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Reconstructing quantum theory from its possibilistic operational formalism

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Abstract

We develop a possibilistic semantic formalism for quantum phenomena from an operational perspective. This semantic system is based on a Chu duality between preparation processes and yes/no tests, the target space being a three-valued set equipped with an informational interpretation. A basic set of axioms is introduced for the space of states. This basic set of axioms suffices to constrain the space of states to be a projective domain. The subset of pure states is then characterized within this domain structure. After having specified the notions of properties and measurements, we explore the notion of compatibility between measurements and of minimally disturbing measurements. We achieve the characterization of the domain structure on the space of states by requiring the existence of a scheme of discriminating yes/no tests, necessary condition for the construction of an orthogonality relation on the space of states. This last requirement about the space of states constrain the corresponding projective domain to be ortho-complemented. An orthogonality relation is then defined on the space of states and its properties are studied. Equipped with this relation, the ortho-poset of ortho-closed subsets of pure states inherits naturally a structure of Hilbert lattice. Finally, the symmetries of the system are characterized as a general subclass of Chu morphisms. We prove that these Chu symmetries preserve the class of minimally disturbing measurements and the orthogonality relation between states. These symmetries lead naturally to the ortho-morphisms of Hilbert lattice, defined on the set of ortho-closed subsets of pure states.

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Notes

  1. i.e., the initial preparation followed by the operation/test step as a global preparation process for subsequent tests.

  2. We note that the description of the preparation/measurement process should then exploit some tools of recursion theory.

  3. Here, and in the following, we will adopt the following basic definition of the word ’semantic’ recalled by Reichenbach: “Modern logic distinguishes between object language and metalanguage; the first speaks about physical objects, the second about statements, which in turn are referred to objects. The first part of the meta-language, syntax, concerns only statements, without dealing with physical objects; this part formulates the structure of statements. The second part of the metalanguage, semantics, refers to both statements and physical objects. This part formulates, in particular, the rules concerning truth and meaning of statements, since these rules include a reference to physical objects. The third part of the meta-language, pragmatics, includes a reference to persons who use the object language.”[64]

  4. Note that, in practice, the observer is rather led to infer a ’mixture’ on the basis of his limited knowledge about this sample.

  5. This formalism has its origins in the pioneering works of Mackey [49], Ludwig [46,47,48] and Kraus [45]. See [42] for a recent review.

  6. The description of quantum theory in this framework then must deal with the problem of defining the notions of consistency, completeness and irredundancy for the set of control tests that define an element of the quantum space of states.

  7. In Von Neumann’s formalism, this point means: the quantum state of the system is not an eigenstate of the associated operator.

  8. For the sake of the usual quantum formalism, quantum probabilities can be recovered from this formalism, under the assumption of the existence of some well-behaved measurements, using Gleason’s theorem [34] (see [55] for a historical and pedagogical presentation of these elements).

  9. In Von Neumann’s quantum mechanics, each entity is associated with a complex Hilbert space H. A state \(\psi \) of this entity is defined by a ray \(\nu (\psi )\) in H, and an observable is defined by a self-adjoint operator on H. In particular, a yes/no test \({ \mathfrak {t}}\) is represented by an orthogonal projector \(\Pi _{ \mathfrak {t}}\) or equivalently by the closed subspace \({ \mathfrak {A}}_{ \mathfrak {t}}\) defined as the range of \(\Pi _{ \mathfrak {t}}\). The answer “yes” or “no” is obtained with certainty for the yes/no test \({ \mathfrak {t}}\), if and only if the state \(\psi \) is such that \(\nu (\psi )\) is included in \({ \mathfrak {A}}_{ \mathfrak {t}}\) or in the orthogonal of \({ \mathfrak {A}}_{ \mathfrak {t}}\), respectively. Birkhoff and von Neumann proposed to focus not on the structure of the Hilbert space itself, but on the structure of the set of closed subspaces of H. The mathematical structure associated with the set of quantum propositions defined by the closed subspaces of H is not a Boolean algebra (contrary to the case encountered in classical mechanics). By shifting the attention to the set of closed subspaces of H instead of H, the possibility is open to build an operational approach to quantum mechanics, because the basic elements of this description are yes/no experiments.

  10. ’It is our contention that the realistic view implicit in classical physics need not be abandoned to accommodate the contemporary conceptions of quantum physics. All that must be abandoned is the presumption that each set of experiments possesses a common refinement (that is, the experiments are compatible). As we shall argue, this in no way excludes the notion of physical systems existing exterior to an observer, nor does it imply that the properties of such systems depend on the knowledge of the observer.’ [30, p.813]

  11. The objects of this category are the natural space of states in quantum mechanics, i.e., the Hilbert spaces of dimension greater than two, and the morphisms are the orbits under the U(1) group action on semi-unitary maps (i.e. unitary or anti-unitary), which are the relevant symmetries of Hilbert spaces from the point of view of quantum mechanics.

  12. The centrality of the notion of Chu spaces for quantum foundations had been noted already by V. Pratt [60].

  13. To clarify the fundamental difference in nature between Reichenbach Quantum Logic and Mainstream Quantum Logic the reader is invited to consult [35].

  14. In the rest of this paper we refer to this construction, based on a three-valued Chu space, as a ’possibilistic’ approach to distinguish it from the ’probabilistic’ one.

  15. The finite character of the tested collection of prepared samples renders any notion of relative frequency of the outcomes ’meaningless’.

  16. If the observer is certain of the positive result after having performed a given yes/no test on a finite number of similarly prepared samples, a negative result obtained for any newly tested sample will lead the observer to revise that prediction and to consider this yes/no test as being ’indeterminate’ for this preparation.

  17. See [12] for a reference paper and [61][62] for a basic presentation.

  18. In this section, we are concerned with the duality aspect and the situation of Chu morphisms will be treated later.

  19. These designations are reminiscent of the basic fact that Chu spaces are generalizations of topological spaces. However this distinction is largely obsolete, as soon as the Chu space construction establishes a duality between these two sets.

  20. Here we mean that the test is effectuated on the considered sample, according to the procedure defined by a yes/no test associated with this property, and the ’answer’ received by the observer is ’positive’.

  21. Note the distinction made by W. Pauli between measurements of the first and second kind : ’On the other hand it can also happen that the system is changed but in a controllable fashion by the measurement - even when, in the state before the measurement, the quantity measured had with certainty a definite value. In this method, the result of a repeated measurement is not the same as that of the first measurement. But still it may be that from the result of this measurement, an un-ambiguous conclusion can be drawn regarding the quantity being measured for the concerned system before the measurement. Such measurements, we call the measurements of the second kind.’ [53, p.75] To be concrete: (i) the determination of the position of a particle by a test of the presence of the particle in a given box appears to be a measurement of the first kind, (ii) the determination of the momentum of a particle by the evaluation of the ’impact’ of this particle on a given detector appears to be a measurement of the second kind.

  22. A basic solution to this problem has been formalized by C. Piron [58]. This construction relies on an orthomodular lattice structure introduced on the space of properties (Note: we do not expect any such a construction in our perspective). In Piron’s vocabulary,

    two properties are compatible as soon as they form a boolean sub-algebra in the orthomodular lattice of properties (this requirement about the sub- boolean structure is a remnant of the particular structure on the space of properties in the classical situation).[55, p.295]

  23. These measurements played a fundamental role in Mackey’s traditional axiomatic approach to quantum theory [18].

  24. See [44] for the original results on non-contextuality in quantum mechanics.

  25. we adopt the following traditional notation \((u{\parallel }_{\mathfrak {L}} x) :\Leftrightarrow (u \not \sqsubseteq _{{}_{ \mathfrak {L}}} x \;\; \textrm{and}\;\;u \not \sqsupseteq _{{}_{ \mathfrak {L}}} x)\)

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    Eric Buffenoir

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Buffenoir, E. Reconstructing quantum theory from its possibilistic operational formalism. Quantum Stud.: Math. Found. 10, 115–159 (2023). https://doi.org/10.1007/s40509-022-00286-w

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