Tensor networks and graphical calculus for open quantum systems

Details Tensor networks and graphical calculus for open quantum systems Tensor networks and graphical calculus for open quantum systems

2011-11-29

arxiv.org/abs/1111.6950v1

Physics

Quantum Physics

21 pages, 53 figures

Scientific paper

We develop a graphical calculus for completely positive maps and in doing so cast the theory of open quantum systems into the language of tensor networks. We tailor the theory of tensor networks to pictographically represent the Liouville-superoperator, Choi-matrix, process-matrix, Kraus, and system-environment representations for the evolution of open-system states, to expose how these representations interrelate, and to concisely transform between them. Several of these transformations have succinct depictions as wire bending dualities in our graphical calculus --- reshuffling, vectorization, and the Choi-Jamiolkowski isomorphism. The reshuffling duality between the Choi-matrix and superoperator is bi-directional, while the vectorization and Choi-Jamiolkowski dualities, from the Kraus and system-environment representations to the superoperator and Choi-matrix respectively, are single directional due to the non-uniqueness of the Kraus and system-environment representations. The remaining transformations are not wire bending duality transformations due to the nonlinearity of the associated operator decompositions. Having new tools to investigate old problems can often lead to surprising new results, and the graphical calculus presented in this paper should lead to a better understanding of the interrelation between CP-maps and quantum theory.

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