Date of Award

8-31-2026

Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Physics, Applied

First Advisor

Christopher A. Fuchs

Abstract

Informationally complete measurements provide a bridge between the operational and geometric structure of quantum theory. They allow quantum states to be reconstructed from measurement probabilities. This thesis studies informationally complete measurements from two complementary perspectives: their physical implementation and their role in the probabilistic reconstruction of quantum theory.

The first part develops a symmetry-driven Naimark extension for rank-one Weyl- Heisenberg covariant Projective Operator-Valued Measures (POVMs) in arbitrary finite dimension. Since every element of such a POVM is generated from a single fiducial state by the Weyl-Heisenberg displacement operators, the full measurement inherits a strong covariance structure. We show that this structure can be used to reduce the construction of the Naimark unitary from a general $(d^2\times d^2)$ problem to the choice of a single $(d\times d)$ unitary matrix whose first row is fixed by the fiducial state. The resulting Naimark unitary has a block-circulant form, admits a Fourier block diagonalization, and yields a simple optical multiport implementation. We also show that the same construction has an equivalent generalized Bell-basis interpretation: preparing the ancilla in the complex conjugate fiducial state and measuring the joint system in a generalized Bell basis realizes the desired Weyl-Heisenberg covariant POVM.

The second part turns from implementation to reconstruction. In the QBist approach, a quantum state represents an agent's probability assignments, and the Born rule becomes a normative relation among those probabilities. Using a special type of POVM known as symmetric, informationally complete (SIC)-POVM as a reference measurement, quantum states can be represented as probability vectors, and the Born rule takes the form of the Urgleichung. The qplex framework abstracts the geometry of these SIC probability vectors. We use this framework to study bipartite correlations by expressing joint expectation values as inner products between $C$-vectors associated with Alice's and Bob's measurement settings. This reformulates Bell inequalities as geometric optimization problems over the centers, norms, and relative orientations of probability-derived vectors. For the Clauser- Horne--Shimony--Holt (CHSH) inequality, the common inner-product structure of the $C$-vectors is strong enough to recover the Tsirelson bound of $2\sqrt2$. In contrast, for the three-outcome Collins--Gisin--Linden--Massar--Popescu (CGLMP) inequality, the corresponding complex $C$-vector geometry allows a violation of up to $2+2\sqrt(3)/3 \approx 3.1547$ versus the quantum maximum of $\approx 2.8729$, thereby exhibiting super-quantum correlations. Thus general qplex constraints reproduce an important quantum feature in the two-outcome case but are not sufficient to recover all quantum correlation bounds. This contrast identifies multi-outcome Bell scenarios as a useful diagnostic for separating Hilbert-space quantum theory from more general qplex theories.

The thesis therefore connects the practical problem of realizing quantum measurements with the foundational problem of understanding which probabilistic principles single out quantum theory.

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