For any quantum observable (positive operator valued measure (POVM)), we show that its compatible quantum instruments can be identified with certain channels and can be seen as combinations of Lüders operations and channels. We prove that a POVM is rank-1 exactly when its compatible instruments are nuclear and their associate channels are entanglement-breaking. Any instrument can be maximally refined into a rank-1 instrument. We present a characterization for (minimal) pure realizations of instruments and characterize completely the minimal pure measurement models of POVMs. The standard model of quantum measurement theory is generalized for arbitrary observables. Finally, we study post measurement states.
For any quantum observable (positive operator valued measure (POVM)), we show that its compatible quantum instruments can be identified with certain channels and can be seen as combinations of Lüders operations and channels. We prove that a POVM is rank-1 exactly when its compatible instruments are nuclear and their associate channels are entanglement-breaking. Any instrument can be maximally refined into a rank-1 instrument. We present a characterization for (minimal) pure realizations of instruments and characterize completely the minimal pure measurement models of POVMs. The standard model of quantum measurement theory is generalized for arbitrary observables. Finally, we study post measurement states.
Quantum instruments describe outcome probability as well as state change induced by measurement of a quantum system. Incompatibility of two instruments, i. e. the impossibility to realize them simultaneously on a given quantum system, generalizes incompatibility of channels and incompatibility of positive operator-valued measures (POVMs). We derive implications of instrument compatibility for the induced POVMs and channels. We also study relation of instrument compatibility to the concept of non-disturbance. Finally, we prove equivalence between instrument compatibility and postprocessing of certain instruments, which we term complementary instruments. We illustrate our findings on examples of various classes of instruments.
The projection postulate and spectral representation of observables remain central in quantum theory but describe only an idealized measurement limit. Realistic measurements may be inefficient, unsharp, weak, indirect, adaptive, destructive, or informationally incomplete. Positive-operator-valued measures (POVMs) provide the probability rule for processes, whereas quantum instruments specify state transformation. This review traces the hierarchy from von Neumann projective measurement to generalized measurement, emphasizing distinctions among effects, instruments, channels, and implementations. Naimark dilation realizes arbitrary POVMs through unitary coupling to an ancillary meter and projective readout. Stinespring dilation represents completely positive trace-preserving dynamics by unitary evolution on an enlarged system followed by partial trace. A qubit–ancilla interaction yields a partial-measurement instrument, its POVM effects, and its information–disturbance trade-off. It compares projective, generalized, and ancilla-mediated schemes; examines compatibility, instrument incompatibility, and disturbance; and surveys applications to state discrimination, tomography, control, and error correction. Recent developments in dynamic-circuit measurements and noisy-POVM compatibility are integrated with questions concerning measurement back-action and outcome definiteness. The review stresses that a POVM does not uniquely determine state update, dilation theorems establish realizability rather than unique mechanisms, and operational measurement theory does not settle definite outcomes, arguing quantum instruments provide a complete description of measurement.
Until recently, a quantum instrument was defined to be a completely positive operation-valued measure from the set of states on a Hilbert space to itself. In the last few years, this definition has been generalized to such measures between sets of states from different Hilbert spaces called the input and output Hilbert spaces. This article presents a theory of such instruments. Ways that instruments can be combined such as convex combinations, post-processing, sequential products, tensor products and conditioning are studied. We also consider marginal, reduced instruments and how these are used to define coexistence (compatibility) of instruments. Finally, we present a brief introduction to quantum measurement models where the generalization of instruments is essential. Many of the concepts of the theory are illustrated by examples. In particular, we discuss Holevo and Kraus instruments. Quanta 2023; 12: 27–40.
Measurement compatibility asks whether a family of quantum measurements can be simulated by one parent measurement followed by classical post-processing. We extend this measurement compatibility to the notion of \emph{d-compatibility} for measurements, channels, and instruments whose parent passes a d-dimensional quantum system and unlimited classical information to the post-processing. We formulate d-compatibility as a common linear factorization of operator spaces and operator systems and show that d-compatible families form a closed set, even with infinitely many settings and a separable infinite-dimensional input. To certify d-incompatibility, we define the compatibility functional as the smallest product of completely bounded norms over common factorizations through a d-dimensional register. Using factorization theory of operator spaces, we show that measure-and-prepare channels defined in terms of mutually unbiased bases and pinching channels defined in terms of Heisenberg-Weyl operators are d-incompatible in certain regimes of d. We also link d-compatibility of multi-instruments to the preparability of Choi assemblages using channel-state duality, and use this connection to show that for bipartite states the (in)compatibility of instruments can be used to certify Schmidt numbers and the implementability of the instrument using one-way LOCC protocols. Finally, our compatibility functional also bounds the generalized robustness of compatibility, which we interpret through a memory-bounded nontransient preparation game.
In many a traditional physics textbook, a quantum measurement is defined as a projective measurement represented by a Hermitian operator. In quantum information theory, however, the concept of a measurement is dealt with in complete generality and we are therefore forced to confront the more general notion of positive-operator valued measures (POVMs), which suffice to describe all measurements that can be implemented in quantum experiments. We study the (in)compatibility of such POVMs and show that quantum theory realizes all possible (in)compatibility relations among sets of POVMs. This is in contrast to the restricted case of projective measurements for which commutativity is essentially equivalent to compatibility. Our result therefore points out a fundamental feature with respect to the (in)compatibility of quantum observables that has no analog in the case of projective measurements.
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.