Compatible Information as a Natural Information Measure of a Quantum Channel

Arindam Mitra, Debashis Saha, Samyadeb Bhattacharya +1 more

The role of completely positive--indivisibility (CP-indivisibility) and incompatibility as valuable resources for various information-theoretic tasks is widely acknowledged. This study delves into the intricate relationship between CP-divisibility and channel compatibility. Our investigation focuses on the behavior of incompatibility robustness of quantum channels for a pair of generic dynamical maps. We show that the incompatibility robustness of channels is monotonically nonincreasing for a pair of generic CP-divisible dynamical maps. Further, our explicit study of the behavior of incompatibility robustness with time for some specific dynamical maps reveals nonmonotonic behavior in the CP-indivisible regime. Additionally, we propose a measure of CP-indivisibility based on the incompatibility robustness of quantum channels. Our investigation provides valuable insights into the nature of quantum dynamical maps and their relevance in information-theoretic applications.

Masashi Ban

Hidetosi Takahasi

Arindam Mitra, Máté Farkas

Incompatibility of quantum devices is one of the cornerstones of quantum theory, and the incompatibility of quantum measurements and channels has been linked to quantum advantage in certain information-theoretic tasks. In this work, we focus on the less well explored question of the incompatibility of quantum instruments, that is, devices that describe the measurement process in its entirety, accounting for both the classical measurement outcome and the quantum postmeasurement state. In particular, we focus on the recently introduced notion of parallel compatibility of instruments, which has been argued to be a natural notion of instrument compatibility. We introduce, in a manner similar to the case of measurements and channels, the incompatibility robustness of quantum instruments and derive universal bounds on it. We then prove that postprocessing of quantum instruments is a free operation for parallel compatibility. Last, we provide families of instruments for which our bounds are tight and families of compatible indecomposable instruments.

Arindam Mitra, Máté Farkas

Incompatibility of quantum devices is a useful resource in various quantum information theoretical tasks, and it is at the heart of some fundamental features of quantum theory. While the incompatibility of measurements and quantum channels is well studied, the incompatibility of quantum instruments has not been explored in much detail. In this work, we revise a notion of instrument compatibility introduced in the literature that we call traditional compatibility. Then, we introduce the notion of parallel compatibility and show that these two notions are inequivalent. We then argue that the notion of traditional compatibility is conceptually incomplete and prove that, while parallel compatibility captures measurement and channel compatibility, traditional compatibility does not capture channel compatibility. Hence, we propose parallel compatibility as the conceptually complete definition of the compatibility of quantum instruments.

Mark M. Wilde