Optimization of quantum measurement processes has a pivotal role in carrying out better, more accurate or less disrupting, measurements and experiments on a quantum system. Especially, convex optimization, i.e., identifying the extreme points of the convex sets and subsets of quantum measuring devices plays an important part in quantum optimization since the typical figures of merit for measuring processes are affine functionals. In this thesis, we discuss results determining the extreme quantum devices and their relevance, e.g., in quantum-compatibility-related questions. Especially, we see that a compatible device pair where one device is extreme can be joined into a single apparatus essentially in a unique way. Moreover, we show that the question whether a pair of quantum observables can be measured jointly can often be formulated in a weaker form when some of the observables involved are extreme. Another major line of research treated in this thesis deals with convex analysis of special restricted quantum device sets, covariance structures or, in particular, generalized imprimitivity systems. Some results on the structure ofcovariant observables and instruments are listed as well as results identifying the extreme points of covariance structures in quantum theory. As a special case study, not published anywhere before, we study the structure of Euclidean-covariant localization observables for spin-0-particles. We also discuss the general form of Weyl-covariant phase-space instruments. Finally, certain optimality measures originating from convex geometry are introduced for quantum devices, namely, boundariness measuring how ‘close’ to the algebraic boundary of the device set a quantum apparatus is and the robustness of incompatibility quantifying the level of incompatibility for a quantum device pair by measuring the highest amount of noise the pair tolerates without becoming compatible. Boundariness is further associated to minimum-error discrimination of quantum devices, and robustness of incompatibility is shown to behave monotonically under certain compatibility-non-decreasing operations. Moreover, the value of robustness of incompatibility is given for a few special device pairs.
A graph-theoretical framework for quantum mechanics is developed in which observables and symmetry operators are incorporated into a single complete bi-coloured graph. The proposed Symmetry/Observables Quantum Graph (SOQG) treats Hamiltonians, observables, conserved quantities, and unitary symmetry transformations on equal footing as operators acting on a common Hilbert space. The vertices of the graph represent operators, while the colours of the edges are determined by their commutation relations: commuting operators are connected by maroon edges and non-commuting operators by teal edges. The resulting structure constitutes a finite Ramsey graph that provides a combinatorial representation of operator compatibility. The formalism is applied to several representative quantum systems, including a particle in a central potential, a charged particle in a uniform magnetic field, and a particle in a homogeneous external field. In each case, monochromatic cliques identify maximal compatibility sectors containing both observables and symmetry operators, thereby revealing the interplay between conserved quantities and dynamical symmetries. It is further shown that coloured-graph automorphisms generally differ from the physical symmetry group of the underlying quantum system, leading to the notion of second-order graph symmetry. The SOQG is proved to be invariant under simultaneous unitary transformations of all operators and therefore represents intrinsic properties of the operator algebra rather than a particular Hilbert-space representation. The graph-theoretical structure is analysed using Ramsey theory and the Mantel–Turán extremal theorem, which provide rigorous conditions for the appearance of mutually compatible and mutually incompatible operator subsets. The framework is subsequently generalized to parameter-dependent Symmetry/Observables Quantum Graphs (PSOQGs), whose compatibility structure changes as external parameters modify the underlying commutation relations. For magnetic translation operators, the magnetic flux through the elementary translation cell induces a flux-controlled colour-switching transition between commuting and non-commuting compatibility sectors, accompanied by changes in the graph automorphism group. The proposed framework establishes a direct connection between operator theory, symmetry, graph theory, and extremal combinatorics, providing a new structural perspective on quantum systems.
Decoherence, when viewed in the Heisenberg picture, can turn incompatible measurements into jointly measurable ones. This provides an observable-level description of emergent classicality in terms of the incompatibility destroyed by a noise channel. The corresponding loss of incompatibility can be captured by the channel's \emph{compatibility region}---the observables in a probe class that become jointly measurable with every noisy observable. The geometry and volume of this region provide a refined operational way to compare noise channels. We apply this framework to decoherence channels, each of which acts by Schur multiplication with extreme points of the convex set of unit-diagonal positive semidefinite matrices. Restricting to extremals eliminates convex mixing and reveals subtle phase-dependent decoherence effects thar are not captured by damping rates or conventional coherence quantifiers. Such channels have a rigid dilation structure described by a rank-one operator frame, which reduces joint measurability to a positivity test, making the compatibility region analytically tractable, and its volume computable from the frame's Gram matrix. If the frame can be normalised to a minimal informationally complete (MIC) POVM, the compatibility region acquires a geometric interpretation in the associated probability representation of the quantum state space, familiar with Qbism. Among maximal-rank extremals, those associated with symmetric informationally complete (SIC) POVMs maximise the compatibility volume and hence destroy the greatest amount of incompatibility, providing an operational characterisation of the special role of SICs in terms of joint measurability. This leads to a new formulation of the well-known SIC existence problem as a joint measurability question for noisy mutually unbiased bases.