A generalization of the Choi isomorphism with application to open quantum systems

Completely positive transformations play an important role in the description of state changes in quantum mechanics, including the time evolution of open quantum systems. One useful tool to describe them is the so-called Choi isomorphism, which maps completely positive transformations to positive semi-definite matrices. Accordingly, there are numerous proposals to generalize the Choi isomorphism. In the present paper, we show that the 1976 paper of Gorini, Kossakowski and Sudarshan (GKS) already holds the key for a further generalization and study the resulting GKS isomorphism. As an application, we compute the GKS matrix of the time evolution of a general open quantum system up to second order in time.

💡 Research Summary

The paper revisits the fundamental role of completely positive (CP) maps in quantum mechanics, especially in describing the dynamics of open quantum systems, and introduces a new mathematical framework – the GKS isomorphism – that generalizes the well‑known Choi isomorphism. After a concise review of the historical development of CP maps, the authors recall the Stinespring theorem, the Kraus representation, and the original Choi construction, emphasizing that the Choi matrix depends on a chosen orthonormal basis of the system Hilbert space.

The core contribution is to show that the 1976 Gorini‑Kossakowski‑Sudarshan (GKS) paper already contains the ingredients for a broader isomorphism. By selecting an arbitrary orthonormal basis ({F_\alpha}) of the Hilbert‑Schmidt operator space (A = L(\mathcal H)), any linear map (E: A \to A) can be expanded as
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