Abstract In this paper, we provide a logic to reason about the compatibility structure and contextuality of quantum measurements, using inputs from the Foulis-Randall approach to operational quantum logic and the framework of team semantics. A central feature of quantum systems is that not all physical properties can be observed simultaneously. Thus, we formalize the notion of ‘co-measurability’, which captures exactly when two experimental propositions can be confirmed or refuted by a single measurement setup. We introduce the ‘Logic of Co-measurable Experimental Propositions’ ( $$L_{\mathrm C}$$ L C ), which models experimental programs as teams of measurement outcomes. Within this framework, we construct a fragment $$L_{\mathrm C}^*$$ L C ∗ where the conjunction of two propositions is defined to hold only if they are co-measurable. We show that this restriction naturally recovers the algebraic structure of standard quantum logic. It entails the failure of the distributive laws, which is a key feature of quantum logic. The resulting framework unifies classical and quantum scenarios, offering a logical tool for analyzing contextuality in generalized experimental settings.
Abstract In this paper, we provide a logic to reason about the compatibility structure and contextuality of quantum measurements, using inputs from the Foulis-Randall approach to operational quantum logic and the framework of team semantics. A central feature of quantum systems is that not all physical properties can be observed simultaneously. Thus, we formalize the notion of ‘co-measurability’, which captures exactly when two experimental propositions can be confirmed or refuted by a single measurement setup. We introduce the ‘Logic of Co-measurable Experimental Propositions’ ( LC L C ), which models experimental programs as teams of measurement outcomes. Within this framework, we construct a fragment LC∗ L C ∗ where the conjunction of two propositions is defined to hold only if they are co-measurable. We show that this restriction naturally recovers the algebraic structure of standard quantum logic. It entails the failure of the distributive laws, which is a key feature of quantum logic. The resulting framework unifies classical and quantum scenarios, offering a logical tool for analyzing contextuality in generalized experimental settings.
We introduce logics of dependence and independence over polyteams (that is, databases with multiple relation schemes), in a similar way as in arXiv:1704.02158v3 [cs.LO]. On this basis, we develop a logical framework in which we can express (in formulas) relevant properties about the statistics of a repeated quantum experiment. In this framework, we formalize Bell's theorem, and the Kochen-Specker theorem as failure of logical entailments, as it was suggested in arXiv:2107.10817v3 [math.LO]. The key feature of polyteam logics in action here is their power to express both local and global properties, with respect to the set of possible measurements. The point of this is at least twofold. Firstly, as in arXiv:1102.0264v7 [quant-ph], it gives a high-level (information-theoretic) understanding of these physical phenomena, that is not intrinsic to the physical objects involved. Secondly, since logical entailment is often (partially) axiomatizable, this framework incites us to tackle the question of contextuality computationally. We also define semiring-weighted polyteams along with logics on them, and use them to recover the probabilistic versions of the contextuality theorems mentioned above.
Contextuality is a key signature of quantum non-classicality, which has been shown to play a central role in enabling quantum advantage for a wide range of information-processing and computational tasks. We study the logic of contextuality from a structural point of view, in the setting of partial Boolean algebras introduced by Kochen and Specker in their seminal work. These contrast with traditional quantum logic à la Birkhoff and von Neumann in that operations such as conjunction and disjunction are partial, only being defined in the domain where they are physically meaningful. We study how this setting relates to current work on contextuality such as the sheaf-theoretic and graph-theoretic approaches. We introduce a general free construction extending the commeasurability relation on a partial Boolean algebra, i.e. the domain of definition of the binary logical operations. This construction has a surprisingly broad range of uses. We apply it in the study of a number of issues, including: - establishing the connection between the abstract measurement scenarios studied in the contextuality literature and the setting of partial Boolean algebras; - formulating various contextuality properties in this setting, including probabilistic contextuality as well as the strong, state-independent notion of contextuality given by Kochen-Specker paradoxes, which are logically contradictory statements validated by partial Boolean algebras, specifically those arising from quantum mechanics; - investigating a Logical Exclusivity Principle, and its relation to the Probabilistic Exclusivity Principle widely studied in recent work on contextuality as a step towards closing in on the set of quantum-realisable correlations; - developing some work towards a logical presentation of the Hilbert space tensor product, using logical exclusivity to capture some of its salient quantum features.
Due to the failure of the classical principles of bivalence and verifunctionality, the logic of experimental propositions relative to quantum systems cannot be interpreted in Boolean algebras. However, we cannot say neither that this logic is captured by orthomodular lattices, as claimed by many authors along the line of Birkhoff's and von Neumann's standard approach. For the alleged violation of distributivity is based on the possibility of combining statements relative to complementary contexts, which does not refer to any experience and, consequently, has no meaning. Indeed, quantum logic should be interpreted in partial, transitive Boolean algebras whose compatibility relation limits the application of the connectives within each of its Boolean sub-algebras, which refer to partial, classical descriptions. Moreover, this approach of quantum logic makes it possible to deal with composite systems, which was not possible to do within the standard approach, and then to deal with the fundamental notion of quantum entanglement. The latter notion can be represented by a series of axioms of the object language that restrict the set of experimental statements bearing on a composite system, while its close link to the notion of complementarity can be expressed in the metalanguage.
Quantum measurements can be incompatible, i.e., they can fail to be jointly measurable. Recently, a weaker notion of joint-measurability, called partial joint-measurability, was proposed by Masini et al. in [Quantum 8, 1574 (2024)]. In this work, we further generalize this notion to the setting where only a subset of the outcomes of each measurement is required to be jointly determined by classical variables. We provide two mathematical formulations of partial joint-measurability and show that, like full joint-measurability, it can be decided by solving a single semidefinite program. We prove that in the case of an untrusted measurement device, an adversary Eve, limited to classical side information, can perfectly guess the outcomes of the measurement device if and only if the set of measurements is partially jointly measurable. We derive analytical thresholds on the detection efficiency below which generic measurements become partially jointly measurable. Such bounds directly yield limits on the robustness of device-independent and semi-device-independent quantum cryptographic protocols against detection inefficiency. In particular, our results highlight the importance of a careful treatment of postselection in security analyses.
Abstract Quantum contextuality represents a fundamental form of nonclassicality in quantum mechanics. To provide a more complete characterization of nonclassical properties in quantum systems, we adopt a logical perspective and propose a mathematical framework based on exclusive partial Boolean algebras. This framework enables a unified description of contextuality and nonclassicality across finite general, quantum, and classical systems. We establish a unified and minimal classical counterpart for any finite general system. Within this framework, we formalize major categories of quantum contextuality, demonstrating that: 12 projectors suffice to generate Kochen–Specker scenarios; 10 projectors suffice to witness state-independent contextuality; and 3 observables suffice to witness quantum contextuality. Finally, we prove that contextuality is a sufficient but not necessary condition for nonclassicality.
The term proposition usually denotes in quantum mechanics (QM) an element of (standard) quantum logic (QL). Within the orthodox interpretation of QM the propositions of QL cannot be associated with sentences of a language stating properties of individual samples of a physical system, since properties are nonobjective in QM. This makes the interpretation of propositions problematical. The difficulty can be removed by adopting the objective interpretation of QM proposed by one of the authors (semantic realism, or SR, interpretation). In this case, a unified perspective can be adopted for QM and classical mechanics (CM), and a simple first order predicate calculus L(x) with Tarskian semantics can be constructed such that one can associate a physical proposition (i.e., a set of physical states) with every sentence of L(x). The set Pf of all physical propositions is partially ordered and contains a subset PTf of testable physical propositions whose order structure depends on the criteria of testability established by the physical theory. In particular, PTf turns out to be a Boolean lattice in CM, while it can be identified with QL in QM. Hence the propositions of QL can be associated with sentences of L(x), or also with the sentences of a suitable quantum language LTQ(x), and the structure of QL characterizes the notion of testability in QM. One can then show that the notion of quantum truth does not conflict with the classical notion of truth within this perspective. Furthermore, the interpretation of QL propounded here proves to be equivalent to a previous pragmatic interpretation worked out by one of the authors, and can be embodied within a more general perspective which considers states as first order predicates of a broader language with a Kripkean semantics.