Quantum theory typically presupposes a fixed factorisation of the total Hilbert space into subsystems, or an equivalent fixed choice of commuting subalgebras. This convention hard-codes a classical notion of boundary between "System A" and "System B". Recent work has formalised subsystem ambiguity by defining partitions directly at the level of C*-subalgebras, including non-tensor-factor subsystems, and by studying representability and the role of centres. In parallel, work under the label "quantum mereology" studies how subsystems can emerge top-down from spectral structure or from changes of tensor product structure. The aim here is different. We define RCC-style region relations directly from unitary invariants of pairs of sharp region projectors, so that boundary indefiniteness is a relation-level observable rather than a chosen decomposition. A quantum mereotopology is developed in which "regions" are represented by projection operators and their topological relations are defined intrinsically, without background space and without a privileged tensor product structure. The central object is the commutator of region projectors, whose spectrum and operator norms quantify the extent to which a putative boundary is definite or indefinite. A quantum extension of region connection reasoning is proposed in which the commuting regime yields a classical Boolean core of region relations, while a noncommuting regime captures superposed and dynamically entangled partitions through an explicit interface sector. This yields a coordinate-free separability criterion expressed purely in terms of region relations, and it provides a framework in which boundary fuzziness becomes an observable notion tied to measurement compatibility.
Quantum theory typically presupposes a fixed factorisation of the total Hilbert space into subsystems, or an equivalent fixed choice of commuting subalgebras. This convention hard-codes a classical notion of boundary between "System A" and "System B". Recent work has formalised subsystem ambiguity by defining partitions directly at the level of C*-subalgebras, including non-tensor-factor subsystems, and by studying representability and the role of centres. In parallel, work under the label "quantum mereology" studies how subsystems can emerge top-down from spectral structure or from changes of tensor product structure. The aim here is different. We define RCC-style region relations directly from unitary invariants of pairs of sharp region projectors, so that boundary indefiniteness is a relation-level observable rather than a chosen decomposition. A quantum mereotopology is developed in which "regions" are represented by projection operators and their topological relations are defined intrinsically, without background space and without a privileged tensor product structure. The central object is the commutator of region projectors, whose spectrum and operator norms quantify the extent to which a putative boundary is definite or indefinite. A quantum extension of region connection reasoning is proposed in which the commuting regime yields a classical Boolean core of region relations, while a noncommuting regime captures superposed and dynamically entangled partitions through an explicit interface sector. This yields a coordinate-free separability criterion expressed purely in terms of region relations, and it provides a framework in which boundary fuzziness becomes an observable notion tied to measurement compatibility.
An algebraic approach to quantum non-separability is applied to the case of two qubits. It is based on the partition of the algebra of observables into independent subalgebras and the tensor product structure of the Hilbert space is not exploited. Even in this simple case, such a general formulation has some advantages. Using algebraic formalism, we can explicitly show the relativity of the notion of entanglement to the observables measured in the system and characterize separable and non-separable pure states. As a universal measure of non-separability of pure states, we propose to take the so-called total correlation. This quantity depends on the state as well as on the algebraic partition. Its numerical value is given by the norm of the corresponding correlation matrix.
One of the most basic notions in physics is the partitioning of a system into subsystems and the study of correlations among its parts. In this Letter, we explore this notion in the context of quantum reference frame (QRF) covariance, in which this partitioning is subject to a symmetry constraint. We demonstrate that different reference frame perspectives induce different sets of subsystem observable algebras, which leads to a gauge-invariant, frame-dependent notion of subsystems and entanglement. We further demonstrate that subalgebras which commute before imposing the symmetry constraint can translate into noncommuting algebras in a given QRF perspective after symmetry imposition. Such a QRF perspective does not inherit the distinction between subsystems in terms of the corresponding tensor factorizability of the kinematical Hilbert space and observable algebra. Since the condition for this to occur is contingent on the choice of QRF, the notion of subsystem locality is frame dependent.
Abstract Quantum Entanglement under B_U: Nonlocal Equivalent Settlement, Tensor Non-Factorization, and Mother-Structure Alignment redefines quantum entanglement within the framework of single-universe boundary consistency. The paper argues that entanglement is not a mysterious communication between two already independent particles, nor a pointwise inner product between two spatial locations in L^2(U). It is the manifestation of one global state across multiple projection interfaces. Formally, the total readable field space may be represented as H_{\mathrm{Total}}=L^2(U,d\mu), while two projected subsystems are described by H_A=L^2(U_A,d\mu_A), H_B=L^2(U_B,d\mu_B), and H_{AB}=H_A\otimes H_B. Entanglement occurs when \Psi_{AB}\in H_{AB} cannot be factorized as \psi_A\otimes\psi_B. The nonlocal relation is then read through local measurement operators O_A,O_B by the correlation function C_{AB}(O_A,O_B), rather than through a signal or spatial path. Under B_U, this correlation is interpreted as a boundary-constrained kernel \mathcal{K}_{B_U,M}(O_A,O_B;\Psi_{AB}), determined by one mother-structure M, one fixed boundary B_0, and one settlement surface. The central claim is that entangled systems are super-distance equivalent settlement points: separated in projection, aligned in total reality.
Decompositional theories describe the ways in which a global physical system can be split into subsystems, facilitating the study of how different possible partitions of a same system interplay, e.g. in terms of inclusions or signalling. In quantum theory, subsystems are usually framed as sub-C* algebras of the algebra of operators on the global system. However, most decompositional approaches have so far restricted their scope to the case of systems corresponding to factor algebras. We argue that this is a mistake: one should cater for the possibility for non-factor subsystems, arising for instance from symmetry considerations. Building on simple examples, we motivate and present a definition of partitions into an arbitrary number of parts, each of which is a possibly non-factor sub-C* algebra. We discuss its physical interpretation and study its properties, in particular with regards to the structure of algebras' centres. We prove that partitions, defined at the C*-algebraic level, can be represented in terms of a splitting of Hilbert spaces, using the framework of routed quantum circuits. For some partitions, however, such a representation necessarily retains a residual pseudo-nonlocality. We provide an example of this behaviour, given by the partition of a fermionic system into local modes.
This work develops a spectral interpretation of quantum entanglement, photons, and photonic quantum gates within the framework of Internal Spectral Theory (IST). IST starts from a pre-geometric relational spectral structure in which neither physical space, time, nor quantum mechanics is assumed at the fundamental level. Hilbert-space representations arise only at a subsequent readability stage, where spectral projectors, observables, and effective quantum states can be defined. Within this hierarchy, quantum entanglement is investigated as an effective manifestation of a deeper non-factorizable spectral organization. The paper examines how non-separability in the represented regime can inherit structural constraints from the underlying spectral system, while maintaining a clear distinction between fundamental non-factorization and ordinary quantum entanglement. The photonic sector is described through a distinguished low-energy spectral subspace in the effective theory. Photonic quantum operations are then characterized by the preservation of this logical subspace. For a projector Pph onto the admissible photonic sector, an ideal operation satisfies [U,Pph]=0 while departures from this condition generate leakage outside the logical space. This provides a quantitative framework for analysing the robustness of photonic gates and architectures. The construction is connected to the broader IST emergence chain: relational spectral structure → readable represented regime → nucleation → crystallization → effective quantum description → emergent three-dimensional space → separate temporal evolution → local 3D+t physics. The results concerning gate invariance and leakage bounds are operator-theoretic within the stated effective assumptions. The identification of the photon sector with a specific fundamental spectral structure, and the complete derivation of quantum entanglement from the pre-geometric IST kernel, remain open problems. The framework therefore provides both a concrete mathematical criterion for photonic operations and a set of falsifiable directions for further investigation.
In quantum information theory, standard notions typically presuppose a fixed partition of the world into subsystems, encoded in a tensor product structure of the Hilbert space. Yet, in some scenarios, the subsystem structure is not uniquely fixed. A case in point is the description of subsystems relative to different quantum reference frames (QRFs), where the tensor product structure, although fixed in a specific QRF, changes across different QRFs. Such a relativity of subsystems underlies many interesting features of QRFs, such as the relativity of entanglement, entropy, and coherence. It also raises a broader question: how do we compare the information across different subsystem perspectives, especially when the corresponding subsystem algebras overlap? Here, we propose a notion of quantum mutual information across different QRF perspectives, which captures the correlations between subsystems relative to different frames of reference. To do so, we focus on an algebraic definition of subsystems. Studying in detail the union and overlap between subsystem algebras relative to different QRFs, we find that the usual definition of mutual information in terms of the chain law overcounts the information associated to the overlap. Based on this insight, we propose a modified chain law and show that the resulting notion of mutual information satisfies subadditivity and connects naturally to other measures found in the literature [1]. Our investigations not only provide a way to quantify the relation between different QRF perspectives from an information-theoretic point of view but also sheds light on some important features of quantum information in absence of a fixed tensor product structure.