This survey paper introduces several topics in quantum information theory, especially positive operator valued measures (POVMs) and entangled states. POVMs are needed to mathematically express measurements in the quantum world, and there are two special POVMs, named symmetric informationally complete POVMs and mutually unbiased bases. These two POVMs have very beautiful symmetry. We give the definitions and examples of these POVMs and discuss the existence or non-existence problem of these POVMs. Entanglement is some kind of relationship between two or more quanta. When several quanta have entanglement, the state of the quanta is called an entangled state. In this paper, quantum teleportation is introduced as an application using entangled states. Moreover, we investigate the problem of determining whether a given quantum state is an entangled state.
This survey paper introduces several topics in quantum information theory, especially positive operator valued measures (POVMs) and entangled states. POVMs are needed to mathematically express measurements in the quantum world, and there are two special POVMs, named symmetric informationally complete POVMs and mutually unbiased bases. These two POVMs have very beautiful symmetry. We give the definitions and examples of these POVMs and discuss the existence or non-existence problem of these POVMs. Entanglement is some kind of relationship between two or more quanta. When several quanta have entanglement, the state of the quanta is called an entangled state. In this paper, quantum teleportation is introduced as an application using entangled states. Moreover, we investigate the problem of determining whether a given quantum state is an entangled state.
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.
We present a comprehensive technical review of quantum measurement theory, emphasizing the mathematical structures underlying the measurement problem and their role in modern quantum information science. Beginning with spectral theory and the Born rule, we develop the theory of positive operator-valued measures (POVMs), Kraus operators, and completely positive channels as essential generalizations of projective measurements. We discuss how decoherence theory addresses the preferred-basis and interference problems while leaving the outcome problem formally unresolved, motivating interpretational responses ranging from collapse models to many-worlds scenarios. Throughout, we emphasize the applicability of POVM theory to quantum state discrimination, quantum cryptography, and quantum error correction. The review prioritizes physical motivation and mathematical hints over detailed proofs, suitable for researchers transitioning into quantum foundations or quantum information theory.
We identify five selected open problems in the theory of quantum information, which are rather simple to formulate, are well studied in the literature, but are technically not easy.As these problems enjoy diverse mathematical connections, they offer a huge breakthrough potential.The first four concern existence of certain objects relevant for quantum information, namely a family of symmetric informationally complete generalized measurements in an infinite sequence of dimensions, mutually unbiased bases in dimension six, measurements saturating multiparameter Cramér-Rao bound and bound entangled states with negative partial transpose.The fifth problem requires checking whether a certain state of a two-ququart system is two-copy distillable.
A broad class of informationally complete symmetric measurements is introduced. It can be understood as a common generalization of symmetric, informationally complete positive operator-valued measures and mutually unbiased bases. Additionally, it provides a natural way to define two more families of mutually unbiased symmetric measurement operators in any finite dimension. We show a general method of their construction, together with an example of an optimal measurement. Finally, we analyze the properties of symmetric measurements and provide applications in entropic relations and entanglement detection.
We identify five selected open problems in the theory of quantum information, which are rather simple to formulate, were well-studied in the literature, but are technically not easy. As these problems enjoy diverse mathematical connections, they offer a huge breakthrough potential. The first four concern existence of certain objects relevant for quantum information, namely a family of symmetric informationally complete generalized measurements in an infinite sequence of dimensions, mutually unbiased bases in dimension six, absolutely maximally entangled states for four subsystems with six levels each and bound entangled states with negative partial transpose. The fifth problem requires checking whether a certain state of a two-ququart system is 2-copy distillable. An award for solving each of them is announced.
Abstract Quantum measurements based on mutually unbiased bases, mutually unbiased measurements and general symmetric informationally complete positive operator valued measurements, which are commonly used in quantum information and quantum computation. In this paper, by a broad class of informationally complete symmetric measurements and correlation matrices, some separability criteria are proposed for different quantum systems that are more robust and comprehensive than the corresponding criteria previously introduced L M Lai, T Li, S M Fei, and Z X Wang, (2018 Quantum Inf. Process., 17, 314); J W Shang, A Asadian, and H J Zhu, (2018 Phys. Rev. A 98, 022309). Furthermore, we extend the separability criteria of tripartite quantum systems to many-body systems. The proposed entanglement criteria may further enrich the relationship between quantum entanglement and measurement.