On taking a positive operator-valued measure (POVM), we introduce nonclassical measurements, together with nonclassical phenomena, in a manner similar to that applied to nonclassical states. The quantitative measure of nonclassicality for the measurements is also defined, and any projective measurement is shown to be nonclassical. When the regularization process is applied to a projective measurement, it becomes a POVM measurement, and its nonclassical measure decreases to vanish at the end.
On taking a positive operator-valued measure (POVM), we introduce nonclassical measurements, together with nonclassical phenomena, in a manner similar to that applied to nonclassical states. The quantitative measure of nonclassicality for the measurements is also defined, and any projective measurement is shown to be nonclassical. When the regularization process is applied to a projective measurement, it becomes a POVM measurement, and its nonclassical measure decreases to vanish at the end.
In the heart of every physical theory lies measurement theory. While classical measurements are passive probes, QMs can be viewed as active physical processes that shape the available information. Depending on the measurement type, different aspects of the information about a system are retrieved. All possible measurements in quantum physics are described by mathematical objects called Positive Operator-Valued Measures (POVMs). The most familiar standard measurements are called Projective-Valued Measures (PVMs) and make up a subset of POVMs. The complementary subset of POVMs is non-projective. However, some non-projective POVMs can be expressed as a probabilistic mixture of PVMs. Such POVMs are called projectively simulable, while non-simulable POVMs are genuinely non-projective. These can be made projectively simulable by adding noise to the measurements. If a genuinely non-projective POVM requires a large amount of noise before it becomes projectively simulable, it is more non-projective than a POVM that requires less noise. Hence, the degree of non-projectiveness is determined by the required noise strength. The act of measuring a quantum system changes the quantum state. While quantum measurements only take into account the readable classical output value, quantum instruments account for both the classical outcome and the post-measurement state. Hence, instruments are associated with measurements, and in an analogous way, projective and projectively simulable instruments can be defined. For many purposes, non-projective POVMs are more useful than PVMs, and the more non-projective a POVM is, the greater the advantage of using them over PVMs. Therefore, the aim of the thesis is to investigate the genuinely non-projective POVMs. More precisely, the degree of non-projectiveness in qubit POVMs is studied. This is done by convex optimization, specifically by solving semidefinite programs. Furthermore, the thesis addresses the open problem of finding the most non-projective qubit quantum instrument by a witness-based see-saw algorithm.
We introduce an operational measure of nonclassicality for qubit systems based on the violation of Kolmogorov consistency conditions in sequential measurements. In contrast to previous work by Milz et al.~\cite{Milz-2020}, who characterized classicality via NCGD maps, and Sakuldee et al.~\cite{Sakuldee-2022a, Sakuldee-2022b}, who studied quantum correlations under measurement disturbance, our witness explicitly quantifies nonclassicality in terms of the POVM unsharpness parameter az. For projective measurements on an initially diagonal state, the witness vanishes identically, showing that such measurements cannot reveal nonclassicality. However, by generalizing to positive operator-valued measures (POVMs), we find that non-projective measurements can reveal nonclassicality even for diagonal initial states. We derive explicit expressions for the witness for general initial states, including coherences, and show that its maximum value is 1/4, which is a new result achieved for unbiased POVMs, maximal dephasing, and equal initial populations. Our results connect Kolmogorov consistency, Leggett-Garg inequalities, and POVMs, providing an experimentally accessible tool for detecting nonclassicality in qubit systems with potential applications in quantum technology certification.
We derive a measurement operator corresponding to a quantum nondemolition (QND) measurement of an atomic ensemble. The quantum measurement operator takes the form of a positive operator valued measure (POVM) and is valid for arbitrary interaction times, initial coherent state amplitudes, and final photon measurement outcomes. We analyze the dependence on various parameters and show that the effect of the QND measurement for short interaction times is to apply a Gaussian modulation of the initial state wavefunction. We derive approximate expressions for the POVM in various limits, such as the short interaction time regime and projective measurement limit. Several examples are shown, which show how spin squeezing and Schrodinger cat states can be generated using the measurement.
Abstract In this chapter we first review projective or von Neumann measurements from the point of view of the projection postulate and the Lüders rule, before turning to more general measurements modelled using positive operator-valued measures (POVMs). After examining their mathematical description, we discuss symmetric informationally complete (SIC) POVMs and Zauner’s conjecture, as well as the representation of POVMs as projective measurements on a larger Hilbert space, a result called Naimark dilation. We then a model for non-ideal projective measurements and analyse the properties of such measurements in relation to the properties unbiasedness, faithfulness, and non-invasiveness of ideal projective measurements. Finally, we consider the problem of distinguishing non-orthogonal quantum states and examine how the use of POVMs can be beneficial for this task.
Introduction As the decades of the twentieth century rolled on, quantum mechanics (QM) became more and more sophisticated and mathematical. Understanding what this theory means intuitively was confounded not only by nonclassical concepts such as wave–particle duality and quantum interference, but also by issues to do with observation. The Newtonian classical mechanics (CM) paradigm of reality, wherein reductionist laws of physics describe observer-independent dynamics of systems under observation (SUOs) with observer-independent properties, was found to be inadequate. Quantum theorists were confronted with the measurement problem , which attempts to understand, explain, and rationalize the laws of QM that underpin the processes of observation that go on in the laboratory. They are not the same as those of CM in several puzzling respects. Historically, the first sign of the measurement problem was Planck's quantization of energy (Planck, 1900b) and the second was Bohr's veto on radiation damping in hydrogen (Bohr, 1913). These occurred in the first quarter of the twentieth century, a period in physics often referred to as Old Quantum Mechanics . Another indicator that intuition was inadequate was Born's interpretation of the squared modulus of the Schrödinger wave function as a probability density (Born, 1926). That interpretation has everything to do with observers and observation, because probability without an observer is a vacuous concept. Eventually, the projection-valued measure (PVM) formalism emerged, championed by von Neumann in an influential book on the mathematical formulation of QM (von Neumann, 1955). Subsequently, pioneers such as Ludwig (Ludwig, 1983a,b) and Kraus (Kraus, 1974, 1983) refined the theory into the general positive operator-valued measure (POVM) formalism that we shall discuss and use. The quantized detector network (QDN) approach to quantum experiments is most naturally expressed in the PVM formalism, as QDN focuses on the individual detectors in the laboratory. However, the POVM formalism is more general than the PVM formalism, giving a description of multiple detector processes similar to QDN. This raises the question of how the two approaches, QDN and POVM, are related. The aim of this chapter is to explore this relationship. Before we review the PVM and POVM formalisms, we review some essential mathematical concepts.