We construct a resource theory of sharpness for finite-dimensional positive operator-valued measures (POVMs), where the sharpness−non−increasing operations are given by quantum preprocessing channels and convex mixtures with POVMs whose elements are all proportional to the identity operator. As required for a sound resource theory of sharpness, we show that our theory has maximal (i.e., sharp) elements, which are all equivalent, and coincide with the set of POVMs that admit a repeatable measurement. Among the maximal elements, conventional non-degenerate observables are characterized as the canonical ones. More generally, we quantify sharpness in terms of a class of monotones, expressed as the EPR–Ozawa correlations between the given POVM and an arbitrary reference POVM. We show that one POVM can be transformed into another by means of a sharpness-non-increasing operation if and only if the former is sharper than the latter with respect to all monotones. Thus, our resource theory of sharpness is complete , in the sense that the comparison of all monotones provides a necessary and sufficient condition for the existence of a sharpness-non-increasing operation between two POVMs, and operational , in the sense that all monotones are in principle experimentally accessible.
We construct a resource theory of sharpness for finite-dimensional positive operator-valued measures (POVMs), where the sharpness−non−increasing operations are given by quantum preprocessing channels and convex mixtures with POVMs whose elements are all proportional to the identity operator. As required for a sound resource theory of sharpness, we show that our theory has maximal (i.e., sharp) elements, which are all equivalent, and coincide with the set of POVMs that admit a repeatable measurement. Among the maximal elements, conventional non-degenerate observables are characterized as the canonical ones. More generally, we quantify sharpness in terms of a class of monotones, expressed as the EPR–Ozawa correlations between the given POVM and an arbitrary reference POVM. We show that one POVM can be transformed into another by means of a sharpness-non-increasing operation if and only if the former is sharper than the latter with respect to all monotones. Thus, our resource theory of sharpness is complete , in the sense that the comparison of all monotones provides a necessary and sufficient condition for the existence of a sharpness-non-increasing operation between two POVMs, and operational , in the sense that all monotones are in principle experimentally accessible.
A resource theory of quantum measurement can be addressed in terms of quantum coherence and measurement sharpness, respectively. The former analyzes the off-diagonal structure of POVM elements in a predetermined basis while the latter analyzes the deviation from trivial, state-independent, measurements. We establish a direct connection between the two resource theories by identifying measurement sharpness as the maximal coherence that is achievable under all possible unitary changes of the reference basis. For a broad class of POVMs whose elements share a common eigenbasis, we show that the maximal distance-based coherence of measurement coincides exactly with the corresponding distance-based sharpness monotone. We further extend this equivalence, with element-additive distances, to POVMs whose elements admit a common mutually unbiased basis structure. These results provide a measurement-theoretic analogue of the maximal-coherence \& purity correspondence for quantum states. We also show that the maximal coherence of measurement is faithful with respect to trivial measurements and is monotonic under fuzzifying operations for dichotomic measurements, as well as under mixed-unitary and unitarily covariant preprocessing channels. Finally, we illustrate the operational meaning and limitations of the equivalence through qubit POVMs, single-photon phase sensing, and noisy photon-number resolving detection. In particular, the maximal Fisher information in a Mach-Zehnder interferometer is shown to be determined by the squared maximal coherence of the measurement, while in an imperfect photon-number resolving detector the maximal coherence behaves as a proper sharpness monotone, unlike conventional PVM-based unsharpness measures.
Coherence is a cornerstone of quantum theory and a prerequisite for the advantage of quantum technologies. In recent work, the notion of coherence with respect to a general quantum measurement, i.e., positive operator-valued measure (POVM), was introduced and embedded into a resource-theoretic framework that generalizes the standard resource theory of coherence. In particular, POVM-incoherent (free) states and operations were established. In this work, we explore features of this framework which arise due to the rich structure of POVMs compared to projective measurements. Moreover, we introduce a rigorous, probabilisitic framework for POVM-based coherence measures and free operations. This leads to the introduction of strongly monotonic resource measures that neatly generalize well-known standard coherence measures. Finally, we show that the relative entropy of POVM coherence is equal to the cryptographic randomness gain, providing an important operational meaning to the concept of coherence with respect to a general measurement.
While there is general consensus on the definition of incompatible POVMs, moving up to the level of instruments one finds a much less clear situation, with mathematically different and logically independent definitions of incompatibility. Here we close this gap by introducing the notion of q − c o m p a t i b i l i t y , which unifies different notions of POVMs, channels, and instruments incompatibility into one hierarchy of resource theories of communication between separated parties. The resource theories that we obtain are c o m p l e t e , in the sense that they contain complete families of free operations and monotones providing necessary and sufficient conditions for the existence of a transformation. Furthermore, our framework is fully o p e r a t i o n a l , in the sense that free transformations are characterized explicitly, in terms of local operations aided by causally-constrained directed classical communication, and all monotones possess a game-theoretic interpretation making them experimentally measurable in principle. We are thus able to pinpoint exactly what each notion of incompatibility consists of, in terms of information-theoretic resources.
Resource theories provide a general framework for the characterization of properties of physical systems in quantum mechanics and beyond. Here we introduce methods for the quantification of resources in general probabilistic theories (GPTs), focusing in particular on the technical issues associated with infinite-dimensional state spaces. We define a universal resource quantifier based on the robustness measure, and show it to admit a direct operational meaning: in any GPT, it quantifies the advantage that a given resource state enables in channel discrimination tasks over all resourceless states. We show that the robustness acts as a faithful and strongly monotonic measure in any resource theory described by a convex and closed set of free states, and can be computed through a convex conic optimization problem. Specializing to continuous-variable quantum mechanics, we obtain additional bounds and relations, allowing an efficient computation of the measure and comparison with other monotones. We demonstrate applications of the robustness to several resources of physical relevance: optical nonclassicality, entanglement, genuine non-Gaussianity, and coherence. In particular, we establish exact expressions for various classes of states, including Fock states and squeezed states in the resource theory of nonclassicality and general pure states in the resource theory of entanglement, as well as tight bounds applicable in general cases.
We identify an operational principle that singles out Projection-Valued Measures (PVMs) among general Positive Operator-Valued Measures (POVMs), bridging the modern quantum measurement theory and the traditional formulation based on projective measurements of physical observables. We reformulate Ozawa's intersubjectivity condition, which requires inter-observer agreement of the measurement outcomes, in a quantitative manner within the framework of generalized probabilistic theories. We prove that (i) a POVM is a PVM if and only if its every coarse-graining is intersubjective, and (ii) a system is classical if and only if intersubjectivity is preserved under any coarse-graining, establishing a complete characterization of the physical observables and the classical theory. Furthermore, measurements with intersubjectivity are sufficiently rich for the informational tasks of state tomography and state discrimination, testifying to its operational significance in quantum and beyond information processing.
Quantum measurements lie at the core of quantum information processing, enabling tasks ranging from state reconstruction and parameter estimation to sensing and computation. As quantum technologies scale and become increasingly precise, a central question emerges: what does it mean for a quantum measurement to be good? This thesis addresses this question by developing quantitative, operational, and physically grounded frameworks for assessing the quality and informational power of quantum measurements.The first part of the thesis focuses on the intrinsic quality of quantum measurements independent of any particular input state. Adopting the formalism of quantum resource theories, measurements are treated as resources ordered under classical post-processing, which cannot increase informational content. Restricting attention to informationally complete measurements, we employ frame theory to introduce a novel resource monotone termed completeness stability, defined as the minimum eigenvalue of a suitably scaled frame operator. We establish its operational significance by showing that the average error in quantum state tomography and the average variance in shadow tomography are tightly bounded by the inverse of the completeness stability. Maximizing this monotone identifies weighted complex projective 2-designs, including symmetric informationally complete POVMs and mutually unbiased bases, as optimal measurement configurations.The second part of the thesis extends these ideas to quantum parameter estimation, where measurement performance necessarily depends on correlations between the measurement and the encoded quantum state. By introducing a state-dependent inner product on operator space, we develop an operator-theoretic formulation in which both the classical and quantum Fisher information arise naturally from a single frame operator associated with the measurement and the reference state. Within this framework, the spectral properties of the frame operator provide a complete characterization of estimation performance. The maximum and minimum eigenvalues identify the best- and worst-case parameter-encoding directions, respectively, yielding sharp bounds on the achievable ratio of classical and quantum Fisher information. This establishes a direct connection between informational completeness, estimation robustness, and resource-theoretic monotonicity.The final part of the thesis returns to foundational questions concerning quantum measurement and arrival time. Using Bohmian mechanics, we develop a microscopic, first-principles account of the Hong–Ou–Mandel effect, a paradigmatic two-particle interference phenomenon. By analyzing particle trajectories guided by the wave function, we show how bosonic bunching and fermionic anti-bunching emerge dynamically from the guiding equation without invoking wave-function collapse. Beyond reproducing the Hong–Ou–Mandel dip, we demonstrate how experimental parameters such as barrier strength, phase velocity, and path asymmetry influence both coincidence probabilities and temporal correlations.Taken together, this thesis provides a unified treatment of quantum measurement quality across operational, metrological, and dynamical regimes. By bridging resource theory, frame theory, Fisher information, and trajectory-based quantum dynamics, it offers new insights into the structure, limitations, and physical interpretation of quantum measurements in both applied and foundational contexts.