We extend the Intrinsic Observability Framework (IOF) by introducing a symmetry structure underlying observer-dependent descriptions of quantum systems. In previous works, probabilistic outcomes were formulated as a consequence of restricted access to the full quantum state, and observable predictions were shown to be invariant under admissible transformations of a fixed decomposition. In the present work, we relax the assumption of a fixed decomposition and allow the partition of the Hilbert space into observable and hidden sectors to depend on the observer. To relate such descriptions, we introduce Access-Transformation Symmetry (ATS), which characterizes invariance under transformations between observer-dependent decompositions. We show that, although the assignment of observable and hidden degrees of freedom may vary between observers, all observable predictions—such as probability distributions and expectation values—remain invariant. This establishes a consistent framework in which observer-dependent representations coexist with observer-independent empirical content. The present formulation reveals that observation is inherently asymmetric, while the underlying access structure exhibits a symmetry under transformations between observers. In this way, the IOF is extended from a structural framework to one that incorporates relational aspects of observability, while preserving its empirical consistency.
We extend the Intrinsic Observability Framework (IOF) by introducing a symmetry structure underlying observer-dependent descriptions of quantum systems. In previous works, probabilistic outcomes were formulated as a consequence of restricted access to the full quantum state, and observable predictions were shown to be invariant under admissible transformations of a fixed decomposition. In the present work, we relax the assumption of a fixed decomposition and allow the partition of the Hilbert space into observable and hidden sectors to depend on the observer. To relate such descriptions, we introduce Access-Transformation Symmetry (ATS), which characterizes invariance under transformations between observer-dependent decompositions. We show that, although the assignment of observable and hidden degrees of freedom may vary between observers, all observable predictions—such as probability distributions and expectation values—remain invariant. This establishes a consistent framework in which observer-dependent representations coexist with observer-independent empirical content. The present formulation reveals that observation is inherently asymmetric, while the underlying access structure exhibits a symmetry under transformations between observers. In this way, the IOF is extended from a structural framework to one that incorporates relational aspects of observability, while preserving its empirical consistency.
We introduce a relational extension of the Intrinsic Observability Framework (IOF), termed Relational Access Symmetry (RAS), in which the roles of observable and hidden subspaces are exchanged between distinct observational structures. While the previously introduced Access Transformation Symmetry (ATS) preserves observational statistics under redefinitions of subsystem decomposition, it does not alter the underlying observational standpoint. In contrast, RAS represents a transformation between observational structures themselves, under which what is accessible in one description becomes inaccessible in another, and vice versa. We formulate RAS as a relational exchange symmetry between observable and hidden subspaces, implemented via a unitary mapping between alternative tensor decompositions of the total Hilbert space. Unlike ATS, RAS does not require invariance of measurement outcomes; rather, it imposes a structural covariance under the exchange of observational roles. As a consequence, observable and hidden subspaces must be isomorphic, enforcing an intrinsic balance between accessible and inaccessible degrees of freedom. This result provides a new structural perspective on the hidden sector in quantum theory: instead of being arbitrary or merely residual, the hidden subspace acquires a constrained and structured character, determined by its relational equivalence to the observable subspace. RAS thus elevates the hidden sector from an unspecified complement to a symmetry-related counterpart, offering a refined structural understanding of observability within the IOF.
We investigate the origin of probabilistic descriptions in quantum theory within the Intrinsic Observability Framework (IOF). In this framework, the Hilbert space of a system is intrinsically decomposed into observable and hidden degrees of freedom, and effective observable descriptions are obtained by restricting the full description to the observable sector. We propose that any access to a quantum system necessarily involves a restriction of the full state. This principle implies that reduced descriptions are not contingent on environmental interactions or incomplete knowledge, but arise as a structural consequence of the act of access itself. As a result, mixed states and probabilistic outcomes emerge naturally from the intrinsic limitation on observable degrees of freedom. Within this perspective, quantum probabilities are interpreted not as manifestations of intrinsic randomness or hidden deterministic variables, but as features of accessible representations of the full state. The appearance of stochastic behavior is thus tied to the structure of observability. We further argue that, while the decomposition into observable and hidden sectors is not strictly unique, all physically observable predictions must remain invariant under admissible transformations. This establishes a consistent framework in which structural freedom coexists with empirical determinacy. The present formulation clarifies the role of access in the emergence of probabilistic descriptions and provides a structural basis for understanding quantum probabilities independently of decoherence.
We investigate the invariance structure of observable predictions in the Intrinsic Observability Framework (IOF), building on the Access-Induced Reduction Principle (ARP) introduced in Ref. [1]. While IOF allows for non-unique decompositions of a quantum system into observable and hidden degrees of freedom, physical predictions must remain invariant under admissible transformations of such decompositions. We formalize admissible transformations as those that preserve observable probability distributions derived from reduced states. Furthermore, we clarify the distinction between invariant statistical structures and non-invariant configurations of individual measurement outcomes. This establishes a consistent framework in which structural freedom coexists with empirical determinacy.
Operational decomposition has been introduced within the Intrinsic Observability Framework (IOF) as the structural process through which observation establishes the operational assignment of the latent structure within the observable and hidden domains. However, once multiple observations independently establish their respective operational decompositions, an important question arises: why do these independently established operational assignments remain mutually coherent rather than producing incompatible operational realities? In the present work, we address this problem by introducing the Global Consistency Rule (GCR) for operational decomposition. The GCR states that independently established operational decompositions need not be identical, but their mutual relations must satisfy common consistency conditions that preserve the coherence of operational reality. From this perspective, Relational Access Symmetry (RAS) is interpreted as an ideal relational consistency principle governing the organization of observational structures, whereas Access Transformation Symmetry (ATS) provides the operational consistency realized among physically accessible decompositions. Together, these complementary principles provide the structural basis of the GCR and constrain operational decomposition without eliminating its observational and accessibility dependence. The present formulation does not modify the mathematical formalism or empirical predictions of quantum mechanics. Instead, it provides a structural interpretation of why independently established operational decompositions can differ while remaining mutually coherent. The proposed framework therefore extends the IOF from a theory describing the establishment of operational decomposition to one in which the admissible relationships among multiple operational decompositions are explicitly constrained by a global consistency rule, allowing them to collectively constitute a coherent operational reality.
We propose a framework, referred to as the Intrinsic Observability Framework (IOF), in which the probabilistic structure of quantum mechanics arises from an intrinsic decomposition of the Hilbert space into observable and hidden degrees of freedom. The total system is described by a pure state evolving unitarily, while observable states are obtained by restricting the description to the observable sector via a partial trace over the hidden sector. A key assumption of the present approach is that this decomposition is a fundamental and non-contextual feature of the system, rather than a modeling choice or a consequence of environmental interactions. The hidden degrees of freedom introduced here are not interpreted as unmeasured physical variables, but as components that are fundamentally excluded from the space of observable quantities. Within the IOF, mixed states and probabilistic outcomes emerge from incomplete access to the full state, without invoking intrinsic randomness, external environments, or branching worlds. This distinguishes the present formulation from decoherence-based approaches, hidden-variable theories, and many-worlds interpretations. The proposed framework provides an alternative structural perspective on the origin of quantum probabilities while preserving the standard unitary formalism. We discuss its conceptual implications and outline directions for future work.
The Intrinsic Observability Framework (IOF) distinguishes structural decomposition from operational decomposition as complementary levels of organization. Structural decomposition establishes the observable and hidden domains as the architecture of observability, whereas operational decomposition assigns the latent structure to these domains under a given observational interaction. The present work extends this framework by introducing two complementary forms of totality: structural totality and operational totality. Structural totality represents the completeness of the structural architecture established through structural decomposition, consisting of the observable and hidden domains. Operational totality, in contrast, represents the completeness of the latent structure as operationally assigned across these domains through operational decomposition. The distinction therefore concerns different objects of completeness: structural totality refers to the architecture within which assignment is possible, whereas operational totality refers to the content completely assigned within that architecture. Operational totality is consequently neither an incomplete portion nor a degraded representation of structural totality. Different observational interactions may establish different operational totalities within a common structural totality. Such observer dependence does not imply incompleteness or mutually isolated operational worlds. Each operational totality is complete with respect to its own operational assignment, while the admissible relations among independently established operational totalities are constrained by the Global Consistency Rule (GCR). The resulting framework therefore distinguishes structural completeness, operational completeness, and global consistency, providing an organizational interpretation of how distinct observer-dependent totalities can coexist while collectively supporting a shared and coherent operational reality without modifying the mathematical formalism or empirical predictions of quantum mechanics.