Projective measurements in quantum theory have a very simple algebraic definition, but their information theoretic significance is quite elusive. Here we introduce a simple order relation based on the concentration of Fisher information, which complements the familiar data-processing order. Under this order relation, the information theoretic significance of projective measurements stands out immediately. Notably, projective measurements are exactly those quantum measurements whose extracted Fisher information is as concentrated as possible, which we call Fisher-sharp measurements. We also introduce the concept of sharpness index and show that it is completely determined by the finest projective measurement among the coarse graining of a given measurement.
Projective measurements in quantum theory have a very simple algebraic definition, but their information theoretic significance is quite elusive. Here we introduce a simple order relation based on the concentration of Fisher information, which complements the familiar data-processing order. Under this order relation, the information theoretic significance of projective measurements stands out immediately. Notably, projective measurements are exactly those quantum measurements whose extracted Fisher information is as concentrated as possible, which we call Fisher-sharp measurements. We also introduce the concept of sharpness index and show that it is completely determined by the finest projective measurement among the coarse graining of a given measurement.
A quantum measurement is Fisher symmetric if it provides uniform and maximal information on all parameters that characterize the quantum state of interest. Using (complex projective) 2-designs, we construct measurements on a pair of identically prepared quantum states that are Fisher symmetric for all pure states. Such measurements are optimal in achieving the minimal statistical error without adaptive measurements. We then determine all collective measurements on a pair that are Fisher symmetric for the completely mixed state and for all pure states simultaneously. For a qubit, these measurements are Fisher symmetric for all states. The minimal optimal measurements are tied to the elusive symmetric informationally complete measurements, which reflects a deep connection between local symmetry and global symmetry. In the study, we derive a fundamental constraint on the Fisher information matrix of any collective measurement on a pair, which offers a useful tool for characterizing the tomographic efficiency of collective measurements.
Starting from a simple estimation problem, here we propose a general approach for decoding quantum measurements from the perspective of information extraction.By virtue of the estimation fidelity only, we provide surprisingly simple characterizations of rank-1 projective measurements, mutually unbiased measurements, and symmetric informationally complete measurements.Notably, our conclusions do not rely on any assumption on the rank, purity, or the number of measurement outcomes, and we do not need bases to start with.Our work demonstrates that all these elementary quantum measurements are uniquely determined by their information-extraction capabilities, which are not even anticipated before.In addition, we offer a new perspective for understanding noncommutativity and incompatibility from tomographic performances, which also leads to a universal criterion for detecting quantum incompatibility.Furthermore, we show that the estimation fidelity can be used to distinguish inequivalent mutually unbiased bases and symmetric informationally complete measurements.In the course of study, we introduce the concept of (weighted complex projective) 1/2-designs and show that all 1/2-designs are tied to symmetric informationally complete measurements, and vice versa.
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.
Abstract By comparing measurement-induced classical Fisher information of parameterized quantum states with quantum Fisher information,we study the notion of Fisher concord (as abbreviation of the concord between the classical and the quantum Fisher information), which is an information-theoretic measure of quantum states and quantum measurements based on both classical and quantum Fisher information. Fisher concord is defined by multiplying the inverse square root of quantum Fisher information matrix to measurement-induced classical Fisher information matrix on both sides, and quantifies the relative accessibility of parameter information from quantum measurements (alternatively, the efficiency of quantum measurements in extracting parameter information). It reduces to the ratio of the classical Fisher information to quantum Fisher information in any single parameter scenario. In general, Fisher concord is a symmetric matrix which depends on both quantum states and quantum measurements. Some basic properties of Fisher concord are elucidated. The significance of Fisher concord in quantifying the interplay between classicality and quantumness in parameter estimation and in characterizing the ef- ficiency of quantum measurements are illustrated through several examples, and some information conservation relations in terms of Fisher concord are exhibited.
Abstract We consider the impact that temporal correlations in the measurement statistics can have on the achievable precision in a sequential metrological protocol. In this setting, and for a single quantum probe, we establish that it is the transitions between the measurement basis states that plays the most significant role in determining the precision, with the resulting conditional Fisher information being interpretable as a rate of information acquisition. Projective measurements are shown to elegantly demonstrate this in two disparate estimation settings. Firstly, in determining the temperature of an environment and, secondly, to ascertain a parameter of the system Hamiltonian. In both settings we show that the sequential estimation approach can provide a useful method to enhance the achievable precision.
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.