Quantum Measurement—The Emergence of POVMs and State Transformers

Abstract

After setting the stage for the present approach and introducing basic notions (like the important notion of pointer states) in Sect. 2.1, measurement (like) processes will be formally defined in section 2.2. From this definition, basic operators of the quantum theory of measurement are immediately inferred, most importantly linear operators, which transform initial states into non-normalized final states and which we shall call state transformers. These operators then give rise to positive operator valued measures (POVMs), encoding the statistics of these state transitions arising from the Born rule. In the special case of projective measurements, the associated POVM (which is then a projection valued measure (PVM)) can be compactly encoded in a single selfadjoint operator referred to as an observable operator, which corresponds to the usual observables of textbooks. The reader who is more interested in later results of this work may read up to this Sect. 2.2 to learn the basic notions and formalism used and then proceed directly to the desired sections. In Sect. 2.3, the conceptual question as to whether quantum measurement can always reveal preexisting properties of the measured system is examined and answered in the negative by a version of the Kochen–Specker theorem, based on a Gedankenexperiment considering a composition of three spin measurements (essentially the GHZ experiment). If the measurements are performed at spacelike separation, the very same argument yields a version of the famous Bell theorem, stating that the empirically well verified quantum predictions are irreconcilable with any attempt to explain nature by only local direct causes (Sect. 2.3.2). Besides the nonlocality of nature, the crucial insight of these arguments is that the wave function dynamics—in particular the reduction of the wave function (collapse) in measurement (like) processes—cannot be interpreted as describing only our limited information (ignorance) or the like, but has to be taken seriously on a dynamical level. In Sect. 2.4, we shall go back to the general definition of measurement (like) processes in Sect. 2.2, distinguishing and technically analysing different kinds of such processes. Important distinctions will be projective/non-projective and reproducible/non-reproducible (reproducibility refers here to the outcome value upon immediate repetition). The simplest class are projective reproducible measurements, of which ideal measurements (the kind of quantum measurements usually presented in textbooks) are a distinguished representative. Generic implementations of non-projective measurements, given by indirect and approximate measurement schemes, will be developed and discussed. After discussing more concrete implementations of quantum measurements (the von Neumann measurement scheme, etc.) in Sects. 2.5 and 2.6, we shall develop the modern operational formulation from the viewpoint of the present approach in section 2.7. This formulation includes trace preserving and trace reducing completely positive maps (CPMs) acting on density operators, quantum channels and instruments, purification of mixed states, first and second Kraus representations of CPMs, Stinespring representation of quantum channels, and Naimark representation of POVMs.