A finite public comparison doctrine supplies finite distinctions between states, not a metric. This paper shows how such finite comparison data nevertheless determine a canonical positive quadratic comparison form. Given a finite-dimensional real comparison-germ quotient \(E_{\mathrm{pub}}\) with its chosen public comparison-difference normalization, form the closed centrally symmetric convex hull \(K_{\mathrm{pub}}\) of those difference vectors. When \(K_{\mathrm{pub}}\) is compact and spans \(E_{\mathrm{pub}}\), the John--L\"owner extremal ellipsoid theorem supplies a unique minimal-volume ellipsoid enclosing \(K_{\mathrm{pub}}\). Its unit balldefines a positive definite quadratic form \(g_{\mathrm{pub}}\) on \(E_{\mathrm{pub}}\). The construction is functorial with respect to public symmetries: every linear automorphism preserving the comparison body \(K_{\mathrm{pub}}\) preserves the quadratic form \(g_{\mathrm{pub}}\). Thus the metric is not postulated. It is the canonical quadratic envelope of the finite public distinguishability body. For the maximal comparison doctrine on a finite set, the construction yields the standard Euclidean form on the zero-sum comparison space, with the scale fixed by the public difference radius. This is the affine comparison geometry underlying the classical simplex before probabilities are added. A final outlook explains how the same convex-body mechanism interfaces with quantum comparison bodies and information geometry.
A finite public comparison doctrine supplies finite distinctions between states, not a metric. This paper shows how such finite comparison data nevertheless determine a canonical positive quadratic comparison form. Given a finite-dimensional real comparison-germ quotient (E_{\mathrm{pub}}) with its chosen public comparison-difference normalization, form the closed centrally symmetric convex hull (K_{\mathrm{pub}}) of those difference vectors. When (K_{\mathrm{pub}}) is compact and spans (E_{\mathrm{pub}}), the John--L"owner extremal ellipsoid theorem supplies a unique minimal-volume ellipsoid enclosing (K_{\mathrm{pub}}). Its unit balldefines a positive definite quadratic form (g_{\mathrm{pub}}) on (E_{\mathrm{pub}}). The construction is functorial with respect to public symmetries: every linear automorphism preserving the comparison body (K_{\mathrm{pub}}) preserves the quadratic form (g_{\mathrm{pub}}). Thus the metric is not postulated. It is the canonical quadratic envelope of the finite public distinguishability body. For the maximal comparison doctrine on a finite set, the construction yields the standard Euclidean form on the zero-sum comparison space, with the scale fixed by the public difference radius. This is the affine comparison geometry underlying the classical simplex before probabilities are added. A final outlook explains how the same convex-body mechanism interfaces with quantum comparison bodies and information geometry.
We study how much of the Hilbert-space and Born-rule structure of quantum theory can be obtained from operational primitives that involve records but not probabilities. The setting is a regime in which coherent alternatives recombine without leaving a stable record. We obtain three results of different kinds. Geometry. From composable contributions, record-inert reversible controls and amplitude scalings, under named conditions, we obtain a real inner-product event geometry. The existence of the inner product is equivalent to bounded mixing and is assumed in that form. In a worked interferometer with one mode per channel, the metric is not an input: metric-free port swaps and phase flips fix the form up to scale, and boundedness only forces the other controls to preserve it. Its uniqueness up to scale, the orthogonality of distinct outcomes, and the representation of events by orthogonal projections are derived. Obstruction. This geometry, even with full symmetry about the preparation and exact dyadic standards, does not select the quadratic weight: the preparation mixed with the maximally mixed state survives. The structural operations exhibited here certify only values on which the quadratic weight equals the counting weight. On restricted event families where Gleason’s theorem is unavailable, a calibration order oriented by the quadratic magnitude selects it: given exact standards, a state is the quadratic one exactly when it respects all such brackets. Whether such an orientation can be earned is the open problem. Given such brackets and refinement only to a finite depth K, every admissible state lies within 2^-K of the quadratic weight, without the Archimedean step. On the full projection lattice, Gleason’s theorem with null support, or sphere-transitive symmetry with null support, gives other conditional routes. Bridge. A predictive probability kernel is introduced separately. Its identification with the real quadratic weight ||P_A ψ||² / ||ψ||² is conditional on kernel-admission hypotheses, one of which is equivalent to the Born value; the complex form needs phase invariance in addition. The hypotheses select a class of contexts, and none is shown to be forced. For each we state the physical feature it expresses and a sufficient condition where one is known, and we work out one apparatus in which the geometry-stage hypotheses hold together. For all but one we give a context in which it fails (for the founding conditions, one that leaves the regime); for the remaining one, TCC-1, we give an illustration. This paper is the first in a series built on a common framework, Structural Ledger Consistency (in preparation).
The companion paper on effective composition characterises, for a finite substrate without a supplied subsystem factorisation, the possibilistic support layer completely and the preparation layer for two reference classes, and explicitly leaves open the states-and-effects layer of the resulting composition contract. We resolve that layer's finite, mono-system, and bipartite-composition questions, keeping the general preparation layer and the tensor product exactly as open as the companion paper left them. The primitive family arising from cylindrical conditioning of a strictly positive, rank-one reference measure is not itself convex: it is finite while its own convex hull is a continuum, unconditionally, for any local outcome set of size at least two. An explicit classical-randomisation postulate closes it onto the full classical probability simplex, independently of the reference measure's rank; rank governs composition of the primitive family, not the shape of its closure once randomisation is added. Simplicial geometry fixes the mathematical representation of affine effects as vectors in a cube, but not their physical availability as measurements; a weak coarse-graining axiom already yields the entire Boolean skeleton of sharp effects for free, while the full continuum of effects requires a further, dual randomisation postulate on the measurement side. Under two further, distinct, and logically independent axioms — one for a common classical control resource, one for the physical existence of a composed bipartite effect — the pointwise product of two local effects is proved unique, not postulated, and local tomography holds exactly when the local effect families linearly span their ambient functional spaces, equivalently when they separate their local states and contain the unit effect in their linear span. All results are finite, exact, and reproduced by committed scripts using rational arithmetic. Interpretive reading (structural, not a result): the finite hypotheses examined here force, successively and for free, a classical, simplicial operational skeleton — convex states once mixing is supplied, a Boolean effect algebra once coarse-graining is supplied, and exact local tomography once spanning local families are supplied — under precisely identified imports, and produce neither a quantum state geometry nor the Born rule. Locating exactly how much operational structure a finite admissibility contract yields for free, and where every further richness must instead be imported, is the intended service of this result to the non-injective foundations programme: it marks the boundary this classical skeleton cannot cross on its own, and hands that boundary, precisely typed, to whatever module or phase law is proposed next.
We show that the strongly symmetric spectral convex compact sets are precisely the normalized state spaces of finite-dimensional simple Euclidean Jordan algebras and the simplices. Spectrality is the property that every state has a convex decomposition into perfectly distinguishable pure states; strong symmetry is transitivity, for each integer N, of the affine automorphism group of the state space on lists of N perfectly distinguishable pure states. Additional assumptions combine with this theorem to give simple characterizations of finite-dimensional complex quantum state space. Important aspects of quantum and classical thermodynamics and of query complexity have been generalized to classes of general probabilistic theories (GPTs) satisfying natural postulates including or implying spectrality and strong symmetry; our result shows that these apply to a narrower class of theories than might have been hoped. Sorkin's notion of irreducibly k-th order interference has been studied in the GPT framework and looked for in experiments. Our result shows that the assumption of no higher-order (k > 2) interference, used along with spectrality and strong symmetry to characterize the same class of Jordan-algebraic convex sets by Barnum, Mueller, and Ududec in arXiv:1403.4147, was superfluous. It also implies that Lee and Selby's extension, on the assumption that interference has fixed maximal degree k, of the important order square root of N lower bound on the quantum black-box query complexity of searching N possibilities for one having a desired property (which is achieved by Grover's quantum algorithm), to a class of theories satisfying certain postulates allowing the formulation of a generalized notion of query algorithm, actually applies in the Jordan-algebraic setting where higher-order interference is not possible.
It is usually taken for granted that the natural mathematical framework for quantum mechanics is the theory of Hilbert spaces, where pure states of a quantum system correspond to complex vectors of unit length. These vectors can be combined to create more general states expressed in terms of positive semidefinite matrices of unit trace called density operators. A density operator tells us everything we know about a quantum system. In particular, it specifies a unique probability for any measurement outcome. Thus, to fully appreciate quantum mechanics as a statistical model for physical phenomena, it is necessary to understand the basic properties of its set of states. Studying the convex geometry of quantum states provides important clues as to why the theory is expressed most naturally in terms of complex amplitudes. At the very least, it gives us a new perspective into thinking about structure of quantum mechanics. \n \n \nThis thesis is concerned with the structure of quantum state space obtained from the geometry of the convex set of probability distributions for a special class of measurements called symmetric informationally complete (SIC) measurements. In this context, quantum mechanics is seen as a particular restriction of a regular simplex, where the state space is postulated to carry a symmetric set of states called SICs, which are associated with equiangular lines in a complex vector space. The analysis applies specifically to 3-dimensional quantum systems or qutrits, which is the simplest nontrivial case to consider according to Gleason's theorem. It includes a full characterization of qutrit SICs and includes specific proposals for implementing them using linear optics. The infinitely many qutrit SICs are classified into inequivalent families according to the Clifford group, where equivalence is defined by geometrically invariant numbers called triple products. The multiplication of SIC projectors is also used to define structure coefficients, which are convenient for elucidating some additional structure possessed by SICs, such as the Lie algebra associated with the operator basis defined by SICs, and a linear dependency structure inherited from the Weyl-Heisenberg symmetry. After describing the general one-to-one correspondence between density operators and SIC probabilities, many interesting features of the set of qutrits are described, including an elegant formula for its pure states, which reveals a permutation symmetry related to the structure of a finite affine plane, the exact rotational equivalence of different SIC probability spaces, the shape of qutrit state space defined by the radial distance of the boundary from the maximally mixed state, and a comparison of the 2-dimensional cross-sections of SIC probabilities to known results. Towards the end, the representation of quantum states in terms of SICs is used to develop a method for reconstructing quantum theory from the postulate of maximal consistency, and a procedure for building up qutrit state space from a finite set of points corresponding to a Hesse configuration in Hilbert space is sketched briefly.
We propose a geometric formulation of quantum indeterminacy based on polar duality between convex bodies representing position and momentum data. A pair (X,P) of centrally symmetric convex bodies is said to satisfy the indeterminacy condition when the h-polar of X is included in P. We show that this condition naturally arises from the geometry of quantum blobs and is closely related to Hardy's uncertainty principle and the Donoho--Stark inequalities. Using symplectic capacities and John ellipsoids, we associate a canonical covariance ellipsoid with every quantum polar pair and prove that it satisfies the quantum condition. The Robertson--Schrödinger inequalities follow as a consequence. This provides a non-statistical formulation of quantum indeterminacy from which the usual uncertainty relations emerge. which the usual uncertainty relations emerge.
Complex multiplication, coherent addition, and quadratic readout are related ingredients of quantum calculation, but they answer different reconstruction questions. We separate them on a supplied reciprocal carrier. The centralizer of a complex structure on a real two-plane is the complex algebra; its invertible transports are nevertheless not closed under the addition needed for destructive interference at a selected output. A one-outcome dephaser and a reversible phase kick show that certainty of a public outcome does not imply nondisturbance. Within an explicitly assumed positive quadratic response class, calibrated zero-leakage comparisons on a connected tree select the all-ones Gram kernel. We prove quantitative stability estimates and retain the distinction between unconditional record-local operations, selective conditioning, and a controlled inverse interaction. A local normalized-weight ansatz, arbitrary anchored orthogonal refinement, and additive coarse-graining of recorded outcomes force weights proportional to quadratic size. Equal binary refinement alone does not: a positive continuous increasing log-periodic function supplies a counterexample, while binary and ternary refinement at every size, with continuity, recover linearity in size. Real rotations implement the requisite refinements and controlled tagging without full complex control. Fixed-complement preservation and reciprocal reversal admit different generator sectors; independently switchable real mixers on a connected graph plus one addressable local phase generate the unitary Lie algebra. A common scalar phase does not. Supplementary finite action–phase and history examples expose the remaining calibration assumptions. The contribution is an assumption-controlled synthesis of conditional selectors and obstructions, not the introduction of the standard component constructions. The carrier, comparison response, control resources, tensor interpretation, and empirical probability interface are not derived from primitive complementarity.