What Survives a Change of Quantum Reference Frame? Coherence Thresholds for the Spectral, Holonomic, and Topological Content of Mixed-State Quantum Geometry
A reduced density matrix conditioned on an internal quantum reference frame (QRF) carries, through its mixed-state quantum geometric tensor, a Fisher–Bures metric and an Uhlmann gauge field. We determine which parts of this structure are frame independent, separating passive frame changes (reparametrizations and observer-local unitaries) from active ones (a change of the internal frame itself) within the perspective-neutral framework. Under passive changes the spectrum and the eigenbundle Chern numbers are invariant for every smooth single-valued transformation, the metric and the Uhlmann holonomy only for reparametrizations and constant unitaries. Under an active change between ideal frames the observer’s marginal obeys an exact shifted-overlap law for every relational state; its charge distribution is frame invariant, which gives the entanglement plus-coherence invariant of Cepollaro et al. in closed form. For a product relative state the law reduces to a partial twirl (relativization), under which the metric can only contract. For a two level observer the geometry is then graded by one inter-frame coherence |κ|: the Bures area is π|κ|, the Chern numbers survive for every |κ| > 0, and the equatorial Uhlmann Z2 holonomy only for |κ| > √3/2, the known thermal critical point reached here by coherence instead of temperature. For bounded generators the observer’s definability is itself frame relative.
A reduced density matrix conditioned on an internal quantum reference frame (QRF) carries, through its mixed-state quantum geometric tensor, a Fisher–Bures metric and an Uhlmann gauge field. We determine which parts of this structure are frame independent, separating passive frame changes (reparametrizations and observer-local unitaries) from active ones (a change of the internal frame itself) within the perspective-neutral framework. Under passive changes the spectrum and the eigenbundle Chern numbers are invariant for every smooth single-valued transformation, the metric and the Uhlmann holonomy only for reparametrizations and constant unitaries. Under an active change between ideal frames the observer’s marginal obeys an exact shifted-overlap law for every relational state; its charge distribution is frame invariant, which gives the entanglement plus-coherence invariant of Cepollaro et al. in closed form. For a product relative state the law reduces to a partial twirl (relativization), under which the metric can only contract. For a two level observer the geometry is then graded by one inter-frame coherence |κ|: the Bures area is π|κ|, the Chern numbers survive for every |κ| > 0, and the equatorial Uhlmann Z2 holonomy only for |κ| > √3/2, the known thermal critical point reached here by coherence instead of temperature. For bounded generators the observer’s definability is itself frame relative.
We observe that the ingredients a relational construction already requires contain a gauge structure, and that its group is the one the frame cannot see. LetM= Gbig/H be a homogeneous space. Three independently established facts close into a statement when placed together. An operational quantum reference frame is a covariant POVM on a homogeneous space Σ ∼= G/Hx, and the relativization map expressing a system observable relative to it is defined only on the Hx-invariant part of the system algebra. The canonical connection of a reductive homogeneous space is h-valued: the projection of the Maurer–Cartan form onto the isotropy algebra. And the isotropy directions carry zero Fisher–Bures weight, which, that metric being the maximal monotone one, is the first statement in the vocabulary of the geometry. Four consequences. Gauge invariance is not imposed on the relational description; it is the condition under which an observable can be expressed relative to a frame at all. The Cartan decomposition splits what the frame cannot resolve from what it measures. The structure is non-abelian whenever the isotropy is, and for Sp(2n,R)/U(n) the isotropy is U(n). And the size of that isotropy and the multiplicities of its content are fixed by the field content, NB = 2 dimG+nH, a gauge sector carrying multiplicity two because the adjoint is real and a charged scalar one because the charge is complex. We then measure what the gauge algebra’s commutant inside the isotropy retains of it: the double centraliser is the full type-preserving algebra L λ u(dλ), of dimension 82 against the gauge algebra’s 12. The commutant does not remember the gauge group; it remembers the isotypic decomposition.No gauge group is derived here, and nothing is identified with anything physical.
The perspective-neutral formulation of quantum reference frames (QRFs) treats observers as quantum systems and describes physics relationally from within the composite system. While frame-change maps and frame-invariant resource sums are theoretically understood, their impact on circuit-based quantum information processing has largely remained unexplored. We formulate QRF transformations as circuit compilation rules and, for systems with finite Abelian symmetry described by the regular representation, derive a gate-level dictionary that maps local operations in one frame to their images in another. This yields a group-theoretic classification of gates where symmetry-commuting operators remain local, up to frame-dependent phases, while generic gates are promoted to controlled entangling operations in which the original frame acts as a control register. The resulting frame-dependence entangling-gate count defines a relational circuit complexity where the cost of a computation depends on the internal reference frame of the observer. We instantiate the framework in a three-qubit model and show that the QRF unitary acts as a lossless converter between a purity-based local coherence and concurrence, preserving their invariant sum and giving a concrete realization of the relativity of entanglement. We implement the corresponding circuits on an IBM Quantum superconducting platform, using full state tomography to reconstruct the redistribution of resources between internal frames. The hardware data reproduce the predicted conversion of local coherence into entanglement, and the observed deviations from exact conservation quantify the effect of realistic device noise on relational quantum protocols.
In the absence of external relata, internal quantum reference frames (QRFs) appear widely in the literature on quantum gravity, gauge theories and quantum foundations. Here, we extend the perspective-neutral approach to QRF covariance to general unimodular Lie groups. This is a framework that links internal QRF perspectives via a manifestly gauge-invariant Hilbert space in the form of "quantum coordinate transformations", and we clarify how it is a quantum extension of special covariance. We model the QRF orientations as coherent states which give rise to a covariant POVM, furnishing a consistent probability interpretation and encompassing non-ideal QRFs whose orientations are not perfectly distinguishable. We generalize the construction of relational observables, establish a variety of their algebraic properties and equip them with a transparent conditional probability interpretation. We import the distinction between gauge transformations and physical symmetries from gauge theories and identify the latter as QRF reorientations. The "quantum coordinate maps" into an internal QRF perspective are constructed via a conditioning on the QRF's orientation, generalizing the Page-Wootters formalism and a symmetry reduction procedure. We find two types of QRF transformations: gauge induced "quantum coordinate transformations" as passive unitary changes of description and symmetry induced active changes of relational observables from one QRF to another. We reveal new effects: (i) QRFs with non-trivial orientation isotropy groups can only resolve isotropy-group-invariant properties of other subsystems; (ii) in the absence of symmetries, the internal perspective Hilbert space "rotates" through the kinematical subsystem Hilbert space as the QRF changes orientation. Finally, we invoke the symmetries to generalize the quantum relativity of subsystems before comparing with other approaches. [Abridged]
This paper is a duplicated. Updated and complete version available at https://doi.org/10.5281/zenodo.21591128" I determine the behaviour, under a change of internal quantum reference frame (QRF), of the geometry and gauge structure obtained from the mixed-state quantum geometric tensor (msQGT) of a reduced density matrix ρObs. The setting is the Relational–Informational Model (RIM), in which the emergent metric and gauge field are the real and imaginary sectors of the msQGT of an observer’s state, defined relationally within a Wheeler–DeWitt universe; I take that identification as given and ask only how the resulting structure transforms between internal observers. Since ρObs is a subsystem state conditioned on a QRF, it inherits the frame relativity of subsystems established by the perspective-neutral QRF programme, which I make precise. In the ideal regime, where the frame-change map is a unitary, the observer’s spectral degeneracy pattern—which fixes the emergent gauge group—and the topological charges of the Uhlmann curvature are frame-invariant, while the metric and gauge potentials are frame-covariant; I verify the topological invariance numerically for spin-1/2 and spin-1 monopoles, and show that it holds only for smooth, single-valued frame changes (singular, Dirac-string changes inject charge). In the physical regime, where the relevant constraint generator is bounded below, the factorisation condition of Ali Ahmad et al. fails in some frames, so that whether ρObs—and hence the emergent gauge content—is even definable becomes frame-relative. The new element relative to existing QRF results is that the msQGT endows the QRF-relative reduced state with an information-geometric (Uhlmann/Yang–Mills) structure whose integer-valued topological charges constitute a new class of QRF invariant, distinct from—and complementary to—the known continuous resource invariant (the conserved entanglement-plus-coherence sum), which carries no gauge structure. The upshot is a sharp division: the objective content of the emergent structure is topological, its metric/gauge representation is perspectival, and in the physical regime even the observer’s existence as a subsystem is frame-relative.
For full-rank density matrices the state space is contractible, the Uhlmann bundle is topologically trivial, and the holonomy admits no quantized invariants; although the Uhlmann phase is genuinely geometric, this topological poverty has kept it from serving as a robust physical diagnostic of mixed-state matter. Mixed states of fixed rank below the Hilbert-space dimension, however, are ubiquitous: reduced states of constrained subsystems, states confined to invariant sectors, and states supported on decoherence-free subspaces can have a support smaller than the Hilbert space, a support that can vary with parameters. Their geometry is far richer: the support carries genuine curvature, non-Abelian holonomy, and Chern topology. We formulate Uhlmann's theory directly on the manifold of rank-k density matrices: minimal purifications make the fixed-rank stratum the base of a principal U(k)-bundle whose Uhlmann connection is uniquely determined by a Sylvester equation, and in an eigenframe this connection takes a closed form reducing to the Berry, Wilczek--Zee, and faithful Uhlmann connections at k=1, at equal weights, and at k=N. The fixed-rank bundle inherits the topology of the Grassmannian, and non-factorizable higher Chern topology requires failure of the global eigenline splitting, which in the present spectral setting requires degeneracy, minimally realized by a rank-2 Yang monopole whose quantized second Chern number links the geometry to the four-dimensional quantum Hall response. Solvable models, from a genuinely non-Abelian holonomy to a dissipatively driven orbit whose Uhlmann holonomy is steered by the reservoir through an elliptic-integral spectral dressing, illustrate the content: the supporting subspace carries the topology, the spectral weights shape the transport, and Uhlmann holonomy provides a direct geometric probe of the resulting mixed-state structures.
This paper is the thirty-first in a sequence developing an interpretive framework that redescribes physical reality as a causally consistent history of information updates rather than as a collection of independently existing objects. It is the third panel — the capstone — of the eighth grouping, a foundational-physics trilogy. Its subject is spacetime localization: where and when a record or system is — its position, orientation, relative phase, and time. The claim is that localization is real but not authoritatively global. There is no Global Frame: no physically privileged reference system that fixes absolute location and orientation, a privileged global simultaneity, a classical frame standing outside quantum theory, and one physically privileged, globally valid perspectival gauge-fixing. Localization is relational: positions and times are defined relative to a declared quantum reference frame (QRF), itself a quantum system that may be superposed and entangled with what it localizes (Giacomini, Castro-Ruiz, & Brukner, 2019; Loveridge, Miyadera, & Busch, 2018). What keeps this objective rather than subjective is a strict typing of cross-frame comparison: a single gauge-invariant relational (Dirac) observable has one invariant prediction, its perspectival representations intertwined by a frame-change map (case A); distinct relational observables form a relationally consistent web (case B); and where two frames share no common refinement their claims are declared incomparable (case C). "Objective" means invariance of the physical predictions under the declared symmetry group and the frame-change groupoid over the admissible frame family, to a declared tolerance — never frame-independence in an absolute sense. The flagship R-A is a dynamics-independent kinematic existence witness: a finite-dimensional discrete-ring model exhibits one physical state and two admissible ideal QRFs under which the localization profile and the entanglement structure differ — a sharp product relative to one particle; an entangled state with a delocalized, mixed marginal relative to another — while the distribution of the same relational observable is preserved. R-B splits into a domain covariance on shared refinements (ideal QRFs compose and agree on shared Dirac predictions) and a model-dependent global-perspective obstruction: where a globally valid relational gauge-fixing fails, as in known Gribov-type models (Vanrietvelde, Höhn, & Giacomini, 2023), no single perspective covers the whole physical state space — which is neither the same as, nor a consequence of, the absence of a privileged frame, and does not preclude a perspective-neutral description. R-C places Paper 8's relational time as the time-translation special case, with its finite-resolution-clock, conditional-covariance, and record–clock qualifications, separating the retired claim (no global clock) from the deferred one (relativistic foliation with dynamical geometry). R-D identifies and transports the frame-representable localization sector of certain Paper-30 grains, partially discharging that paper's grain-coherence deferral while leaving interaction, fragmentation, redundancy, and history-family structure grain-relative. This completes the trilogy's dismantling of the Global Bookkeeper while keeping geometry fixed: the quantum-mechanics–gravity interface is prepared, not entered. Localization is frame-relative but objective: no absolute frame, no globally gauge-fixed perspective claimed beyond what models establish, and no anarchy. ALAC Paper 31. Third panel (capstone) of the eighth grouping — a foundational-physics trilogy (29 Public Quantum Records -> 30 Classicality Without a Global Classicalizer -> 31 Spacetime Localization Without a Global Frame). Part of the Consistency-Constrained Information History (Akashic Ledger Architecture Cosmology) series. Presents no new quantum mechanics; results are interpretive (L2) on imported physics (L1).