account, noncausality pointing towards temporal nonlocality can be given a variety of underlying descriptions depending on the exact way to interpret the process matrix formalism.The last chapter focused precisely on this particular point, namely on the various ways to understand process matrices and causal nonseparability in a realist context. In order to explore the potential impact of causal nonseparability on spacetime,we shifted from the notion of (indefinite) causal structure to (indefinite) spatiotemporal ones. This shift was allowed under a set of reasonable assumptions regarding the properties of a physical spacetime manifold and the connection between an operational and relativistic notions of causal relations.While different readings were suggested for indefiniteness of spatiotemporal relations, we insisted in particular on an objective understanding appealing to the concept of metaphysical indeterminacy. It was argued that such an approach could prove useful in a more general theoretical context such as quantum gravity, while being already partly supported in standard quantum mechanics.
Quantum physics provides us with a formalism with which we can try to understand the microscopic world. However, the underpinning concepts of the basic principles of quantum mechanics itself are surprisingly ill understood. Even the ubiquitous concept of the wavefunction collapsing post measurement, lives outside the formal theory of quantum mechanics and must be taken as a postulate. The Schr\"odinger equation, which governs the evolution of quantum systems, does not describe the system during the measurement process. This is referred to as the measurement problem, and has been debated since the inception of quantum theory. Many different interpretations of quantum mechanics try to elucidate what happens during wavefunction collapse, such as the famous many-worlds interpretation. However, these interpretations either give rise to the same physical outcome or simply are not testable, and so cannot be physically differentiated from one another. We attempt to gain insight into the measurement problem through experiments involving sequential measurements, both in the weak and strong regime. We find surprising outcomes such as the persistence of time-ordering asymmetry in the weak measurement limit, and the survival of coherence in the strong measurement limit.
This chapter stresses what may be called conventional or standard textbook quantum mechanics, that is, to the working hypothesis that quantum mechanics is complete. Some remarkable features of the general quantum formalism, often grouped under the general name "nonseparability," rather drastically limit the domain of validity of the idea. The two notions referred to in the title of this chapter are therefore, in a way, antagonistic. The state operator concept is of some use for investigating the properties of ensembles of subsystems of larger systems, sometimes uttered claim that its introduction makes it possible to remove "nondivisibility by thought" from quantum physics is definitely erroneous. When J. S. Bell's theorem appeared it was immediately clear that some expressions had to be comed in order to express in words its bearing and significance.
We present a formulation of quantum mechanics based on a logic representing some aspects of the behaviour of the measurement process. With such an approach, we make no direct mention of quantum states, and thus avoid the problems associated to this rather evasive notion. We then study some properties of the models of this logic, and deduce some characteristics that any model (and hence any formulation of quantum mechanics compatible with its prediction and relying on a notion of measurement) should verify. The main results we obtain are that in the case of a Hilbert space of dimension at least 3, no model can lead to the prediction with certainty of more than one atomic outcome. Moreover, if the Hilbert space is finite dimensional, then we are able to precisely describe the structure of the predictions of any model of our logic. As a consequence, we finally show that all the models of our logic make exactly the same predictions regarding whether a given sequence of outcomes is possible or not.