Mathematics of Quantum Uncertainty

Dear Colleagues,

The uncertainty principle is a cornerstone of one of the fundamental theories of modern physics, quantum mechanics. Somewhat surprisingly, an important aspect of the principle—Heisenberg’s famous error-disturbance relation—has, until recent years, led a “life in the shadows”. It has taken until the late 1990s before first attempts were made at finding precise formulations of trade-off relations for the errors in joint measurements of incompatible quantities, and at understanding the role of measurement disturbance. The underlying issues that prevented an earlier development are at once of a mathematical and conceptual nature. Firstly, it has taken several decades before mathematical tools for the representation of unsharp and approximate measurements were developed; these tools include the theory of operator-valued measures and their (measurement) dilations. Secondly, the concept of joint measurability had to be extended to encompass sets of noncommuting observables. Finally, a notion of approximation of one observable by another had to be developed along with appropriate measures of error. Meanwhile there are a variety of approaches for such formalizations, yielding new insights such as a deep connection between measurement uncertainty and preparation uncertainty, but also resulting in an ongoing controversy over what constitutes a good quantum generalization of the time-honored Gaussian root-mean-square deviation. The framework of operational quantum theory, now routinely used in quantum information theory, provides the basis for the ongoing search of novel forms of tight preparation and measurement uncertainty relations that are open to experimental testing and may be expected to inform ultimate quantum bounds for high-precision measurement protocols and support the inception of new quantum information tasks, such as cryptographic protocols that utilize entropic uncertainty relations adapted to the presence of quantum memories.

Needless to say, the investigation of quantum uncertainty is not restricted to uncertainty relations, but also addresses related structural aspects of quantum mechanics including, inter alia, the study of the notion of incompatibility of observables and of theories, limitations of measurements due to symmetry, connections between incompatibility and nonlocality, and the relationship between joint measurability and approximate cloning.

The purpose of this Special Issue is to establish a collection of articles that reflect the latest mathematical and conceptual developments in the field of quantum uncertainty and explore the scope for applications in areas such as quantum cryptography, quantum control, and quantum metrology.

Prof. Dr. Paul Busch
Dr. Takayuki Miyadera
Dr. Teiko Heinosaari
Guest Editors

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